Multiple hybrid phase transition: Bootstrap percolation on complex networks with communities

¹ School of Mathematics, Southwest Jiaotong University - Chengdu 610031, China
² Levich Institute and Physics Department, City College of the City University of New York New York, NY 10031, USA
³ Institute for Molecular Engineering, University of Chicago - Chicago, IL 60637, USA
⁴ School of Systems Science, Beijing Normal University - Beijing 100875, China

^(a) chenlj@bnu.edu.cn
^(b) yanqing.hu.sc@gmail.com

Received: 18 April 2014
Accepted: 25 July 2014

Abstract

Bootstrap percolation is a well-known model to study the spreading of rumors, new products or innovations on social networks. The empirical studies show that the community structure is ubiquitous among various social networks. Thus, studying the bootstrap percolation on the complex networks with communities can bring us new and important insights into the spreading dynamics in social networks. This has attracted a lot of scientists' attention recently. In this letter, we study the bootstrap percolation on Erdős-Rényi networks with communities and observed second-order, hybrid (both second and first order) and multiple hybrid phase transitions, which is rare in natural system. Moreover, we have analytically solved this system and obtained the phase diagram, which is further justified well by the corresponding simulations.