Promotion/Inhibition Effects in Networks: A Model with Negative Probabilities

TL;DR

This paper introduces a novel method for inferring network edge-weights in biological systems with promotion and inhibition effects, using a framework of negative probabilities and a generalized Sinkhorn algorithm.

Contribution

It develops a new inverse problem approach for sign-indefinite adjacency matrices employing negative probabilities and extends the Sinkhorn algorithm for this purpose.

Findings

Framework applicable to gene co-expression networks.

Algorithm efficiently computes edge-weights with negative probabilities.

Method generalizes existing matrix scaling techniques.

Abstract

Biological networks often encapsulate promotion/inhibition as signed edge-weights of a graph. Nodes may correspond to genes assigned expression levels (mass) of respective proteins. The promotion/inhibition nature of co-expression between nodes is encoded in the sign of the corresponding entry of a sign-indefinite adjacency matrix, though the strength of such co-expression (i.e., the precise value of edge weights) cannot typically be directly measured. Herein we address the inverse problem to determine network edge-weights based on a sign-indefinite adjacency and expression levels at the nodes. While our motivation originates in gene networks, the framework applies to networks where promotion/inhibition dictates a stationary mass distribution at the nodes. In order to identify suitable edge-weights we adopt a framework of ``negative probabilities,'' advocated by P.\ Dirac and R.\ Feynman, and we set up a likelihood formalism to obtain values for the sought edge-weights. The proposed optimization problem can be solved via a generalization of the well-known Sinkhorn algorithm; in our setting the Sinkhorn-type ``diagonal scalings'' are multiplicative or inverse-multiplicative, depending on the sign of the respective entries in the adjacency matrix, with value computed as the positive root of a quadratic polynomial.