Complex contagions for topological data analysis of networks

Social and biological contagions are often strongly influenced by the spatial embeddedness of networks. In some cases, such as in the Black Death, they can spread as a wave through space. In many modern contagions, however, long-range edges -- e.g., due to airline transportation or communication media -- allow clusters of a contagion to arise in distant locations. We study these competing phenomena for the Watts threshold model (WTM) of complex contagions on empirical and synthetic noisy geometric networks, which are networks that are embedded in a manifold and which consist of both short-range and long-range,"noisy"edges. Our approach involves using the WTM to construct contagion maps that use multiple contagions to map the nodes as a point cloud, which we analyze using tools from high-dimensional data analysis and computational homology. For contagions that predominantly exhibit wavefront propagation, we identify a noisy geometric network's underlying manifold in the point cloud, highlighting our approach as a tool for inferring low-dimensional structure. Our approach makes it possible to obtain insights to aid in the modeling, forecast, and control of spreading processes, and it simultaneously leads to a novel technique for manifold learning in noisy geometric networks.

Significance Statement: The spread of social and biological contagions is strongly influenced by the spatial embeddedness of networks. In some cases, such as in the Black Death, contagions spread like a wave through space. In many modern contagions, however, long-range edges-which can represent, e.g., airline transportation or communication media-allow clusters of a contagion to arise in distant locations. We study this phenomenon by analyzing the Watts threshold model of complex contagions on noisy geometric networks, in which we augment a spatially-embedded network with long-range edges. We adapt methodology from the study of data topology and homology to investigate the qualitative and quantitative features of contagion dynamics on networks. Our results illustrate an unexpected and fascinating connection between dynamical systems and manifold learning.
analysis for a class of noisy geometric networks called noisy ring lattices, for which we analyze the presence versus absence of WFP and ANC. We concomitantly investigate the speed of WFP and the occurrence frequency of ANC. In Fig. 1, we illustrate WFP, ANC, and their dependence on the contagion threshold T .  [52] with uniform threshold T on a noisy geometric network containing geometric edges along a manifold (in this case, a two-dimensional lattice) and nongeometric edges that introduce shortcuts in the network. We study two phenomena in the evolution of contagion clusters (shaded areas): wavefront propagation (WFP) describes the outward expansion of a contagion cluster's boundary, and the appearance of a new contagion cluster (ANC) occurs when a contagion spreads exclusively along non-geometric edges (dashed arrow). (Bottom panels) One can examine WFP and ANC through the node activation times (i.e., the times at which nodes adopt the contagion), which is dependent on T . For small T , frequent ANC leads to rapid dissemination of a contagion. For moderate T , little to no ANC occurs and WFP leads to slow dissemination. For large T , there is no spreading. For a given network, activation times collected across N realizations of contagion (with varying initial conditions) map network nodes to a point cloud in R N via what we call a WTM map. We study the extent to which the dynamics follows a network's embedding in an underlying metric space by analyzing the geometry, dimensionality, and topology of the WTM map.
We also address an important but poorly-understood question: To what extent do spreading dynamics follow the manifold on which a network is embedded? To this end, we introduce a mapping of network nodes as a point cloud based on WTM contagions. By analogy to diffusion maps [11] and similar ideas in dimension reduction [51,3,13], we use the term WTM maps for these maps. We examine the geometry, topology, and dimensionality of WTM maps and compare these features to those of the underlying manifold of the noisy geometric network on which the dynamics occur. This approach aligns our work with the fields of manifold learning and dimension reduction [51,13,3,11,17,50], which focus on data analysis and complement our dynamical-systems approach. Additionally, because WTM maps are based on nonconservative dynamics (e.g., for the WTM, the number of nodes that have adopted a contagion is nondecreasing, because nodes cannot subsequently un-adopt it) [19], they provide a contrast to techniques that are based on conservative dynamics like ordinary random walks and diffusion. Thus, the motivation for our work is twofold: (i) introduce a methodology for studying the extent to which contagions follow the underlying structure of a noisy geometric network; and (ii) introduce a methodology for inferring lowdimensional (e.g., manifold) structure in networks based on WTM contagions.

Noisy Geometric Networks
Noisy geometric networks are a class of networks that arise from geometric networks [2] but also include non-geometric, "noisy" edges. Consider a set V of network nodes that have intrinsic locations {w (i) } i∈V in a metric space. We will restrict our attention to nodes that lie on a manifold M that is embedded in an ambient space A (i.e., w (i) ∈ M ⊂ A). We use the term "node-to-node distance" to refer to the distance between nodes in this embedding space A, which we equip with the Euclidean norm · 2 (although one can also use other metrics [31]). To form a synthetic noisy geometric network, we construct such a nodal embedding and add two families of edges: (1) a set of geometric edges E (G) , such that (i, j) ∈ E (G) if and only if nodes i and j are sufficiently close to one another on the manifold M; and (2) a set of non-geometric edges E (NG) , which we select using some random process for pairs of nodes (i, j), where i = j and (i, j) ∈ E (NG) . In Figs. 2(a,b), we show examples of such a construction, where we add non-geometric edges uniformly at random. In Fig. 2(c), we show an alternative construction that is motivated by applications in dimension reduction [51,3,11,17].  Figure 2: Noisy geometric networks in which geometric edges (blue) connect nodes that are nearby on an underlying manifold and non-geometric edges (red) connect distant nodes, which we show for three manifolds: (a) a ring embedded as a circle in R 2 ; (b) a spherical surface (2D) in R 3 ; and (c) a bounded plane (2D) embedded (nonlinearly) in R 3 in a configuration known as the "swiss roll" [51]. In panels (a) and (b), we add non-geometric edges uniformly at random. In panel (c), we add noise to the nodes' locations in the ambient space and we place edges between nodes that are nearby in the ambient space. In this case, we interpret edges between nodes that are nearby with respect to the ambient space but not the manifold as non-geometric edges.
As an illustrative example, consider the noisy ring lattice in Fig. 2(a), which is similar to the Newman-Watts variant of the Watts-Strogatz small-world model [53,40,37]. Specifically, we consider N nodes that are uniformly spaced along the unit circle in R 2 . We then add geometric edges so that every node i is connected to its d (G) nearest-neighbor nodes. (Note that there are no self-edges.) We then add d (NG) non-geometric edges to each node and connect the ends of edges (i.e., the stubs) uniformly at random while avoiding self-edges and multi-edges. The resulting network is a (d (G) + d (NG) )-regular network that contains N d (G) /2 geometric edges and N d (NG) /2 non-geometric edges. We can thus specify this network using three parameters: N , d (G) , and d (NG) . Note that our construction assumes that N and d (G) are even.
In Fig. 2(a), we show such a network with N = 20 and (d (G) , d (NG) ) = (4,2). In Sec. 4 of the Supporting Information (SI), we discuss additional models of noisy geometric networks on a ring manifold.

Watts Threshold Model (WTM)
Following Ref. [52], we define a WTM contagion as follows. Given an unweighted network (which we represent using an adjacency matrix A) with a set of nodes V and a set of edges E, we let η i (t) denote the state of node i ∈ V at time t, where η i (t) = 1 and η i (t) = 0 indicates adoption and non-adoption, respectively. We initialize a contagion at time t = 0 by choosing a set of nodes S ⊂ V and setting η i (0) = 1 for i ∈ S and η i (0) = 0 for all other nodes. We refer to S as the "contagion seed." We consider synchronous updating in discrete time [44], such that a node i that has not already adopted the contagion at time t [i.e., η i (t) = 0] will adopt it during the next time step [i.e., η i (t + 1) = 1] if and only if f i > T , where denotes the fraction of neighbors that are infected and d i = j A ij is the degree of node i. We repeat this process until the system reaches an equilibrium point at some time t * < N (i.e., no further adoptions occur). For each node i, we let x (i) denote the node's "activation time," which is the time t at which the node adopts the contagion. A node's adoption of the contagion is an irreversible event (i.e., there is no un-adoption for this model), so the dynamics are monotonic in the sense that the subset I(t) ⊆ V of infected nodes at time t is non-decreasing with time (i.e., I(t) ⊆ I(t + 1)). One can thus interpret the contagion as a "filtration" of the network nodes V (see Refs. [27,15] and our discussion in Sec. 2 of the SI). Finally, we note that our formulation of WTM contagions differs slightly from its original version [52], which allows node-specific thresholds and uses the adoption condition f i ≥ T i .

WTM Maps
A WTM mapping is a nonlinear mapping of nodes in a network to a point cloud in a metric space based on activation times for several realizations of a WTM contagion. In practice, we consider N realizations of a WTM contagion for an N -node network, and we initialize the j-th realization with a contagion seed S (j) = {j} ∪ {k|A jk = 0} that includes node j and its network neighbors. We use the term "cluster seeding" to describe this type of initial condition. We denote the activation time of node i for realization j by x i , we also define a "reflected version" V → {y (i) } and a "symmetric version" V → {z (i) } of a WTM map.
The above construction is convenient because it allows us to think of the activation time x (i) j as a notion of distance from node j to node i (i.e., it describes the time that is required for a contagion to travel from node j to i). To illustrate this point, suppose that contagion seeds were taken to be S (j) = {j} (for j ∈ {1, . . . , N }), and suppose that we construct the WTM map V → {x (i) } i∈V for T = 0. In this case, the activation timex i exactly recovers the length of the shortest path between nodes i and j, and this in turn defines a metric on the discrete space V [16]. In fact, the N × N matrixX = [x (1) , . . . ,x (N ) ] is a dissimilarity matrix, which is central to many algorithms for dimension reduction [50,12] (including Isomap [51]). Letting T > 0 and still assuming that eachx (i) j is finite, we show in Sec. 2 of the SI that the symmetric WTM map induces a metric on V. (More generally, we show that a set of "filtrations" induces a metric under certain conditions.) When T > 0, however, it is common to reach equilibrium points of the system that do not saturate the network with contagion, which implies that x (i) j = ∞ for some i, j ∈ V. This poses a problem, because WTM maps are well-defined only for finite activation times x (i) j . We therefore promote the spread of WTM contagions by initializing them with cluster seeding (rather than node seeding) [20]. This relaxes the requirement that WTM maps induce a metric on the node set V, but one still obtains a directed notion of distance. (In other words, x (j) i and x (i) j reflect contagions that move in opposite directions, and they can differ from each other).

Bifurcation Analysis
To guide our study of WTM maps, we first provide a bifurcation analysis of WTM contagions on noisy ring lattices [see Fig. 2(a)]. In particular, we investigate the dependence of WFP and ANC on the contagion threshold T and on the network parameters d (G) , d (NG) , and N . We consider small contagions on large networks: q(t) N , where q(t) denotes the number of nodes that have adopted the contagion at time t. We examine these results for time t = 0, although similar conclusions follow from examining contagions for t > 0. See Sec. 1 of the SI for further discussion.
In Fig. 3, we illustrate WTM contagions at time t = 0 on a noisy ring lattice with d (G) = 4, d (NG) = 1, and N = 40. The light blue nodes are in the contagion seed S = s ∪ {k|A sk = 0}, which is centered at node s ∈ V. Because node s is incident to both geometric and non-geometric edges, the contagion is initialized with 1 + d (NG) = 2 contagion clusters. (We denote these clusters by C 1 and C 2 .) Note that C 1 is more likely to grow via WFP than C 2 . The orange nodes are the ones that can adopt the contagion via WFP for cluster C 1 . The magnification on the right illustrates a clockwise WFP in which nodes a and b can potentially become infected at time t = 1. Nodes can also adopt the contagion via transmission exclusively across non-geometric edges; this results in ANC. See the dark blue nodes and dashed edges.  Figure 3: A WTM contagion on a noisy ring lattice in which each node has d (G) = 4 geometric and d (NG) = 1 non-geometric edges. We initialize the contagion at time t = 0 by setting node s and its network neighbors (light blue nodes) as infected. At t = 0, there are two contagion clusters (C 1 and C 2 ). At t = 1, additional nodes can adopt the contagion by one of two methods: WFP entails the spread of the contagion across geometric edges. (See the oranges nodes, orange edges, and the magnification on the right. The orange nodes are in the "boundary" of cluster C 1 .) ANC can occur when the contagion spreads exclusively across non-geometric edge. (See the dark blue nodes and dashed edges.) We first consider WFP in the clockwise direction for cluster C 1 . (See the magnification in Fig. 3.) Nodes a, b, and c are exposed, respectively, to 2, 1, and 0 nodes that have adopted the contagion, so their fractions of neighbors that are infected are f a = 2/5, f b = 1/5, and f c = 0/5. Note that we assume that the nongeometric edges for nodes a, b, and c are incident to nodes that are not infected (i.e., which have not adopted the contagion). Because f i > T for node i to adopt the contagion, one of three situations can occur at time t = 1: (i) if 0 ≤ T < 1/5, then nodes a and b adopt the contagion; (ii) if 1/5 ≤ T < 2/5, then node a adopts the contagion; and (iii) if 2/5 ≤ T , then the contagion cluster does not increase in size by WFP. Note that node c cannot adopt the contagion by WFP at time t = 1 for any T ≥ 0. We find that WFP is governed by the critical thresholds where a wavefront propagates at a speed of k nodes per time step for T ∈ T , there is no WFP. Importantly, Eq. (1) indicates a baseline description of WFP, as we have assumed that none of the nongeometric neighbors of nodes in the boundary (i.e., nodes a and b) have become infected. Recalling that q(t) denotes the number of infected nodes at time t, we show in Sec. 1 of the SI that this occurs with probability (NG) , which is approximately equal to 1 when q(t) N and d (NG) N . However, the presence of nodes in the boundary with infected non-geometric neighbors can accelerate WFP. In particular, when the contagion size is large [i.e., when q(t) ≈ N ], we expect WFP to propagate at the accelerated speed of d (G) /2 nodes per time step.
We now turn to ANC, which occurs when a contagion spreads exclusively across non-geometric edges. Using the fact that a node can have up to d (NG) infected non-geometric neighbors, we find the following sequence of critical thresholds: The dynamics changes abruptly at critical values of T , so ANC phenomena for any T ∈ T are similar to each other, but there are abrupt changes at the endpoints of the interval. The ANC phenomena depend on the probability that a node has k infected non-geometric neighbors at time t. As we discuss in Sec. 1 of the SI, this probability is approximately given by In Fig. 4(a), we show a bifurcation diagram that summarizes the WTM dynamics for various values of the contagion threshold T and ratio α = d (NG) /d (G) of non-geometric edges to geometric edges. The solid and dashed curves, respectively, describe Eq. (1) and Eq. (2) for k = 0. That is, T and T (ANC) 0 = α/(α + 1), which intersect at (α, T ) = (1/2, 1/3) and yield four regimes of contagion dynamics that we characterize by the presence versus absence of WFP and ANC. In Fig. 4(b), we plot Eqs. (1) and (2) with other k values for d (G) = 6, where we note that lower curves correspond to larger k. Observe that increasing T for fixed α leads to slower WFP and less frequent ANC. In particular, for (d (G) , d (NG) ) = (6, 2) (which implies that α = 1/3), we find four qualitatively different regimes of WFP and ANC traits (see the regions that we label I-IV). In Figs. 4(c,d), we illustrate dynamics from these regimes by choosing T ∈ {0.05, 0.2, 0.3, 0.45} and plotting the size q(t) of the contagion [see    (1) and (2) for other values of k further describe WFP and ANC. We show them for for d (G) = 6, and we note that the curves become lower with increasing k. Fixing (d (G) , d (NG) ) = (6, 2), which yields α = 1/3, we find four contagion regimes (which we label using the symbols I-IV), where increasing T corresponds to slower WFP and less frequent ANC. (c) For T ∈ {0.05, 0.2, 0.3, 0.45}, we plot the contagion size q(t) versus time t for one realization of a WTM contagion with cluster seeding [i.e., q(0) = 1 + d (G) + d (NG) = 9]. We observe, as expected, that the growth rate decreases with T . In particular, for regime III (e.g., T = 0.3), the contagion spreads strictly via WFP, which initially spreads at a rate of 1 node per time step (both clockwise and counterclockwise along the ring) but eventually accelerates to d (G) /2 nodes per time step. (d) We plot the number of contagion clusters C(t) versus t. As expected, C(t) only increases above its initial value of C(0) = 1 + d (NG) = 3 for regimes I and II (for which T < T ). There is no spreading in regime IV.
the network [i.e., q(t) → N ] very rapidly due in part to the appearance of many contagion clusters early in the contagion process. For T = 0.2, the contagion saturates the network relatively rapidly due to the appearance of some new contagion clusters. For T = 0.3, the contagion saturates the network slowly, as no new contagion clusters appear and the contagion spreads only via WFP. For T = 0.45, the contagion does not saturate the network, as neither WFP nor ANC occur.

Analyzing WTM Contagions Through WTM Maps
In this section, we analyze symmetric WTM maps V → {z (i) } for noisy ring lattices in several ways: geometrically, topologically, and in terms of dimensionality. Such point cloud analytics identify parameter regimes in which characteristics of a network's underlying manifold also appear in the WTM maps, and they thereby offer a methodology to assess the extent to which a contagion exhibits WFP (along the network's underlying manifold) versus ANC. These results are in excellent agreement with our theoretical results in Eqs. (1) and (2).
In Fig. 5, we study WTM maps for a noisy ring lattice with N = 200 and (d (G) , d (NG) ) = (6, 2). We give each node i an intrinsic location w (i) = [cos(2πi/N ), sin(2πi/N )] T on the unit circle M = {(a, b)|a 2 +b 2 = 1}. In Fig. 5(a), we illustrate the point clouds {z (i) } i∈V ∈ R N that result from WTM maps with thresholds T ∈ {0.05, 0.2, 0.3, 0.45}, which correspond to the four regimes of contagion dynamics that are predicted by Eqs. (1) and (2). [See labels I-IV in Fig. 4(b).] To visualize the N -dimensional point clouds {z (i) } i∈V , we use principle component analysis (PCA) to project onto R 2 [12,50,51]. The color of each node at location w (i) and point z (i) reflects the activation time for node i during one realization of the WTM contagion that we use to generate the WTM map. In particular, dark blue nodes (points) indicate the contagion seed under cluster seeding. Gray nodes (points) never adopt the contagion and thus have activation times that are infinite. For practical purposes, we set these activation times to be 2N rather than ∞ (see Sec. 6 of the SI for additional discussion). Regime III is the regime for which the point cloud {z (i) } appears to best resemble (up to rotation) the nodes' intrinsic locations {w (i) }. This is expected, as this regime corresponds to WFP and no ANC. (In other words, the contagion follows the network's underlying manifold M.) In Fig. 5(c), we summarize the characteristics of WTM maps for different thresholds T ∈ [0, 0.6]. For each threshold, we analyze the point cloud's (top) geometry, (center) dimensionality, and (bottom) topology. Specifically, we compare the geometry of {z (i) } [see panel (a)] to that of the nodes' locations {w (i) } ∈ M [see panel (b)] by computing a Pearson correlation coefficient ρ to compare the node-to-node distances for the two point clouds (i.e., z (i) −z (j) 2 and w (i) −w (j) 2 for (i, j) ∈ V ×V). We study the dimensionality by examining the residual variance [51,12] of the point cloud {z (i) } and computing the smallest dimension such that we lose less than 5% of the variance when projecting to a lower dimension using PCA [51,12]. We refer to this dimension as the "embedding dimension" P . We study the topology by examining the persistence diagram of a Vietoris-Rips filtration that is generated by the point cloud {z (i) } [27,15]. In particular, we define ∆ = l 1 − l 2 , where l 1 and l 2 are lifetimes of the most persistent and second most persistent one-dimensional topological features in the Vietoris-Rips filtration. Large values of ∆ indicate the presence of a single dominant 1-cycle (i.e., ring) in the point cloud. As expected by our analysis, for regime III (which exhibits WFP but no ANC), we identify characteristics of the manifold M in the point clouds that result from WTM maps. Namely, for regime III, the point cloud has similar geometry (i.e., indicated by large ρ), embedding dimension (i.e., indicated by P = 2), and topology (i.e., indicated by large ∆) as the network's underlying manifold M. See the Methods and Materials section and Sec. 3 of the SI for additional discussion.
In Fig. 6, we analyze WTM maps applied to noisy ring lattices for various values of α = d (NG) /d (G) . We show results for N = 200, d (G) = 20, various T , and various d (NG) . For each point cloud, we study (a) geometry through ρ, (b) dimensionality through P , and (c) topology through ∆. As we show using the solid and dashed curves, respectively, the transitions between the qualitatively different regions of these properties closely resemble the bifurcation structure from Eqs. (1) and (2) with k = 0. In particular, when we observe WFP but no ANC, we find that we consistently identify the geometry, embedding dimension, and topology  I  III  IV  II  I  III IV  II   0 (a) We show point clouds {z (i) } ∈ R N for WTM maps with uniform thresholds T ∈ {0.05, 0.2, 0.3, 0.45}, which correspond, respectively, to regimes I-IV in Fig. 4(b). For visualization purposes, we show the twodimensional projections of the N -dimensional point clouds after applying principle component analysis (PCA) [12,50]. (b) One realization of contagion used to construct the WTM maps in panel (a). The color of each point in panel (a)-and corresponding node in panel (b)-indicates the node's activation time. Nodes in the contagion seed are dark blue, and nodes that never adopt the contagion are gray. (c) We analyze point clouds that result from WTM maps with respect to three criteria: (top) geometry, (center) dimensionality, and (bottom) topology. We study the geometry by considering the node-to-node distances (in the metric of the embedding space, which in this case is the Euclidean norm) and computing the Pearson correlation coefficient ρ between these distances for the WTM map {z (i) } ∈ R N [see panel (a)] and the original node We study the dimensionality through its embedding dimension P , which we calculate by studying the residual variance when reducing dimensionality with PCA [51,12]. We study the topology by examining persistent homology when applying a Vietoris-Rips filtration to the point cloud [27,15], where ∆ = l 1 − l 2 denotes the difference in lifetimes for the two most persistent onedimensional (1D) features (i.e., 1-cycles). The vertical dashed lines indicate the predicted bifurcations in contagion dynamics from Eqs. (1) and (2) [see Fig. 4(b)]. Note that there are activation times that are infinite for T ≥ T (WFP) 0 = 3/8 (shaded region). As expected for regime III, the geometry, dimensionalitym and topology of the point cloud recovers that of the ring manifold M (as indicated by large correlation ρ ≈ 1), an estimated embedding dimension of P = 2, and a large ∆ (which identifies a single dominant 1-cycle, or a ring, in the point cloud). See the Methods section and the SI for further discussion.
of the underlying manifold of the noisy ring lattice using the WTM map. When we observe both WFP and ANC, the extent to which the contagion adheres to the network's underlying manifold depends on α and T , and we can quantify this extent using the point cloud measures ρ, P , and ∆. We illustrate this result further in Fig. 6(d) by fixing α = 1/3 and plotting ρ, P , and ∆ as a function of the threshold T . We show results for (d (G) , d (NG) ) = (6, 2) (blue dashed curves) and (d (G) , d (NG) ) = (24, 8) (red solid curves). Observe that the latter curve is smoother than the former one. The latter curve yields values of ρ, P , and ∆ that better reflect the underlying ring manifold M. By contrast, increasing the number of nodes N increases the contrast between the region that predominantly exhibits WFP and the other regions (see Sec 5.1 of the SI).
To give some perspective on the performance of WTM maps for identifying a noisy geometric network's underlying manifold even in the presence of many non-geometric edges, the arrows in Fig. 6(d) indicate the values of ρ, P , and ∆ for a mapping of nodes based on shortest paths, which one can construe as a variant of the dimension-reduction algorithm Isomap [51] (which we apply to an unweighted network rather than to   The arrows indicate the ρ, P , and ∆ values that we obtain for the embedding of nodes based on shortest paths, which (as discussed in the text) one can construe as a variant of the dimension-reduction algorithm Isomap [51]. a point cloud). Specifically, we map V → {x (i) } with T = 0 (as we discussed in the section "WTM Maps" on page 3). In Sec. 5 of the SI, we describe additional results comparing a WTM map to Isomap and a Laplacian eigenmap [3] for generalizations of the noisy ring lattice by allowing (1) the node locations to be a random sampling of points on the unit circle and (2) heterogeneity in their geometric and non-geometric degrees.

Complex Contagions on a London Transit Network
In addition to synthetic networks, we study WTM maps for a London transit network. Nodes represent intersections of known latitude and longitude, geometric edges represent roads (from data used in Ref. [32]), and non-geometric edges represent metropolitan lines (from data used in Ref. [47]). We present our results on this network in detail in Sec. 6 of the SI, and we summarize them here. Our central finding is that the qualitative dynamical regimes that we observe for synthetic noisy geometric networks also occur in the London transit network. In particular, we find that the agreement between the geometry of point clouds resulting from WTM maps and the known node locations {w (i) } (i.e., as measured by ρ) depends on the WTM threshold T . Interestingly, we find that the small node degrees (e.g., d i = 2.59) and the significant heterogeneity (e.g., with respect to node locations, node degrees, and the length of roads) in the London transit network causes a WTM contagion to be extremely sensitive to the value of T for only a few of the contagion seeds S (j) . Nevertheless, as we discuss in the SI, such minority cases have a significant effect on WTM maps.

Conclusions
Many empirical networks include a combination of geometric edges between nearby nodes and non-geometric, long-range edges [2]. Such situations can arise when nodes are restricted by their locations in a physical space (such as in a city) or in terms of latent underlying spaces [24,31,16,48,39,49,8,51,3,11,17,50]. When considering a spreading process on a noisy geometric network, where applications range from identifying influential people in a social network [28] to administering immunizations [45,42], it is important to understand the extent to which a contagion follows such underlying structure. To address this question, we conducted a detailed investigation using the Watts threshold model (WTM) of complex contagions (with uniform threshold T ) on noisy geometric networks. The spreading dynamics exhibit both wavefront propagation (WFP) that follows the underlying manifold structure of a network as well as the appearance of new clusters (ANC) of contagions in distant locations. To investigate the extent to which a WTM contagion adheres to a network's underlying manifold, we introduced the notion of WTM maps, which map nodes as a point cloud based on several realizations of a contagion on a network. We found that when contagion predominantly spreads by WFP, WTM maps recover the topology, geometry, and dimensionality of a network's underlying manifold even in the presence of many non-geometric, "noisy" edges.
Our approach relies on the idea that the time required for a contagion to travel from one node to another provides a proxy for the distance between the nodes along the underlying manifold of a noisy geometric network. Using dynamics as a proxy for distance is not a new concept and is widely utilized for nonlinear dimension reduction (e.g., in diffusion maps [11]). An important distinction, however, is that our methodology is based on non-conservative dynamics (in particular, on complex contagions) rather than on conservative dynamics such as random walks and diffusion. It is known that considering conservative dynamics versus non-conservative dynamics gives very different answers for questions like which nodes are most important [19] and what network structures constitute bottlenecks to such dynamics [33] (which is closely related to which network structures yield dense "communities" of nodes [26]). Comparing WTM maps to Laplacian eigenmaps [3] and Isomaps [51] (see Sec. 5 of the SI) illustrates that nonconservative and conservative dynamics also lead to differences in nonlinear dimension reduction.
Recent research on network epidemiology underscores the importance of the perspective that we have taken in the present paper. For example, Brockmann and Helbing [4] defined node-to-node distances based on a stochastic model for contagions that takes into account human mobility patterns in the worldwide airline network, and they reported that such a notion of distance did a good job of predicting global contagions. We have used a simple model of complex contagions to illustrate what we think is a very promising approach to understanding (both biological and social) contagion phenomena, and it will be important to extend these ideas using other dynamical systems that describe spreading processes.

Geometry of WTM Maps
To quantify the similarity of the geometry of a WTM map to that of the nodes on the underlying manifold of a noisy geometric network, we calculate the Pearson correlation coefficient ρ to relate node-to-node distances for the WTM map (e.g., z (i) − z (j) 2 ) to those for the actual node locations (i.e., w (i) − w (j) 2 ). We conduct this comparison with respect to the dimension of the ambient spaces in which points lie (i.e., R N for {x (i) } and R 2 for {w (i) }). See Sec. 3.1 of the SI for further details.

Dimensionality of WTM Maps
We estimate the embedding dimension P of a WTM map by studying p-dimensional projections of the WTM map obtained via PCA [12,50] for different values of p = 1, 2, . . . . For each projection, we calculate the residual variance R p = 1 − (ρ (p) ) 2 [51], where ρ (p) denotes the Pearson correlation coefficient that relates the geometric similarity between the p-dimensional projection and the unprojected WTM map. We define the embedding dimension P as the smallest dimension p such R p < 0.05. See Sec. 3.2 of the SI for further discussion.

Topology of WTM Maps
To assess the presence versus absence of a ring topology in a WTM map, we study the persistent homology of the WTM map by examining a Vietoris-Rips filtration [27,15] using the software package Perseus [35]. We calculate persistent 1D features (i.e., 1-cycles) and record the difference ∆ of lifetimes over all such features. See Sec. 3.3 of the SI for further details. , and contracts from AFOSR and DARPA. We thank James Gleeson, Ezra Miller, Hal Schenck, and Sayan Mukherjee for helpful discussions. We used the London road data from Ref. [32] and the London metropolitan data from Ref. [47].

Additional Theory for the Noisy Ring Lattice
In this section, we conduct additional analysis of the Watts threshold model (WTM) on noisy ring lattices, which we introduced on page 2 of the main text (see the section called "Noisy Geometric Networks") and which we describe further in Sec. 4.1 of the present document.

Wavefront Propagation (WFP)
In the main manuscript, we found that wavefront propagation (WFP) has the following sequence of critical thresholds: where d (G) and d (NG) , respectively, denote a node's geometric and non-geometric degree for the noisy ring lattice with N nodes. (A node's "geometric degree" is its number of geometric stubs, and a node's "nongeometric degree" is its number of non-geometric stubs.) A wavefront propagates with a speed of at least k nodes per time step for T ∈ T , there is no WFP. For a contagion that consists of a single cluster that is expanding by WFP in two directions along a noisy ring lattice, the size q(t) of the contagion (i.e., the number of nodes that have adopted the contagion) for time t ∈ {0, 1, 2, . . . } has a lower bound of where and the factor of 2 accounts for WFP in both directions along the ring. We derived Eq. (2) for WFP in the absence of ANC. It is possible, however, that some nodes have both geometric and non-geometric neighbors that have adopted a contagion. This accelerates WFP. For large networks (i.e., N 1), we find that such acceleration occurs infrequently early in a WTM contagion but rather frequently towards the end of a contagion (i.e., just before q(t) → N , which is when a contagion saturates a network). In Sec. 1.2, we show that the probability that a node in the boundary of a contagion has an infected non-geometric neighbor is O(q(t)/N ). Therefore, WFP is improbable (because q(t) is small, but N is large) in the early stages of a contagion on a large network. When t = 0, for example, we have N . However, during the late stages of a contagion (i.e., q(t) ≈ N ), accelerated WFP is very likely at every time step. Therefore for small q(t), Eq. (2) is both a lower bound for q(t) and an approximation for it. In general, the speed of a WFP increases with time until it reaches an upper bound of d (G) /2 nodes per time step. This bound corresponds to the situation in which all nodes that are incident by geometric edges to one side of a contagion cluster become infected during each time step. Note that there is no acceleration of WFP when k (WFP) = d (G) /2, as the wavefront is already propagating at its fastest rate.

Appearance of New Contagion Clusters (ANC)
Equation [2] in the main manuscript gives a sequence of critical thresholds for the appearance of new contagion clusters (ANC): , a node will adopt a contagion if at least d (NG) − k non-geometric neighbors , the contagion cannot spread exclusively by exposure to the contagion via non-geometric edges. In this section, we show by considering spreading exclusively on the subgraph of non-geometric edges that the rate of ANC of a WTM contagion increases as the contagion threshold T decreases.
We first consider the probability that a given node has exactly k infected non-geometric neighbors, given that q(t) of the N nodes are infected at time step t. First, let's consider the case k = 1, in which a node i has exactly one infected non-geometric neighbor. Given that node i has d (NG) non-geometric edges (which we label as e 1 , . . . , e d (NG) ), there are d (NG) possible outcomes such that k = 1. For example, e 1 is incident to an infected node and the remaining edges are incident to uninfected nodes, e 2 is incident to an infected node and the remaining edges are incident to uninfected nodes, and so on. The probability that edge e 1 (which we recall that we placed uniformly at randomly) is incident to an infected node is q(t) N −1 , as there are q(t) such potential infected nodes and there are N − 1 other nodes (because there are no self-edges). Given that edge e 1 is incident to an infected node, the probability that edge e 2 is incident to an uninfected node is Given that edges e 1 and e 2 are, respectively, incident to an infected node and an uninfected node, the probability that edge e 3 is incident to an uninfected node is . We can continue similarly for the other nodes. Taking into account that there are d (NG) possible outcomes in which the d (NG) edges are incident to exactly one infected node, the probability that a node has exactly one infected non-geometric neighbor is More generally, the probability that a node has exactly k infected non-geometric neighbors is For fixed d (NG) and q(t) = O(N ), Eq. (6) simplifies in the limit N → ∞ to We now estimate the expected growth of a WTM contagion that spreads exclusively by ANC, g(t). In other words, we neglect exposures to the contagion from geometric edges, as we are assuming that they do not contribute to spreading. Defining it follows that the minimum number of non-geometric neighbors that need to be infected for a node i to adopt the WTM contagion is d (NG) − k (ANC) . Using Eq. (6) above result, we estimate that the expected contagion growth satisfies where we have calculated this expectation over the ensemble of noisy ring lattices. We again stress that Eq. (9) estimates the size of a WTM contagion for ANC independent of WFP and does not account for the joint effect of spreading via both geometric and non-geometric edges. It therefore provides a lower bound for the size of the contagion, q(t), for the regime exhibiting ANC but no WFP.

Activation Times Define a Metric
In this section, we prove that the set of activation times for a WTM contagion with threshold T on a network induces a metric on the node set V = {1, 2, . . . , N }. Letx (i) j denote the activation time (which we assume to be finite for all node pairs (i, j) ∈ V × V) for node i for a contagion initialized with the seed node {j}.
However, rather than showing this result for the specific case of a WTM contagion, we will prove a more general result using the observation that the growing set of infected nodes during one realization of a WTM contagion defines a filtration of the node set V. We will therefore prove that any "complete" and "consistent" set of filtrations (see the definitions below) on a finite set V induces a metric on V. Subsequently, because the filtrations that result from realizations of a WTM contagion on a given network with contagion seeds {j} for j ∈ {1, . . . , N } satisfy the conditions of completeness and consistency, it follows that

Complete and Consistent Filtrations
Before proving that the set of activation times-and, more generally, any "complete" and "consistent" set of "filtrations"-leads to a metric, we give a few necessary definitions.

Definition: Filtration.
Consider a sequence of sets N t for t ∈ {1, 2, . . . }. The sequence of sets is called a filtration [6,7] if it has the property that N t ⊆ N t+1 for all t.
Definition: Completeness. Let V be a finite set with cardinality |V| = N . We define a set of filtrations to be complete if there are N filtrations of the following form: for every j ∈ V, there exists a filtration such that Note that the filtration {N t (j)} consists of nested sets N t (j), where the innermost set is the element {j} and the outermost set (i.e., the t * j -th set) is the full set of indices, V.

Definition: Consistency.
Let V be a finite set with cardinality |V| = N , and suppose that one has a complete set of filtrations as defined above. We define the set of filtrations to be consistent if for any two filtrations {N t (i)} and {N t (j)}, the following is true: where the indices t and τ can be different from each other. Specifically, Eq. (11) implies that the two (11) is true for all i, j ∈ V.

Filtration-Induced Metrics
Theorem: A Metric Induced by Filtrations.
Let V be a finite set with cardinality |V| = N , and let N t (j) denote sets that define a complete and consistent set of filtrations on V. Additionally, let t j defines a metric on the set V.
However, we know by definition that j = N 0 (j) [and i = N 0 (i)], so it must be the case that i = j. It is trivial to show that m(i, j) = m(j, i). Finally, we will complete the proof by showing that m(i, j) satisfies the triangle inequality: m(i, j) ≤ m(i, k) + m(k, j). This step is a bit more complicated, and it relies on the completeness and consistency of the set of filtrations. Using the definition of m(i, j), it suffices to show that t k , we will prove that a ≤ b + c. Because the result is trivial when b ≥ a due to the non-negativity of c, we can assume that a > b. By definition, it must be the case that , which uses the completeness of the filtrations. Using the consistency property, it follows that , and so on. Repeating this procedure demonstrates that N c (k) ⊆ N b+c (j). Noting that i ∈ N c (k), it follows that i ∈ N b+c (j). It follows, in turn, that t (i) j ≤ b + c, which is equivalent to a ≤ b + c (i.e., the desired result).

Lemma: A Metric Induced by WTM Contagions.
Consider a network with node set V and edge set E that consists of a single connected component. Letx (i) j denote the activation time of node i for a WTM contagion with seed {j}. As before, we assume thatx j is a metric on the node set V.
Proof: It suffices to show that N realizations of a WTM contagion with the set of contagion seeds S (j) = {j} (for j ∈ V) produces a complete and consistent set of filtrations on V. It will be convenient to use the notation t We first prove completeness. Let N t (j) denote the set of nodes for realization j that have adopted the contagion by time t. Note that N 0 (j) = S (j) = {j} for each j. Additionally, N t (j) ⊆ N t+1 (j) for any t as nodes cannot unapt a contagion during a time step. Therefore, the sequence {N t (j)} t * j t=0 yields a filtration of the node set V that satisfies Eq. (10). It follows that the set of filtrations of the form is a complete set of filtrations. We now prove consistency. Consider two realizations of a WTM contagion on a single network. Let N t i (i) ⊂ V denote the set of nodes that are adopters at time t i for the i-th realization, and let N t j (j) ⊂ V denote the set of nodes that are adopters at time t j in the j-th realization. To have consistency, it must be true that . Suppose that t i and t j are times such that N t i (i) ⊆ N t j (j), and consider the spreading that occurs for a WTM contagion during one time step. By definition, the update rule for each node is identical across all realizations of a WTM contagion. (In other words, for a node k the fraction of infected neighbors f k must surpass T for adoption.) Additionally, for any node k ∈ V, increasing the infection size can only increase f k . Hence, if for some node k, f k > T when nodes N t i (i) are infected, then f k > T is also true if we instead consider a superset of N t i (i) to be infected. Thus, the set N t i +1 (i) of adopters at time step (t = t i + 1) must satisfy N t i +1 (i) ⊆ N t j +1 (j). The N realizations of a WTM contagion with seeds S (j) = {j} (where j ∈ {1, . . . , N }) thus produce to a complete and consistent set of filtrations on V for which t j . It follows that the activation times define a metric on the node set V.

Extended Discussion of Point Cloud Analyses
In the main manuscript, we introduced the idea of a WTM map, which embeds a network's nodes as a (potentially) high-dimensional point cloud. Specifically, we let x (i) j denote the activation time of node i for a WTM contagion that we initialize with cluster seeding centered at node j, and we define the vectors In the section entitled "WTM Maps" on page 3 of the main text, we introduced three versions of a WTM mapping: In this section, we restrict our discussion to regular WTM maps, but one can apply the same techniques to any of the three WTM maps to obtain insights about WTM contagions-in particular, on the extent to which the spreading dynamics follow a network's underlying manifold. It can also reveal network structure (e.g., potentially hidden or unknown low-dimensional manifold structure), which one can in turn study by examining topological, geometric, and dimensionality properties of the resulting point cloud. In this section, we provide further details on our methodology for analyzing WTM maps. We discuss the following three items: 1. The Pearson correlation coefficient ρ, which we use to investigate a point cloud's geometry.
2. The embedding dimension P , which we use to investigate a point cloud's dimensionality. 3. The difference ∆ = l 1 − l 2 in lifetimes of the two most persistent 1-cycles [i.e., one-dimensional (1D) holes], which we use to investigate a point cloud's topology.

Analysis of Geometry
Studying the geometry of a WTM map (e.g., V → {x (i) } i∈V ) can reveal the extent to which the geometry of a WTM contagion compares to the underlying geometry for a network. In our study, we restrict our attention to noisy geometric networks in which the nodes V have intrinsic locations {w (i) } i∈V ∈ M on a manifold M ⊂ R p that lies in the p-dimensional ambient space R p (which we equip with a metric from the Euclidean norm w 2 = p j=1 w 2 j ). In particular, we investigate the extent to which the distance between two nodes after the WTM map relates to the intrinsic distance between those nodes in the embedding space. This intrinsic distance is given by the metric For the point cloud that results from the WTM map {x (i) } ∈ R N , we also use the Euclidean norm. The distance between nodes i and j is thus given by Given two distances m and m (WTM) for each pair of nodes (i, j), we compute the Pearson correlation coefficient between them over all unordered pairs i and j. Because i = j for distinct nodes, there are N (N − 1)/2 such pairs. Note that calculating Eq. (14) requires the activation time x (i) j to be finite for all nodes i and realizations j of a WTM contagion. This, however, is not the case whenever there is a node that never becomes activated.
} is too large for the example of the noisy ring lattice.) For practical purposes, for any numerical simulation in which node i eventually becomes infected. In Sec. 6 we discuss other methods for handling the case with activation times of infinity.

Analysis of Dimensionality
We study the dimensionality of WTM maps by exploring embedding dimension. Given the point cloud {x (i) } i∈V ∈ R N , we let {x (i) (p)} (for each i) denote the linear projection onto R p that we obtain from principal component analysis (PCA) [5,15]. Let ρ (p) denote the Pearson correlation coefficient that relates node-to-node distances m (WTM) from the original point cloud to those in the projected point cloud, which are given by Specifically, ρ (p) is given by Eq. (14) under the variable substitution We estimate the embedding dimension as the smallest dimension P such that the residual variance is (strictly) less than 0.05. That is, P = min{p|R p < 0.05}. For our calculations of embedding dimension, we only calculate dimensions up to P = 20, as this simplifies the computational overhead of calculating P . Our motivation for this simplification (besides reducing computational cost) is that we are particularly interested in determining whether or not P is close to the known embedding dimension (e.g., P = 2 for the unit circle in R 2 ).

Analysis of Topology
In this section, we explain how to analyze the topology of a point cloud {x (i) } i∈V ∈ R N . We will present this discussion generally for a point cloud U = {u i } n i=1 ∈ R N (i.e., there are n points u (i) in N dimensions), and note that for the analysis of a WTM map one can assume, for example, u (i) = x (i) and n = N .
A set U has a very simple topology. If u (i) = u (j) for i = j, then U consist of N distinct connected components that correspond to the points {u (i) }. There are no 1-cycles in U. To infer the topology, if present, of an underlying manifold that gives rise to a point cloud, we study its topology on different spatial scales. In particular, we are interested in the topology of the sets for different values of r ∈ [0, ∞). That is, we study the topology of sets that we construct as the union of balls centered at points u (i) ∈ U with radius r. Note that U (0) = U. We use the Euclidean norm, but it is also possible to use other norms. Figure 1: We study the topology of a point cloud U by examining the persistent homology that is induced by a Vietoris-Rips filtration. This entails examining simplicial complexes that are created by forming, for every set of points, a simplex (e.g., an edge, a triangle, a tetrahedron, etc.) whose diameter is at most r.
Increasing r from 0 and considering how a simplicial complex evolves yields a filtration. In panel (a), we show point cloud U = {u (i) } given by a noisy sample of the unit circle. Note that in this example there are n = 10 points in N = 2 dimensions. In panels (b)-(d), we show U (r) for r ∈ {0.22, 0.6, 0.85}. One can approximate the homology of U (r) using a Vietoris-Rips complex that is given by the nodes, edges, and triangles that we show in the panels. The first 1-cycle in U (r) occurs at r = 0.22. It is a result of the noisy sampling, and it is filled in almost immediately. The dominant 1-cycle (i.e., the 1-cycle that corresponds to the ring and persists across many spatial scales) is shown in panel (c) and is born at r = 0.5. It persists until r = 0.81. Identifying a single persistent 1-cycle indicates that the point cloud lies on a ring manifold.
We start with an example. In Fig. 1, we show a noisy point cloud that we sample from a ring manifold. In particular, we have sampled the points uniformly from a unit circle in R 2 , and we add a small amount of noise to their locations in the embedding space R 2 . When r = 0, there are 10 distinct connected components that corresponding to the points. As we increase r, four of the components merge and create a 1-cycle [see Fig 1(b)]. As we continue to increase r, this 1-cycle becomes filled in very soon after its birth. After it is filled in, another 1-cycle appears when r = 0.5 [see Fig 1(c)]. This 1-cycle persists for a larger range of r values than the first 1-cycle, and it appears to correspond to the ring manifold that underlies the point cloud. This illustrates that one can study the topology of a point cloud by examining 1-cycles that persist across different spatial scales. To make this statement more quantitative, we will employ tools from persistent homology [4,7].
For every set U (r) , one can assign homology groups H c (U (r) ), where c ∈ {0, 1, 2, . . . }. The rank β c of the group H c (U (r) ) is the number of c-dimensional topological features that are present in U (r) . In particular, β 0 counts the number of connected components, β 1 counts the number of 1-cycles (which one can construe as a 1D hole or loop), and β 2 counts the number of cavities [i.e., two-dimensional (2D) holes]. The fact that U (r) ⊆ U (r ) for r ≤ r is very important. As we discussed in Sec. 2.1, a sequence of sets with this property is called a "filtration." Thus, for any sequence  Fig. 1. It contains two points, which correspond to the two observed 1-cycles. One point (the red diamond) indicates a 1-cycle that persists over a long range of spatial scales. Its lifetime is thus large. The second point (the yellow square) indicates another 1-cycle. Its small lifetime l 2 = r d (2) − r b (2) indicates that it dies a short time after it is born, so it does not persist over many spatial scales. The large difference ∆ = l 1 − l 2 in the top two lifetimes indicates that the point cloud contains a single dominant 1-cycle and offers strong evidence that the point cloud lies on a ring manifold.
In the present paper, we are interested in understanding the birth and death of 1-cycles of U (r) as we vary r. The quantity β 1 encodes such information, which one can summarize by drawing a persistence diagram. In Fig. 2, we show the β 1 persistence diagram for the point cloud in Fig. 1. The diagram contains two points, which correspond to the two 1-cycles that we discussed previously. The horizontal ("birth") axis of the point is the value of r at which the 1-cycle corresponding to this point first appears in U (r) , and the vertical ("death") axis indicates when the 1-cycle is filled in. Enumerating the points i = 1, 2, . . . , for every point i which has coordinates (r b (i), r d (i)) in the persistence diagram, we define the "lifetime" l i = r d (i) − r b (i), and we denote the set of lifetimes of all points by L = {l 1 , l 2 , . . . , } (which we order such that l 1 ≥ l 2 ≥ . . . ). Topological features with longer lifetimes (i.e., ones that are more persistent) indicate more dominant features in a point cloud. In our example, there is one point with a very short lifetime that corresponds to a 1-cycle that arises for a single spatial scale due to the noisy sampling. The other point has a much larger lifetime, which indicates that its associated 1-cycle persists across many spatial scales. We thereby identify the ring structure of the sampled manifold. For the purpose of identifying whether or not a point cloud lies on a ring manifold, we summarize persistence diagrams by using the difference ∆ = l 1 − l 2 between the most persistent lifetimes. Large values of ∆ correspond to persistence diagrams that consist of a single point with a large lifetime, as we expect for a point cloud that lies on a ring manifold.
In practice, computing the persistent homology of a set U (r) is complicated. However, the so-called "Nerve Theorem" [3] guarantees that the homology of U (r) is the same as the homology of a correspondinǧ Cech complex, which simplifies analysis but is computationally expensive to construct. Therefore, we study an approximation of theČech complex that is known as the Vietoris-Rips complex. For a given point cloud U = {u (1) , u (2) , . . . , u (n) } ∈ R N and r ∈ R, the Vietoris-Rips complex VR (r) is a complex that consists of the simplices (u (s 1 ) , u (s 2 ) , . . . , u (s k ) ) such that u (s i ) − u (s j ) 2 ≤ r for all s i and s j . In the present paper, we are interested only in identifying the 1-cycles in VR (r) so it is sufficient for us to use only 0-simplices (i.e., points), 1-simplices (i.e., line segments), and 2-simplices (i.e., triangles).
To compute persistent homology, we use the software package Perseus [10] (version 3.0 Beta), and we also check some of our results using the javaPlex Persistent Homology Library [14]. To construct Vietoris-Rips filtrations VR (r) for a given point cloud resulting from a WTM map (e.g., {x (i) }), we use Eq. (13) to define distances between points. As an input to Perseus, we use the dissimilarity matrix in which the entry in the i-th row and j-th column encodes the distance between nodes i and j given by Eq. (13).
In Fig. 3, we study β 1 persistence diagrams for point clouds that result from the application of WTM maps to noisy ring lattices. We thereby reveal the absence versus presence of 1-cycles in the point cloud.  Fig. 3 represents the point that corresponds to the most persistent 1-cycle. The second-most persistent 1-cycle is indicated by a yellow square, and the remaining points in the persistence diagram are shown by white circles. If there is only one dominate 1-cycle, then the separation between the red diamond and the other points is large. To measure this separation, we calculate ∆ = l 1 − l 2 , where l 1 and l 2 are the lifetimes of the dominant and the second most dominate 1-cycle, respectively. The background coloration reflects the value of ∆. To construct a filtration using various values of r, we consider 100 evenly-spaced values of r that range from 0 up to the maximum distance distance r max max i,j∈V ||z (i) − z (j) || 2 between any two points. For plotting purposes, we normalize all r values by r max , so that ∆ ∈ [0, 1]. It follows that ∆ ≈ 1 indicates the presence of the ring topology, whereas small values of ∆ indicates its absence. For a given point cloud, we apply a Vietoris-Rips filtration to yield the β 1 persistence diagram that summarizes the multiscale 1D features (i.e., 1-cycles or loops). In each persistence diagram, we use a red diamond to mark the most persistent 1-cycle, a yellow square to mark the second most persistent 1-cycle, and white circles to indicate the remaining 1-cycles we find. Note that the lifetime l i of a given point i corresponds to the height above the diagonal (see the blue lines). We shade the background color of each persistence diagram according to the difference ∆ = l 1 − l 2 between the two largest lifetimes. Note that ∆ ∈ [0, 1] due to normalization (see text). The magnitude of ∆ provides strong evidence regarding whether or not a given point cloud lies along a 1D ring topology. In the main manuscript, we thus summarize our topological analysis with the parameter ∆ [see Fig. 6(c) in the main manuscript]. Note that we do not do any calculations (see the black squares) for WTM maps in which some node has an activation time of infinity [i.e., when there is at least one pair (i, j) such that z

Generalizations of the Noisy Ring Lattice
In the main manuscript, we analyzed the WTM on noisy ring lattices. In this section, we review our construction of noisy ring lattices and introduce three additional families of noisy geometric networks that use an underlying a ring manifold. In these families, we introduce heterogeneity into the nodes' geometric and non-geometric degrees, which we now denote, respectively, by d for a given node i. We denote their means over the nodes by d , respectively. We therefore adjust our definition of the ratio α to denote the ratio of the mean non-geometric degree to the mean geometric degree: For the network families we define in Sec. 4.1, it is equivalent to state that α denotes the number of nongeometric edges to the number of geometric edges in a given network.

Families of Noisy Geometric Networks on a Ring Manifold
We now define four families of noisy geometric networks on a ring manifold. We label these families as (a), (b), (c), and (d). In Fig. 4, we illustrate an example network for each family and plot its corresponding adjacency matrix and degree distribution.
• Family (a). To generate the noisy ring lattice that we studied in the main manuscript, we place N nodes evenly spaced on the unit circle in R 2 so that each node i has location w (i) = [cos(θ i ), sin(θ i )] T with θ i = 2πi/N . We then add geometric edges between all node pairs (i, j) ∈ V × V that are nearest neighbors, so that each node i has exactly d is necessarily even because of symmetry. We then assign non-geometric edges randomly using (a slight modification of) the configuration model [2] so that each node has exactly d non-geometric edges. As in the configuration model, we connect ends of edges (i.e., "stubs") to each other uniformly at random, but we disallow self-edges and multi-edges. Note that our implementation of the configuration model is slightly modified from the original version so that we can guarantee that the set of geometric edges is disjoint from the set of non-geometric edges. Specifically, if we propose a candidate edge between two nodes that would lead to a disallowed situation (i.e., it would lead to a self-edge, multi-edge, or an edge that is already a geometric edge), then we discard the candidate edge, and we propose a new candidate edge as prescribed by the configuration model. The resulting network is a k-regular network (with k = d /2 geometric edges and N d (NG) i /2 non-geometric edges. The geometric edges form a deterministic backbone (as in the Newman-Watts variant [11,12] of the Watts-Strogatz model [17]), whereas we obtain non-geometric edges through a probabilistic process.
• Family (b). The first generalization of the noisy ring lattice in family (a) is to allow heterogeneity in the number of non-geometric edges that are incident to a given node i (i.e., in its non-geometric degree d ). The total number of non-geometric edges is still equal to the constant N d but we now distribute them uniformly at random among the N ·(N −1−d (G) ) 2 possible edge locations that are unoccupied by geometric edges. Hence, the subgraph that consists only of non-geometric edges is an Erdős-Rényi (ER) network when N d (G) [2]. The distribution of non-geometric degrees is thus a binomial distribution that is centered at d = 2 for four families of noisy geometric networks on a ring manifold: family (a), the noisy ring lattice (which we also discuss in the main manuscript), for which nodes are evenly spaced and have constant geometric and non-geometric degrees; family (b), for which the nodes are evenly spaced, have constant geometric degrees, and have heterogeneous non-geometric degrees; family (c), for which we sample the node locations from the ring using a random process (see the text), and the nodes have heterogeneous geometric degree and constant non-geometric degrees; model (d) for which we sample the node locations from the ring using the aforementioned random process, and the nodes have heterogeneous geometric and non-geometric degrees. The top row depicts example networks, where blue solid and red dashed lines indicate geometric and non-geometric edges. The center row depicts the corresponding adjacency matrices, where one can observe the the blue pixels indicating geometric edges align along the diagonal, whereas the red pixels indicating non-geometric edges arise randomly. The bottom row depicts the corresponding distributions for the geometric (red), non-geometric (blue), and total (grey) degrees. Note that for families (a) and (b) the geometric degrees are identical, d • Family (c). Our second generalization of (a) is to allow heterogeneity in the node locations on the unit circle in R 2 , which in turn leads to heterogeneity in the number of geometric edges that are incident to a given node i (and hence in its geometric degree d (G) i ). To make such a generalization in a tunable manner, we assign the node locations (or, equivalently, angles {θ i } in the case of the unit circle) to be evenly spaced as for family (a), and we then perturb these by a random variable δθ i , so that the location for each node i is given by [cos(θ i + δθ i ), sin(θ i + δθ i )] T . We consider Gaussian-distributed random variable δθ i ∼ N 0, (s 2π N ) 2 , where one can vary s to adjust the amount of heterogeneity in node location along the ring manifold. The choice s = 0 recovers the original node locations, and s → ∞ corresponds to sampling locations on the unit circle uniformly at random. Unless we specify otherwise, we use s = 1/2. To generate geometric edges, we choose an > 0 and place edges between all pairs of nodes i and j such that |θ i − θ j | < . To compare networks from family (c) to networks from families (a) and (b), for which the nodes have the identical geometric degree d , we choose the parameter so that each network in family (c) has exactly N d

Perturbed Bifurcation Results
Equations (1) and (2)  }, that are (potentially) specific to that node. Consequently, the nodes can exhibit qualitatively dissimilar contagion dynamics with respect to WFP and ANC. For example, for a given threshold T , some nodes can have geometric and non-geometric degrees that support WFP but no ANC, whereas other nodes can have degrees that support both WFP and ANC. Nevertheless, one can construe the bifurcation analysis that we developed for family (a) as an approximate bifurcation analysis for the other families. In this light, note that if the degree heterogeneities are sufficiently small compared to the mean degrees, then we still identify four different qualitative regimes of WTM contagion dynamics that are marked by the absence versus presence of WFP and ANC. (The boundaries that separate these regimes are only slightly perturbed from what we found for family (a).
More precisely, for each node i let δ denote the difference between its geometric degree and the mean. Similarly, let δ denote the difference between its nongeometric degree and the mean. Restricting our attention to the critical thresholds given by Eqs.
Expressions (17) and (18) is small]. We therefore interpret our bifurcation analysis for network family (a) as an approximate bifurcation analysis for network families (b)-(d). Moreover, this interpretation is expected to be increasingly accurate as the mean degrees become larger relative to the heterogeneity in degrees.
In Fig. 5, we plot curves that indicate the node-specific critical thresholds given by Eqs. (17) and (18)  Note for all panels that there exist parameter regimes in which the nodes support similar contagion phenomena (i.e., WFP and no ANC, WFP and ANC, no WFP and ANC, or no WFP and no ANC) even through their degrees are heterogeneous. Therefore, one can construe the set of curves that one obtains for multiple values of δ (G) i and δ (NG) i as a "thickening" of the boundary between regions of qualitatively different dynamics. In other words, as we vary parameters, we see that the transitions between regions of different dynamics can occur with different timing for different nodes in a network. Note, however, that this interpretation does not take into account the distribution of node degrees, as we have only shown the critical threshold curves in Fig. 5 for degrees that are near the mean degrees (i.e., for |δ  If the perturbed node degrees are bounded and sufficiently small compared to the nodes' degrees (e.g., we |δ (G) | ≤ 2 and |δ (NG) | ≤ 2 in the present figure), then we still obtain four qualitatively different contagion regimes for all nodes, which we again characterize by the presence versus absence of WFP and ANC. However, because of the heterogeneity of the nodes' degrees, transitions between these regimes in the (T, α) parameter plane occur at different values for different nodes. That is, the boundaries between the WTM contagion regimes have "thickened." We note for any fixed |δ (G) |, |δ (NG) | > 0 that the perturbed curves approach those that correspond to δ (G) = δ (NG) = 0 as the mean geometric and non-geometric degrees increase.

Complex Contagions on Synthetic Networks on a Ring Manifold
In this section, we provide results for numerical experiments in which we study the geometry, topology, and dimensionality of point clouds that result from the application of symmetric WTM maps V → {z (i) } to noisy geometric networks generated by network families (a)-(d). We thereby reveal the extent to which a WTM contagion exhibits WFP that follows the underlying ring manifold (i.e., the extent to which spreading occurs across a network subgraph that contains exclusively geometric edges) versus ANC. In particular, WFP is more prevalent than ANC when one can identify the properties of the underlying manifold in the point cloud that results from a WTM map.
To give some perspective for these numerical experiments, we compare our results for point clouds produced by WTM maps to results from two well-known methods of mapping network nodes as a point cloud: a Laplacian eigenmap [1] and Isomap [15]. In particular, we consider a 2D Laplacian eigenmap in which we map each node i to [v is the eigenvector that corresponds to the j-th eigenvalue λ j of the unnormalized Laplacian matrix L (i.e., Lv (j) = λ j v (j) ) and we have ordered the eigenvalues so that 0 = λ 1 < λ 2 ≤ λ 3 ≤ · · · ≤ λ N . The unnormalized Laplacian matrix has the form L is the total degree of node i and A is the adjacency matrix. As we discussed on page 3 of the main text (see the section entitled "WTM Maps"), Isomap entails mapping network nodes based on the shortest paths between nodes. It corresponds to a WTM map with T = 0 if we initialize the contagions with node seeding rather than cluster seeding. As we will see, when assessing the extent to which point clouds that result from WTM maps resemble the underlying ring manifold, we typically find a range of threshold values for which the geometry, dimensionality, and topology of the manifold is more apparent in WTM maps than for Laplacian eigenmap and Isomap algorithms. For other threshold values, the manifold is less apparent for WTM maps than for the other methods.
It is important to note that Laplacian eigenmaps and Isomap were introduced originally for the purpose of nonlinear dimension reduction rather than for network analysis. That is, they were introduced to map a high-dimensional point cloud to a network and then to map that network to a low-dimensional point cloud. Therefore, applying a Laplacian eigenmap or Isomap directly to a network-especially one that is unweighted-is different from what they were designed to do. In particular, for networks that arise from high-dimensional data-e.g., ones with nodes that are connected to each other by applying a k-nearestneighbor algorithm-one often weights network edges based on distances in the original, high-dimensional point cloud. Incorporating such additional information can, of course, improve the results of dimension reduction.

Numerical Results for Geometry
In this section, we compare the geometry of symmetric WTM maps for networks in the families (a)-(d) through a Pearson correlation coefficient ρ, as discussed in Sec. 3.1. We also investigate the effects of varying the mean geometric and non-geometric degrees and the network size N on WTM maps (when we hold other parameters constant). We show our results in Figs. 6-8, for which panels (a)-(d), respectively, give our results for network families (a)-(d). Unless we indicate otherwise, in these and subsequent figures, results are shown for one network from each family.
In Fig. 6, we plot ρ for point clouds resulting from symmetric WTM maps for the (T, α) parameter plane. The solid and dashed curves yield approximate bifurcation curves, which we obtain by Eqs. (17) and (18)  . From comparing the four panels to each another, we find that the agreement between T (WFP) 0 and the observed shifts in ρ decreases as the node degrees become more heterogeneous. In particular, the transition that is imposed by T (WFP) 0 appears to shift to smaller values of T , so the heterogeneities that we introduce in network families (b)-(d) mostly affect the regime in which T ≈ T (WFP) 0 . However, the qualitative behavior of WTM contagions in the (T, α) parameter plane is similar for all four families of networks.
In Fig. 7, we study the effect on ρ for symmetric WTM maps when we increase the mean node degrees, d . Fixing α = 1/3, we plot ρ as a function of the threshold T . This amounts to examining vertical cross sections from the four panels in Fig. 6. We study the effect of varying mean node degree by showing results for ( d . We also show results for a 2D Laplacian eigenmap [1] (horizontal dashed lines) and Isomap [15] (horizontal dotted lines) applied to our (unweighted) networks. Compare ρ values for the WTM maps with various mean degrees, and note that increasing the mean degrees smoothens the dependence of ρ versus T . Specifically, for smaller T values (e.g., for T < 0.3) the discontinuous jumps in ρ become smaller as the mean degrees increase. Interestingly, increasing heterogeneity in the node degrees also smoothens the ρ versus T curves. For example, the curves are smoother for network families (b)-(d) than for family (a). Note also that in all panels, we observe an abrupt drop in ρ for T ≈ T . Finally, note in all panels that ρ in general increases as we increase the mean degrees, and we observe similar increases in ρ for the Laplacian-eigenmap and Isomap algorithms. Thus, in this series of experiments, increasing mean node degree improves the ability of the maps to translate the geometry of the underlying manifold to the resulting point cloud.
In Fig. 8, we study the geometry of symmetric WTM maps by plotting ρ versus T for networks of various sizes N . We fix d ] to illustrate how ρ depends on N . As N increases, we observe that ρ systematically decreases for WTM maps that correspond to contagions in which WFP is not the dominant phenomenon. However, for WTM maps in which WFP dominates (e.g., when T ∈ [0.2, 0.25]), we find that ρ remains above 0.85. This provides strong evidence that, for this parameter regime, WTM maps translate the geometry of the underlying ring manifold to the resulting point cloud for a wide range of network sizes (with the other parameters held constant). Importantly, one does not obtain such independence with network size when using a Laplacian eigenmap or Isomap. In those cases, we find that ρ systematically decreases as N increases (with the other parameters held constant).
Ring with constant d  Figure 7: We study the geometry of symmetric WTM maps by calculating a Pearson correlation coefficient ρ as a function of WTM threshold T to compare node-to-node distances for the WTM map {z (i) } ∈ R N to those for the node locations {w (i) } ∈ R 2 on the ring manifold. Panels (a)-(d), respectively, illustrate results for network families (a)-(d), and they amount to vertical cross sections of the corresponding contour plots in Fig. 6 (i.e., for a constant value of α), although we show results for several choices of mean node degrees: ( d In each panel, we study WTM maps on a noisy ring network with N = 1000 nodes with α = 1/3. We also plot ρ for a 2D Laplacian eigenmap [1] (dashed lines) and for the Isomap algorithm [15] (dotted lines). In all panels and for all mapping algorithms, increasing mean node degree tends to increase ρ, so the ability of the maps to translate the underlying ring manifold's geometry to a point cloud improves with increasing mean node degree for these experiments. Note also that the curves for the largest mean degree (magenta × symbols) remain more consistent across the panels. In all panels and for all mapping algorithms, increasing the network size N tends to decrease ρ, except for WTM maps that are characterized by WFP and little (or no) ANC (see the discussion in the text).

Numerical Results for Dimensionality
In this section, we examine the dimensionality of point clouds that result from symmetric WTM maps that we apply to networks on the ring manifold by studying their embedding dimension P . As discussed in Sec. 3.2, we define the embedding dimension P as the smallest dimension p such that the residual variance R p for the projection onto R p is small. In practice, we use PCA for such projections, and we specify "small" as being (strictly) less than 0.05. (In other words, we lose less than 5% of the variance after the projection.) We show our results in Figs. 9-10, for which panels (a)-(d), respectively, give our results for network families (a)-(d).
In Fig. 9, we plot P in the (T, α) parameter plane for networks with N = 200 nodes, mean geometric degree d . We also plot the approximate bifurcation curves given by Eqs. (17) and (18) that panel (a) is similar to the plot in Fig. 6(b) of the main manuscript. We observe in all panels that WTM maps for the contagion regime that we expect to exhibit WFP but no ANC yields point clouds {z (i) } with a small embedding dimension of P ≈ 2. This result is unsurprising, because the ring manifold is exactly the unit circle in R 2 . Note that this low dimensionality also persists into the regime that we expect to exhibit both WFP and ANC, although the embedding dimension P in general increases as one moves away from the regime exhibit WFP and no ANC.
In Fig. 10, we continue our investigation of the dimensionality of point clouds that result from the application of symmetric WTM maps to networks on the ring manifold by showing their embedding dimension P as a function of threshold T . One can construe the curves of P versus T as a vertical cross section of the contour plots in Fig. 9, although we now consider several different choices of mean node degrees: We also show values (horizontal dotted lines) of P versus T for the point clouds that we obtain by applying Isomap [15] to the networks. We obtain horizontal lines because Isomap does not include any dependence on T . We do not investigate the dimensionality of the 2D Laplacian eigenmaps, as we fix their dimension to 2 in our study.
Note that the curves in panels of Fig. 10 are rather similar to each other. In particular, we consistently observe in all panels the smallest embedding dimension P for the regime in which we expect a WTM contagion to exhibit WFP without ANC [i.e., for T ∈ (1/4, 3/8)]. Additionally, we consistently identify the correct embedding dimension (i.e., P = 2) for this regime provided that the mean degrees are sufficiently large (e.g., see the magenta × symbols). For smaller mean degrees, we still observe that P is small for a similar range of threshold T . However, in general the curves of P versus T suggest that smaller mean degrees lead to larger embedding dimensions in this numerical experiment. Examining P for Isomap (in which we map nodes based on shortest paths), we observe that the embedding dimension P is always at least 10 in this experiment. Additionally, the embedding dimension P for Isomap appears to decrease systematically as the mean degrees increase. Thus, using shortest paths to map nodes for network families (a)-(d) leads to point clouds with a dimensionality that is higher than P = 2. However, it might be possible to recover the correct embedding dimension of the ring manifold when the mean degrees are sufficiently large (keeping all other parameters fixed). Finally, note that P ≤ 20 in all panels. Recall that this is the maximum value of P that we can observe because it is the largest dimension that we examine.  Figure 9: We examine the dimensionality of point clouds that result from symmetric WTM maps that we apply to networks on the ring manifold by studying their embedding dimension P = min{p|R p < 0.05}, where R p denotes the residual variance for the projection onto R p (see the discussion in the text). We plot P in the (T, α) parameter plane for networks with N = 200 nodes, mean geometric degree d and various values of the non-geometric degree d (NG) i . As before, panels (a)-(d), respectively, illustrate results for network families (a)-(d). We also plot by solid and dashed lines, respectively, theoretical critical threshold values given by Eqs. (17) and (18) In all panels, we see that WTM maps for the contagion regime that we predict to be characterized by WFP but no ANC corresponds to point clouds with an embedding dimension of P ≈ 2, which agrees with the fact that the ring manifold is embedded in R 2 . This low-dimensional structure persists into the regime that we predict has both WFP and ANC, although the embedding dimension P increases as one moves away from regime exhibiting WFP and no ANC.  Figure 10: We study the dimensionality of point clouds resulting from symmetric WTM maps by showing their embedding dimension P as a function of T for networks with N = 200 and α = 1/3. One can construe these curves of P versus T as vertical cross sections for the contour plots in Fig. 9, although we show results for several choices of mean node degrees: ( d (18,6) (magenta × symbols). As before, panels (a)-(d) correspond to network families (a)-(d). In all panels, we identify the correct dimension (i.e., P = 2) for the regime that we expect WTM contagion to exhibit WFP without ANC [i.e., T ∈ (1/4, 3/8)] if the mean node degrees are sufficiently large (see magenta × symbols). The curves for the other mean degrees also consistently depict small values of P for a similar range of threshold T . We also plot P versus T for the point clouds that we obtain by mapping the nodes based on shortest paths, as in Isomap [15] (horizontal dotted lines). For these experiments, P ≥ 10 in all panels, although it appears to decrease systematically with increasing mean node degrees.

Numerical Results for Topology
In this section, we study the topology of point clouds resulting from symmetric WTM maps applied to noisy geometric networks on the ring manifold. As discussed in Sec. 3.3, we examine the difference between the largest lifetimes ∆ = l 1 − l 2 for 1D features (i.e., 1-cycles). We determine the persistence of these 1-cycles across spacial scales using a Vietoris-Rips filtration of the point cloud [6,4], which we normalize so that ∆ ∈ [0, 1]. We show our results in Figs. 11-12, for which panels (a)-(d), respectively, give our results for network families (a)-(d).
In Fig. 11, we plot ∆ for the (T, α) parameter plane. We show results for networks with N = 200 nodes, which have a mean geometric degree of d . For all four network families, ∆ is largest for WTM maps that correspond to the contagion regime that we predict to be characterized by WFP without ANC. By comparing the panels, we see that the identifiability of the underlying ring topology (as indicated by ∆ ≈ 1) decreases as we increase the heterogeneity in the nodes' degrees. For example, panel (a) includes parameter values (T, α) for which ∆ > 0.9, but ∆ < 60 in panel (d) for the same area of the parameter plane (T, α).
In Fig. 12, we continue our investigation of the topology of the point clouds resulting from symmetric WTM maps by fixing α = 1/3 and plotting ∆ as a function of T . One can construe these curves of ∆ versus T curves as examining vertical cross sections from the panels in Fig. 11, although we show results for several choices of mean degrees: ( d As before, the networks are constructed with N = 200 nodes. Observe in Fig. 12 that ∆ tends to decrease as the heterogeneity of the network increases. For example, the values of ∆ in panels (b) and (c) tend to be smaller than those in panel (a), and the ∆ values in panel (d) tend to be even smaller than those in panels (b) and (c). This makes it harder to successfully identify the ring topology in the point clouds. This is most evident for the curves that correspond to ( d (G) i , d (NG) i ) = (6, 2) (red triangles). Although we observe large values of ∆ in panels (a) and (b) for the point clouds for the contagion regime that we predict to exhibit WFP without ANC [i.e., T ∈ (1/4, 3/8), we find that the values of ∆ for this regime are much smaller in panels (c) and (d). We also note that ∆ does not depend on mean node degrees in a simple manner in this experiment. In network family (a), for example, when comparing the ∆ versus T curve for ( d , we see that larger ∆ values are observed when increasing the mean degrees. However, restricting out attention to the range T ∈ (1/4, 3/8), the ∆ versus T curve for ( d (24,8) yields ∆ values that are smaller than that for ( d (12,4). This nontrivial behavior might be due to the relatively small differences in magnitude between the mean node degrees and the network size (N = 200) that we use for this experiment. We also computed ∆ for Isomap for this experiment, but we found that ∆ ≈ 0 in all cases. We thus omit these results from Fig. 12.
Ring with constant d  Figure 11: We study the topology of symmetric WTM maps by calculating the difference between the largest lifetimes ∆ = l 1 − l 2 for 1D features (i.e., 1-cycles), which we normalize so that ∆ ∈ [0, 1]. Panels  Figure 12: We study the topology of symmetric WTM maps by plotting ∆ as a function of T for N = 200 and α = 1/3. As before, panels (a)-(d) correspond, respectively, to network families (a)-(d). One can construe the curves of ∆ versus T as vertical cross sections of the contour plots in Fig. 11, although we now consider several different choices of mean node degrees: ( d Note that in general, introducing heterogeneity decreases our ability to identify the ring topology in the point cloud with ∆. For example, note that the ∆ values in panels (b) and (c) are smaller than those in panel (a), and the ∆ values in panel (d) are smaller than those in panels (b) and (c). Also note in panel (c) and panel (d) that when the mean degrees are too small (e.g., see red triangles), ∆ ≈ 0 for all thresholds T and thus we don't find evidence of the ring topology for these point clouds.

Sampling the Ring Manifold
In our numerical experiments thus far, we have investigated symmetric WTM maps for four families of noisy geometric networks on a ring manifold. In particular, network families (c) and (d) allow heterogeneity in the node locations along the ring manifold, which corresponded to choosing non-evenly spaced angles {θ i } along the unit circle to place the nodes. Specifically, each node i has an associated angle θ i = 2πi N + δθ i , where we draw δθ i ∼ N (0, (s 2π N ) 2 ) from a Gaussian distribution with standard deviation s 2π N . Note that 2π N is the spacing between the N nodes if they are evenly spaced along the ring. Hence, by varying the parameter s one can tune the level of heterogeneity in node location and thus the heterogeneity of the geometric degrees {d Recall that s → ∞ corresponds to sampling locations on the unit circle uniformly at random. In our previous experiments, we let s = 1/2 for network families (c) and (d). In this section, we investigate the effect of varying s. Because s > 0 introduces heterogeneity in the geometric degrees, we consider both the case in which the non-geometric degrees are identical and the case in which they are heterogeneous. Thus, one can construe the networks that we now consider as generalizations of network families (c) and (d).
In Fig. 13, we show results for the (left column) geometry, (center column) dimensionality, and (right column) topology of symmetric WTM maps, where we fix α = 1/3 and N = 200 and vary the threshold T . We consider networks with N = 200 nodes, a mean geometric degree of d Increasing network heterogeneity by increasing s has a significant effect on the structure of the point clouds resulting from symmetric WTM maps. For example, we see in panels (a) and (d) that increasing s shifts the abrupt drop-off in the Pearson correlation coefficient ρ, which originally occurs near its expected value of T (WFP) 0 = 3/8, to progressively smaller values of T . In fact, we see in all panels that increasing s causes the curves of ρ versus T to shift to the left. Additionally, in panels (a) and (d), we see for sufficiently large s that there is a regime in which ρ is small for all threshold values T . In panels (b) and (e), we still obtain regimes in which the WTM maps are low-dimensional (i.e., P ≈ 2). However, as s increases, the range of T values for which P indicates low dimensionality becomes smaller and shifts to the left. In panels (c) and (f), one can also observe that the ability to identify the ring topology becomes more difficult with increasing s. In panel (c), one obtains large ∆ when T ∈ (1/4, 3/8) for s = 0, which provides strong evidence that the point cloud lies on a ring manifold. For small s (e.g., 0 ≤ s ≤ 3/2), one also can observe large ∆, however the range of thresholds T producing large ∆ becomes smaller and shifts to the left. However, when s is large (e.g., s = ∞), ∆ remains small for all threshold values T in panels (c). There is even less evidence of the ring topology in panel (f), as ∆ remains small for all values of s and T .

Summary of Experiments with Synthetic Networks
We have conducted an extensive investigation of the geometry, dimensionality, and topology of symmetric WTM maps for several families of noisy geometric networks on a ring manifold. We now briefly summarize our results. We demonstrated that the structure (e.g., geometry, dimensionality, and topology) of WTM maps depends strongly on the network parameters (e.g., the number of nodes N , geometric degrees {d (G) i }, and non-geometric degrees {d (G) i }) and the contagion threshold T . Consequently, the extent to which a WTM contagion exhibits wavefront propagation (WFP) versus the appearance of new clusters (ANC) of contagions also depends on these parameters. Additionally, our bifurcation analysis did a very good job of predicting which parameter regimes have similar point cloud structures. This is particularly evident in the (T, α) parameter plane in Figs. 6, 9, and 11, where we observed that the geometry, dimensionality, and topology of WTM maps align well with our theoretical predictions for the occurrence of bifurcations in the dynamics of a WTM contagion. We found such agreement even for networks with heterogeneity in geometric degrees, nongeometric degrees, and/or node locations along the ring manifold. As we discussed in Sec. 4.2, we interpret our bifurcation analysis for the noisy ring lattice as an approximate bifurcation analysis for the networks with more heterogeneous structures. We also observed that (as expected) accuracy of this approximation increases as the mean node degrees increase. However, the accuracy of this approximation is sensitive to a variety of factors-including the threshold T , network size N , and the particular type of heterogeneity in a network. In many our numerical experiments, we compared the structure of point clouds resulting from WTM maps to those that result from a 2D Laplacian eigenmap [1] and Isomap [15], which respective map the network nodes based on diffusion dynamics and shortest paths. Our approach provides a nice complement to these methods.

Complex Contagions on a London Transit Network
The primary goal of our work has been to develop the notion of a WTM map and to demonstrate the utility of using such maps for examining WTM contagions on noisy geometric networks. Specifically, we conducted a detailed examination that contrasts WFP along geometric edges versus ANC due to the presence of nongeometric, "noisy" edges. We have focused on synthetic networks-and, in particular, on noisy geometric networks on a ring manifold-and we conducted a bifurcation analysis to guide our study. However, one can use WTM maps on far more general types of networks such as noisy geometric networks that are constructed from empirical data. This allows two important applications for real systems: (1) one can study the extent to which a contagion on a network exhibits spatial phenomena such as wavefront propagation (WFP) versus non-spatial phenomena such as the appearance of new clusters (ANC); and (2) one can infer (potentially) unknown low-dimensional structure in a network. In this section, we highlight these ideas for an empirical network that describes transit infrastructure in part of London.

The London Transit Network
As illustrated in Fig. 14(a), we study WTM contagions on a London transit network that includes both roads (which we interpret as short-range, geometric edges) and metro lines (which we interpret as longrange, non-geometric edges). The nodes V = {1, . . . , N } (where N = 2217) in the network correspond to intersections, and we obtained the edges from Refs. [9] (road data) and [13] (metro data). To merge these data sets, which both include latitudinal and the longitudinal coordinates, we place the locations of metro stations at the nearest intersection of roads. Thus, the nodes V consist of two sets: (1) nodes P ⊂ V that correspond to both metro stations and intersections and thus have both geometric and non-geometric edges; and (2) nodes V \ P that correspond to intersections and have only geometric edges (i.e., roads). Because the network of metro lines in Ref. [13] covers a much larger spatial area than the road network in Ref. [9], we include only metro stations in the convex hull of the road network. (There are |P| = 11 such stations.) In Fig. 14(b), we show the frequency of the nodes' total degrees and the mean is d i ≈ 2.59. In Fig. 14(c), we show the frequency of the edge lengths {χ ij }, where χ ij = m(i, j) is the Euclidean distance between locations w (i) and w (j) [see Eq. (12)] for each edge (i, j) ∈ E. In practice, we give node i the intrinsic location 1 and w (i) 2 denote, respectively, the intersection's latitudinal and longitudinal coordinate. We normalize each set of coordinates to have unit variance [see Fig. 14(a)]. In general, such a projection from a patch on the surface of a sphere (e.g., the Earth's surface) to a 2D plane might not be justified. However, the effect on geometry of this projection is negligible in this case due to the very small size of the patch. (The length and width of this part of London is much smaller than the Earth's radius.) , which we take from Ref. [9], are roads between intersections; and nongeometric edges (red), which we take from Ref. [13], give connections between metro stations. Some nodes (i ∈ P ⊂ V, where |P| = 11) correspond to both intersections and metro stations, whereas other nodes (i ∈ V \ P) correspond to intersections only. All nodes i ∈ V have intrinsic locations {w (i) } based on their latitudes and longitudes. (b) Frequency of the nodes' total degrees {d i } (i.e., d i = d where the mean is d i ≈ 2.59. (c) Frequency of the edge lengths {χ ij }, where χ ij = m(i, j) is the Euclidean distance between locations w (i) and w (j) for each edge (i, j) ∈ E [see Eq. (12)].
Before analyzing WTM contagions and WTM maps for the London transit network, let's consider the following experiment. In Fig. 15, we illustrate that the extent to which a WTM contagion adheres to the networks' underlying manifold-Earth's surface-can be very sensitive to a variety of factors, including the contagion seed and the WTM threshold T . We plot the London transit network and color each node i ∈ V according to its activation time x (i) j for a single contagion that we initialize with cluster seeding centered at a node j, which we take to be near the Bond Street Station. In panels (a) and (b), we show {x (i) j } for nodes i ∈ V with thresholds of T = 0.02 and T = 0.18, respectively. Note for T = 0.02 that the contagion spreads via both roads and metro lines, so the contagion includes ANC. By contrast, for T = 0.18, the contagion does not spread across the metro lines; rather, it spreads via WFP along the roads. As we shall see, this extreme sensitivity to the threshold T for the behavior of WTM contagions is not typical for all contagion seeds. Nevertheless, we find that such rare cases can have a large impact on WTM maps applied to the network.

Numerical Results for the Geometry of WTM Maps
In this section, we study the geometry of WTM contagions on the London transit network by examining the geometry of WTM maps. As before, we study geometry through the Pearson correlation coefficient ρ given by Eq. (14). We do not study the dimensionality and topology because of the large computational time that it would entail.
To guide our investigation, we first study the equilibrium sizes of contagions (i.e., the number of infected nodes after the contagion stops spreading [8]). Our motivation is as follows: Recall that for WTM maps to be well-defined, all activation times {x (i) j } must be finite. In our numerical experiments for synthetic networks, we therefore focused on this situation (e.g., see the main text and Sec. 5 of the present document), and we chose to handle activation times that were infinite by setting them to be 2N . Even with the restriction to finite activation times, we found a rich set of diverse qualitative dynamics. However, for the London transit network, the most interesting WTM maps occur for threshold values T that involve activation times of infinity. For this example, we must account for activation times of infinity more carefully to be able to study WTM contagions with WTM maps in such situations.
We thus begin by studying the equilibrium sizes of WTM contagions that we initialize with cluster seeding centered at each node i ∈ V. Specifically, for a given threshold T , we study the size C (target) i of each node i's "target node set" (which we define as the set of nodes {j} such that x (j) i is finite) and the size C (source) i of its "source node set" (which we define as the set of nodes {j} such that x (i) j is finite). In other words, node j is in the target node set for node i if a contagion that is initialized at node i eventually spreads to node j, and node j is in the source node set for node i if a contagion that is initialized at node j eventually spreads to node i.
In Fig. 16(a), we show histograms of the frequencies of C 3} that we initialize with cluster seeding. As expected, the WTM contagions infect almost all (or all) of the nodes when T is small, whereas they spread to just a small number of nodes (or even 0 nodes) when T is sufficiently large. For example, observe for most nodes that C that neither initialize large contagions nor adopt many contagions. In the above classification, we arbitrarily take N/2 to be the boundary between "large" and "small" for both types of division. (By convention, the "small" quantities above are less than or equal to N/2, and the large quantities are strictly greater than N/2.) In Fig. 16(b), we examine the fraction of nodes in each class as a function of T . For sufficiently small T (e.g., T < 0.1), almost all network nodes are in class (1) and almost all WTM contagions saturate the entire network. However, for large T (e.g., T > 0.35), all nodes are in class (4) because no contagions spread if the threshold is sufficiently large. The transition between these classes is interesting. Specifically, observe for the approximate range T ∈ [0.1, 0.2) that a small fraction of nodes moves from class (1) to class (2). Moreover, for the approximate range T ∈ [0.2, 0.25), class (2) and class (3) each contain only a small fraction of the nodes. Class (4) remains empty until T ≥ 0.25, and it them grows as we increase T until all nodes are in category (4) for T 0.35.
In Fig. 16(c), we plot the Pearson correlation coefficient ρ to compare the geometry of nodes' original locations {z (i) } to point clouds that result from WTM maps given by Eq. (14). We show results for the regular, reflected, and symmetric versions of the WTM map. (See the section "WTM Maps" on page 3 of the main manuscript.) We consider two different methods for handling the activation times of infinity for the source node sets. Note that nodes tend to have either very large or very sizes of the target and source node sets, so we assign nodes into four classes: (1) large C For each version, we handle the activation times of infinity in two ways: we either (1) set these activation times to be 2N and consider the full matrix of activation times ("full") as we proceeded with our studies of synthetic networks; or (2) we neglect these values and examine only the remaining submatrix of activation times ("part") after removing appropriate rows and columns. For the values of T for which nodes are exclusively in classes (1) and (2) (i.e., for T in the approximate range [0.1, 0.2]), we find that ρ increases for the WTM maps when we neglect the activation times of infinity. For the WTM maps in which we set the activation times of infinity to 2N , we see a drop in ρ for T 0.1.
[which necessarily arise whenever nodes are in classes (2)-(4)]. We either set the activation times to 2N and investigate the full matrix of activation times, or we consider only finite activation times by using only the associated submatrix of activation times (after removing appropriate rows and columns containing activation times that are infinite). To illustrate our analysis, let's consider the latter case for the mapping V → {x (i) }. We project each point x (i) ∈ R N onto R n with n ≤ N by ignoring the dimensions that correspond to WTM contagions initialized at nodes in class (2). In practice, this corresponds to considering the point cloud {x (i) }, wherex (i) = P x (i) and the n × N projection matrix P has entries [P ] ij i = 1, where i ∈ {1, . . . , n}, the set {j 1 , j 2 , . . . , j n } indicates the nodes that are not in class (2), and all other entries [P ] ij are equal to 0. For the reflected WTM map, we consider the mapping {i} → {y (i) } only for nodes i that are not in class (2). Finally, for the symmetric WTM map, we consider the mapping {i} → {ẑ (i) }, withẑ (i) = P z (i) , and we only consider nodes i that are not in class (2). Note for the reflected and symmetric WTM maps that we calculate the Pearson correlation coefficients ρ for the mapped points only. As expected, ρ for the WTM maps depends significantly on T , and one can observe that shifts in ρ are well-aligned with changes in C . The approximate range of thresholds T ∈ [0.1, 0.2) is particularly interesting, as we observe that values of ρ for WTM maps increase when we neglect the activation times that are infinite.
These larger ρ values, in turn, indicate an improved agreement between the nodes' original locations and the geometry of the point clouds that result from the WTM maps. By contrast, for WTM maps in which we handle activation times of infinity by setting them to 2N , we find that the ρ values are smaller for T 0.1 than they are for T 0.1. (f) For each node i, we plot the mean length of the shortest paths from that node to the remaining nodes versus its metro proximity ψ i . Note that isolated nodes, which are by definition distant from metro stations, are also distant from other nodes.
We now attempt to gain some insight into which nodes we assign to classes (1)-(4). In Fig. 17, we investigate the importance of the nodes' metro proximities {ψ i }, where ψ i denotes the length of a shortest path on the London transit network from node i to a metro station (i.e., ψ i = 0 for nodes that are metro stations, ψ i = 1 for their neighbors, and so on). We consider nodes that are at least 20 edges from any metro station to be "isolated." In the top row, for a given metro proximity ψ i , we plot the fraction of nodes at that proximity in each of the four classes. Panels (a), (b), and (c), respectively, give results for threshold values of T = 0.16, T = 0.18, and T = 0.2, which are characteristic of the range of T in which we observe large ρ values for the WTM maps [see Fig. 16(c)]. Note for T = 0.16 that almost all nodes are in class (1), but several are in class (2). Nodes in the latter set are each located ψ i = 2 edges from metro stations, and a WTM contagion tends not to spread very far when we initialize it with cluster seeding centered at such nodes. For T = 0.18, we again find that some nodes are in class (2), whereas the majority of nodes are in class (1). However, the nodes in class (2) are either 2-3 edges from a metro station or they are "isolated" nodes, which are distant from all other nodes (including, by definition, metro stations). For T = 0.2, we find that nodes are in classes (1)- (3). As before, nodes in class (2) are either 2-3 edges from a metro station or are "isolated." The nodes in class (3) are "isolated." In the bottom row of Fig. 17, we show properties of the metro proximities {ψ i } for the London transit network. In panel (d), we show the frequency of nodes at a given metro proximity ψ i , and we note that most nodes are 5-20 edges from a metro station. In panel (e), we give a scatter plot of the nodes' total degrees {d i } versus their metro proximities {ψ i }. Note that the metro stations (for which ψ i = 0) have large degrees relative to the other nodes; their mean is 5 versus 2.59. In panel (f), we show that "isolated" nodes, which by definition are distant from metro stations, are also distant to the other nodes in the London transit network. Specifically, for each node i, we plot the mean length of the shortest path from it to the remaining nodes {j} ∈ V versus its metro proximity ψ i . Notes with large ψ i are also more distant (on average) to the other nodes. It is therefore appropriate to use the term "isolated" to describe these nodes.
Combining the results from Figs. 16 and 17, we find when we ignore the activation times that are infinite that WTM maps have larger values of ρ when T is in the approximate interval [0.1, 0.2] than when T takes other values. The activation times of infinity result from the existence of a few nodes {i} such that WTM contagions that we initialize with cluster seeding centered at those nodes tend not to spread very far. These nodes are in class (2), and they are often either 2-3 edges from metro stations or are often "isolated" nodes.

"Egocentric" Analysis of Geometry
Thus far, we have studied geometry through the Pearson correlation coefficient ρ given by Eq. (14). As we discussed in Sec. 3.1, ρ describes the correlation between node-to-node distances {m(i, j)} for the intrinsic locations {w (i) } [see Eq. (12)] and node-to-node distances {m (WTM) (i, j)} for the point clouds {x (i) }, {y (i) }, or {z (i) } that result from a WTM map [see Eq. (13)]. We calculate this correlation using the N (N − 1)/2 unordered pairs of nodes (i, j) ∈ V × V (where i = j), and one can interpret it as comparing the geometry of these two point clouds at a "network level." To gain further insight, we now compare the geometry of the two point clouds at a "node level" by computing "egocentric" correlation coefficients that consider only node-to-node distances that involve a particular node i. Specifically, we study a set of Pearson correlation coefficients {ρ i (T )} for i ∈ V.
We introduce the egocentric correlation coefficientρ i (T ) for the regular WTM map V → {x (i) }, and we note that one can apply it to any version of a WTM map. For each node i, we study the Pearson correlation coefficientρ i (T ) that relates node-to-node distances {m(i, j)} from node i to all nodes {j} ∈ V with respect to the intrinsic locations {w (i) } [see Eq. (12)] to the node-to-node distances {m (WTM) (i, j)} from node i to all nodes {j} ∈ V for a point cloud {x (i) } that results from a WTM map [see Eq.
where the bar above a variable indicates taking its mean for all nodes j ∈ V. Note the strong similarity between Eq. (19) and Eq. (14); the only difference is that the summations in Eq. (19) are over j rather than over both j and i. In Fig. 18, we study egocentric correlation coefficients {ρ i (T )} for WTM maps on the London transit network for two values of the threshold T . In the top panels, we show results for the mapping V → {x (i) }; in the bottom panels, we show results for the mapping V → {y (i) }. For both mappings, we handle the activation times of infinity by neglecting them (as described in Sec. 6.2). In the left column, we plot egocentric Pearson correlations {ρ i (T )} versus the metro proximities {ψ i } for threshold values T ∈ {0.1, 0.18}.
We show values only for nodes that are not in class 2. Note that the larger value of T tends to have a largerρ i (T ) in both panels (a) and (d). We highlight this feature further in the center column by plotting {ρ i (0.18) −ρ i (0.1)} versus ψ i . The solid and dashed curves denote the mean values for a given ψ i . In the right column of Fig. 18, we plot the frequencies of observed values {ρ i (0.18) −ρ i (0.1)}, and we note that appear to have heavy tails: {ρ i (0.18) −ρ i (0.1)} is small and positive for most nodes j ∈ V, but there are some nodes for which {ρ i (0.18) −ρ i (0.1)} is rather large.

Summary of Experiments with the London Transit Network
We studied WTM contagions on a London transit network in which nodes are intersections that are connected by either roads (which we interpreted as geometric edges) or metro lines (which we interpreted as non-geometric edges). Similar to our study of WTM contagions on synthetic networks, we found that WFP and ANC arise for WTM contagions on this empirical network, and these properties depend significantly on the contagion threshold T . We studied these phenomena by analyzing the geometry of WTM maps, and we observed that the geometry of point clouds that result from WTM maps agree better with the geometry of nodes' intrinsic locations on Earth's surface for values of T in the approximate range [0.1, 0.2] than for other values of T . To obtain this result, we examined situations with activation times of infinity in two different ways: (1) setting those times to be 2N as for the main manuscript and the synthetic examples in 5; and (2) ignoring these values in our subsequent calculations. For the London transit network, we found the latter approach to be more useful. Our investigation led us to assign nodes into four classes based on their ability to initiate large-scale contagions and consistently adopt contagions, and our calculations yielded an interesting connection between the proximity of nodes to metro stations and their behavior with respect to WTM contagions.