We develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree. Framing the problem in terms of optimisation of a quadratic form on the sphere and introducing a fictitious temperature, we employ the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by population dynamics. This derivation allows us to identify and unpack the individual contributions to the top eigenvector's components coming from nodes of degree k. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, and are further crosschecked on the cases of random regular graphs and sparse Markov transition matrices for unbiased random walks. arXiv:1903.09852v2 [cond-mat.stat-mech] 25 Oct 2019 eigenpair problem for a differential operator [8]. The top eigenpair is also relevant in signal reconstruction problems employing algorithms based on the spectral method [9]. In the context of graph theory, the eigenvectors of both adjacency and Laplacian matrices are employed to solve combinatorial optimisation problems, such as graph 3-colouring [10] and to develop clustering and cutting techniques [11][12][13]. In particular, the top eigenvector of graphs is intimately related to the "ranking" of the nodes of the network [14]. Indeed, beyond the natural notion of ranking of a node given by its degree, the relevance of a node can be estimated from how "important" its neighbours are. The vector expressing the importance of each node is exactly the top eigenvector of the network adjacency matrix. Google PageRank algorithm works in a similar way [15,16]: the PageRanks vector is indeed the top eigenvector of a large Markov transition matrix between web pages.When the matrix J is random and symmetric with i.i.d. entries, analytical results on the statistics of the top eigenpair date back to the classical work by Füredi and Komlós [17]: the largest eigenvalue of such matrices follows a Gaussian distribution with finite variance, provided that the moments of the distribution of the entries do not scale with the matrix size. This result directly relates to the largest eigenvalue of Erdős-Rényi (E-R) [18] adjacency matrices in the case when the probability p for two nodes to be connected does not scale with the matrix size N . This result has been then extended by Janson [19] in the case when p is large. However, in our analysis we will be mostly dealing with the sparse case, i.e. when p = c/N , with c being the constant mean degree of nodes (or equivalently, the mean number of nonzero elements per row of the corresponding adjacency matrix). In this sparse regime, Krivelevich and Sudakov [20] proved a theorem stating that for any constant c the largest eigenvalue of Erdős-Rényi graph diverges slowly with N as log N/ log log N . To ensure that the largest eigenvalue remains ∼ O(1), the nodes with very large degree must be prun...

We review the problem of how to compute the spectral density of sparse symmetric random matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects.

We develop a formalism to compute the statistics of the second largest eigenpair of weighted sparse graphs with N ≫ 1 nodes, finite mean connectivity and bounded maximal degree, in cases where the top eigenpair statistics is known. The problem can be cast in terms of optimisation of a quadratic form on the sphere with a fictitious temperature, after a suitable deflation of the original matrix model. We use the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by an improved population dynamics algorithm enforcing eigenvector orthogonality on-the-fly. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, focussing on the cases of random regular and Erdős–Rényi (ER) graphs. We further analyse the case of sparse Markov transition matrices for unbiased random walks, whose second largest eigenpair describes the non-equilibrium mode with the largest relaxation time. We also show that the population dynamics algorithm with population size N P does not actually capture the thermodynamic limit N → ∞ as commonly assumed: the accuracy of the population dynamics algorithm has a strongly non-monotonic behaviour as a function of N P, thus implying that an optimal size N P ⋆ = N P ⋆ ( N ) must be chosen to best reproduce the results from numerical diagonalisation of graphs of finite size N.