Distribution of centrality measures on undirected random networks via cavity method

The Katz centrality of a node in a complex network is a measure of the node's importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like and undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its "global" character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-cavity algorithm to solve linear systems on a tree-like structure, we show that (i) the Katz centrality of a single instance can be computed recursively in a very fast way, and (ii) the probability $P(K)$ that a random node in the ensemble of undirected random graphs has centrality $K$ satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erd\H{o}s-R\'enyi and scale-free networks in the locally tree-like regime, with excellent agreement. The distributions display a crossover between multimodality and unimodality as the mean degree increases, where distinct peaks correspond to the contribution to the centrality coming from nodes of different degrees. We also provide an approximate formula based on a rank-$1$ projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.