Clique is a mesoscale measurement of system and widely exists in various real-world networks. The roles of cliques on information spreading dynamics have not been systematically investigated. We here propose an information spreading model on networks with cliques, where an informed node makes all nodes in the same clique be infected simultaneously. We exactly solve the critical behavior of the proposed information spreading dynamics and find that there is a double transition when the homogeneous or weak heterogeneous network has a relatively small fraction of nodes belonging to the clique and a relatively large clique size. The first transition exhibits a continuous phase transition and the critical exponent is the same as that of the information spreading on networks without cliques. However, the second transition is not a continuous phase transition in the classic sense. For strong heterogeneous networks, the system always has one continuous phase transition. The present work provides an explanation of the cliques on critical phenomena of information spread dynamics, which is markedly different from the results of cascading-failure dynamics.

• Received 4 June 2018

DOI:https://doi.org/10.1103/PhysRevE.98.052312