Semi-Blind Inference of Topologies and Dynamical Processes Over Dynamic Graphs

A task of major practical importance in network science is inferring the graph structure from noisy observations at a subset of nodes. Available methods for topology inference typically assume that the process over the network is observed at all nodes. However, application-specific constraints may prevent acquiring network-wide observations. Alleviating the limited flexibility of existing approaches, this work advocates structural models for graph processes and develops novel algorithms for joint inference of the network topology and processes from partial nodal observations. Structural equation models (SEMs) and structural vector autoregressive models (SVARMs) have well documented merits in identifying even directed topologies of complex graphs; while SEMs capture contemporaneous causal dependencies among nodes, SVARMs further account for time-lagged influences. A batch solver is proposed that iterates between inferring directed graphs that "best" fit the sequence of observations, and estimating the network processes at reduced computational complexity by leveraging tools related to Kalman smoothing. To further accommodate delay-sensitive applications, an online joint inference approach is put forth that even tracks time-evolving topologies. Furthermore, we specify novel conditions for identifying the network topology given partial observations. We prove that the required number of observations for unique identification reduces significantly when the network structure is sparse. Numerical tests with synthetic as well as real datasets corroborate the effectiveness of the proposed approach.