Noninvasive Approximation of Linear Dynamic System Networks Using Polytrees

Abstract

We consider the problem of approximating a complex network of linear dynamic systems via a simpler network, with the goal of highlighting the most significant connections. Indeed, paradoxically, an approximate network with fewer edges could be more informative in terms of how a system operates than a more accurate representation including a large number of “weak” links. Broadly, this article explores the meaning of approximating a network belonging to a certain class using another network belonging to a subset of its class (the set of approximators). We posit that any network approximation algorithm is expected to satisfy at least a congruity property. By congruity, we mean that if the approximated network belongs to the set of approximators, then the algorithm should map it into itself. From a technical perspective, we choose a class of dynamic networks with a directed tree (polytree) structure as a set of approximators and analytically derive a technique, which asymptotically satisfies the congruity property when the observation horizon approaches infinity. Also, we test such a technique using high-frequency financial data. Financial data provide a challenging benchmark since they are not expected to meet the theoretical assumptions behind our methodology, such as linearity or stationarity.