Spectral density of the non-backtracking operator

The non-backtracking operator was recently shown to provide a significant improvement when used for spectral clustering of sparse networks. In this paper we analyze its spectral density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero spectral density, that is stable outside a circle of radius √ ρ, where ρ is the leading eigenvalue of the non-backtracking operator. We observe a second-order phase transition at the edge of this circle, between a zero and a non-zero spectral density. The fact that this phase transition is absent in the spectral density of other matrices commonly used for spectral clustering provides a physical justification of the performances of the non-backtracking operator in spectral clustering. Introduction. – Clustering and community detection are central tasks in the study of social, biological, and technological networks. Sparse networks, where the average degree of every node is a constant independent on the size of the network, are arguably the most relevant for applications, and at the same time the most challenging for clustering. Spectral methods are among the most widely used for this task. They are conceptually simply based on the computation of principal eigenvalues and eigenvectors of an operator associated with the network [1]. Most commonly this operator is the adjacency matrix, the Laplacian (symmetrized and/or normalized), the random walk matrix, or the modularity matrix. The spectrum of these matrices generically decomposes into a bulk of noninformative eigenvalues, and some informative eigenvalues separated from the bulk by a gap. The eigenvectors corresponding to the informative eigenvalues are correlated with the cluster structure. However, on sparse networks, spectral clustering based on these commonly used matrices does not perform as well as for instance methods based on Bayesian inference that can perform well even when the tails of the bulk of the spectrum flood the informative eigenvalue [2]. Recently, the author of [3] proposed the so-called nonbacktracking operator for spectral clustering and conjectured that this method is optimal: it is able to find clusters for large random clustered networks (in the stochastic block model) as long as it is information theoretically possible. The non-backtracking matrix B, associated with an undirected graph, encodes adjacency between directed edges. Its element Bi→j,k→l is one if the edge i→ j flows into the edge k → l, i.e. j = k and i 6= l, and zero otherwise. The authors of [3] give theoretical and numerical evidence that apart from the informative eigenvalues the spectrum of this matrix is confined to the circle of radius the square root of the average excess degree of the network, not presenting the so-called Liftshitz tails [2] that spoil the performance of spectral clustering for the other matrices mentioned above. In order to understand better the performance of spectral clustering it is crucial to understand in detail the spectral properties of the associated operators on random graphs. Analytical results for spectral densities of sparse random graphs are largely based on the method of replicas and cavity and were mostly developed and studied for symmetric random matrices [4–8]. The result most relep-1 ar X iv :1 40 4. 77 87 v1 [ co nd -m at .d is -n n] 3 0 A pr 2 01 4