Maximizing graph probability under conditionally exponential models

TL;DR

This paper explores maximizing network reliability within the framework of exponential random graph models, translating complex probabilistic network design problems into solvable linear systems and demonstrating effective optimization methods.

Contribution

It introduces a novel approach to reliability maximization in ERGMs by converting combinatorial problems into linear constraints, enabling efficient optimization.

Findings

Successfully formulated reliability maximization as linear systems.

Provided analytical solutions and numerical experiments.

Demonstrated the approach's effectiveness with computer implementations.

Abstract

Designing reliable networks consists in finding topological structures, which are able to successfully carry out desired processes and operations. When this set of activities performed within a network are unknown and the only available information is a probabilistic model reflecting topological network features, highly probable networks are regarded as "reliable", in the sense of being consistent with those probabilistic model. In this paper we are studying the reliability maximization, based on the Exponential Random Graph Model (ERGM), whose statistical properties has been widely used to capture complex topological feature of real-world networks. Under such models the probability of a network is maximized when specified structural properties appear in the network. However, the search of maximally reliable (highly probable) networks might result in difficult combinatorial optimization problems and an important goal of this work is to translate them into solvable systems of linear constraints. Analytical and numerical results are provided, using exact optimization techniques and efficient computer implementation.