Non-backtracking Matrix Research Articles

Backtracking activation impacts the criticality of excitable networks

Networks of excitable elements are widely used to model real-world biological and social systems. The dynamic range of an excitable network quantifies the range of stimulus intensities that can be robustly distinguished by the network response, and is maximized at the critical state. In this study, we examine the impacts of backtracking activation on system criticality in excitable networks consisting of both excitatory and inhibitory units. We find that, for dynamics with refractory states that prohibit backtracking activation, the critical state occurs when the largest eigenvalue of the weighted non-backtracking (WNB) matrix for excitatory units, , is close to one, regardless of the strength of inhibition. In contrast, for dynamics without refractory state in which backtracking activation is allowed, the strength of inhibition affects the critical condition through suppression of backtracking activation. As inhibitory strength increases, backtracking activation is gradually suppressed. Accordingly, the system shifts continuously along a continuum between two extreme regimes—from one where the criticality is determined by the largest eigenvalue of the weighted adjacency matrix for excitatory units, , to the other where the critical state is reached when is close to one. For systems in between, we find that and at the critical state. These findings, confirmed by numerical simulations using both random and synthetic neural networks, indicate that backtracking activation impacts the criticality of excitable networks.

On the stability of multilayer Boolean networks under targeted immunization

In this paper, we study targeted immunization in a multilayer Boolean network model for genetic regulatory networks. Given a specific set of nodes immune to perturbations, we find that the stability of a multilayer Boolean network is determined by the largest eigenvalue of the weighted non-backtracking matrix of corresponding aggregated network. Aimed to minimize this largest eigenvalue, we developed the metric of multilayer collective influence (MCI) to quantify the impact of immunizing individual nodes on the stability of the system. Compared with other competing heuristics, immunizing nodes with high MCI scores can stabilize an unstable multilayer network with higher efficiency on both synthetic and real-world networks. Moreover, despite that coupling nodes can exert direct influence across multiple layers, they are found to exhibit less importance as measured by the MCI score. Our work reveals the mechanism of maintaining the stability of multilayer Boolean networks and provides an efficient targeted immunization strategy, which can be potentially applied to the location of pathogenesis of diseases and the development of targeted therapy.

Recovering Asymmetric Communities in the Stochastic Block Model

We consider the sparse stochastic block model in the case where the degrees are uninformative. The case where the two communities have approximately the same size has been extensively studied and we concentrate here on the community detection problem in the case of unbalanced communities. In this setting, spectral algorithms based on the non-backtracking matrix are known to solve the community detection problem (i.e., do strictly better than a random guess) when the signal is sufficiently large namely above the so-called Kesten-Stigum threshold. In this regime and when the average degree tends to infinity, we show that if the community of a vanishing fraction of the vertices is revealed, then a local algorithm (belief propagation) is optimal down to Kesten-Stigum threshold and we quantify explicitly its performance. Below the Kesten-Stigum threshold, we show that, in the large degree limit, there is a second threshold called the spinodal curve below which, the community detection problem is not solvable. The spinodal curve is equal to the Kesten-Stigum threshold when the fraction of vertices in the smallest community is above $p^*=\frac{1}{2}-\frac{1}{2\sqrt{3}}$ , so that the Kesten-Stigum threshold is the threshold for solvability of the community detection in this case. However when the smallest community is smaller than $p^*$ , the spinodal curve only provides a lower bound on the threshold for solvability. In the regime below the Kesten-Stigum bound and above the spinodal curve, we also characterize the performance of best local algorithms as a function of the fraction of revealed vertices. Our proof relies on a careful analysis of the associated reconstruction problem on trees which might be of independent interest. In particular, we show that the spinodal curve corresponds to the reconstruction threshold on the tree.

Recovering a hidden community beyond the Kesten–Stigum threshold in O(|E|log*|V|) time

Abstract Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime n1-o(1) ≤ K ≤ o(n). A critical parameter is the effective signal-to-noise ratio λ = K2(p - q)2 / ((n - K)q), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log*n + O(1) iterations, with the total time complexity O(|E|log*n), where log*n is the iterated logarithm of n. Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying a power iteration to the nonbacktracking matrix of the graph is shown to attain weak recovery if and only if λ > 1. In addition, the BP algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all K ≥ (n / logn) (ρBP + o(1)), where ρBP is a function of p / q.

Non-Backtracking Centrality Based Random Walk on Networks

Random walks are a fundamental tool for analyzing realistic complex networked systems and implementing randomized algorithms to solve diverse problems such as searching and sampling. For many real applications, their actual effect and convenience depend on the properties (e.g. stationary distribution and hitting time) of random walks, with biased random walks often outperforming traditional unbiased random walks (TURW). In this paper, we present a new class of biased random walks, non-backtracking centrality based random walks (NBCRW) on a network, where the walker prefers to jump to neighbors with high non-backtracking centrality that has some advantages over eigenvector centrality. We study some properties of the non-backtracking matrix of a network, on the basis of which we propose a theoretical framework for fast computation of the transition probabilities, stationary distribution, and hitting times for NBCRW on the network. Within the paradigm, we study NBCRW on some model and real networks and compare the results with those corresponding to TURW and maximal entropy random walks (MERW), with the latter being biased random walks based on eigenvector centrality. We show that the behaviors of stationary distribution and hitting times for NBCRW widely differ from those associated with TURW and MERW, especially for heterogeneous networks.

Optimal stabilization of Boolean networks through collective influence.

Boolean networks have attracted much attention due to their wide applications in describing dynamics of biological systems. During past decades, much effort has been invested in unveiling how network structure and update rules affect the stability of Boolean networks. In this paper, we aim to identify and control a minimal set of influential nodes that is capable of stabilizing an unstable Boolean network. For locally treelike Boolean networks with biased truth tables, we propose a greedy algorithm to identify influential nodes in Boolean networks by minimizing the largest eigenvalue of a modified nonbacktracking matrix. We test the performance of the proposed collective influence algorithm on four different networks. Results show that the collective influence algorithm can stabilize each network with a smaller set of nodes compared with other heuristic algorithms. Our work provides a new insight into the mechanism that determines the stability of Boolean networks, which may find applications in identifying virulence genes that lead to serious diseases.

Dynamic range maximization in excitable networks.

We study the strategy to optimally maximize the dynamic range of excitable networks by removing the minimal number of links. A network of excitable elements can distinguish a broad range of stimulus intensities and has its dynamic range maximized at criticality. In this study, we formulate the activation propagation in excitable networks as a message passing process in which a critical state is reached when the largest eigenvalue of the weighted non-backtracking matrix is close to one. By considering the impact of single link removal on the largest eigenvalue, we develop an efficient algorithm that aims to identify the optimal set of links whose removal will drive the system to the critical state. Comparisons with other competing heuristics on both synthetic and real-world networks indicate that the proposed method can maximize the dynamic range by removing the smallest number of links, and at the same time maintaining the largest size of the giant connected component.

Spectral estimation of the percolation transition in clustered networks.

There have been several spectral bounds for the percolation transition in networks, using spectrum of matrices associated with the network such as the adjacency matrix and the nonbacktracking matrix. However, they are far from being tight when the network is sparse and displays clustering or transitivity, which is represented by existence of short loops, e.g., triangles. In this paper, for the bond percolation, we first propose a message-passing algorithm for calculating size of percolating clusters considering effects of triangles, then relate the percolation transition to the leading eigenvalue of a matrix that we name the triangle-nonbacktracking matrix, by analyzing stability of the message-passing equations. We establish that our method gives a tighter lower bound to the bond percolation transition than previous spectral bounds, and it becomes exact for an infinite network with no loops longer than 3. We evaluate numerically our methods on synthetic and real-world networks, and discuss further generalizations of our approach to include higher-order substructures.

Nonbacktracking expansion of finite graphs.

Message passing equations yield a sharp percolation transition in finite graphs, as an artifact of the locally treelike approximation. For an arbitrary finite, connected, undirected graph we construct an infinite tree having the same local structural properties as this finite graph, when observed by a nonbacktracking walker. Formally excluding the boundary, this infinite tree is a generalization of the Bethe lattice. We indicate an infinite, locally treelike, random network whose local structure is exactly given by this infinite tree. Message passing equations for various cooperative models on this construction are the same as for the original finite graph, but here they provide the exact solutions of the corresponding cooperative problems. These solutions are good approximations to observables for the models on the original graph when it is sufficiently large and not strongly correlated. We show how to express these solutions in the critical region in terms of the principal eigenvector components of the nonbacktracking matrix. As representative examples we formulate the problems of the random and optimal destruction of a connected graph in terms of our construction, the nonbacktracking expansion. We analyze the limitations and the accuracy of the message passing algorithms for different classes of networks and compare the complexity of the message passing calculations to that of direct numerical simulations. Notably, in a range of important cases, simulations turn out to be more efficient computationally than the message passing.

Fragmenting networks by targeting collective influencers at a mesoscopic level.

A practical approach to protecting networks against epidemic processes such as spreading of infectious diseases, malware, and harmful viral information is to remove some influential nodes beforehand to fragment the network into small components. Because determining the optimal order to remove nodes is a computationally hard problem, various approximate algorithms have been proposed to efficiently fragment networks by sequential node removal. Morone and Makse proposed an algorithm employing the non-backtracking matrix of given networks, which outperforms various existing algorithms. In fact, many empirical networks have community structure, compromising the assumption of local tree-like structure on which the original algorithm is based. We develop an immunization algorithm by synergistically combining the Morone-Makse algorithm and coarse graining of the network in which we regard a community as a supernode. In this way, we aim to identify nodes that connect different communities at a reasonable computational cost. The proposed algorithm works more efficiently than the Morone-Makse and other algorithms on networks with community structure.

Localized eigenvectors of the non-backtracking matrix

In the case of graph partitioning, the emergence of localized eigenvectors can cause the standard spectral method to fail. To overcome this problem, the spectral method using a non-backtracking matrix was proposed. Based on numerical experiments on several examples of real networks, it is clear that the non-backtracking matrix does not exhibit localization of eigenvectors. However, we show that localized eigenvectors of the non-backtracking matrix can exist outside the spectral band, which may lead to deterioration in the performance of graph partitioning.

Distinct types of eigenvector localization in networks.

The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes’ centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution P(q) ~ q−γ, localization occurs on the largest hub if γ > 5/2; for γ < 5/2 a new type of localization arises on a mesoscopic subgraph associated with the shell with the largest index in the K-core decomposition. Similar evidence for the existence of distinct localization modes is found in the analysis of real-world networks. Our results open a new perspective on dynamical processes on networks and on a recently proposed alternative measure of node centrality based on the non-backtracking matrix.

Influence maximization in complex networks through optimal percolation.

The whole frame of interconnections in complex networks hinges on a specific set of structural nodes, much smaller than the total size, which, if activated, would cause the spread of information to the whole network, or, if immunized, would prevent the diffusion of a large scale epidemic. Localizing this optimal, that is, minimal, set of structural nodes, called influencers, is one of the most important problems in network science. Despite the vast use of heuristic strategies to identify influential spreaders, the problem remains unsolved. Here we map the problem onto optimal percolation in random networks to identify the minimal set of influencers, which arises by minimizing the energy of a many-body system, where the form of the interactions is fixed by the non-backtracking matrix of the network. Big data analyses reveal that the set of optimal influencers is much smaller than the one predicted by previous heuristic centralities. Remarkably, a large number of previously neglected weakly connected nodes emerges among the optimal influencers. These are topologically tagged as low-degree nodes surrounded by hierarchical coronas of hubs, and are uncovered only through the optimal collective interplay of all the influencers in the network. The present theoretical framework may hold a larger degree of universality, being applicable to other hard optimization problems exhibiting a continuous transition from a known phase.

Nonbacktracking operator for the Ising model and its applications in systems with multiple states

The nonbacktracking operator for a graph is the adjacency matrix defined on directed edges of the graph. The operator was recently shown to perform optimally in spectral clustering in sparse synthetic graphs and have a deep connection to belief propagation algorithm. In this paper we consider nonbacktracking operator for Ising model on a general graph with a general coupling distribution and study the spectrum of this operator analytically. We show that spectral algorithms based on this operator is equivalent to belief propagation algorithm linearized at the paramagnetic fixed point and recovers replica-symmetry results on phase boundaries obtained by replica methods. This operator can be applied directly to systems with multiple states like Hopfield model. We show that spectrum of the operator can be used to determine number of patterns that stored successfully in the network, and the associated eigenvectors can be used to retrieve all the patterns simultaneously. We also give an example on how to control the Hopfield model, i.e., making network more sparse while keeping patterns stable, using the nonbacktracking operator and matrix perturbation theory.

Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs

A nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge. The nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length. It has been used recently in the context of community detection and has appeared previously in connection with the Ihara zeta function and in some generalizations of Ramanujan graphs. In this work, we study the largest eigenvalues of the nonbacktracking matrix of the Erdős–Renyi random graph and of the stochastic block model in the regime where the number of edges is proportional to the number of vertices. Our results confirm the “spectral redemption conjecture” in the symmetric case and show that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold.

Predicting percolation thresholds in networks.

We consider different methods, which do not rely on numerical simulations of the percolation process, to approximate percolation thresholds in networks. We perform a systematic analysis on synthetic graphs and a collection of 109 real networks to quantify their effectiveness and reliability as prediction tools. Our study reveals that the inverse of the largest eigenvalue of the nonbacktracking matrix of the graph often provides a tight lower bound for true percolation threshold. However, in more than 40% of the cases, this indicator is less predictive than the naive expectation value based solely on the moments of the degree distribution. We find that the performance of all indicators becomes worse as the value of the true percolation threshold grows. Thus, none of them represents a good proxy for the robustness of extremely fragile networks.

The non-backtracking spectrum of the universal cover of a graph

A non-backtracking walk on a graph, H H , is a directed path of directed edges of H H such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency matrix, B B , indexed by H H ’s directed edges and related to Ihara’s Zeta function. We show how to determine B B ’s spectrum in the case where H H is a tree covering a finite graph. We show that when H H is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of B B ’s spectrum, the corresponding Green function has “periodic decay ratios”. The existence of such a “ratio system” can be effectively checked and is equivalent to being outside the spectrum. We also prove that the spectral radius of the non-backtracking walk operator on the tree covering a finite graph is exactly g r \sqrt {\mathrm {gr}} , where g r \mathrm {gr} is the cogrowth of B B , or growth rate of the tree. This further motivates the definition of the graph theoretical Riemann hypothesis proposed by Stark and Terras. Finally, we give experimental evidence that for a fixed, finite graph, H H , a random lift of large degree has non-backtracking new spectrum near that of H H ’s universal cover. This suggests a new generalization of Alon’s second eigenvalue conjecture.

Localization and centrality in networks.

Eigenvector centrality is a common measure of the importance of nodes in a network. Here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to concentrate on a small number of nodes in the network. In this regime the measure is no longer useful for distinguishing among the remaining nodes and its efficacy as a network metric is impaired. As a remedy, we propose an alternative centrality measure based on the nonbacktracking matrix, which gives results closely similar to the standard eigenvector centrality in dense networks where the latter is well behaved but avoids localization and gives useful results in regimes where the standard centrality fails.

Percolation on sparse networks.

We study percolation on networks, which is used as a model of the resilience of networked systems such as the Internet to attack or failure and as a simple model of the spread of disease over human contact networks. We reformulate percolation as a message passing process and demonstrate how the resulting equations can be used to calculate, among other things, the size of the percolating cluster and the average cluster size. The calculations are exact for sparse networks when the number of short loops in the network is small, but even on networks with many short loops we find them to be highly accurate when compared with direct numerical simulations. By considering the fixed points of the message passing process, we also show that the percolation threshold on a network with few loops is given by the inverse of the leading eigenvalue of the so-called nonbacktracking matrix.

A nonbacktracking matrix decomposition algorithm for routing on Clos networks

A number of matrix decomposition schemes have been previously reported for routing on Clos switching networks. These schemes occasionally fail to find the right decomposition, unless backtracking is used. It is shown that a partition may occur during the decomposition process, and that this is the underlying reason these algorithms fail for some decompositions. A parallel algorithm which can recognize when a partition exists and set up the Clos network without backtracking is given. >