Bootstrap Percolation on Complex Networks with Community Structure

X iv :1 40 1. 46 80 v3 [ ph ys ic s. so cph ] 22 J an 2 01 4 Bootstrap Percolation on Complex Networks with Community Structure Chong Wu1, Shenggong Ji 1, 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 practi c lly interesting and fundamental question. In this paper we study the bootstrap percolation on a single net work with community structure, in which we initiate the bootstrap process by activating different fra ction 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 communityi in the final stateSi and its giant component size Sgci are theoretically obtained as functions of the initial fractions of active nodesfi. We show that such functions undergo multiple discontinuou s 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 transit ions 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 dia gram of the total number of jumps with respect to the inner-degree s of the two communities on Erdős-Rényi networks. If their inner-degrees are comparable or one of wh ich is small, the system exhibits at most one discontinuous jump; otherwise it undergoes two discontinu ous transitions. The number of discontinuous transitions reveals the internal structure of the network.