Network Learning with Directional Sign Patterns

TL;DR

This paper introduces a novel framework for learning signed and directed network relations from aggregate data using an extended Schr"odinger bridge approach and an efficient Sinkhorn-type algorithm, applicable to high-order networks.

Contribution

It develops a new formalism generalizing Schr"odinger bridges to incorporate directionality and high-order interactions, enabling fast convergence in learning signed adjacency matrices from limited data.

Findings

Effective algorithm for signed network inference

Applicable to high-order and directed networks

Demonstrated on synthetic and real-world data

Abstract

Complex systems can be effectively modeled via graphs that encode networked interactions, where relations between entities or nodes are often quantified by signed edge weights, e.g., promotion/inhibition in gene regulatory networks, or encoding political of friendship differences in social networks. However, it is often the case that only an aggregate consequence of such edge weights that characterize relations may be directly observable, as in protein expression of in gene regulatory networks. Thus, learning edge weights poses a significant challenge that is further exacerbated for intricate and large-scale networks. In this article, we address a model problem to determine the strength of sign-indefinite relations that explain marginal distributions that constitute our data. To this end, we develop a paradigm akin to that of the Schr\"odinger bridge problem and an efficient Sinkhorn type algorithm (more properly, Schr\"odinger-Fortet-Sinkhorn algorithm) that allows fast convergence to parameters that minimize a relative entropy/likelihood criterion between the sought signed adjacency matrix and a prior. The formalism that we present represents a novel generalization of the earlier Schr\"odinger formalism in that marginal computations may incorporate weights that model directionality in underlying relations, and further, that it can be extended to high-order networks -- the Schr\"odinger-Fortet-Sinkhorn algorithm that we derive is applicable all the same and allows geometric convergence to a sought sign-indefinite adjacency matrix or tensor, for high-order networks. We demonstrate our framework with synthetic and real-world examples.