Cascading Failures in Interdependent Networks with Multiple Supply-Demand Links and Functionality Thresholds

TL;DR

This paper introduces a realistic model of interdependent networks where nodes require a minimum number of supply nodes to stay functional, analyzing how different failure thresholds affect network robustness and phase transitions.

Contribution

It develops a new model incorporating supply thresholds and internal failure conditions, extending previous all-or-nothing failure models for interdependent networks.

Findings

Multiple internal failure conditions lead to similar nontrivial results.

The model exhibits both discontinuous and continuous phase transitions.

Analytical solutions match stochastic simulations for large networks.

Abstract

Various social, financial, biological and technological systems can be modeled by interdependent networks. It has been assumed that in order to remain functional, nodes in one network must receive the support from nodes belonging to different networks. So far these models have been limited to the case in which the failure propagates across networks only if the nodes lose all their supply nodes. In this paper we develop a more realistic model for two interdependent networks in which each node has its own supply threshold, i.e., they need the support of a minimum number of supply nodes to remain functional. In addition, we analyze different conditions of internal node failure due to disconnection from nodes within its own network. We show that several local internal failure conditions lead to similar nontrivial results. When there are no internal failures the model is equivalent to a bipartite system, which can be useful to model a financial market. We explore the rich behaviors of these models that include discontinuous and continuous phase transitions. Using the generating functions formalism, we analytically solve all the models in the limit of infinitely large networks and find an excellent agreement with the stochastic simulations.