Approximating the largest eigenvalue of network adjacency matrices

Predicting epidemic thresholds on complex networks: limitations of mean-field approaches

Over the last decade considerable research effort has been invested in an attempt to understand the dynamics of viruses as they spread through complex networks, be they the networks in human population, computers or otherwise. The efforts have contributed to an understanding of epidemic behavior in random networks, but were generally unable to accommodate specific nonrandom features of the network's actual topology. Recently, though still in the context of the mean field theory, Chakrabarti et al. (2008) proposed a model that intended to take into account the graph's specific topology and solve a longstanding problem regarding epidemic thresholds in both random and nonrandom networks. Here we review previous theoretical work dealing with this problem (usually based on mean field approximations) and show with several relevant and concrete counter examples that results to date breakdown for nonrandom topologies.

Synchronization from second order network connectivity statistics

We investigate how network structure can influence the tendency for a neuronal network to synchronize, or its synchronizability, independent of the dynamical model for each neuron. The synchrony analysis takes advantage of the framework of second order networks, which defines four second order connectivity statistics based on the relative frequency of two-connection network motifs. The analysis identifies two of these statistics, convergent connections, and chain connections, as highly influencing the synchrony. Simulations verify that synchrony decreases with the frequency of convergent connections and increases with the frequency of chain connections. These trends persist with simulations of multiple models for the neuron dynamics and for different types of networks. Surprisingly, divergent connections, which determine the fraction of shared inputs, do not strongly influence the synchrony. The critical role of chains, rather than divergent connections, in influencing synchrony can be explained by their increasing the effective coupling strength. The decrease of synchrony with convergent connections is primarily due to the resulting heterogeneity in firing rates.

Decreasing the spectral radius of a graph by link removals

The decrease of the spectral radius, an important characterizer of network dynamics, by removing links is investigated. The minimization of the spectral radius by removing m links is shown to be an NP-complete problem, which suggests considering heuristic strategies. Several greedy strategies are compared, and several bounds on the decrease of the spectral radius are derived. The strategy that removes that link l=i~j with largest product (x(1))(i)(x(1))(j) of the components of the eigenvector x(1) belonging to the largest adjacency eigenvalue is shown to be superior to other strategies in most cases. Furthermore, a scaling law where the decrease in spectral radius is inversely proportional to the number of nodes N in the graph is deduced. Another sublinear scaling law of the decrease in spectral radius versus the number m of removed links is conjectured.

Effects of network topology, transmission delays, and refractoriness on the response of coupled excitable systems to a stochastic stimulus

We study the effects of network topology on the response of networks of coupled discrete excitable systems to an external stochastic stimulus. We extend recent results that characterize the response in terms of spectral properties of the adjacency matrix by allowing distributions in the transmission delays and in the number of refractory states and by developing a nonperturbative approximation to the steady state network response. We confirm our theoretical results with numerical simulations. We find that the steady state response amplitude is inversely proportional to the duration of refractoriness, which reduces the maximum attainable dynamic range. We also find that transmission delays alter the time required to reach steady state. Importantly, neither delays nor refractoriness impact the general prediction that criticality and maximum dynamic range occur when the largest eigenvalue of the adjacency matrix is unity.

Sample-to-sample fluctuations in real-network ensembles

Network modeling based on ensemble averages tacitly assumes that the networks meant to be modeled are typical in the ensemble. Previous research on network eigenvalues, which govern a range of dynamical phenomena, has shown that this is indeed the case for uncorrelated networks with minimum degree ≥ 3. Here, we focus on real networks, which generally have both structural correlations and low-degree nodes. We show that: (i) the ensemble distribution of the dynamically most important eigenvalues can be not only broad and far apart from the real eigenvalue but also highly structured, often with a multimodal rather than a bell-shaped form; (ii) these interesting properties are found to be due to low-degree nodes, mainly those with degree ≤ 3, and network communities, which is a common form of structural correlation found in real networks. In addition to having implications for ensemble-based approaches, this shows that low-degree nodes may have a stronger influence on collective dynamics than previously anticipated from the study of computer-generated networks.

The dynamics of network coupled phase oscillators: an ensemble approach

We consider the dynamics of many phase oscillators that interact through a coupling network. For a given network connectivity we further consider an ensemble of such systems where, for each ensemble member, the set of oscillator natural frequencies is independently and randomly chosen according to a given distribution function. We then seek a statistical description of the dynamics of this ensemble. Use of this approach allows us to apply the recently developed ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)] to the marginal distribution of the ensemble of states at each node. This, in turn, results in a reduced set of ordinary differential equations determining these marginal distribution functions. The new set facilitates the analysis of network dynamics in several ways: (i) the time evolution of the reduced system of ensemble equations is much smoother, and thus numerical solutions can be obtained much faster by use of longer time steps; (ii) the new set of equations can be used as a basis for obtaining analytical results; and (iii) for a certain type of network, a reduction to a low dimensional description of the entire network dynamics is possible. We illustrate our approach with numerical experiments on a network version of the classical Kuramoto problem, first with a unimodal frequency distribution, and then with a bimodal distribution. In the latter case, the network dynamics is characterized by bifurcations and hysteresis involving a variety of steady and periodic attractors.

Maximal-entropy random walks in complex networks with limited information

Maximization of the entropy rate is an important issue to design diffusion processes aiming at a well-mixed state. We demonstrate that it is possible to construct maximal-entropy random walks with only local information on the graph structure. In particular, we show that an almost maximal-entropy random walk is obtained when the step probabilities are proportional to a power of the degree of the target node, with an exponent α that depends on the degree-degree correlations and is equal to 1 in uncorrelated graphs.

Predicting criticality and dynamic range in complex networks: effects of topology

The collective dynamics of a network of coupled excitable systems in response to an external stimulus depends on the topology of the connections in the network. Here we develop a general theoretical approach to study the effects of network topology on dynamic range, which quantifies the range of stimulus intensities resulting in distinguishable network responses. We find that the largest eigenvalue of the weighted network adjacency matrix governs the network dynamic range. When the largest eigenvalue is exactly one, the system is in a critical state and its dynamic range is maximized. Further, we examine higher order behavior of the steady state system, which predicts that networks with more homogeneous degree distributions should have higher dynamic range. Our analysis, confirmed by numerical simulations, generalizes previous studies in terms of the largest eigenvalue of the adjacency matrix.

Thresholds for epidemic spreading in networks

We study the threshold of epidemic models in quenched networks with degree distribution given by a power-law. For the susceptible-infected-susceptible model the activity threshold λ(c) vanishes in the large size limit on any network whose maximum degree k(max) diverges with the system size, at odds with heterogeneous mean-field (HMF) theory. The vanishing of the threshold has nothing to do with the scale-free nature of the network but stems instead from the largest hub in the system being active for any spreading rate λ>1/√k(max) and playing the role of a self-sustained source that spreads the infection to the rest of the system. The susceptible-infected-removed model displays instead agreement with HMF theory and a finite threshold for scale-rich networks. We conjecture that on quenched scale-rich networks the threshold of generic epidemic models is vanishing or finite depending on the presence or absence of a steady state.

Hub synchronization in scale-free networks

Heterogeneity in the degree distribution is known to suppress global synchronization in complex networks of symmetrically coupled oscillators. Scale-free networks display a great deal of heterogeneity, containing a few nodes, termed hubs, that are highly connected, while most nodes receive only a few connections. Here, we show that a group of synchronized nodes may appear in scale-free networks: hubs undergo a transition to synchronization while the other nodes remain unsynchronized. This general phenomenon can occur even in the absence of global synchronization. Our results suggest that scale-free networks may have evolved to complement various levels of synchronization.

Connectivity, cycles, and persistence thresholds in metapopulation networks

Synthesising the relationships between complexity, connectivity, and the stability of large biological systems has been a longstanding fundamental quest in theoretical biology and ecology. With the many exciting developments in modern network theory, interest in these issues has recently come to the forefront in a range of multidisciplinary areas. Here we outline a new theoretical analysis specifically relevant for the study of ecological metapopulations focusing primarily on marine systems, where subpopulations are generally connected via larval dispersal. Our work determines the qualitative and quantitative conditions by which dispersal and network structure control the persistence of a set of age-structured patch populations. Mathematical modelling combined with a graph theoretic analysis demonstrates that persistence depends crucially on the topology of cycles in the dispersal network which tend to enhance the effect of larvae "returning home." Our method clarifies the impact directly due to network structure, but this almost by definition can only be achieved by examining the simplified case in which patches are identical; an assumption that we later relax. The methodology identifies critical migration routes, whose presence are vital to overall stability, and therefore should have high conservation priority. In contrast, "lonely links," or links in the network that do not participate in a cyclical component, have no impact on persistence and thus have low conservation priority. A number of other intriguing criteria for persistence are derived. Our modelling framework reveals new insights regarding the determinants of persistence, stability, and thresholds in complex metapopulations. In particular, while theoretical arguments have, in the past, suggested that increasing connectivity is a destabilizing feature in complex systems, this is not evident in metapopulation networks where connectivity, cycles, coherency, and heterogeneity all tend to enhance persistence. The results should be of interest for many other scientific contexts that make use of network theory.

Effect of node-degree correlation on synchronization of identical pulse-coupled oscillators

We explore the effect of correlations between the in and out degrees of random directed networks on the synchronization of identical pulse-coupled oscillators. Numerical experiments demonstrate that the proportion of initial conditions resulting in a globally synchronous state (prior to a large but finite time) is an increasing function of node-degree correlation. For those networks observed to globally synchronize, both the mean and standard deviation of time to synchronization are decreasing functions of node-degree correlation. Pulse-coupled oscillator networks with negatively correlated node degree often exhibit multiple coherent attracting states, with trajectories performing fast transitions between them. These effects of node-degree correlation on dynamics of pulse-coupled oscillators are consistent with aspects of network topology (e.g., the effect of node-degree correlation on the eigenvalues of the Laplacian matrix) that have been shown to affect synchronization in other contexts.

Spectral properties of networks with community structure

In this paper, we discuss the eigenspectra of networks with community structure. It is shown that in many cases, the spectrum of eigenvalues of the adjacency matrix of a network with community structure gives a clear indication of the number of communities in the network. In particular, for a network with N nodes and N_(c) communities, there will typically be N_(c) eigenvalues that are significantly larger than the magnitudes of all the other (N-N_(c)) eigenvalues. We discuss this property as well as its use and limitations for determining N_(c) .

Approximating the largest eigenvalue of the modified adjacency matrix of networks with heterogeneous node biases

Motivated by its relevance to various types of dynamical behavior of network systems, the maximum eigenvalue lambdaA of the adjacency matrix A of a network has been considered and mean-field-type approximations to lambdaA have been developed for different kinds of networks. Here A is defined by Aij=1 (Aij=0) if there is (is not) a directed network link to i from j. However, in at least two recent problems involving networks with heterogeneous node properties (percolation on a directed network and the stability of Boolean models of gene networks), an analogous but different eigenvalue problem arises, namely, that of finding the largest eigenvalue lambdaQ of the matrix Q, where Qij=qiAij and the "bias" qi may be different at each node i. (In the previously mentioned percolation and gene network contexts, qi is a probability and so lies in the range 0<or=qi<or=1.) The purposes of this paper are to extend the previous considerations of the maximum eigenvalue lambdaA of A to lambdaQ, to develop suitable analytic approximations to lambdaQ, and to test these approximations with numerical experiments. In particular, three issues considered are (i) the effect of the correlation (or anticorrelation) between the value of qi and the number of links to and from node i, (ii) the effect of correlation between the properties of two nodes at either end of a network link ("assortativity"), and (iii) the effect of community structure allowing for a situation in which different q values are associated with different communities.

The effect of network topology on the stability of discrete state models of genetic control

Boolean networks have been proposed as potentially useful models for genetic control. An important aspect of these networks is the stability of their dynamics in response to small perturbations. Previous approaches to stability have assumed uncorrelated random network structure. Real gene networks typically have nontrivial topology significantly different from the random network paradigm. To address such situations, we present a general method for determining the stability of large Boolean networks of any specified network topology and predicting their steady-state behavior in response to small perturbations. Additionally, we generalize to the case where individual genes have a distribution of "expression biases," and we consider a nonsynchronous update, as well as extension of our method to non-Boolean models in which there are more than two possible gene states. We find that stability is governed by the maximum eigenvalue of a modified adjacency matrix, and we test this result by comparison with numerical simulations. We also discuss the possible application of our work to experimentally inferred gene networks.