Abstract
Physical networks are made of nodes and links that are physical objects embedded in a geometric space. Understanding how the mutual volume exclusion between these elements affects the structure and function of physical networks calls for a suitable generalization of network theory. Here, we introduce a networkofnetworks framework where we describe the shape of each extended physical node as a network embedded in space and these networks are bound together by physical links. Relying on this representation, we introduce a minimal model of network growth and we show for a general class of physical networks that volume exclusion induces heterogeneity in both node volume and degree, with the two becoming correlated. These emergent properties strongly affect the dynamics on physical networks: by calculating their Laplacian spectrum as a function of the coupling strength between the nodes we show that degreevolume correlations suppress the role of hubs as early spreaders in diffusive dynamics. We apply the networkofnetworks framework to describe several real systems and find properties analog to the minimal model networks. The prevalence of these properties points towards general growth mechanisms that do not depend on the specifics of the systems.
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Introduction
The building blocks of physical networks are extended objects that do not intersect each other, resulting in nontrivial geometric layouts^{1}, link entanglement^{2} and emergent correlations between physical and network structure^{3}. Yet, these works model nodes as localized spheres connected by extended tubelike links, an assumption that does not necessarily reflect the structure of most realworld physical networks. In the connectome, for example, nodes represent neurons with nontrivial dendritic shapes, and links are pointlike synapses^{4}. A similar picture emerges for molecular networks such as the cytoskeleton, mitochondrial networks, or fiber materials, where nodes are extended molecular strands and bonds between them are localized^{5,6,7}, as well as in the woodwideweb, where extended tree roots and mycelia connect to form a complex underground network^{8,9}. Therefore, the spheretube paradigm often falls short of describing physical networks, calling for a more general framework to cope with the complex shape of nodes and links.
In this work, we develop a networkofnetworks representation of physical networks that is able to capture arbitrary node shapes^{10,11} and allows us to characterize both structural and dynamical properties of networks. Relying on the networkofnetworks framework, we introduce a model that grows physical networks from fractal segments. Analytically solving the model, we show that physicality induces heterogeneity in both the physical and the network properties of the nodes and that the two become strongly correlated. These correlations also affect the dynamics on the networks: generalizing the combinatorial Laplacian to physical networks^{12,13,14}, we show that fast dynamical modes associated to hubs (and corresponding to the tail of the Laplacian spectra) are suppressed by the emergent correlations between node volume and degree. The usefulness of the mathematical tools we develop in this paper goes beyond the specifics of the model, and we demonstrate this by applying our framework to several real physical networks, including a recently collected data set describing more than ~ 20, 000 neurons of the adult fruit fly’s brain^{15}. In doing so, we identify positive node degreevolume correlations similar to our minimal growth model, and we show that these have an analog effect on the Laplacian spectrum of the connectome. The fact that degreevolume correlations emerge in a minimal model while also prevalent in real systems suggests a general mechanism behind such correlations that does not depend on the complex details of the growth of real networks.
Results
Networkofnetworks representation
We aim to describe physical networks embedded in some substrate or medium. In its most general form, the substrate is represented by a graph \({{{{{{{\mathcal{S}}}}}}}}\), and each physical node i is an extended object occupying a subgraph \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\subset {{{{{{{\mathcal{S}}}}}}}}\). To capture volume exclusion, we do not allow nodes to overlap, i.e., \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\cap {{{{{{{{\mathcal{V}}}}}}}}}_{j}=\varnothing\) for i ≠ j. Two nodes i and j may form a link (i, j) if they occupy adjacent sites in \({{{{{{{\mathcal{S}}}}}}}}\). The physical layout \({{{{{{{\mathcal{P}}}}}}}}\) of the network is a networkofnetworks, i.e., it is the union of physical nodes, where each node is a network itself, together with the bonds forming the connections between the nodes (Fig. 1a). The layout \({{{{{{{\mathcal{P}}}}}}}}\) is a physical realization of the combinatorial network \({{{{{{{\mathcal{G}}}}}}}}\), where node i of \({{{{{{{\mathcal{G}}}}}}}}\) corresponds to the physical node \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\), and nodes i and j are connected if there is a bond between \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\) and \({{{{{{{{\mathcal{V}}}}}}}}}_{j}\) in \({{{{{{{\mathcal{P}}}}}}}}\) (Fig. 1b). Though the substrate \({{{{{{{\mathcal{S}}}}}}}}\) can represent any available space, here we focus on substrates that are d dimensional cubic lattices with linear size L and periodic boundary conditions. Note that network representations of this kind are employed in the graph drawing literature with the focus on algorithms that embed a given combinatorial network into \({{{{{{{\mathcal{S}}}}}}}}\)^{16}. Here, we are interested in physical networks \({{{{{{{\mathcal{P}}}}}}}}\) growing in \({{{{{{{\mathcal{S}}}}}}}}\), the emergent relation between \({{{{{{{\mathcal{P}}}}}}}}\) and \({{{{{{{\mathcal{G}}}}}}}}\), and its consequences on the dynamics on the network.
Network growth
To study the effect of physicality on network evolution, we define a model of network growth relying on the networkofnetworks representation. We start with an empty \({\mathcal {S}}\) and we place a single physical node \({{{{\mathcal{V}}}}_{0}}\) occupying a subset of the sites. We add the rest of the nodes iteratively: At time step t we add a new node \({{{{{{{{\mathcal{V}}}}}}}}}_{t}\) that is seeded at a random unoccupied site and grows until it hits an already existing node \({{{{{{{{\mathcal{V}}}}}}}}}_{s}\) and a link (t − s) is formed. The growth of node \({{{{{{{{\mathcal{V}}}}}}}}}_{t}\) is driven by some random or deterministic process; and we assume that the physical nodes are characterized by a fractal dimension d_{f} ∈ [1, d]^{17,18}. We add N physical nodes or until all of \({{{{{{{\mathcal{S}}}}}}}}\) is occupied; in the latter case we call the physical network saturated.
Since the total volume of the network increases over time, later nodes hit the network at higher rates, and the typical size of nodes decreases. Hence, we expect that nodes added early have a higher degree than nodes added in the final stages of the network evolution, both because they are larger and they have more time to collect connections. This suggests that to analytically characterize the evolution of the physical network two ingredients have to be considered: (i) network growth, i.e., nodes are added sequentially to the system and (ii) externally limited node growth, i.e., the nodes grow until they hit the already existing network. We show that these two ingredients lead to the emergence of power law combinatorial networks with degree exponents γ≤3.
We start the analytical treatment of the model by estimating the probability p_{ij} that two randomly placed physical nodes \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\) and \({{{{{{{{\mathcal{V}}}}}}}}}_{j}\) intersect. If the two boxes containing the physical nodes have side length l_{i} ≫ l_{j}, respectively, and the larger node \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\) intersects the box containing the smaller node \({{{{{{{{\mathcal{V}}}}}}}}}_{j}\), then, by dimension count, the two nodes overlap with positive probability if d_{f}≥d/2. We can tile the lattice with \({(L/{l}_{j})}^{d}\) boxes with side length l_{j}, and the number of such boxes intersected by \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\) is \(\sim {({l}_{i}/{l}_{j})}^{{d}_{{{{{{{{\rm{f}}}}}}}}}}\). Therefore the intersection probability is
where \({v}_{i}= {{{{{{{{\mathcal{V}}}}}}}}}_{i} \sim {l}_{i}^{{d}_{{{{{{{{\rm{f}}}}}}}}}}\) is the volume of node i. If, however, d_{f} < d/2 and l_{j}≤l_{i} ≪ L, then the nodes avoid each other with high probability. In this case, the intersection probability will have the meanfield behavior, wellapproximated by the probability of selecting the sites of \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\) and \({{{{{{{{\mathcal{V}}}}}}}}}_{j}\) uniformly from \({{{{{{{\mathcal{S}}}}}}}}\), i.e., \({p}_{ij}^{{{{{{{{\rm{MF}}}}}}}}} \sim {v}_{i}{v}_{j}/{L}^{d}\), which is independent of d_{f} and agrees with Eq. (1) for d_{f} = d/2.
Using the same boxcounting argument that led to Eq. (1), the probability that a node added at time t intersects any existing node s < t is approximately given by \({\sum }_{s < t}{p}_{st}={v}_{t}^{d/{d}_{{{{{{{{\rm{f}}}}}}}}}1}{V}_{t1}/{L}^{d}\), where V_{t−1} is the total volume of nodes s < t. A key observation is that the size of node t increases until it hits the existing network, meaning that v_{t} increases until ∑_{s<t}p_{st} ≈ 1, allowing us to estimate the volume of node t as
Equation (2) allows us to express the evolution of the expected total volume via the recursion V_{t+1} = v_{t+1} + V_{t} with initial condition V_{0} = v_{0}. Using a continuous time approximation, we obtain
where c is a constant depending on v_{0}. A natural choice for the latter is that the first node spans the entire available space, i.e., \({v}_{0} \sim {L}^{{d}_{{{{{{{{\rm{f}}}}}}}}}}\), in which case c is independent of L. Equation (3) predicts that N_{sat}, the number of nodes when the network saturates, scales as N_{sat} ~ L^{d}, meaning that the average node volume 〈v〉 remains constant in the L → ∞ large system limit. Therefore, the physical layout \({{{{{{{\mathcal{P}}}}}}}}\) is optimal in the sense that no physical representation of a combinatorial network of N_{sat} nodes can fit into a smaller volume than ~ N_{sat} ~ L^{d}. It is noteworthy that the model achieves this bound despite the fact that the nodes grow randomly.
In light of Eq. (3), we can now calculate the expected degree of the physical nodes in the combinatorial network \({{{{{{{\mathcal{G}}}}}}}}\). In the continuous time approximation, the volume of the newly added node v_{t} is provided by the time derivative of V_{t}, i.e., \({v}_{t} \sim {\left(t/{L}^{d}\right)}^{{d}_{{{{{{{{\rm{f}}}}}}}}}/d}\). Hence, following Eq. (1), the expected degree of node t after the addition of N nodes is
where the proportionality is valid for t ≪ N. This means that the volume occupied by large nodes (i.e., nodes that were added early) in the physical layout is proportional to their degree in the combinatorial network.
We finally calculate the complementary cumulative degree distribution \(P(k)=1\frac{1}{N}\mathop{\sum }\nolimits_{t:{k}_{t}\ge k}^{N}1\), finding that P(k) ~ k^{−(γ−1)} with exponent
For d_{f}≤d < 2d_{f} the degree exponent falls in the range 2≤γ < 3. In the meanfield regime, the degree exponent can be obtained by substituting d/d_{f} with 2, yielding γ_{MF} = 3. Note that the upper critical dimension of the physical network depends on the kinetic growth of the nodes. For example, growing nodes along a straight trajectory in a random direction generates nodes with d_{f} = 1; therefore, the networks fall in the meanfield regime d_{f}≤d/2 even for embedding dimension d = 2.
In the above model, each physical node grows starting from a random location following some growth process. We stress that our calculations hold for a general class of node growth algorithms, the crucial assumption being that if the boxes around two random walk pieces intersect, then with uniformly positive probability the trajectories also intersect, which implies a level of isotropy of node growth. As a counterexample, consider nodes that always grow in one direction along one of the axes. Such nodes will run parallel to each other, avoiding intersection, hence the resulting network will be a collection of disconnected chains. If, however, the nodes grow along straight lines but in random directions, thus restoring isotropy on average, then the resulting network has a power law degree distribution (SI Sec. S1.4).
Numerical simulations
To test our analytical predictions, we numerically generate physical networks where nodes grow according to random walk trajectories. Specifically, we generate nodes using looperased random walks (LERWs), i.e., a trajectory that evolves as a simple random walk in which any loop is erased as soon as it is formed^{19,20,21,22}. Here, we focus on the LERW, as it represents a tractable model ofnonselfintersecting random trajectories with wellunderstood nontrivial critical properties. Its critical properties are studied both in the mathematics and physics literature^{23,24,25,26}; for example, their fractal dimension in d = 2 is d_{f} = 5/4^{22}, in d = 3 it is d_{f} ≃ 1.6236(4)^{27,28}, while its upper critical dimension is d_{u} = 4 where d_{f} = 2 with a logarithmic correction^{29}. (See “METHODS” for further details.) We remark that our predictions are not specific to LERWs, in Sec. S1 of the Supplementary Information, we study various alternative growth processes.
Knowing the fractal dimensions of LERWs allows us to directly verify the predictions of Eqs. (3)–(5):
Volume evolution
Equation (3) predicts that the total volume of the physical network evolves as \({V}_{t} \sim {t}^{1{d}_{{{{{{{{\rm{f}}}}}}}}}/d}\). Figure 2a shows the excellent agreement between the theoretical predictions and numerical simulations. Note that, as expected, in the meanfield regime d≥4 the network volume follows the classic diffusion growth V_{t} ~ t^{1/2}. Figure 2b further corroborates the predicted scaling of the number of nodes at saturation, i.e., N_{sat} ~ L^{d}.
Degreevolume correlations
A second prediction is the emergence of degreevolume correlations, capturing the interplay between the physical layout \({{{{{{{\mathcal{P}}}}}}}}\) and the combinatorial network \({{{{{{{\mathcal{G}}}}}}}}\). In particular, Eq. (4) predicts a linear proportionality between the node volume v_{i} and degree k_{i}, and we again find excellent agreement with simulations for all the tested dimensions (Fig. 2c).
Power law emergence
As a final test, we verify the emergence of power law scaling in the degree distribution of the combinatorial networks \({{{{{{{\mathcal{G}}}}}}}}\). As anticipated in Eq. (5), the degree exponent depends on both the dimensionality of the embedding substrate, d, and the fractal dimension of the nodes, d_{f}. Figure 2d shows that numerical simulations confirm the predicted degree exponent γ = 1 + d/d_{f} for dimensions d < 4, while the meanfield exponent γ_{MF} = 3 is found for d≥4. In traditional models of combinatorial networks, heterogeneity typically arises from preferential attachment or some other optimization process. Our model is based on random growth without any explicit preference to create highly connected nodes; therefore, the uniform attachment tree may be considered as the nonphysical counterpart of our model. Uniform attachment yields exponential degree distribution, hence the power law distribution observed here is a direct consequence of volume exclusion, which, together with the dynamic network growth, induces effective preferential attachment.
Physical network Laplacian
The layout \({{{{{{{\mathcal{P}}}}}}}}\) is a physical realization of the combinatorial network \({{{{{{{\mathcal{G}}}}}}}}\). Traditional studies of dynamics on physical networks ignore the layout \({{{{{{{\mathcal{P}}}}}}}}\) and focus only on the role of \({{{{{{{\mathcal{G}}}}}}}}\), thus prompting the question: does modeling dynamics on \({{{{{{{\mathcal{G}}}}}}}}\) accurately capture dynamics on physical networks? To explore this, we study the spectral properties of \({{{{{{{\mathcal{P}}}}}}}}\) and show that physical nodes emerge as functional units through timescale separation, yet even in this limit the structure of \({{{{{{{\mathcal{P}}}}}}}}\) continues to affect the dynamics. We focus on the Laplacian spectrum^{12} since it influences the behavior of several dynamical processes on networks^{30} including diffusion^{31,32}, synchronization^{33} and it underlies the definition of several informationtheoretic tools to analyze the multiscale functioning of networks^{10,14,34,35,36,37}.
We study the problem by invoking, once again, the networkofnetworks representation. In our setup, we assign a weight to each connection in \({{{{{{{\mathcal{P}}}}}}}}\) such that links within physical nodes have weight 1 and links connecting two physical nodes have weight w, capturing that in real physical networks bonds between nodes are often qualitatively different than those within nodes. The weighted Laplacian matrix of \({{{{{{{\mathcal{P}}}}}}}}\) occupying V sites is then \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}={{{{{{{{\bf{D}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}{{{{{{{{\bf{A}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}\), where \({{{{{{{{\bf{A}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}\) is the V × V weighted adjacency matrix and \({{{{{{{{\bf{D}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}\) is a diagonal matrix such that \({[{{{{{{{{\bf{D}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}]}_{{{{{{{{\rm{ss}}}}}}}}}={\sum }_{u}{[{{{{{{{{\bf{A}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}]}_{{{{{{{{\rm{su}}}}}}}}}\) is the sum of the weights of the links adjacent to site s in \({{{{{{{\mathcal{P}}}}}}}}\). If we now set w = 0, the networkofnetworks falls apart and each physical node becomes a separate connected component, resulting in a blockdiagonal Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}(0)={{{{{{{\rm{diag}}}}}}}}({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{{\mathcal{V}}}}}}}}}_{1}},{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{{\mathcal{V}}}}}}}}}_{2}},\ldots,{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{{\mathcal{V}}}}}}}}}_{N}})\), where \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{{\mathcal{V}}}}}}}}}_{i}}\) is the Laplacian of the physical node \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\). The Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}(0)\) has N zero eigenvalues corresponding to the N blocks (i.e., the physical nodes), hence we can assign an eigenvector u_{i}(w = 0) to the ith node such that \({[{{{{{{{{\bf{u}}}}}}}}}_{i}(0)]}_{s}=1/\sqrt{{v}_{i}}\) if site s is within node i, otherwise \({[{{{{{{{{\bf{u}}}}}}}}}_{i}(0)]}_{s}=0\), where \({v}_{i}= {{{{{{{{\mathcal{V}}}}}}}}}_{i}\) is the volume of node i. Since linear combinations of these vectors are also eigenvectors, we can write the zero eigenvectors of \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}\) as \({{{{{{{\bf{u}}}}}}}}(0)={{{{{{{{\bf{M}}}}}}}}}{\tilde{{{{{{{{\bf{u}}}}}}}}}}\), where M is the N × V membership matrix such that \({[{{{{{{{\bf{M}}}}}}}}]}_{si}=1/\sqrt{{v}_{i}}\) if site s is part of node i, otherwise [M]_{si} = 0, and \(\tilde{{{{{{{{\bf{u}}}}}}}}}\) is any normalized N dimensional vector.
We can gain insights about the spectral properties of \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}\) by working in the weak coupling regime w ≪ 1 and relying on perturbation theory. Following a treatment similar to the one adopted to study diffusion in multilayer networks^{38,39,40}, we consider w a small perturbation and write \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}(w)={{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}(0)+w{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}^{{\prime} }\), where \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}^{{\prime} }\) is the Laplacian matrix of the subnetwork of \({{{{{{{\mathcal{P}}}}}}}}\) formed by the bonds between physical nodes. The characteristic equation, up to first order in w, becomes then
Perturbations around λ(0) = 0 lead to N eigenvalues that are \({{{{{{{\mathcal{O}}}}}}}}(w)\), while the rest of the eigenvalues are constant in leading order (Fig. 3a). This means that on the 1/w timescale, diffusionlike dynamics on the physical network are captured by the N slow eigenmodes. We obtain these from Eq. (6) (see “METHODS”), yielding
where \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) is the N × N Laplacian matrix of the combinatorial network \({{{{{{{\mathcal{G}}}}}}}}\) and V is an N × N diagonal matrix such that its diagonal elements are [V]_{ii} = v_{i}. We call the volumenormalized Laplacian the physical network Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{phys}}}}}}}}}={{{{{{{{\bf{V}}}}}}}}}^{1/2}{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}{{{{{{{{\bf{V}}}}}}}}}^{1/2}\).
Equation (7) is a key relation for understanding the dynamics on physical networks since it allows to characterize the dynamics on \({{{{{{{\mathcal{P}}}}}}}}\) on the timescale 1/w in a coarsegrained way: after integrating out the fast modes corresponding to eigenvalues λ(w) ≫ w, the state of each physical node \({{{{{{{{\mathcal{V}}}}}}}}}_{i}\) is given by a single variable, while the coupling between the nodes is provided by the combinatorial network \({{{{{{{\mathcal{G}}}}}}}}\). However, the combinatorial Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) is not sufficient to capture the dynamics, and we must normalize \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) by the volume of the nodes, as shown in Eq. (7). This means that physical networks with the same combinatorial network but different layout can have drastically different dynamical properties. For example, if nodes have approximately the same size, i.e., v_{i} ≈ V/N, then the physical layout only affects the overall timescale, otherwise the Laplacian spectrum is determined by \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\). If, however, node sizes are heterogeneously distributed, normalizing by volume will also have a heterogeneous effect on the eigenvalues.
Application to the physical network growth model
We showed above that physical networks generated by our network growth model are characterized by heterogeneous nodevolume distribution and proportionality between the degree and the volume of nodes (Fig. 2). To probe the effect of this emergent correlation, we shuffle the volume of the nodes of a LERW physical network to remove the correlation between network and physical structure. We then compare the spectrum of the volumenormalized Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{phys}}}}}}}}}={{{{{{{{\bf{V}}}}}}}}}^{1/2}{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}{{{{{{{{\bf{V}}}}}}}}}^{1/2}\) to its randomized version \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{phys}}}}}}}}}^{{{{{{{{\rm{rand}}}}}}}}}={{{{{{{{\bf{V}}}}}}}}}_{{{{{{{{\rm{rand}}}}}}}}}^{1/2}{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}{{{{{{{{\bf{V}}}}}}}}}_{{{{{{{{\rm{rand}}}}}}}}}^{1/2}\) and to the Laplacian spectrum of the combinatorial network \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\). Figure 3b shows that the spectrum of \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) has a heavy tail characterized by the same γ exponent of Eq. (5), as expected for combinatorial networks with power law degree distributions^{12}. Adding heterogeneous but uncorrelated node sizes does not influence the tail while taking into account the degreevolume correlation of nodes removes the heavy tail and leads to a rapidly decaying spectrum. In power law networks, the eigenvector corresponding to the largest eigenvalue λ_{N} of \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) is typically concentrated on the node with the largest degree^{41,42}. In our model, the largest degree node also has the largest volume; therefore normalizing by node volume \({{{{{{{{\bf{V}}}}}}}}}^{1/2}{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}{{{{{{{{\bf{V}}}}}}}}}^{1/2}\) significantly lowers λ_{N}. Since node sizes are heterogeneously distributed, with high probability, we associate volume ~ 1 to the highest degree node after randomization. Hence, the eigenvalue λ_{N} of \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) is largely unaffected by the randomized normalization (Fig. 3c). At the other end of the spectrum, controlling the longtime mixing of the dynamics, the eigenvector associated with the algebraic connectivity λ_{2} typically spans the entire network. Figure 3d shows that taking node volumes into account slows the dynamics down; however, degreevolume correlations do not significantly affect λ_{2}.
Note that positive degreevolume correlations, responsible for the suppression of the tail of the Laplacian spectrum, naturally arise in minimumvolume physical realizations of combinatorial networks. Any combinatorial network \({{{{{{{\mathcal{G}}}}}}}}\) has many possible physical realizations \({{{{{{{\mathcal{P}}}}}}}}\), a minimum volume realization is a \({{{{{{{\mathcal{P}}}}}}}}\) that minimizes the total volume of the network. Consider node \(i\in {{{{{{{\mathcal{G}}}}}}}}\) with degree k_{i}; in any possible \({{{{{{{\mathcal{P}}}}}}}}\), the physical realization of node i must have volume at least proportional to k_{i}, otherwise it is unable to support k_{i} connections. Therefore, we expect positive degreevolume correlations in minimumvolume physical layouts. This means that any physical network generation process that minimizes total volume – either explicitly or as an emergent property, like in our model – is characterized by positive degreevolume correlations and hence that the spectrum of Q_{phys} is similarly affected by physicality as in our model.
Real physical networks
We identified the degreevolume correlations and the profile of the Laplacian spectrum as important features of physical networks that can emerge even in the simplest models. To measure these properties, we do not need a detailed description of the layout of a physical system – we only need the combinatorial network and a list of the node volumes, allowing us to describe very large and complex physical networks. As a case study, we investigate a recently published data set providing the threedimensional layout of more than 20,000 neurons of the brain of an adult fruit fly and the location of more than 13 million synapses connecting them (Fig. 4a)^{15}. Although our simple growth model does not attempt to capture the myriad of complex mechanisms shaping brain development, we find that the fruit fly brain is characterized by similar emergent properties as the model networks. Figure 4b shows, for example, that the multiplicityweighted node degree, i.e., the number of synapses a neuron has, can be approximated by a power law γ_{ff} ≈ 2.3, albeit with an exponential cutoff^{43,44}. We also find a strong positive correlation between the weighted degree and the volume of the nodes (Fig. 4c).
To compare the spectrum of the combinatorial Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) and the physical network Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{phys}}}}}}}}}={{{{{{{{\bf{V}}}}}}}}}^{1/2}{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}{{{{{{{{\bf{V}}}}}}}}}^{1/2}\), we measure volume in units such that the mean node volume is unity, i.e., 〈v〉 = 1 (see “METHODS” for further details). Calculating the leading eigenvalues of \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) and Q_{phys}, we find that \({\lambda }_{N}^{{{{{{{{\mathcal{G}}}}}}}}}/{\lambda }_{N}^{{{{{{{{\rm{phys}}}}}}}}}\approx 32.7\), indicating that degreevolume correlations greatly suppress the modes of the dynamics that spread the fastest, similarly to model networks. This is further supported by Fig. 4d, showing again that physicality suppresses the tail of the spectrum.
To further probe the role of degreevolume correlations, we calculate the leading eigenvectors \({\tilde{{{{{{{{\bf{u}}}}}}}}}}_{N}\) of \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) and Q_{phys}. Figure 4e, g show that, as expected for heterogeneous combinatorial networks, \({\tilde{{{{{{{{\bf{u}}}}}}}}}}_{N}^{{{{{{{{\mathcal{G}}}}}}}}}\) is concentrated on the largest hub \({i}_{{{{{{{{\mathcal{G}}}}}}}}}\) in the network, and the weight of the eigenvector decays exponentially as the geodesic distance from \({i}_{{{{{{{{\mathcal{G}}}}}}}}}\) in \({{{{{{{\mathcal{G}}}}}}}}\). This means that, without taking physicality into account, the largest degree node is also the earliest spreader of diffusive dynamics. For the physical Laplacian Q_{phys} we find a different picture: \({\tilde{{{{{{{{\bf{u}}}}}}}}}}_{N}^{{{{{{{{\rm{phys}}}}}}}}}\) is again concentrated on a single node i_{phys}; this node, however, is not the largest hub. The leading eigenvector instead is centered on a node that balances high degree and low volume: node i_{phys} is the 159th largest degree node and is at the top 15 percentile of the volume distribution. In fact, node i_{phys} is the node that maximizes the degreevolume ratio, i.e., i_{phys} = argmax_{i}k_{i}/v_{i}. This means that degreevolume correlations not only slow down spreading dynamics in physical networks, but also change the identity of the early spreaders.
Here we chose to focus on the fruit fly brain network as it represents one of the largest and most detailed maps of physical networks available; however, our framework is not specific to neural networks. In the Sec. S2 of the Supplementary Information, we analyze four additional real systems: a network describing the cavities of a porous material, a neural network of a nematode, a river network, and a vascular network. In each case, we find positive degreevolume correlations and that these correlations suppress the tail of the Laplacian spectra. The fact that the physical and network properties of nodes become intertwined in such a diverse set of real networks, and also in the simplest models, indicates a general mechanism behind the emergence of degreevolume correlations that do not depend on the details of the individual networks.
Discussion
Physical networks are complex networks that have a complex threedimensional layout. The networkofnetworks framework naturally lends itself to representing these systems: representing nodes as physically embedded networks allows us to capture arbitrary node shapes and complex wiring. Here, we relied on the networkofnetworks framework to characterize both model and real physical networks. We identified correlations between node degree and volume as a prevalent feature of physical networks: We analytically showed that degreevolume correlations emerge in a minimal network growth model, in fact, we provided arguments that such correlations naturally arise through any growth process that minimizes network volume. We also showed that positive degreevolume correlations are generally present in real systems. These correlations have important consequences on dynamics unfolding on physical networks: the tail of the physical Laplacian spectrum is suppressed by the large volume of hubs. More broadly, these results vividly demonstrate that traditional methods of network science focusing on combinatorial networks cannot fully describe physical networks and that their threedimensional layout must be accounted for.
Our work opens new avenues for physical network research in several ways. First, by establishing the connection between physical networks and networkofnetworks, we allow future work to leverage the rich literature of multilayer networks to characterize physical systems^{10,11}. For example, multilayer centrality measures can be used to quantify the importance of physical nodes^{45,46,47}. Second, previous work on physical networks relies on methods that require a full description of their spatial layout and, therefore, are often limited to systems of a few hundred nodes^{1,2,3}. In contrast, the quantities we studied can be measured relying on the combinatorial network and a list of node volumes, allowing the characterization of largescale physical networks without the need of the full threedimensional layout. For example, we can tune the volume of the nodes to systematically study how physical layout affects the Laplacian spectrum. Finally, the simple growth model and its analytical description can serve as the starting point for the exploration of additional growth mechanisms that characterize neural networks and other physical networks. For example, future work may study branched nodes, longrange interactions that guide the growth of physical nodes, or the expansion of available space by modeling the evolution of the underlying substrate.
Methods
Looperased random walks
In our network growth model, we can generate physical nodes with any stochastic or deterministic process that produces a growing fractal embedded in \({{\mathbb{Z}}}^{d}\). Standard selfavoiding walks are traditionally used to model polymers obeying volume exclusion and, therefore, represent a natural choice to model node growth^{17}. However, the naïve kinetic version of the selfavoiding walk traps itself in two and three dimensions at finite length^{21}, making it a poor candidate for constructing large physical networks. Instead, we focus on looperased random walks (LERW): a LERW evolves as a simple random walk, except when it intersects itself, we delete the loop that it created and continue the walk^{20}. This guarantees that the final physical node does not intersect itself and that the walk never gets trapped. Alternatively, the LERW can be defined as a special case of Laplacianrandom walks, where transition probabilities are defined by a harmonic function^{48,49}. This alternative construction does not require deleting loops, hence is more realistic as a growth model. The LERW has attractive mathematical properties making it amenable to analytical treatment. For example, Wilson’s algorithm uses iterative LERWs to construct a uniform spanning tree (UST) of any graph^{24}. In fact, the physical network our algorithm constructs is a UST of the \({{{{{{{\mathcal{S}}}}}}}}\) substrate together with a partition identifying the nodes. Future work may exploit this connection between USTs and LERW physical networks, together with known results in dimensions d = 2 and d > 4^{22,50}, to rigorously prove some of the results presented here.
Perturbation of the physical Laplacian
To obtain the slow eigenmodes, we match the firstorder terms of Eq. (6) and substitute \({{{{{{{\bf{u}}}}}}}}(0)={{{{{{{{\bf{M}}}}}}}}}{\tilde{{{{{{{{\bf{u}}}}}}}}}}\), so that
Multiplying from the left by the transpose of the membership matrix M we get
The ith row of M^{T} is the trivial eigenvector u_{i}(w = 0) corresponding to physical node i; therefore \({{{{{{{{\bf{M}}}}}}}}}^{T}{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{P}}}}}}}}}(0)\) is all zeros and M^{T}M is the N × N identity matrix, leading to Eq. (7) in the text.
The fruit fly connectome
We study the Hemibrain data set, which describes a portion of the central brain of the fruit fly, Drosophila melanogaster^{15}. The physical layout of the connectome is provided by the detailed threedimensional shape of each neuron and the location of the synapses between them. The corresponding combinatorial network contains 21,662 nodes representing neurons and 13,603,750 links representing synapses. Synaptic partners are connected through approximately 5 synapses on average, and the maximum number of synapses between two neurons is 6039. In our calculations, we treat the combinatorial network as a weighted and undirected network, where the weight of the link (i, j) is equal to the number of synapses between neurons i and j. Note that we only require the combinatorial network and the volume of each node for our calculations; therefore, the detailed physical layout of the connectome is, in fact, not needed.
Note that the Hemibrain data set covers a large portion of, but not the entire, fruit fly brain. Since degree and volume are local properties of the nodes, we expect that the results presented here would not change significantly if the entire connectome were to be considered.
Degree distribution
We find that the weighted degree distribution has a heavy tail, which can be approximated by a power law with γ_{ff} ≈ 2.3 for degrees ≥1058 with an exponential cutoff; the power law fit, however, cannot be distinguished from a lognormal fit on the same range^{43,44}.
Laplacian spectrum
Comparing the spectrum of the combinatorial Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) and the volumenormalized Laplacian \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\rm{phys}}}}}}}}}={{{{{{{{\bf{V}}}}}}}}}^{1/2}{{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}{{{{{{{{\bf{V}}}}}}}}}^{1/2}\) carries a level of ambiguity: \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) does not depend on the node volumes, while changing the unit of volume multiplies the spectrum of Q_{phys} by a constant. To meaningfully compare the two spectra, (i) we think of \({{{{{{{{\bf{Q}}}}}}}}}_{{{{{{{{\mathcal{G}}}}}}}}}\) as a physical Laplacian where all nodes have unit volume, and (ii) we set the mean node volume in Q_{phys} to unity, i.e., 〈v〉 = 1. With this choice of units, any difference in the eigenvalues is due to the heterogeneous distribution of node volumes in the physical network and not to a global shift caused by the choice of units.
Reporting summary
Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.
Data availability
Data to reproduce the figures is available at https://github.com/posfaim/physnets_as_netonets.
Code availability
Code to generate random networks and reproduce the figures is available at https://github.com/posfaim/physnets_as_netonets^{51}.
References
Dehmamy, N., Milanlouei, S. & Barabási, A.L. A structural transition in physical networks. Nature 563, 676–680 (2018).
Liu, Y., Dehmamy, N. & Barabási, A.L. Isotopy and energy of physical networks. Nat. Phys. 17, 216–222 (2021).
Pósfai, M. et al. Impact of physicality on network structure. Nat. Phys. 20, 142–149 (2024).
Bullmore, E. & Sporns, O. Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10, 186–198 (2009).
Viana, M. P. et al. Mitochondrial fission and fusion dynamics generate efficient, robust, and evenly distributed network topologies in budding yeast cells. Cell Syst. 10, 287–297 (2020).
Fletcher, D. A. & Mullins, R. D. Cell mechanics and the cytoskeleton. Nature 463, 485–492 (2010).
Picu, C. R. Network Materials: Structure and Properties. (Cambridge University Press, Cambridge, 2022).
Simard, S. W. et al. Net transfer of carbon between ectomycorrhizal tree species in the field. Nature 388, 579–582 (1997).
Steidinger, B. S. et al. Climatic controls of decomposition drive the global biogeography of foresttree symbioses. Nature 569, 404–408 (2019).
De Domenico, M. et al. Mathematical formulation of multilayer networks. Phys. Rev. X 3, 041022 (2013).
Bianconi, G. Multilayer Networks: Structure and Function. (Oxford University Press, Oxford, 2018).
Van Mieghem, P. Graph Spectra for Complex Networks. (Cambridge University Press, Cambridge, 2010).
Castellano, C. & PastorSatorras, R. Relating topological determinants of complex networks to their spectral properties: structural and dynamical effects. Phys. Rev. X 7, 041024 (2017).
Villegas, P., Gili, T., Caldarelli, G., and Gabrielli, A. Laplacian renormalization group for heterogeneous networks. Nat. Phys.19, 445–450, (2023).
Scheffer, L. K. et al. A connectome and analysis of the adult drosophila central brain. Elife 9, e57443 (2020).
Tamassia, R. editor. Handbook of Graph Drawing and Visualization. (CRC Press, Boca Raton, Fl, (2013).
Vicsek, T. Fractal Growth Phenomena. 2nd Edn (World Scientific, Singapore,1992).
Bunde, A. and Havlin, S. (eds). Fractals and Disordered Systems. (Springer Berlin, Heidelberg (2012).
de Gennes, P.G. Exponents for the excluded volume problem as derived by the wilson method. Phys. Lett. A 38, 339–340 (1972).
Lawler, G. F. A selfavoiding random walk. Duke Math. J. 47, 655–693 (1980).
Pietronero, L. Survival probability for kinetic selfavoiding walks. Phys. Rev. Lett. 55, 2025 (1985).
Schramm, O. Scaling limits of looperased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000).
Niemeyer, L., Pietronero, L. & Wiesmann, H. J. Fractal dimension of dielectric breakdown. Phys. Rev. Lett. 52, 1033 (1984).
Wilson, D. B. Generating random spanning trees more quickly than the cover time. In Proceed of the 28th ACM Theory of computing, 296–303. https://doi.org/10.1145/237814.237880 (1996).
Lawler, G. F., Schramm, O. & Werner, W. Conformal invariance of planar looperased random walks and uniform spanning trees. Ann. Probab. 32, 939–995 (2004).
Wiese, K. & Fedorenko, A. A. Field theories for looperased random walks. Nucl. Phys. B 946, 114696 (2019).
Agrawal, H. & Dhar, D. Distribution of sizes of erased loops of looperased random walks in two and three dimensions. Phys. Rev. E 63, 056115 (2001).
Grassberger, P. Scaling of looperased walks in 2 to 4 dimensions. J. Stat. Phys. 136, 399–404 (2009).
Wilson, D. B. Dimension of the looperased random walk in three dimensions. Phys. Rev. E 82, 062102 (2010).
Barrat, A., Barthelemy, M., and Vespignani, A. Dynamical Processes on Complex Networks. (Cambridge University Press, Cambridge, 2008).
Masuda, N., Porter, M. A. & Lambiotte, R. Random walks and diffusion on networks. Phys. Rep. 716, 1–58 (2017).
De Domenico, M. & Biamonte, J. Spectral entropies as informationtheoretic tools for complex network comparison. Phys. Rev. X 6, 041062 (2016).
Arenas, A., DíazGuilera, A., Kurths, J., Moreno, Y. & Zhou, C. Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008).
Boguna, M. et al. Network geometry. Nat. Rev. Phys. 3, 114–135 (2021).
Ghavasieh, A., Stella, M., Biamonte, J. & De Domenico, M. Unraveling the effects of multiscale network entanglement on empirical systems. Commun. Phys. 4, 129 (2021).
Villegas, P., Gabrielli, A., Santucci, F., Caldarelli, G. & Gili, T. Laplacian paths in complex networks: Information core emerges from entropic transitions. Phys. Rev. Res. 4, 033196 (2022).
Ghavasieh, A. & De Domenico, M. Generalized network density matrices for analysis of multiscale functional diversity. Phys. Rev. E 107, 044304 (2023).
Gomez, S. et al. Diffusion dynamics on multiplex networks. Phys. Rev. Lett. 110, 028701 (2013).
SoleRibalta, A. et al. Spectral properties of the laplacian of multiplex networks. Phys. Rev. E 88, 032807 (2013).
Radicchi, F. & Arenas, A. Abrupt transition in the structural formation of interconnected networks. Nat. Phys. 9, 717–720 (2013).
PastorSatorras, R. & Castellano, C. Distinct types of eigenvector localization in networks. Sci. Rep. 6, 18847 (2016).
Hata, S. & Nakao, H. Localization of laplacian eigenvectors on random networks. Sci. Rep. 7, 1–11 (2017).
Clauset, A., Shalizi, CosmaRohilla & Newman, MarkE. J. Powerlaw distributions in empirical data. SIAM Rev. 51, 661–703 (2009).
Alstott, J., Bullmore, E. D. & Plenz, D. powerlaw: a python package for analysis of heavytailed distributions. PloS one 9, e85777 (2014).
Halu, A., Mondragón, RaúlJ., Panzarasa, P. & Bianconi, G. Multiplex pagerank. PloS one 8, e78293 (2013).
SoléRibalta, A., De Domenico, M., Gómez, S., and Arenas, A. Centrality rankings in multiplex networks. In Proceedings of the 2014 ACM conference on Web science, 149–155. https://doi.org/10.1145/2615569.2615687 (2014).
Iacovacci, J., Rahmede, C., Arenas, A. & Bianconi, G. Functional multiplex pagerank. Europhys. Lett. 116, 28004 (2016).
Lyklema, J. W., Evertsz, C. & Pietronero, L. The Laplacian random walk. EPL (Europhys. Lett.) 2, 77 (1986).
Lawler, G. F. Looperased selfavoiding random walk and the Laplacian random walk. J. Phys. A: Math. Gen. 20, 4565 (1987).
Bhupatiraju, S., Hanson, J., and Járai, A. A. Inequalities for critical exponents in ddimensional sandpiles. https://doi.org/10.48550/arXiv.1602.06475 (2017).
Pósfai, Márton posfaim/physnets_as_netonets: physical networks as networkofnetworks. https://doi.org/10.5281/zenodo.11140782, (2024).
Acknowledgements
I.B., M.P., and Á.T. were funded by ERC grant No. 810115DYNASNET. ÁT and SÖS acknowledge partial support from the Icelandic Research Fund, grant No. 239736051. GP was funded by the ERC Consolidator Grant No. 772466NOISE.
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M.P. developed and performed the numerical simulations. M.P. and I.B. performed the data analysis. G.P., Á.T., S.Ö.S., I.B., and M.P. contributed to the analytical results and the conceptual design of the study. M.P. was the lead writer of the manuscript.
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Pete, G., Timár, Á., Stefánsson, S.Ö. et al. Physical networks as networkofnetworks. Nat Commun 15, 4882 (2024). https://doi.org/10.1038/s41467024492278
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DOI: https://doi.org/10.1038/s41467024492278
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