  Directed networks are pervasive both in nature and engineered systems, often
underlying the complex behavior observed in biological systems, microblogs and
social interactions over the web, as well as global financial markets. Since
their structures are often unobservable, in order to facilitate network
analytics, one generally resorts to approaches capitalizing on measurable nodal
processes to infer the unknown topology. Structural equation models (SEMs) are
capable of incorporating exogenous inputs to resolve inherent directional
ambiguities. However, conventional SEMs assume full knowledge of exogenous
inputs, which may not be readily available in some practical settings. The
present paper advocates a novel SEM-based topology inference approach that
entails factorization of a three-way tensor, constructed from the observed
nodal data, using the well-known parallel factor (PARAFAC) decomposition. It
turns out that second-order piecewise stationary statistics of exogenous
variables suffice to identify the hidden topology. Capitalizing on the
uniqueness properties inherent to high-order tensor factorizations, it is shown
that topology identification is possible under reasonably mild conditions. In
addition, to facilitate real-time operation and inference of time-varying
networks, an adaptive (PARAFAC) tensor decomposition scheme which tracks the
topology-revealing tensor factors is developed. Extensive tests on simulated
and real stock quote data demonstrate the merits of the novel tensor-based
approach.
