A continuous-time network evolution model describing $N$-interactions

We have introduced a new continuous-time network evolution model. We have described cooperation, so we have considered the cliques of nodes. The evolution of the network was based on cliques of nodes of the network and was governed by a branching process. The basic properties of the evolution process were described. Asymptotic theorems were proved for the number of cliques having a fixed size and the degree of a fixed node. The generating function was calculated, and then the probability of extinction was obtained. For the proof, advanced results of multi-type branching processes were used. Besides precise mathematical proofs, simulation examples also supported our results.