# Generalized model for $k$-core percolation and interdependent networks

## TL;DR

This paper introduces a unified analytical model combining $k$-core percolation and interdependent networks, revealing complex phase transitions including novel two-stage transitions and tricritical points, validated through simulations.

## Contribution

It develops the first comprehensive analytical framework for combined $k$-core percolation and interdependent networks, uncovering new phase transition phenomena.

## Key findings

- Rich phase diagram with first, second, and two-stage transitions.
- Identification of tricritical lines merging into a two-stage transition region.
- Analytical derivation of phase boundaries and critical exponents.

## Abstract

Cascading failures in complex systems have been studied extensively using two different models: $k$-core percolation and interdependent networks. We combine the two models into a general model, solve it analytically and validate our theoretical results through extensive simulations. We also study the complete phase diagram of the percolation transition as we tune the average local $k$-core threshold and the coupling between networks. We find that the phase diagram of the combined processes is very rich and includes novel features that do not appear in the models studying each of the processes separately. For example, the phase diagram consists of first and second-order transition regions separated by two tricritical lines that merge together and enclose a novel two-stage transition region. In the two-stage transition, the size of the giant component undergoes a first-order jump at a certain occupation probability followed by a continuous second-order transition at a lower occupation probability. Furthermore, at certain fixed interdependencies, the percolation transition changes from first-order $\rightarrow$ second-order $\rightarrow$ two-stage $\rightarrow$ first-order as the $k$-core threshold is increased. The analytic equations describing the phase boundaries of the two-stage transition region are set up and the critical exponents for each type of transition are derived analytically.