# Quantifying the Total Effect of Edge Interventions in Discrete Multistate Networks

## TL;DR

This paper develops polynomial formulas to quantify the global impact of edge interventions in discrete multistate networks, especially Boolean networks, using canalizing functions, and validates these formulas with random network simulations.

## Contribution

It introduces a polynomial normal form for nested canalizing functions and formulas to count transition changes due to edge deletions, advancing the analysis of intervention effects in network models.

## Key findings

- Formulas accurately estimate maximum transition changes upon edge deletion.
- Canalizing structure influences the number of affected transitions.
- Upper bounds are validated and characterized in random networks.

## Abstract

Developing efficient computational methods to assess the impact of external interventions on the dynamics of a network model is an important problem in systems biology. This paper focuses on quantifying the global changes that result from the application of an intervention to produce a desired effect, which we define as the total effect of the intervention. The type of mathematical models that we will consider are discrete dynamical systems which include the widely used Boolean networks and their generalizations. The potential interventions can be represented by a set of nodes and edges that can be manipulated to produce a desired effect on the system. We use a class of regulatory rules called nested canalizing functions that frequently appear in published models and were inspired by the concept of canalization in evolutionary biology. In this paper, we provide a polynomial normal form based on the canalizing properties of regulatory functions. Using this polynomial normal form, we give a set of formulas for counting the maximum number of transitions that will change in the state space upon an edge deletion in the wiring diagram. These formulas rely on the canalizing structure of the target function since the number of changed transitions depends on the canalizing layer that includes the input to be deleted. We also present computations on random networks to compare the exact number of changes with the upper bounds provided by our formulas. Finally, we provide statistics on the sharpness of these upper bounds in random networks.