Heterogeneity in outcomes of repeated instances of percolation experiments

Abstract

We investigate the heterogeneity of outcomes of repeated instances of percolation experiments in complex networks using a message passing approach to evaluate heterogeneous, node dependent probabilities of belonging to the giant or percolating cluster, i.e. the set of mutually connected nodes whose size scales linearly with the size of the system. We evaluate these both for large finite single instances, and for synthetic networks in the configuration model class in the thermodynamic limit. For the latter, we consider both Erdős-Rényi and scale free networks as examples of networks with narrow and broad degree distributions respectively. For real-world networks we use an undirected version of a Gnutella peer-to-peer file-sharing network with N = 62, 568 nodes as an example. We derive the theory for multiple instances of both uncorrelated and correlated percolation processes. For the uncorrelated case, we also obtain a closed form approximation for the large mean degree limit of Erdős-Rényi networks.

The Fate of Articulation Points and Bredges in Percolation

“…3 and 4. Other prominent features seen in the heat maps are not captured by these families of curves, as they require an analysis of local environments of nodes and edges beyond first coordination shells, as discussed and exploited before in [24,33] to rationalize prominent features in pdfs of percolation probabilities.…”

“…In summary, we demonstrated that the message passing approach to percolation -apart from its original purpose to compute heterogeneous node-dependent percolation probabilities [25,33] -can also be utilized to evaluate heterogeneous node dependent probabilities of vertices in a complex networks to be APs as well as het-erogeneous edge dependent probabilities of pairs of neighboring vertices in a network to be connected by a bredge.…”

“…For networks exhibiting 'narrow' degree distributions in the sense that the the standard deviation of the degrees is negligibly small in comparison to the mean degree, it is possible to derive closed form approximations of the results above [25,33]. The obvious candidate to consider is the Poisson degree distribution of ER graphs with large mean degree k = c, for which the standard deviation σ k = √ c is small compared to the mean for c ≫ 1.…”

Generalized k-cores of networks under attack with limited knowledge

“…By convention in statistics physics, we will resort to mean-field theory [2,23], which allows us to appreciate the average network structure characteristics in the large graph size limit. An orthogonal dimension to network robustness is the stability, which looks into the common nodes shared by a functional component such as giant component [31,32] and Gk-core [20] when a network undergoes repeated independent percolation processes. A higher number of common nodes indicates a stronger stability of the network as these nodes are likely to be retained regardless of a specific adverse event.…”