  Contagions such as the spread of popular news stories, or infectious
diseases, propagate in cascades over dynamic networks with unobservable
topologies. However, "social signals" such as product purchase time, or blog
entry timestamps are measurable, and implicitly depend on the underlying
topology, making it possible to track it over time. Interestingly, network
topologies often "jump" between discrete states that may account for sudden
changes in the observed signals. The present paper advocates a switched dynamic
structural equation model to capture the topology-dependent cascade evolution,
as well as the discrete states driving the underlying topologies. Conditions
under which the proposed switched model is identifiable are established.
Leveraging the edge sparsity inherent to social networks, a recursive
$\ell_1$-norm regularized least-squares estimator is put forth to jointly track
the states and network topologies. An efficient first-order proximal-gradient
algorithm is developed to solve the resulting optimization problem. Numerical
experiments on both synthetic data and real cascades measured over the span of
one year are conducted, and test results corroborate the efficacy of the
advocated approach.
