Random walks associated with symmetric M-matrices

In this paper we generalize the transition probability matrix for a random walk on a finite network by defining the transition probabilities througha symmetric M-matrix. Usually, the walker jumps from a vertex to a neighboraccording to the probabilities given by the adjacency matrix. Moreover, we can find in the literature the relation between random walks and the normalized laplacian or the combinatorial laplacian that are singular and symmetric M-matrices. Our model takes into consideration not only the probability of transitioning given by the adjacency matrix but also some added probability that depends on a node property. This also includes the probability of remaining in each node, when the M-matrix is not singular. The nodes importance is taking into account by considering the lower eigenvalue and its associated eigenfunction for the given M-matrix. We give expressions for the mean first passage time and Kemeny’s constant for such a random walks in terms of 1-inverses of the considered Mmatrix.This work has been partly supported by the Spanish Research Council (Ministerio de Ciencia e Innovación) under project PID2021-122501NB-I00 and by the Universitat Politècnica de Catalunya under funds AGRUPS-2022 and AGRUPS-2023.Peer ReviewedPostprint (author's final draft

Hubs-attracting laplacian and related synchronization on networks

In this work, we introduce a new Laplacian matrix, referred to as the hubs-attracting Laplacian, accounting for diffusion processes on networks where the hopping of a particle occurs with higher probability from low to high degree nodes. This notion complements the one of the hubs-repelling Laplacian discussed in [E. Estrada, Linear Algebra Appl., 596 (2020), pp. 256-280], that considers the opposite scenario, with higher hopping probabilities from high to low degree nodes. We formulate a model of oscillators coupled through the new Laplacian and study the synchronizability of the network through the analysis of the spectrum of the Laplacian. We discuss analytical results providing bounds for the quantities of interest for synchronization and computational results showing that the hubs-attracting Laplacian generally has better synchronizability properties when compared to the classical one, with a low occurrence rate for the graphs where this is not true. Finally, two illustrative case studies of synchronization through the hubs-attracting Laplacian are considered

Beyond the rich-club: Properties of networks related to the better connected nodes

35 pages, 16 figures35 pages, 16 figuresMany of the structural characteristics of a network depend on the connectivity with and within the hubs. These dependencies can be related to the degree of a node and the number of links that a node shares with nodes of higher degree. In here we revise and present new results showing how to construct network ensembles which give a good approximation to the degree-degree correlations, and hence to the projections of this correlation like the assortativity coefficient or the average neighbours degree. We present a new bound for the structural cut--off degree based on the connectivity within the hubs. Also we show that the connections with and within the hubs can be used to define different networks cores. Two of these cores are related to the spectral properties and walks of length one and two which contain at least on hub node, and they are related to the eigenvector centrality. We introduce a new centrality measured based on the connectivity with the hubs. In addition, as the ensembles and cores are related by the connectivity of the hubs, we show several examples how changes in the hubs linkage effects the degree--degree correlations and core properties