Multiple Hybrid Phase Transition: Bootstrap Percolation on Complex Networks with Communities

X iv :1 40 1. 46 80 v3 [ ph ys ic s. so cph ] 2 2 Ja n 20 14 Bootstrap Percolation on Complex Networks with Community Structure Chong Wu1, Shenggong Ji1, Rui Zhang1,2,3, Liujun Chen4,∗ Jiawei Chen4, Xiaobin Li1, and Yanqing Hu1† 1School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China 2Levich Institute and Physics Department, City College of the City University of New York, New York, NY 10031, USA 3Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA 4School of Systems Science, Beijing Normal University, Beijing 100875, China (Dated: January 23, 2014) Abstract Real complex networks usually involve community structure. How innovation and new products spread on social networks which have internal structure is a practically interesting and fundamental question. In this paper we study the bootstrap percolation on a single network with community structure, in which we initiate the bootstrap process by activating different fraction of nodes in each community. A previously inactive node transfers to active one if it detects at least k active neighbors. The fraction of active nodes in community i in the final state Si and its giant component size Sgci are theoretically obtained as functions of the initial fractions of active nodes fi. We show that such functions undergo multiple discontinuous transitions; The discontinuous jump of Si or Sgci in one community may trigger a simultaneous jump of that in the other, which leads to multiple discontinuous transitions for the total fraction of active nodes S and its associated giant component size Sgc in the entire network. We have further obtained the phase diagram of the total number of jumps with respect to the inner-degrees of the two communities on Erdős-Rényi networks. If their inner-degrees are comparable or one of which is small, the system exhibits at most one discontinuous jump; otherwise it undergoes two discontinuous transitions. The number of discontinuous transitions reveals the internal structure of the network.