Non-backtracking Walk Research Articles

Generating functions of non-backtracking walks on weighted digraphs: Radius of convergence and Ihara's theorem

It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In Grindrod et al. [13], the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed (unweighted or weighted) graphs, showing that it depends on the number of cycles in the undirectization of the graph. We also consider backtrack-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case. Finally, for weighted directed graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound, and we also prove a version of Ihara's theorem.

Communities in C. elegans connectome through the prism of non-backtracking walks

The fundamental relationship between the mesoscopic structure of neuronal circuits and organismic functions they subserve is one of the major challenges in contemporary neuroscience. Formation of structurally connected modules of neurons enacts the conversion from single-cell firing to large-scale behaviour of an organism, highlighting the importance of their accurate profiling in the data. While connectomes are typically characterized by significant sparsity of neuronal connections, recent advances in network theory and machine learning have revealed fundamental limitations of traditionally used community detection approaches in cases where the network is sparse. Here we studied the optimal community structure in the structural connectome of Caenorhabditis elegans, for which we exploited a non-conventional approach that is based on non-backtracking random walks, virtually eliminating the sparsity issue. In full agreement with the previous asymptotic results, we demonstrated that non-backtracking walks resolve the ground truth annotation into clusters on stochastic block models (SBM) with the size and density of the connectome better than the spectral methods related to simple random walks. Based on the cluster detectability threshold, we determined that the optimal number of modules in a recently mapped connectome of C. elegans is 10, which precisely corresponds to the number of isolated eigenvalues in the spectrum of the non-backtracking flow matrix. The discovered communities have a clear interpretation in terms of their functional role, which allows one to discern three structural compartments in the worm: the Worm Brain (WB), the Worm Movement Controller (WMC), and the Worm Information Flow Connector (WIFC). Broadly, our work provides a robust network-based framework to reveal mesoscopic structures in sparse connectomic datasets, paving way to further investigation of connectome mechanisms for different functions.

Global eigenvalue fluctuations of random biregular bipartite graphs

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class of analytic functions. As a key step in the proof, we obtain a total variation distance bound for the Poisson approximation of the number of cycles and cyclically non-backtracking walks in random biregular bipartite graphs, which might be of independent interest. We also prove a semicircle law for random [Formula: see text]-biregular bipartite graphs when [Formula: see text]. As an application, we translate the results to adjacency matrices of uniformly distributed random regular hypergraphs.

Non-backtracking spectra of weighted inhomogeneous random graphs

We study a model of random graphs where each edge is drawn independently (but not necessarily identically distributed) from the others, and then assigned a random weight. When the mean degree of such a graph is low, it is known that the spectrum of the adjacency matrix A deviates significantly from that of its expected value \mathbb{E} A . In contrast, we show that over a wide range of parameters the top eigenvalues of the non-backtracking matrix B – a matrix whose powers count the non-backtracking walks between two edges – are close to those of \mathbb{E} A , and all other eigenvalues are confined in a bulk with known radius. We also obtain a precise characterization of the scalar product between the eigenvectors of B and their deterministic counterparts derived from the model parameters. This result has many applications, in domains ranging from (noisy) matrix completion to community detection, as well as matrix perturbation theory. In particular, we establish as a corollary that a result known as the Baik–Ben Arous–Péché phase transition, previously established only for rotationally invariant random matrices, holds more generally for matrices A as above under a mild concentration hypothesis.

Non-backtracking walks reveal compartments in sparse chromatin interaction networks

Chromatin communities stabilized by protein machinery play essential role in gene regulation and refine global polymeric folding of the chromatin fiber. However, treatment of these communities in the framework of the classical network theory (stochastic block model, SBM) does not take into account intrinsic linear connectivity of the chromatin loci. Here we propose the polymer block model, paving the way for community detection in polymer networks. On the basis of this new model we modify the non-backtracking flow operator and suggest the first protocol for annotation of compartmental domains in sparse single cell Hi-C matrices. In particular, we prove that our approach corresponds to the maximum entropy principle. The benchmark analyses demonstrates that the spectrum of the polymer non-backtracking operator resolves the true compartmental structure up to the theoretical detectability threshold, while all commonly used operators fail above it. We test various operators on real data and conclude that the sizes of the non-backtracking single cell domains are most close to the sizes of compartments from the population data. Moreover, the found domains clearly segregate in the gene density and correlate with the population compartmental mask, corroborating biological significance of our annotation of the chromatin compartmental domains in single cells Hi-C matrices.

Beyond non-backtracking: non-cycling network centrality measures.

Walks around a graph are studied in a wide range of fields, from graph theory and stochastic analysis to theoretical computer science and physics. In many cases it is of interest to focus on non-backtracking walks; those that do not immediately revisit their previous location. In the network science context, imposing a non-backtracking constraint on traditional walk-based node centrality measures is known to offer tangible benefits. Here, we use the Hashimoto matrix construction to characterize, generalize and study such non-backtracking centrality measures. We then devise a recursive extension that systematically removes triangles, squares and, generally, all cycles up to a given length. By characterizing the spectral radius of appropriate matrix power series, we explore how the universality results on the limiting behaviour of classical walk-based centrality measures extend to these non-cycling cases. We also demonstrate that the new recursive construction gives rise to practical centrality measures that can be applied to large-scale networks.

Graph Powering and Spectral Robustness

Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix $A$, one may look at the spectrum of $\psi(A)$ for a properly chosen $\psi$. The issue is that the spectrum of $A$ might be contaminated by non-informational top eigenvalues, e.g., due to scale` variations in the data, and the application of $\psi$ aims to remove these. Designing a good functional $\psi$ (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, $$\psi(A) = \1((I+A)^r \ge1),$$ where $A$ denotes the adjacency matrix and $r$ is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance $r$, and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd\H{o}s-R\'enyi ensemble, which has no spectral gap, it is shown that graph powering produces a `maximal' spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to \cite{massoulie-STOC}, settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts \cite{massoulie-STOC,Mossel_SBM2,bordenave,colin3}. This is illustrated on a geometric block model that is dense in cliques.

Non-backtracking PageRank: From the classic model to hashimoto matrices

Abstract Non-backtracking centrality was introduced as a way to correct what may be understood as a deficiency in the eigenvector centrality, since the eigenvector centrality in a network can be artificially increased in high-degree nodes (hubs) because a hub is central because its neighbors are central, but these, in turn, are central just because they are hub neighbors. We define the non-backtracking PageRank as a new measure modifying the well-known classic PageRank in order to avoid the possibility of the random walker returning to the node immediately visited (non-backtracking walk). But, as we show, this measure presents a gap and a remarkable difference between the limit of “no penalty for return trips” and the direct calculation of the non-backtracking PageRank. Also, as it is shown in the applications presented, in certain cases this new measure produces notable variations with respect to the classifications obtained by the classic PageRank.

Comparing mixing times on sparse random graphs

Il est naturel de s’attendre a ce que la marche aleatoire sans rebroussement melange plus vite que la marche aleatoire simple, mais jusqu’ici, cela n’etait prouve que dans le cas des graphes reguliers. Pour analyser le cas de graphes irreguliers typiques, soit $G$ un graphe aleatoire a $n$ sommets de degres au moins $3$ et distribues selon une loi a queue exponentielle. On determine le temps de melange partant du pire point de depart pour la marche aleatoire simple sur $G$, et l’on montre qu’avec grande probabilite, cette marche presente le phenomene de cutoff au temps ${\mathbf{h}}^{-1}\log n$, ou ${\mathbf{h}}$ est l’entropie asymptotique de la marche aleatoire simple sur un arbre de Galton–Watson qui est une approximation locale de $G$. (Precedemment, cela n’etait connu que pour des points de depart typiques.) De plus, on montre que ce temps de melange est strictement plus grand que celui de la marche aleatoire sans rebroussement, via une comparison delicate des entropies sur l’arbre de Galton–Watson.

Clustering ensembles of social networks

AbstractRecently there has been significant work in the social sciences involving ensembles of social networks, that is, multiple, independent, social networks such as students within schools or employees within organizations. There remains, however, very little methodological work on exploring these types of data structures. We present methods for clustering social networks with observed nodal class labels, based on statistics of walk counts between the nodal classes. We extend this method to consider only non-backtracking walks, and introduce a method for normalizing the counts of long walk sequences using those of shorter ones. We then present a method for clustering networks based on these statistics to explore similarities among networks. We demonstrate the utility of this method on simulated network data, as well as on advice-seeking networks in education.

On the exponential generating function for non-backtracking walks

We derive an explicit formula for the exponential generating function associated with non-backtracking walks around a graph. We study both undirected and directed graphs. Our results allow us to derive computable expressions for non-backtracking versions of network centrality measures based on the matrix exponential. We find that eliminating backtracking walks in this context does not significantly increase the computational expense. We show how the new measures may be interpreted in terms of standard exponential centrality computation on a certain multilayer network. Insights from this block matrix interpretation also allow us to characterize centrality measures arising from general matrix functions. Rigorous analysis on the star graph illustrates the effect of non-backtracking and shows that localization can be eliminated when we restrict to non-backtracking walks. We also investigate the localization issue on synthetic networks.

The Deformed Graph Laplacian and Its Applications to Network Centrality Analysis

We introduce and study a new network centrality measure based on the concept of nonbacktracking walks, that is, walks not containing subsequences of the form uvu where u and v are any distinct connected vertices of the underlying graph. We argue that this feature can yield more meaningful rankings than traditional walk-based centrality measures. We show that the resulting Katz-style centrality measure may be computed via the so-called deformed graph Laplacian---a quadratic matrix polynomial that can be associated with any graph. By proving a range of new results about this matrix polynomial, we gain insights into the behavior of the algorithm with respect to its Katz-like parameter. The results also inform implementation issues. In particular we show that, in an appropriate limit, the new measure coincides with the nonbacktracking version of eigenvector centrality introduced by Martin, Zhang, and Newman in 2014. Rigorous analysis on star and star-like networks illustrates the benefits of the new approach,...

Non-backtracking walk centrality for directed networks

The theory of zeta functions provides an expression for the generating fu nction of nonbacktracking walk counts on a directed network. We show how this expression can be used to produce a centrality measure that eliminates backtracking walks at no cost. We also show that the radius of convergence of the generating function is determined by the spect rum of a three-by-three block matrix involving the original adjacency matrix. This giv es a means to choose appropriate values of the attenuation parameter. We find that three important a dditional benefits arise when we use this technique to eliminate traversals around the network that are unlikely to be of relevance. First, we obtain a larger range of choices for the attenuation para meter. Second, because the radius of convergence of the generating function is invariant under the remov al of certain types of nodes, we can gain computational efficiencies through reducing the dimension of t he resulting eigenvalue problem. Third, the dimension of the linear system defining the centrali ty measures may be reduced in the same manner. We show that the new centrality measure may be interp reted as standard Katz on a modified network, where self loops are added, and where nonreciproca l edges are augmented with negative weights. We also give a multilayer interpretation, wh ere negatively weighted walks between layers compensate for backtracking walks on the only non-emp ty layer. Studying the limit as the attenuation parameter approaches its upper bound allows us to propose an eigenvector-based nonbacktracking centrality measure in this directed network setting. We find that the two-by-two block matrix arising in previous studies focused on undirected networks must be extended to a new three-by-three block structure to allow for directed edges. We illustrat e the centrality measure on a synthetic network, where it is shown to eliminate a localization effect p resent in standard Katz centrality. Finally, we give results for real networks.

Non-Backtracking Random Walks and a Weighted Ihara’s Theorem

We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara’s Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et al. in [1] that in most cases, a non-backtracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.

Exploring temporal networks with greedy walks

Temporal networks come with a wide variety of heterogeneities, from burstiness of event sequences to correlations between timings of node and link activations. In this paper, we set to explore the latter by using greedy walks as probes of temporal network structure. Given a temporal network (a sequence of contacts), greedy walks proceed from node to node by always following the first available contact. Because of this, their structure is particularly sensitive to temporal-topological patterns involving repeated contacts between sets of nodes. This becomes evident in their small coverage per step as compared to a temporal reference model -- in empirical temporal networks, greedy walks often get stuck within small sets of nodes because of correlated contact patterns. While this may also happen in static networks that have pronounced community structure, the use of the temporal reference model takes the underlying static network structure out of the equation and indicates that there is a purely temporal reason for the observations. Further analysis of the structure of greedy walks indicates that burst trains, sequences of repeated contacts between node pairs, are the dominant factor. However, there are larger patterns too, as shown with non-backtracking greedy walks. We proceed further to study the entropy rates of greedy walks, and show that the sequences of visited nodes are more structured and predictable in original data as compared to temporally uncorrelated references. Taken together, these results indicate a richness of correlated temporal-topological patterns in temporal networks.

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.

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.

Tight lower bound for percolation threshold on an infinite graph.

We construct a tight lower bound for the site percolation threshold on an infinite graph, which becomes exact for an infinite tree. The bound is given by the inverse of the maximal eigenvalue of the Hashimoto matrix used to count nonbacktracking walks on the original graph. Our bound always exceeds the inverse spectral radius of the graph's adjacency matrix, and it is also generally tighter than the existing bound in terms of the maximum degree. We give a constructive proof for existence of such an eigenvalue in the case of a connected infinite quasitransitive graph, a graph-theoretic analog of a translationally invariant system.

Ising model observables and non-backtracking walks

This paper presents an alternative proof of the connection between the partition function of the Ising model on a finite graph G and the set of non-backtracking walks on G. The techniques used also give formulas for spin-spin correlation functions in terms of non-backtracking walks. The main tools used are Viennot's theory of heaps of pieces and turning numbers on surfaces.

The Fermionic Observable in the Ising Model and the Inverse Kac–Ward Operator

We show that the critical Kac-Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac-Ward operator for any planar graph is given by what we call a fermionic generating function. Furthermore, using bounds for the spectral radius and operator norm of the Kac-Ward transition matrix, we provide a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac-Ward operators on isoradial graphs.