  Network science provides valuable insights across numerous disciplines
including sociology, biology, neuroscience and engineering. A task of major
practical importance in these application domains is inferring the network
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. This paper develops algorithms that iterate between
inferring directed graphs that "best" fit the data, 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, conditions for identifying the network topology given
partial observations are specified. It is proved 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 novel approach.
