Abstract
Node copying is an important mechanism for network formation, yet most models assume uniform copying rules. Motivated by observations of heterogeneous triadic closure in real networks, we introduce the concept of a hidden network model—a generative twolayer model in which an observed network evolves according to the structure of an underlying hidden layer—and apply the framework to a model of heterogeneous copying. Framed in a social context, these two layers represent a node’s inner social circle, and wider social circle, such that the model can bias copying probabilities towards, or against, a node’s inner circle of friends. Comparing the case of extreme inner circle bias to an equivalent model with uniform copying, we find that heterogeneous copying suppresses the powerlaw degree distributions commonly seen in copying models, and results in networks with much higher clustering than even the most optimum scenario for uniform copying. Similarly large clustering values are found in real collaboration networks, lending empirical support to the mechanism.
Introduction
Node copying is an important network growth mechanism^{1,2,3,4,5,6,7}. In social networks, copying is synonymous with triadic closure, playing an important role in the emergence of high clustering^{8,9}. In biology, node copying encapsulates duplication and deletion, a key mechanism in the formation of proteininteraction networks^{10,11,12,13,14}.
Despite this range of applications, most node copying models assume uniform, or homogeneous copying, i.e., that the probability of copying any given neighbour of a node is equal. The exact formulation varies widely, but examples include “links are attached to neighbours of [node] j with probability p”^{5}, or “one node [is duplicated]... edges emanating from the newly generated [node] are removed with probability δ”^{11}. Many other models use similar uniform copying rules^{2,3,6,7,8,10,12,15,16,17,18,19,20,21,22,23,24,25,26}.
Homogeneous copying is a sensible base assumption, often aiding a model’s analytical tractability. However, especially in a social context, there are good reasons to believe that node copying may be heterogeneous. As an example, consider the social brain hypothesis, a theory which suggests that the average human has around 150 friends (Dunbar’s number), encapsulating progressively smaller subgroups of increasing social importance^{27,28}. In contrast, large social networks often have an average degree far exceeding Dunbar’s number^{29}, implying that most of these observed friends are only distant acquaintances. In this context, if individual A introduces individual B to one of their friends, C, (i.e., B is copying A’s friend C), we may reasonably expect that C is more likely to be chosen from A’s inner social circle, than A’s wider social circle.
This is directly related to the principle of strong triadic closure: “If a node A has edges to nodes B and C, then the B−C edge is especially likely to form if A’s edges to B and C are both strong ties”^{30}. In weighted networks where tie strength can be equated to edge weight, empirical evidence for the strong triadic closure principle can be inferred by measuring the neighbourhood overlap between two nodes as a function of tie strength^{30}; for example, using mobile communication networks^{31}, or using face to face proximity networks^{32}.
Unfortunately, for many networks tie strength data is unavailable or unknown. In these cases, evidence for asymmetric triadic closure may be inferred through proxy means. For instance, in academic collaboration networks, it has been shown that the ratio of triadic closure varies strongly with the number of shared collaborators between nodes^{33}. Although the average triadic closure ratio is small (typically < 10%), the ratio rapidly increases with the number of shared collaborators. However, these aggregate measures are highly coarsegrained and likely only approximate real closure dynamics.
This motivates the study of simple heterogeneous copying models^{4,5,9,34,35}. Typically these models fall into a small number of distinct categories. In the first, heterogeneity is introduced as a node intrinsic property (e.g., node fitness) in the absence of structural considerations^{5}. In the second, heterogeneity is introduced via group homophily where the probability of triadic closure between nodes A and B is dependent on whether nodes A and B are in the same group or different groups (e.g., researchers from the same academic discipline, as opposed to different disciplines)^{9,35}. However, intragroup copying is typically modelled uniformly. Finally, some models consider heterogeneous copying driven by the network structure around nodes A and B, without introducing node homophily^{4}.
Bhat et al.^{4} define a threshold model where node A introduces node B to one of their friends C. An edge then forms between B and C if the fraction of neighbours common to B and C exceeds some threshold F. The model demonstrates a transition from a state where networks are almost complete for small F, to a state where networks are sparse but highly clustered as F increases past a critical threshold. However, the model is limited in its tractability and has peculiarities such as the observation that fringe communities are almost always complete.
In the current work, our aim is to extend these ideas and introduce a more general framework for heterogeneous node copying based on the concept of hidden strong ties. To do so, we introduce the hidden network model, a framework based on multilayer networks^{36} where layers have identical node structure but different edge structure. The framework lets us build models where local heterogeneity in the rules of network growth is a property of the hidden network structure and not arbitrarily encoded using node intrinsic properties or group homophily. The concept is closely related to other multilayer paradigms including the use of replica nodes to model heterogeneity^{37}, interdependent networks^{38}, and multilayer copying^{21}.
In the remainder of this paper, we define and analytically study the case of extreme heterogeneous copying, the correlated copying model (CCM). The CCM is an adaptation of the uniform copying model introduced by Lambiotte et al.^{3}, see Fig. 1a. In the UCM, a single node, α, is added to the network at time t_{α}, and connects to one target node, β, which is chosen uniformly at random. The formation of an edge between the new node and the target node puts the UCM in the class of corded copying models; Steinbock et al.^{16} refer to the UCM as the corded node duplication model. We label each neighbour of β with the index γ_{j} where j ∈ {1, ⋯ , k^{β}}, and k^{β} is the degree of node β. For each neighbour γ_{j}, the copied edge (α, γ_{j}) is added to the network independently with probability p.
Following the convention of previous copying models, the nodes α and β are sometimes referred to as the daughter and mother nodes respectively. The network is initialised at t = 1 with a single node. If p = 0, no edges are copied resulting in a random recursive tree. If p = 1, the UCM generates a complete graph.
In addition to the extreme copying case (the CCM), we numerically investigate a generalised form of the correlated copying model (GCCM) which interpolates between the UCM and CCM. The GCCM generates a diverse spectrum of network structures spanning both ergodic sparse and nonergodic dense networks, with degree distributions ranging from exponential decay, through stretchedexponentials and powerlaws, to extremely fattailed distributions with anomalous fluctuations. These networks exhibit a broad clustering spectrum from sparse networks with significantly higher clustering than their uniform equivalents to the unusual case where networks are almost complete, but with nearzero clustering. We comment on a selection of real collaboration networks, which, in line with the CCM, exhibit higher clustering than can be explained by uniform copying. This suggests that heterogeneous copying may be an important explanatory mechanism for social network formation.
Results
Hidden network models
We define a hidden network model as the pair of single layer graphs G = (G_{O}, G_{H}), comprising an observed network G_{O} = (V, E_{O}) and a hidden network G_{H} = (V, E_{H}), where V is the set of nodes for both networks and E_{O} and E_{H} are the set of edges for each network. The set V represents the same entities in both G_{O} and G_{H}, with differences lying exclusively in the edge structure between nodes. The key feature of a hidden network model is that the evolution of G_{O} is dependent on G_{H} (or vice versa). Mathematically, this is closely related to interdependent networks^{39}.
Correlated copying model
In the CCM, see Fig. 1b, the observed and hidden networks are initialised with a single node at t = 1. At t = t_{α}, node α is added to both networks and a single target node, β, is chosen uniformly at random. We label the \({k}_{O}^{\beta }\) neighbours of β in G_{O} with the index γ_{j}. Then, in the observed network only, the copied edge (α, γ_{j}) is formed with p_{hid} = 1 if the edge (β, γ_{j}) ∈ E_{H}, p_{obs} = 0 otherwise. The general case with intermediate copying probabilities is discussed in section “General correlated copying model”. No copied edges are added to the hidden network G_{H}. The direct edge (α, β) is added to both G_{O} and G_{H}. The CCM therefore also falls into the class of corded node duplication models. Using the convention of referring to β as the mother node and α as the daughter node, we note that the hidden network consists exclusively of firstorder relations (mother−daughter), whereas edges found only in the observed network correspond to secondorder relations (sister−sister, or grandmother−granddaughter).
G_{H} evolves as a random recursive tree. Unlike the UCM, all copying in G_{O} is deterministic, with the only probabilistic element emerging in the choice of the target node β. For comparative purposes, we define the effective copying probability in the CCM as \({p}_{{{\mbox{eff}}}}=\langle {k}_{H}^{\beta }/{k}_{O}^{\beta }\rangle\), i.e., the fraction of the observed neighbours of node β which are copied by node α.
Framed in a social context, we might think of G_{O} as an observed social network where individuals have many friends, but the quality of those friendships is unknown, with most ties being weak. In contrast, underlying every social network is a hidden structure representing the inner social circle of individuals, where a node is only connected to their closest friends^{28}. Copying in the CCM is biased to this inner circle.
Basic topological properties
The total number of edges in G_{H} scales as E_{H}(t) ~ t, with the average degree given by 〈k_{H}〉 = 2. Using the degree distribution of G_{H}, see below, \(\langle {k}_{H}^{2}\rangle =6\). In the observed network, each time step a single edge is added by direct attachment, and one copied edge is added for each neighbour of the target node in G_{H}, \({k}_{H}^{\beta }\). The average change in the number of edges is therefore \(\langle {{\Delta }}{E}_{O}(t)\rangle =1+\langle {k}_{H}^{\beta }\rangle =1+\langle {k}_{H}\rangle =3\), such that 〈E_{O}(t)〉 ~ 3t and 〈k_{O}〉 = 6.
As an alternative, note that the observed degree of node α can be written as
where the index α, β labels the \({({k}_{H})}_{\alpha }\) unique neighbours of α in G_{H}. Averaging both sides of Eq. (1) over all nodes we find,
where n_{ℓ} is the number of times that the degree of node ℓ appears in the expanded sum. For any tree graph, node ℓ will appear exactly once in Eq. (2) for each of its \({({k}_{H})}_{\ell }\) neighbours. Hence, \({n}_{\ell }={({k}_{H})}_{\ell }\) and \(\langle {k}_{O}\rangle =\langle {k}_{H}^{2}\rangle\). In Supplementary Note 1, Eq. (1) is used to derive \(\langle {k}_{O}^{2}\rangle \approx 62\).
We may naively expect that the effective copying probability is p_{eff} = 〈k_{H}〉/〈k_{O}〉 = 1/3. However, for the CCM, \({p}_{{{\mbox{eff}}}}=\langle {k}_{H}^{\beta }/{k}_{O}^{\beta }\rangle \ne \langle {k}_{H}\rangle /\langle {k}_{O}\rangle\). We have not found a route to calculating this exactly, but simulations suggest p_{eff} ≈ 0.374.
Degree distribution
The hidden network evolves as a random recursive tree which has a limiting degree distribution given by
In Supplementary Note 2, we show that the degree distribution for the observed network can be written as the recurrence
where the final term is the probability that at time t the newly added node has initial degree k_{O} and
with 〈k_{H}∣k_{O}〉 as the average degree of nodes in the hidden network with observed degree k_{O}. Here, the 1 corresponds to edges that are gained from direct attachment, whereas 〈k_{H}∣k_{O}〉 corresponds to edges gained from copying. Although we have not found an exact expression for 〈k_{H}∣k_{O}〉, we can make progress by considering the evolution of individual nodes.
Consider node α added to the network at t_{α}. The initial conditions for node α are
where the final term is the average hidden degree of the target node β. In G_{H}, node α gains edges from direct attachment only. Hence, at t > t_{α},
where H_{n} is the n^{th} harmonic number. In G_{O}, either node α is targeted via direct attachment, or a copied edge is formed from the new node to node α via any of the \({({k}_{H}(t))}_{\alpha }\) neighbours of node α. Hence,
where we have subbed in Eq. (7) and H_{j−1} = H_{j} − 1/j. Evaluating this sum, see Supplementary Note 2, we find
where \({H}_{n}^{(m)}\) is the nth generalised Harmonic number of order m. For t → ∞, \({H}_{t}^{(2)}\to {\pi }^{2}/6\). Hence, for large t we can drop the final two terms and substitute in Eq. (7) to give
Noting, that Eq. (10) is a monotonically increasing function of k_{H} for k_{H} > 1, we assume that we can drop the index α and the time dependence giving the average observed degree of nodes with specific hidden degree as
where \(\langle {\tilde{k}}_{O} {k}_{O}\rangle\) denotes the average initial observed degree of nodes with current degree k_{O}. Finally, we make the approximation that \(\langle {k}_{H} {k}_{O}\rangle \approx {\langle {k}_{O} {k}_{H}\rangle }^{1}\) where the exponent denotes the inverse function. This gives
To proceed, let us solve the degree distribution at k_{O} = 2. Although the average initial condition \(\langle {\tilde{k}}_{O}\rangle =1+\langle {k}_{H}\rangle =3\), in this case \(\langle {\tilde{k}}_{O} 2\rangle =2\). Therefore
giving p_{O}(2) = 1/6. Since \(\langle {\tilde{k}}_{O} {k}_{O}\rangle\) has an almost negligible effect on π_{O}(k_{O}) for k_{O} > 2, for simplicity we set \(\langle {\tilde{k}}_{O} {k}_{O}\rangle =2\). We can now rewrite Eq. (4) as
Although computing this recurrence shows good agreement with simulations, see Fig. 2, we have not found a closedform solution to Eq. (14).
As an approximation, we return to Eq. (9) and note that \({H}_{t1}{H}_{{t}_{\alpha }1}\approx \,{{\mbox{ln}}}\,(t/{t}_{\alpha })\). Substituting this into Eq. (9) and dropping small terms
which inverted gives
We have dropped the expectation value and define t_{α} as the time a node was created such that its degree at time t is approximately k_{O}. Exponentiating each side and taking the reciprocal,
Finally, by substituting this approximation into the cumulative degree distribution we find
which corresponds to a Weibull (stretched exponential) distribution, suppressing the powerlaw scaling observed in the UCM, see Fig. 2.
The approximation for the cumulative degree distribution stems from the observation that, on average, nodes with \(k^{\prime} \; > \; {k}_{O}\) were added to the network at \(t^{\prime} \; < \; {t}_{\alpha }\), whereas nodes with \(k^{\prime} \; < \; {k}_{O}\) were added to the network at \(t^{\prime} \; > \; {t}_{\alpha }\). Note that Eq. (18) is close to the scaling expected from sublinear preferential attachment^{40} with an exponent 1/2.
Clique distribution
In a simple undirected graph, a clique of size n is a subgraph of n nodes that is complete. A clique of size n = 2 is an edge, whereas n = 3 is a triangle. Here we calculate the exact scaling for the number of n cliques, Q_{n}(t), in G_{O}.
Let us first consider the case of triangles. At t = t_{α}, there are two mechanisms by which a new triangle forms:

1.
Direct triangles. The new node, α, forms a direct edge to the target node, β, and forms copied edges to each of the \({k}_{H}^{\beta }\) neighbours of node β, labelled with the index γ_{j}. The combination of the direct edge (α, β), the copied edge (α, γ_{j}), and the existing edge (β, γ_{j}) creates one triangle, (α, β, γ_{j}), for each of the \({k}_{H}^{\beta }\) neighbours.

2.
Induced triangles. If node α forms copied edges to both node γ_{j}, and to node \({\gamma }_{j^{\prime} }\), \(j\;\ne\; j^{\prime}\), the triangle \((\alpha ,{\gamma }_{j},{\gamma }_{j^{\prime} })\) is formed if \(({\gamma }_{j},{\gamma }_{j^{\prime} })\in {E}_{O}\).
Combining these mechanisms, the change in the number of triangles can be written as
where the first and second terms on the right correspond to direct and induced triangles respectively. One new direct triangle is formed for each of the \({k}_{H}^{\beta }\) neighbours of node β, \({{\Delta }}{Q}_{3}^{D}={k}_{H}^{\beta }\). For induced triangles, the copied edge (α, γ_{j}) is only formed if (β, γ_{j}) ∈ E_{H}. Additionally, all pairs of nodes which are nextnearest neighbours in G_{H} must be nearest neighbours in G_{O}. Hence, the edge \(({\gamma }_{j},{\gamma }_{j^{\prime} })\) must exist in the observed network if both γ_{j} and \({\gamma }_{j^{\prime} }\) are copied. As a result, one induced triangle is formed for each pair of copied edges (α, γ_{j}) and \((\alpha ,{\gamma }_{j^{\prime} })\) such that
A visual example of the combinatorics for \({k}_{H}^{\beta }=3\) is shown in Fig. 3.
Extending the triangle argument to general n we can write
where direct cliques are those which include the edge (α, β). For a clique of size n, the number of direct cliques is given by the number of ways in which n − 2 nodes can be chosen from \({k}_{H}^{\beta }\) nodes,
whereas the number of induced cliques is given by the number of ways in which n − 1 nodes can be chosen,
As t → ∞, the average change in clique number is
where p_{H}(k_{H}) is the probability that the randomly chosen target node \({k}_{H}^{\beta }={k}_{H}\). To avoid illdefined binomials, we rewrite Eq. (24) as
where we have combined the two terms into a single binomial. After subbing in p_{H}(k_{H}) and solving the sum,
Consequently, for large t we find the curious result that the number of n cliques scales as
independent of the clique size. In practice, this result only applies for t → ∞. To see this, note that the largest clique in G_{O} at time t is always directly related to the largest degree node in G_{H},
with the largest hidden degree at time t scaling as approximately
We can invert this and ask how large the network is if we observe that the largest observed clique is n. This gives
Hence, the scaling relation in Eq. (27), is only valid for cliques of size n when t ≫ t_{n}. In Supplementary Note 3, we plot the number of cliques in simulations of the CCM as a function of t. For small clique sizes, the scaling in Eq. (27) is clearly apparent early in the evolution of the CCM. However, for moderate and large cliques, the standard deviation in the number of cliques is significantly larger than the average number of cliques, obscuring a clear trend.
Clustering
Transitivity is a global clustering measure defined as
where a twig is any three nodes connected by two edges. The number of twigs is equivalent to the number of star graphs of size 2, S_{2}, where a star graph of size n is a subgraph with 1 central node and n connected neighbours. The number of subgraphs of size 2 is related to the degree distribution by
where we have used the property that p_{O}(k_{O} < 2) = 0. Recalling that 〈k_{O}〉 = 6 and \(\langle {k}_{O}^{2}\rangle \approx 62\), the number of twigs scales as S_{2} ~ 28t, such that
The observed network can be recovered from the hidden network by converting every wedge in G_{H} into a triangle. This can be thought of as complete triadic closure where every possible triangle which can be closed, from the addition of a single edge to the hidden network, is closed. This implies that the CCM has the largest possible transitivity from a single iteration of triadic closure on a random recursive tree.
The local clustering coefficient, cc(α), is defined as the number of edges between the \({({k}_{O})}_{\alpha }\) neighbours of α, normalised by the the number of edges in a complete subgraph of size \({({k}_{O})}_{\alpha }\). For the CCM,
where the first term corresponds to the complete subgraph of the \({({k}_{H})}_{\alpha }\) neighbours of α in G_{H}, and the sum contributes the edges from one complete subgraph formed by node α, β, and β \({({k}_{H})}_{\alpha ,\beta }1\) neighbours, excluding α. The global clustering coefficient, CC(G_{O}), is defined as the average of Eq. (34) over all nodes in the network. In simulations, CC(G_{O}) ≈ 0.771 for large t.
Path lengths
Steinbock et al.^{16} calculate the distribution of shortest path lengths for the UCM (referred to in their paper as the corded node duplication model). Specifically, the authors calculate the probability that two randomly chosen nodes, i and j, will be separated by a shortest path of length ℓ, denoted as \({{{{{{{\mathcal{P}}}}}}}}(L=\ell ;t)\), at time t.
The UCM with p = 0 corresponds to a random recursive tree and is therefore equivalent to G_{H} in the CCM. Hence, for the hidden network, we can lift the path length distribution, \({{{{{{{{\mathcal{P}}}}}}}}}_{H}({L}_{H}=\ell ;t)\), and the mean shortest path, 〈L_{H}(t)〉, from Steinbock et al.^{16}. We can then exploit a convenient mapping to calculate the distribution of shortest path lengths in G_{O} from G_{H}.
Consider two randomly chosen nodes i and j. In G_{H}, there is a unique path (due to its tree structure) from i to j of length \({({\ell }_{H})}_{ij}\). In G_{O}, the enforced triadic closure process means that for every two steps on the path from i to j in G_{H}, an observed edge exists in G_{O}, which acts as a shortcut, reducing the path length by one. Hence, if the path length \({({\ell }_{H})}_{ij}\) is even, the path length in G_{O} is given by ℓ_{O} = ℓ_{H}/2; if the path length is odd ℓ_{O} = (ℓ_{H} + 1)/2. Using this mapping, we can write
If we assume that, for large t, there are an approximately equal number of odd and even shortest paths in G_{H}, the average shortest path length in G_{O} is
where the 1/4 term accounts for the discrepancy in the mapping for odd and even paths.
From Steinbock et al.^{16}, we note that the mean shortest path length for G_{H} scales as
which indicates that the hidden network exhibits the smallworld property^{41}. We have omitted constants which are negligible at large t. Hence, applying the mapping in Eq. (36) and omitting the 1/4 term for simplicity, the mean shortest path length for G_{O} is given by
indicating that the observed network also exhibits the smallworld phenomenon. This mapping is confirmed by simulations.
For interest, we note that for 0 < p < 1, the shortest paths for the UCM are in general not unique; there may be multiple paths between nodes i and j, which are equally short. Unusually for a nontree network, all shortest paths are unique in the CCM.
General correlated copying model
The GCCM, is defined analogously to the CCM, starting with observed and hidden networks initialised at t = 1. Like the UCM and CCM, the GCCM is a corded node duplication model. For practical reasons, we initialise the graph with three nodes which form a complete graph in G_{O}, and a wedge in G_{H}. This ensures that the initial graph contains some edges found in G_{H}, and some edges found only in G_{O}.
At t = t_{α}, node α is added to both networks and a single target node, β, is chosen uniformly at random. We label the \({k}_{O}^{\beta }\) neighbours of β in G_{O} with the index γ_{j}. In the observed network, the copied edge (α, γ_{j}) is formed with probability p_{hid} if the edge (β, γ_{j}) ∈ E_{H} (inner circle copying), and probability p_{obs} otherwise (outer circle copying). The direct edge (α, β) is added to both G_{O} and G_{H}.
The GCCM encapsulates a wide spectrum of heterogeneous copying. Setting p_{hid} = 1 and p_{obs} = 0 reduces the GCCM to the CCM, whereas setting p_{hid} = p_{obs} = p reduces the GCCM to the UCM. We have discussed the social motivation for the case where p_{hid} > p_{obs}, representing a copying bias towards the inner social circle of a node. However, the GCCM can also be tuned to the reverse case where p_{hid} < p_{obs}, resulting in a bias against inner circle nodes. We are not aware of a clear physical motivation for this latter case. However, the structural diversity of these anticorrelated networks warrants their discussion here.
Figure 4 shows numerical results for (a) the effective copying probability, (b) the densification exponent, (c) the average local clustering coefficient, and (d) the transitivity, for the GCCM with 10^{4} nodes.
The effective copying probability corresponds to the fraction of target node neighbours which appear to be copied in the observed network. Formally, we can write the average effective copying probability at time t as
where β is the index of the target node at time t, the first term represents edges copied from node β’s inner circle, and the second term represents edges copied from the outer circle.
The dashed contour in Fig. 4b corresponds to an effective copying probability of 0.5, calculated numerically by averaging over the preceding 10^{4} time steps. We note that p_{eff} = 0 if p_{hid} = p_{obs} = 0 (random tree), p_{eff} = 1 if p_{hid} = p_{obs} = 1 (complete graph), and p_{eff} = p if p_{hid} = p_{obs} = p (UCM). In general, the rise in p_{eff} is faster with increasing p_{obs} than increasing p_{hid}, although for p_{hid} = 0 we find very small p_{eff}, even for large p_{obs}. However, this observation is somewhat deceptive since, if the GCCM is in the dense regime and p_{hid} ≠ p_{obs}, p_{eff} is not stationary. Calculated over longer time frames, we note that the effective copying probability appears to slowly converge to the outer copying probability, p_{eff} → p_{obs}, since for t → ∞, the ratio of the number of edges in the hidden network to the number of edges in the observed network tends to zero. This suggests that the dashed p_{eff} = 0.5 contour will converge to the p_{obs} = 0.5 line as t → ∞.
We test whether the GCCM is in the sparse or dense regime explicitly by tracking the growth in the number of edges in the observed network. Let us define the densification exponent, δ, 0 ≤ δ ≤ 1, using E_{O}(t) ∝ t^{1+δ}, which relates the number of edges in the observed network to the number of nodes t. If δ ≈ 0, the GCCM is sparse. If δ = 1, the GCCM grows as a complete graph. For intermediate values, the GCCM undergoes densification. For the UCM, the transition from the sparse to dense regime is known to take place at p = 0.5^{3}. We have not analytically calculated the transition for the GCCM, but may intuitively expect the transition at p_{obs} = 0.5 since the hidden network is a random tree. This seems to be supported by the numerical values of δ in Fig. 4b, although the transition from zero to nonzero δ is shifted to slightly larger p_{obs} for p_{hid} = 0, and to smaller p_{obs} for p_{hid} = 1; this shift is likely to disappear as t → ∞.
Figures 4c and d shows the average local clustering coefficient, CC(G_{O}), and transitivity (global clustering), \({\tau }_{{G}_{O}}\), for the GCCM. Patterns are similar between the two figures, although local clustering generally exceeds global clustering in the sparse regime. For the UCM it is known that, in the dense regime, \({\tau }_{{G}_{O}}\) slowly converges to zero as t → ∞, unless p = 1^{2}. In contrast, the local clustering appears to remain nonzero.
As expected, clustering is minimised at p_{hid} = p_{obs} = 0 (random tree) and maximised for a complete graph, p_{hid} = p_{obs} = 1. However, in the sparse regime we find that the maximum clustering is found at p_{hid} = 1, p_{obs} = 0 which corresponds to the CCM. Bhat et al.^{2} note that local and global clustering for the UCM is not a monotonically increasing function of the copying probability p, with a local maxima in the sparse regime at nonzero p. This bimodal clustering is also present in the GCCM. In the anticorrelated regime where p_{hid} ≈ 0, we find nearzero clustering values. In particular if p_{hid} = 0 and p_{obs} = 1, we observe the unusual property that δ ≈ 1, such that the network scales as (but is not) a complete graph, yet both the local and global clustering are approximately zero.
Extracting the degree distributions for the GCCM for various p_{hid} and p_{obs} shows similarly diverse behaviour, see Fig. 5. Each distribution is averaged over 100 instances, but points are left deliberately unbinned to illustrate the significant fluctuations observed in the dense regime. For p_{hid} = p_{obs} = 0 (bottom left) the GCCM reduces to a random recursive tree, see Eq. (3). The CCM case with p_{hid} = 1, p_{obs} = 0 (top left) follows Eq. (14), where the tail can be approximated as a stretched exponential. This distribution is also shown in Fig. 2. Along the diagonal where p_{hid} = p_{obs} (UCM), the degree distribution has a powerlaw tail in the sparse regime, and exhibits anomalous scaling in the dense regime (p ≥ 0.5). For p_{hid} = p_{obs} = 1, the GCCM reduces to a complete graph and all nodes have degree t − 1.
For p_{obs} = 0, the powerlaw scaling observed in the UCM is completely suppressed, with a gradual transition from exponential decay to a stretched exponential tail as p_{hid} is increased from 0 to 1. In the sparse regime with p_{obs} ≠ 0, all degree distributions appear fattailed with only small deviations from the powerlaws observed for the UCM. However, unusual scaling is observed for p_{hid} = 0, p_{obs} ≠ 0, where the distributions exhibit initial exponential decay at small k_{O}, attributable to the hidden network, before a second fattailed regime starting at intermediate k_{O}.
In the dense regime, all degree distributions exhibit anomalous scaling, such that individual instances are not selfaveraging. For p_{obs} = 0.75, the tail of the degree distributions is largely consistent across all p_{hid}. However, the probability of finding nodes with small degree is large for p_{hid} = 0, and is gradually suppressed as p_{hid} → 1. These effects are most pronounced for p_{obs} = 1 where the modal degree is 1 for p_{hid} = 0, and t − 1 for p_{hid} = 1, with a gradual transition in between. Throughout this transition, the degree distribution appears almost uniform at p_{hid} = 0.25, where the probability of finding nodes with any given degree is approximately constant up until the large k_{O} limit. However, this effect is only observed when averaging over many instances, with a much smaller degree range observed in individual networks.
It is possible to extend the GCCM further by adding copied edges from G_{O} to the hidden network, G_{H}, with probability q. Results are shown in Supplementary Note 4 for q > 0 where clustering is enhanced if p_{hid} > p_{obs} and suppressed if p_{hid} < p_{obs}, relative to the UCM. In the limiting case of q = 1, the GCCM is independent of p_{obs} and equivalent to the UCM with p = p_{hid}. The p_{hid} = p_{obs} line (UCM) is invariant under changes in q. One potential application of the q ≠ 0 case is for generating random simplicial complexes^{42} by combining the hidden and observed networks into a single structure. Such a construction may be interesting since it explicitly distinguishes between cliques of strong ties, where all nodes are within each other’s inner circle, and cliques of weak ties, see Supplementary Note 5.
Comparing copying models
We have introduced a simple model of heterogeneous node copying, motivated by arguments that triadic closure may not be structurally homogeneous in real networks.
Comparing the CCM, for which we have analytical results, to the UCM with the equivalent effective copying probability (p = p_{eff} = 0.374) we find significant differences in network structure. Both the average local clustering coefficient, CC(G_{O}), and the transitivity, \({\tau }_{{G}_{O}}\), are significantly larger in the CCM than the UCM. The CCM suppresses the powerlaw tail observed in the UCM for the sparse regime, and consequently, the degree variance observed in the CCM is smaller than for the UCM. CCM: σ^{2}(k_{O}) ≈ 26; UCM: σ^{2}(k_{O}) ≈ 192. The CCM also has the unusual property, not found in the UCM, that the growth in the number of cliques of size n scales independently of n as t → ∞. For both the UCM and CCM, the mean shortest path lengths scale as ln(t) indicative of the smallworld property.
The above comparison uses a single effective copying probability, but key differences are robust for variable p in the sparse regime. Specifically, the UCM degree distribution always exhibits a powerlaw tail, and the largest measured clustering coefficients fall below the values seen for the CCM, see Table 1. Relaxing the CCM to the GCCM, we note that for large p_{hid} and small p_{obs}, the measured clustering values regularly exceed those observed in the UCM, with the UCM only reaching similar values far into the dense regime. Given the continuing debate about the ubiquity of powerlaws in real networks^{43}, the observation that powerlaws are suppressed in the GCCM as soon as the UCM symmetry is broken supports the view that powerlaw network scaling is an idealised case which in practice is rarely observed for real networks.
Whether such extreme bias is plausible in real networks is uncertain. However, observations in academic collaboration networks suggest that extreme bias may be possible^{33}. For instance, Kim and Diesner^{33} show that the ratio of triadic closure between two nodes is approximately zero if the number of shared collaborators is zero, rises rapidly as the number of shared collaborators increases, and plateaus at a ratio of one.
A second clue towards heterogeneous copying is the observation of very large clustering values in real networks. A selection of these networks and their clustering coefficients are shown in Table 2. Stressing that both the UCM and GCCM are toy models of node copying, the networks in Table 2 exhibit average local and/ or global clustering far exceeding even the most optimistic values for the UCM. In contrast, the listed clustering values are relatively similar to what may plausibly emerge from heterogeneous copying, although even the clustering observed for the extreme CCM case falls below some of the values shown in Table 2. Future work should go beyond this qualitative analysis and should attempt to measure the degree to which copying symmetry is broken for real networks where these mechanisms are relevant.
Discussion
The UCM, CCM, and GCCM are all examples of corded copying models where an edge forms between a newly added node and the target node which is duplicated. This is in contrast to uncorded duplication models where a new node is formed by copying an existing target node and its neighbours, but an edge is not formed between the new node and the duplicated target node. Corded models are more common in the context of social phenomena and triadic closure, whereas uncorded models are typically more relevant to duplication–divergence processes in protein interaction networks. In the current work, we have focused exclusively on corded models; considering heterogeneous copying in uncorded models^{13,14,23,24} would be an appropriate future extension. Heterogeneous copying could also be studied by extending directed models^{25,26}.
The GCCM and CCM are examples of hidden network models. From a mathematical standpoint, hidden network models can be thought of as a variant of interdependent networks where nodes in one layer have dependencies of nodes in another layer^{38,39}. However, at a conceptual level, hidden network models put an emphasis on how the evolution of network structure can depend on asymmetries not observed in our data.
In this paper, we have focused on copying in social networks, but the ideas naturally extend to other contexts. In economics, our framework may be applied to shareholder networks^{44}, where nodes are connected if they both own a common asset. Here, the hidden network represents the full set of coowned assets, whereas the observed network includes publicly disclosed assets. Similarly, the idea can be applied to cobidding networks in public procurement, where an edge indicates that two companies both placed bids on the same contract. In many jurisdictions, only winning bids (of which there may be multiple) are publicly revealed. Therefore, the observed network may represent the network of winning bids, whereas the hidden network includes all bids. Hidden network models may be a valuable representation in these cases if there are structural reasons for why some data is observed and some data is hidden. For instance, fraudulent behaviour in public procurement has been associated with anomalous structural features in the cobidding network^{45}.
Other examples may be found in ecology, where multilayer networks have been used to represent different interactions between a common set of species^{46,47}. Kéfi et al.^{46} find that the structure of interactions in one layer has significant crossdependencies to the structure of other layers. This mirrors how interlayer dependencies in the CCM are used to break symmetries in the evolution of the observed network. Finally, hidden networks may find general relevance to other fields where interdependent networks have been influential. This may include studies on energy demand management for power grids^{48}, and the emergence of synchronisation in multilayer neuronal models^{49}.
A more unusual application of the hidden network concept is for decomposing complex singlelayer networks into simpler twolayer structures. One such example is secondneighbour preferential attachment; an implementation of the Barabási−Albert model where nodes attach proportionally to the number of nodes within two steps of a target node^{50}. Using our framework, the model is decomposed into an observed network, and a hidden network (in this case referred to as the influence network) where nodes are connected to all nodes which are two or fewer steps away, representing the node’s sphere of influence. Here, secondneighbour preferential attachment is equivalent to conventional firstneighbour preferential attachment followed by a local copying step. Structural heterogeneity that is intrinsic in such a model has profound consequences for the time dependence of network growth^{50}.
Conclusion
We have introduced a general model of heterogeneous copying, implemented using a hidden network model. In the case of extreme copying bias, we have derived analytical results and have demonstrated significant differences to similar models with uniform copying rules. In particular, powerlaw degree distributions observed in uniform copying can be suppressed under heterogeneous copying, and networks are significantly more clustered if copying is biased towards a node’s inner circle. Although a systematic study of copying in real networks is necessary, evidence suggests that heterogeneous copying may be relevant in a social context.
The heterogeneous copying model is just one simple application of a hidden network model. In general, the framework allows us to deconstruct network growth heterogeneities in a nonarbitrary way, focusing on structural rather than node heterogeneity, and poses questions concerning the role of hidden information in network growth. Exploring these questions is a key aim in upcoming work.
Data availability
All data can be generated using the Python code provided.
Code availability
Python code is available at: github.com/MaxFalkenberg/RandomCopying.
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Acknowledgements
I am grateful to Tim S. Evans, Chester Tan, and Kim Christensen for a number of useful discussions, and to Tim S. Evans and Chester Tan for proofreading the paper. I acknowledge a Ph.D. studentship from the Engineering and Physical Sciences Research Council through Grant No. EP/N509486/1.
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Falkenberg, M. Heterogeneous node copying from hidden network structure. Commun Phys 4, 200 (2021). https://doi.org/10.1038/s42005021006941
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DOI: https://doi.org/10.1038/s42005021006941
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