Heterogeneous-k-core versus Bootstrap Percolation on Complex Networks

TL;DR

This paper introduces the heterogeneous-k-core, a generalized network pruning process with variable thresholds, comparing its phase transitions and critical phenomena to bootstrap percolation, highlighting the impact of network structure on these processes.

Contribution

It defines the heterogeneous-k-core, analyzes its phase transitions, and compares it with bootstrap percolation, emphasizing the role of network degree distribution.

Findings

Heterogeneous-k-core exhibits both continuous and hybrid phase transitions.

Network structure significantly influences the emergence of the giant heterogeneous-k-core.

Power-law networks with degree exponent less than 3 show immediate giant core formation for any positive threshold probability.

Abstract

We introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which may be different at each vertex. If a vertex has less than $k_i$ neighbors it is pruned from the network. The heterogeneous-$k$-core is the sub-graph remaining after no further vertices can be pruned. If the thresholds $k_i$ are $1$ with probability $f$ or $k \geq 3$ with probability $(1-f)$, the process forms one branch of an activation-pruning process which demonstrates hysteresis. The other branch is formed by ordinary bootstrap percolation. We show that there are two types of transitions in this heterogeneous-$k$-core process: the giant heterogeneous-$k$-core may appear with a continuous transition and there may be a second, discontinuous, hybrid transition. We compare critical phenomena, critical clusters and avalanches at the heterogeneous-$k$-core and bootstrap percolation transitions. We also show that network structure has a crucial effect on these processes, with the giant heterogeneous-$k$-core appearing immediately at a finite value for any $f > 0$ when the degree distribution tends to a power law $P(q) \sim q^{-\gamma}$ with $\gamma < 3$.