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.
Highlights
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
Concerning the mathematical tools, we follow the line of Móri and Rokob [16], where connections of two units were described by edges and the evolution of the edges was governed by a continuous-time branching process
The novelty of the paper is the usage of a two-type continuous time branching process to describe these two types of interactions
Summary
In [10], the results of [11] on multi-type continuous time branching processes were applied to describe the evolution of the network. Concerning the mathematical tools, we follow the line of Móri and Rokob [16], where connections of two units were described by edges and the evolution of the edges was governed by a continuous-time branching process. We obtain a two-type continuous time branching process, in which an individual can be either an edge or a triangle of the network. The starting individual (that is the ancestor) can be either an edge or a triangle It produces offspring at each time given by the driving branching process. The proofs are based on known general results of multi-type continuous-time branching processes. In that paper some preliminary theoretical results were announced together with some numerical evidence but without mathematical proofs
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