General Clique Percolation in Network Evolution

We introduce a general $(k,l)$ clique community, which consists of adjacent $k$-cliques sharing at least $l$ vertices with $k-1 \ge l \ge 1$. The emergence of a giant $(k,l)$ clique community indicates a $(k,l)$ clique percolation, which is studied by the largest size gap $\Delta$ of the largest clique community during network evolution and the corresponding evolution step $T_c$. For a clique percolation, the averages of $\Delta$ and $T_c$ and the root-mean-squares of their fluctuations have power law finite-size effects whose exponents are related to the critical exponents. The fluctuation distribution functions of $\Delta$ and $T_c$ follow a finite-size scaling form. In the evolution of the Erd\H{o}s-R\'enyi network, there are a series of $(k,l)$ clique percolation with $(k,l)=(2,1),(3,1),(3,2),(4,1),(4,2),(5,1),(4,3)$, and so on. The critical exponents of clique percolation depend on $l$, but are independent of $k$. The universality class of a $(k,l)$ clique percolation is characterized alone by $l$.