Two models of sparse and clustered dynamic networks

Abstract We present two models of sparse dynamic networks that display transitivity—the tendency for nodes 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\geq 0\} $ whose states are graphs with the common set of nodes $ V=\{1 , \dots, n\} $. The transitions are defined as follows. Given $ t $, the node 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_{0}j_{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 decreasing functions of the degrees of common neighbours of nodes $ i $ and $ j $ in $ G_{t} $. Our second network $ G^{\prime}=\{G^{\prime}_{t}=(E^{\prime}_{t},V),t\geq 0\} $ is the affiliation network based on a latent Markov chain $ H=\{H_{t}=(V\cup W, E_{t}),t\geq 0\} $ whose states are bipartite graphs with the bipartition $ V\cup W $, where $ W=\{1 , \dots, m\} $ is an auxiliary set of attributes/affiliations. Nodes $ i_{1},i_{2}\in V $ are adjacent in $ G^{\prime}_{t} $ whenever $ i_{1} $ and $ i_{2} $ have a common neighbour in $ H_{t} $. We analyse geometric properties of both dynamic networks at stationarity and show that networks possess high clustering. They admit tunable degree distribution and clustering coefficients.

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