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

arXiv: https://arxiv.org/abs/2307.07738 · HTML: https://mlsanity.com/paper/2307.07738/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 14 references that are in the map.
- First posted on arXiv 15 Jul 2023.

## Built on by (most cited first, 1 of 1)

- [Data Assimilation for Sign-indefinite Priors: A generalization of Sinkhorn's algorithm](https://mlsanity.com/paper/2308.11791/) (2023)

## Abstract (arXiv)

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.