Universal mechanism for hybrid percolation transitions

Deokjae Lee,B Kahng,Wonjun Choi,J Kertész

doi:10.1038/s41598-017-06182-3

Abstract

Hybrid percolation transitions (HPTs) induced by cascading processes have been observed in diverse complex systems such as k-core percolation, breakdown on interdependent networks and cooperative epidemic spreading models. Here we present the microscopic universal mechanism underlying those HPTs. We show that the discontinuity in the order parameter results from two steps: a durable critical branching (CB) and an explosive, supercritical (SC) process, the latter resulting from large loops inevitably present in finite size samples. In a random network of N nodes at the transition the CB process persists for O(N1/3) time and the remaining nodes become vulnerable, which are then activated in the short SC process. This crossover mechanism and scaling behavior are universal for different HPT systems. Our result implies that the crossover time O(N1/3) is a golden time, during which one needs to take actions to control and prevent the formation of a macroscopic cascade, e.g., a pandemic outbreak.

Highlights

Hybrid percolation transitions (HPTs) induced by cascading processes have been observed in diverse complex systems such as k-core percolation, breakdown on interdependent networks and cooperative epidemic spreading models
We conclude that before nc the I-state nodes are almost entirely generated through the critical branching (CB) tree and the W-state nodes accumulate to an extent of O(N2/3) because the number of W-state nodes is proportional to that of I-state nodes
We disclosed the universal mechanism of the HPT induced by cascade dynamics on ER networks

Summary

OPEN Universal mechanism for hybrid percolation transitions

Hybrid percolation transitions (HPTs) induced by cascading processes have been observed in diverse complex systems such as k-core percolation, breakdown on interdependent networks and cooperative epidemic spreading models. Examples include the k-core percolation[9,10,11,12], and the cascading failure (CF) model on interdependent networks[13,14,15,16] In those systems, as nodes or links are removed one by one above the transition point, the order parameter, the relative size of the giant component decreases continuously, approaches a nonzero value in a critical way at the transition point, where it collapses to zero: A HPT occurs. Refs 19 and 20 pointed out the importance of large loops in the creation of the supercritical process for an epidemic model It is still unclear whether there is a universal mechanism, which explains why, how and when such SC processes occur in the late time regime. We will consider k-core percolation, the threshold model, and the CF model on interdependent networks

Results

Discussion

Additional Information