Multi-Hop Generalized Core Percolation on Complex Networks

Abstract

Recent theoretical studies on network robustness have focused primarily on attacks by random selection and global vision, but numerous real-life networks suffer from proximity-based breakdown. Here we introduce the multi-hop generalized core percolation on complex networks, where nodes with degree less than [Formula: see text] and their neighbors within [Formula: see text]-hop distance are removed progressively from the network. The resulting subgraph is referred to as [Formula: see text]-core, extending the recently proposed [Formula: see text]-core and classical core of a network. We develop analytical frameworks based upon generating function formalism and rate equation method, showing for instance continuous phase transition for [Formula: see text]-core and discontinuous phase transition for [Formula: see text]-core with any other combination of [Formula: see text] and [Formula: see text]. We test our theoretical results on synthetic homogeneous and heterogeneous networks, as well as on a selection of large-scale real-world networks. This unravels, e.g., a unique crossover phenomenon rooted in heterogeneous networks, which raises a caution that endeavor to promote network-level robustness could backfire when multi-hop tracing is involved.

A local algorithm and its percolation analysis of bipartite z-matching problem

“…Variants of the GLR procedure are also adopted in the k -XORSAT problem [22][23][24][25], Boolean networks [26], maximum independent set problem [27], minimum dominating set problem [28,29], and covering problems on hypergraphs [30]. Besides its original definition based on leaves (any node with a degree of 1) on undirected graphs, the GLR procedure can further be generalized on directed graphs [31] and can also be based on k -leaves (any node with a degree < k with k as an integer) on both single and multiplex networks [32][33][34][35].…”