A Continuous-Time Network Evolution Model Describing 2- and 3-Interactions

Abstract

A continuous-time network evolution model is considered. The evolution of the network is based on 2- and 3-interactions. 2-interactions are described by edges, and 3-interactions are described by triangles. The evolution of the edges and triangles is governed by a multi-type continuous-time branching process. The limiting behaviour of the network is studied by mathematical methods. We prove that the number of triangles and edges have the same magnitude on the event of non-extinction, and it is eαt, where α is the Malthusian parameter. The probability of the extinction and the degree process of a fixed vertex are also studied. The results are illustrated by simulations.

Keywords

• network evolution
• random graph
• multi-type branching process
• continuous-time branching process
• 2- and 3-interactions
• Malthusian parameter