Fluctuations in percolation of sparse complex networks

Abstract

We study the role of fluctuations in percolation of sparse complex networks. To this end we consider two random correlated realizations of the initial damage of the nodes and we evaluate the fraction of nodes that are expected to remain in the giant component of the network in both cases or just in one case. Our framework includes a message-passing algorithm able to predict the fluctuations in a single network, and an analytic prediction of the expected fluctuations in ensembles of sparse networks. This approach is applied to real ecological and infrastructure networks and it is shown to characterize the expected fluctuations in their response to external damage.

Attack robustness and stability of generalized k-cores

“…The potential damage and recovery of the network crucially rely on the location of the functional components. [12] showed a pronounced variation of giant component sizes corresponding to two correlated random realizations of percolation, suggesting the changing role of individual nodes in response to different percolation. Stability is recently introduced in [13] as a novel measure to quantify the extent to which the giant connected component of a network consists of the same nodes under multiple independent edge percolation.…”

Stability of a giant connected component in a complex network

“…The dashed line corresponds to S( |q) of mutual GCCs consisting of random sets of nodes. Solid lines correspond to analytic solutions of Eqs (8)(9)(10) …”

“…Our work is motivated by two recent results in percolation theory [9,10] analyzing heterogeneous network responses to different percolation realizations. In particular, Ref.…”

“…Yet, apart from a handful of recent works aiming to understand individual node properties in percolation and epidemic processes [6][7][8][9][10], organization patterns of individual network's functional components are poorly studied. One reason is that the random nature of adverse events coupled with the complexity of relevant networked systems often makes the prediction of functional subgraphs impossible.…”

“…Concurrently, Ref. [10] introduces a framework to quantify GCC fluctuations by analyzing network responses to two random but possibly correlated percolation realizations. In our work we ask a related question: How stable is network's GCC?…”