Large Network Limit Research Articles

On Inference for Modularity Statistics in Structured Networks

This article revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional results in the large-network limit for networks exhibiting nodal community structure. Our work facilitates testing for network differences and can be used in conjunction with existing theoretical guarantees for stochastic blockmodel random graphs. Our results are enabled by recent advances in the study of low-rank truncations of large network adjacency matrices. We provide confirmatory simulation studies and real data analysis pertaining to the network neuroscience study of psychosis, specifically schizophrenia. Collectively, this article contributes to the limited existing literature to date on statistical inference for modularity-based network analysis. Supplemental materials for this article are available online.

The spectrum of covariance matrices of randomly connected recurrent neuronal networks with linear dynamics

A key question in theoretical neuroscience is the relation between the connectivity structure and the collective dynamics of a network of neurons. Here we study the connectivity-dynamics relation as reflected in the distribution of eigenvalues of the covariance matrix of the dynamic fluctuations of the neuronal activities, which is closely related to the network dynamics' Principal Component Analysis (PCA) and the associated effective dimensionality. We consider the spontaneous fluctuations around a steady state in a randomly connected recurrent network of stochastic neurons. An exact analytical expression for the covariance eigenvalue distribution in the large-network limit can be obtained using results from random matrices. The distribution has a finitely supported smooth bulk spectrum and exhibits an approximate power-law tail for coupling matrices near the critical edge. We generalize the results to include second-order connectivity motifs and discuss extensions to excitatory-inhibitory networks. The theoretical results are compared with those from finite-size networks and the effects of temporal and spatial sampling are studied. Preliminary application to whole-brain imaging data is presented. Using simple connectivity models, our work provides theoretical predictions for the covariance spectrum, a fundamental property of recurrent neuronal dynamics, that can be compared with experimental data.

Rare event asymptotics for exploration processes for random graphs

Large deviations for random graph models has been a topic of significant recent research activity. Much work in this area is focused on the class of dense random graph models (number of edges in the graph scale as n2, where n is the number of vertices) where the theory of graphons has emerged as a principal tool in the study of large deviation properties. These tools do not give a good approach to large deviation problems for random graph models in the sparse regime. The aim of this paper is to study an approach for large deviation problems in this regime by establishing large deviation principles (LDP) on suitable path spaces for certain exploration processes of the associated random graph sequence. Exploration processes are an important tool in the study of sparse random graph models and have been used to understand detailed asymptotics of many functionals of sparse random graphs, such as component sizes, surplus, deviations from trees, etc. In the context of rare event asymptotics of interest here, the point of view of exploration process transforms a large deviation analysis of a static random combinatorial structure to the study of a small noise LDP for certain stochastic dynamical systems with jumps. Our work focuses on one particular class of random graph models, namely the configuration model; however, the general approach of using exploration processes for studying large deviation properties of sparse random graph models has broader applicability. The goal is to study asymptotics of probabilities of nontypical behavior in the large network limit. The first key step for this is to establish a LDP for an exploration process associated with the configuration model. A suitable exploration process here turns out to be an infinite-dimensional Markov process with transition probability rates that diminish to zero in certain parts of the state space. Large deviation properties of such Markovian models is challenging due to poor regularity behavior of the associated local rate functions. Our proof of the LDP relies on a representation of the exploration process in terms of a system of stochastic differential equations driven by Poisson random measures and variational formulas for moments of nonnegative functionals of Poisson random measures. Uniqueness results for certain controlled systems of deterministic equations play a key role in the analysis. Next, using the rate function in the LDP for the exploration process we formulate a calculus of variations problem associated with the asymptotics of component degree distributions. The second key ingredient in our study is a careful analysis of the infinite-dimensional Euler–Lagrange equations associated with this calculus of variations problem. Exact solutions of these systems of nonlinear differential equations are identified which then provide explicit formulas for decay rates of probabilities of nontypical component degree distributions and related quantities.

Analytical results for the distribution of cover times of random walks on random regular graphs

We present analytical results for the distribution of cover times of random walks (RWs) on random regular graphs consisting of N nodes of degree c (c ⩾ 3). Starting from a random initial node at time t = 1, at each time step t ⩾ 2 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may visit a new, yet-unvisited node, while in other time steps it may revisit a node that has already been visited before. The cover time T C is the number of time steps required for the RW to visit every single node in the network at least once. We derive a master equation for the distribution P t (S = s) of the number of distinct nodes s visited by an RW up to time t and solve it analytically. Inserting s = N we obtain the cumulative distribution of cover times, namely the probability P(T C ⩽ t) = P t (S = N) that up to time t an RW will visit all the N nodes in the network. Taking the large network limit, we show that P(T C ⩽ t) converges to a Gumbel distribution. We calculate the distribution of partial cover (PC) times P(T PC,k = t), which is the probability that at time t an RW will complete visiting k distinct nodes. We also calculate the distribution of random cover (RC) times P(T RC,k = t), which is the probability that at time t an RW will complete visiting all the nodes in a subgraph of k randomly pre-selected nodes at least once. The analytical results for the distributions of cover times are found to be in very good agreement with the results obtained from computer simulations.

Higher-order simplicial synchronization of coupled topological signals

Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of simplicial complexes. In particular we consider topological signals, i.e., dynamical signals defined on simplices of different dimension, here taken to be nodes and links for simplicity. We show that coupling between signals defined on nodes and links leads to explosive topological synchronization in which phases defined on nodes synchronize simultaneously to phases defined on links at a discontinuous phase transition. We study the model on real connectomes and on simplicial complexes and network models. Finally, we provide a comprehensive theoretical approach that captures this transition on fully connected networks and on random networks treated within the annealed approximation, establishing the conditions for observing a closed hysteresis loop in the large network limit.

Degree correlations in scale-free random graph models

AbstractWe study the average nearest-neighbour degree a(k) of vertices with degree k. In many real-world networks with power-law degree distribution, a(k) falls off with k, a property ascribed to the constraint that any two vertices are connected by at most one edge. We show that a(k) indeed decays with k in three simple random graph models with power-law degrees: the erased configuration model, the rank-1 inhomogeneous random graph, and the hyperbolic random graph. We find that in the large-network limit for all three null models, a(k) starts to decay beyond $n^{(\tau-2)/(\tau-1)}$ and then settles on a power law $a(k)\sim k^{\tau-3}$, with $\tau$ the degree exponent.

Analytical results for the in-degree and out-degree distributions of directed random networks that grow by node duplication

We present exact analytical results for the degree distribution in a directed network model that grows by node duplication (ND). Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability p . In addition, the daughter node forms a directed link to the mother node itself. Thus, the model is referred to as the corded directed-node-duplication (DND) model. The corresponding undirected ND model was studied before and was found to exhibit a power-law degree distribution. We obtain analytical results for the in-degree distribution , and for the out-degree distribution , of the corded DND network at time t. It is found that the in-degrees follow a shifted power-law distribution, so the network is asymptotically scale free. In contrast, the out-degree distribution is a narrow distribution, that converges to a Poisson distribution in the limit of and to a Gaussian distribution in the limit of . Such distinction between a broad in-degree distribution and a narrow out-degree distribution is common in empirical networks such as scientific citation networks. Using these distributions we calculate the mean degree , which converges to in the large network limit, for the whole range of 0 < p < 1. This is in contrast to the corresponding undirected network, which exhibits a phase transition at p = 1/2 such that for p > 1/2 the mean degree diverges in the large network limit. We also present analytical results for the distribution of the number of upstream nodes, , and for the distribution of the number of downstream nodes, , from a random node. We show that the mean values scale logarithmically with the network size. This means that in the large network limit only a diminishing fraction of pairs of nodes are connected by directed paths, unlike the corded undirected ND network that consists of a single connected component. Therefore, the corded DND network is not a small-world network.

Scale-free network clustering in hyperbolic and other random graphs

Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as resilience, navigability and small distances. We introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees. We apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale-free networks with latent geometries and clustering. We show that clustering in the hyperbolic model is non-vanishing and self-averaging, so that a single random graph sample is a good representation in the large-network limit. We also demonstrate the variational principle for some classical random graphs including the preferential attachment model and the configuration model.

Analytical results for the distribution of shortest path lengths in directed random networks that grow by node duplication

We present exact analytical results for the distribution of shortest path lengths (DSPL) in a directed network model that grows by node duplication. Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability $p$. In addition, the daughter node forms a directed link to the mother node itself. Thus, the model is referred to as the corded directed-node-duplication (DND) model. In this network not all pairs of nodes are connected by directed paths, in spite of the fact that the corresponding undirected network consists of a single connected component. More specifically, in the large network limit only a diminishing fraction of pairs of nodes are connected by directed paths. To calculate the DSPL between those pairs of nodes that are connected by directed paths we derive a master equation for the time evolution of the probability $P_t(L=\ell)$, $\ell=1,2,\dots$, where $\ell$ is the length of the shortest directed path. Solving the master equation, we obtain a closed form expression for $P_t(L=\ell)$. It is found that the DSPL at time $t$ consists of a convolution of the initial DSPL $P_0(L=\ell)$, with a Poisson distribution and a sum of Poisson distributions. The mean distance ${\mathbb E}_t[L|L<\infty]$ between pairs of nodes which are connected by directed paths is found to depend logarithmically on the network size $N_t$. However, since in the large network limit the fraction of pairs of nodes that are connected by directed paths is diminishingly small, the corded DND network is not a small-world network, unlike the corresponding undirected network.

Power Network Dynamics on Graphons

Power grids are undergoing major changes, shifting from a few large producers to smart grids built upon renewable energies. Mathematical models for power grid dynamics have to be adapted to capture when dynamic nodes can achieve synchronization to a common grid frequency on complex network topologies. In this paper we study a second-order rotator model in the large network limit. We merge the recent theory of random graph limits for complex networks with approaches to first-order systems on graphons. We prove that there exists a well-posed continuum limit integral equation approximating a large finite-dimensional synchronization model problem from power grid network dynamics. Then we analyze the linear stability of synchronized solutions and prove linear stability. However, on small-world networks we demonstrate that there are topological parameters moving the spectrum arbitrarily close to the imaginary axis leading to potential instability on finite time scales.

The impact of spike-frequency adaptation on balanced network dynamics.

A dynamic balance between strong excitatory and inhibitory neuronal inputs is hypothesized to play a pivotal role in information processing in the brain. While there is evidence of the existence of a balanced operating regime in several cortical areas and idealized neuronal network models, it is important for the theory of balanced networks to be reconciled with more physiological neuronal modeling assumptions. In this work, we examine the impact of spike-frequency adaptation, observed widely across neurons in the brain, on balanced dynamics. We incorporate adaptation into binary and integrate-and-fire neuronal network models, analyzing the theoretical effect of adaptation in the large network limit and performing an extensive numerical investigation of the model adaptation parameter space. Our analysis demonstrates that balance is well preserved for moderate adaptation strength even if the entire network exhibits adaptation. In the common physiological case in which only excitatory neurons undergo adaptation, we show that the balanced operating regime in fact widens relative to the non-adaptive case. We hypothesize that spike-frequency adaptation may have been selected through evolution to robustly facilitate balanced dynamics across diverse cognitive operating states.

Weighted Exponential Random Graph Models: Scope and Large Network Limits

We study models of weighted exponential random graphs in the large network limit. These models have recently been proposed to model weighted network data arising from a host of applications including socio-econometric data such as migration flows and neuroscience. Analogous to fundamental results derived for standard (unweighted) exponential random graph models in the work of Chatterjee and Diaconis, we derive limiting results for the structure of these models as the number of nodes goes to infinity. Our results are applicable for a wide variety of base measures including measures with unbounded support. We also derive sufficient conditions for continuity of functionals in the specification of the model including conditions on nodal covariates. Finally we include a number of open problems to spur further understanding of this model especially in the context of applications.

Rare slips in fluctuating synchronized oscillator networks.

We study rare phase slips due to noise in synchronized Kuramoto oscillator networks. In the small-noise limit, we demonstrate that slips occur via large fluctuations to saddle phase-locked states. For tree topologies, slips appear between subgraphs that become disconnected at a saddle-node bifurcation, where phase-locked states lose stability generically. This pattern is demonstrated for sparse networks with several examples. Scaling laws are derived and compared for different tree topologies. On the other hand, for dense networks slips occur between oscillators on the edges of the frequency distribution. If the distribution is discrete, the probability-exponent for large fluctuations to occur scales linearly with the system size. However, if the distribution is continuous, the probability is a constant in the large network limit, as individual oscillators fluctuate to saddles while all others remain fixed. In the latter case, the network's coherence is approximately preserved.

Centrality Analysis for Modified Lattices

We derive new, exact expressions for network centrality vectors associated with classical Watts-Strogatz style ring plus shortcut networks. We also derive easy-to-interpret approximations that are highly accurate in the large network limit. The analysis helps us to understand the role of the Katz parameter and the PageRank parameter, to compare linear system and eigenvalue based centrality measures, and to predict the behavior of centrality measures on more complicated networks.

Centrality metrics and localization in core-periphery networks

Two concepts of centrality have been defined in complex networks. The first considers the centrality of a node and many different metrics for it have been defined (e.g. eigenvector centrality, PageRank, non-backtracking centrality, etc). The second is related to large scale organization of the network, the core-periphery structure, composed by a dense core plus an outlying and loosely-connected periphery. In this paper we investigate the relation between these two concepts. We consider networks generated via the stochastic block model, or its degree corrected version, with a core-periphery structure and we investigate the centrality properties of the core nodes and the ability of several centrality metrics to identify them. We find that the three measures with the best performance are marginals obtained with belief propagation, PageRank, and degree centrality, while non-backtracking and eigenvector centrality (or MINRES [], showed to be equivalent to the latter in the large network limit) perform worse in the investigated networks.

Giant component sizes in scale-free networks with power-law degrees and cutoffs

Scale-free networks arise from power-law degree distributions. Due to the finite size of real-world networks, the power law inevitably has a cutoff at some maximum degree Δ. We investigate the relative size of the giant component S in the large-network limit. We show that S as a function of Δ increases fast when Δ is just large enough for the giant component to exist, but increases ever more slowly when Δ increases further. This gives that while the degree distribution converges to a pure power law when Δ → ∞, S approaches its limiting value at a slow pace. The convergence rate also depends on the power-law exponent τ of the degree distribution. The worst rate of convergence is found to be for the case , which concerns many of the real-world networks reported in the literature.

Twitter event networks and the Superstar model

Condensation phenomenon is often observed in social networks such as Twitter where one "superstar" vertex gains a positive fraction of the edges, while the remaining empirical degree distribution still exhibits a power law tail. We formulate a mathematically tractable model for this phenomenon that provides a better fit to empirical data than the standard preferential attachment model across an array of networks observed in Twitter. Using embeddings in an equivalent continuous time version of the process, and adapting techniques from the stable age-distribution theory of branching processes, we prove limit results for the proportion of edges that condense around the superstar, the degree distribution of the remaining vertices, maximal nonsuperstar degree asymptotics and height of these random trees in the large network limit.

Mesoscopic structures and the Laplacian spectra of random geometric graphs

Max Planck Institute for the Physics of Complex Systems,No¨thnitzer Strase 38, Dresden D-01187, Germany(Dated: December 3, 2014)We investigate the Laplacian spectra of random geometric graphs (RGGs). The spectra are foundto consist of both a discrete and a continuous part. The discrete part is a collection of Diracdelta peaks at integer values roughly centered around the mean degree. The peaks are mainlydue to the existence of mesoscopic structures that occur far more abundantly in RGGs than innon-spatial networks. The probability of certain mesoscopic structures is analytically calculated forone-dimensional RGGs and they are shown to produce integer-valued eigenvalues that comprise asignificant fraction of the spectrum, even in the large network limit. A phenomenon reminiscent ofBose-Einstein condensation in the appearance of zero eigenvalues is also found.

Binary Willshaw learning yields high synaptic capacity for long-term familiarity memory

In this study, we investigate from a computational perspective the efficiency of the Willshaw synaptic update rule in the context of familiarity discrimination, a binary-answer, memory-related task that has been linked through psychophysical experiments with modified neural activity patterns in the prefrontal and perirhinal cortex regions. Our motivation for recovering this well-known learning prescription is two-fold: first, the switch-like nature of the induced synaptic bonds, as there is evidence that biological synaptic transitions might occur in a discrete stepwise fashion. Second, the possibility that in the mammalian brain, unused, silent synapses might be pruned in the long-term. Besides the usual pattern and network capacities, we calculate the synaptic capacity of the model, a recently proposed measure where only the functional subset of synapses is taken into account. We find that in terms of network capacity, Willshaw learning is strongly affected by the pattern coding rates, which have to be kept fixed and very low at any time to achieve a non-zero capacity in the large network limit. The information carried per functional synapse, however, diverges and is comparable to that of the pattern association case, even for more realistic moderately low activity levels that are a function of network size.

Network growth with preferential attachment for high indegree and low outdegree

We study the growth of a directed transportation network, such as a food web, in which links carry resources. We propose a growth process in which new nodes (or species) preferentially attach to existing nodes with high indegree (in food-web language, number of prey) and low outdegree (or number of predators). This scheme, which we call inverse preferential attachment, is intended to maximize the amount of resources available to each new node. We show that the outdegree (predator) distribution decays at least exponentially fast for large outdegree and is continuously tunable between an exponential distribution and a delta function. The indegree (prey) distribution is poissonian in the large-network limit.