  Many complex systems change their structure over time, in these cases dynamic
networks can provide a richer representation of such phenomena. As a
consequence, many inference methods have been generalized to the dynamic case
with the aim to model dynamic interactions. Particular interest has been
devoted to extend the stochastic block model and its variant, to capture
community structure as the network changes in time. While these models assume
that edge formation depends only on the community memberships, recent work for
static networks show the importance to include additional parameters capturing
structural properties, as reciprocity for instance. Remarkably, these models
are capable of generating more realistic network representations than those
that only consider community membership. To this aim, we present a
probabilistic generative model with hidden variables that integrates
reciprocity and communities as structural information of networks that evolve
in time. The model assumes a fundamental order in observing reciprocal data,
that is an edge is observed, conditional on its reciprocated edge in the past.
We deploy a Markovian approach to construct the network's transition matrix
between time steps and parameters' inference is performed with an
Expectation-Maximization algorithm that leads to high computational efficiency
because it exploits the sparsity of the dataset. We test the performance of the
model on synthetic dynamical networks, as well as on real networks of citations
and email datasets. We show that our model captures the reciprocity of real
networks better than standard models with only community structure, while
performing well at link prediction tasks.
