Two models of sparse and clustered dynamic networks

Abstract

We present two models of sparse dynamic networks that display transitivity - the tendency for vertices sharing a common neighbour to be neighbours of one another. Our first network is a continuous time Markov chain $G=\{G_t=(V,E_t), t\ge 0\}$ whose states are graphs with the common vertex set $V=\{1,\dots, n\}$. The transitions are defined as follows. Given $t$, the vertex pairs $\{i,j\}\subset V$ are assigned independent exponential waiting times $A_{ij}$. At time $t+\min_{ij} A_{ij}$ the pair $\{i_0,j_0\}$ with $A_{i_0j_0}=\min_{ij} A_{ij}$ toggles its adjacency status. To mimic clustering patterns of sparse real networks we set intensities $a_{ij}$ of exponential times $A_{ij}$ to be negatively correlated with the degrees of the common neighbours of vertices $i$ and $j$ in $G_t$. Another dynamic network is based on a latent Markov chain $H=\{H_t=(V\cup W, E_t), t\ge 0\}$ whose states are bipartite graphs with the bipartition $V\cup W$, where $W=\{1,\dots,m\}$ is an auxiliary set of attributes/affiliations. Our second network $G'=\{G'_t =(E'_t,V), t\ge 0\}$ is the affiliation network defined by $H$: vertices $i_1,i_2\in V$ are adjacent in $G'_t$ whenever $i_1$ and $i_2$ have a common neighbour in $H_t$. We analyze geometric properties of both dynamic networks at stationarity and show that networks possess high clustering. They admit tunable degree distribution and clustering coefficients.