Abstract: Undirected hyperbolic graph models have been extensively used as models of scale-free small-world networks with high clustering coefficient. Here we presented a simple directed hyperbolic model, where nodes randomly distributed on a hyperbolic disk are connected to a fixed number m of their nearest spatial neighbours. We introduce also a canonical version of this network (which we call network with varied connection radius''), where maximal length of outgoing bond is space-dependent and is determined by fixing the average out-degree at m. We study local bond length, in-degree and reciprocity in these networks as a function of spacial coordinates of the nodes, and show that the network has a distinct core-periphery structure. We show that for small densities of nodes the overall in-degree has a truncated power law distribution.

We study the inference of a model of dynamic networks in which both communities and links keep memory of previous network states. By considering maximum likelihood inference from single snapshot observations of the network, we show that link persistence makes the inference of communities harder, decreasing the detectability threshold, while community persistence tends to make it easier. We analytically show that communities inferred from single network snapshot can share a maximum overlap with the underlying communities of a specific previous instant in time. This leads to time-lagged inference: the identification of past communities rather than present ones. Finally we compute the time lag and propose a corrected algorithm, the Lagged Snapshot Dynamic (LSD) algorithm, for community detection in dynamic networks. We analytically and numerically characterize the detectability transitions of such algorithm as a function of the memory parameters of the model and we make a comparison with a full dynamic inference.

Although the computational and statistical trade-off for modeling single graphs, for instance using block models, is relatively well understood, extending such results to sequences of graphs has proven to be difficult. In this work, we propose two models for graph sequences that capture: (a) link persistence between nodes across time, and (b) community persistence of each node across time. In the first model, we assume that the latent community of each node does not change over time, and in the second model we relax this assumption suitably. For both of these proposed models, we provide computationally efficient inference algorithms, whose unique feature is that they leverage community detection methods that work on single graphs. We also provide experimental results validating the suitability of the models and the performance of our algorithms on synthetic instances.