  Many real-world processes evolve in cascades over complex networks, whose
topologies are often unobservable and change over time. However, the so-termed
adoption times when blogs mention popular news items, individuals in a
community catch an infectious disease, or consumers adopt a trendy electronics
product are typically known, and are implicitly dependent on the underlying
network. To infer the network topology, a \textit{dynamic} structural equation
model is adopted to capture the relationship between observed adoption times
and the unknown edge weights. Assuming a slowly time-varying topology and
leveraging the sparse connectivity inherent to social networks, edge weights
are estimated by minimizing a sparsity-regularized exponentially-weighted
least-squares criterion. To this end, solvers with complementary strengths are
developed by leveraging (pseudo) real-time sparsity-promoting proximal gradient
iterations, the improved convergence rate of accelerated variants, or reduced
computational complexity of stochastic gradient descent. Numerical tests with
both synthetic and real data demonstrate the effectiveness of the novel
algorithms in unveiling sparse dynamically-evolving topologies, while
accounting for external influences in the adoption times. Key events in the
recent succession of political leadership in North Korea, explain connectivity
changes observed in the associated network inferred from global cascades of
online media.
