The minimal representation of a system with interacting units using Boltzmann machines

Abstract

This paper presents an alternative methodology to find a network model with the least amount of critical bonds necessary to represent the behavior of the interacting elements of a system. The model is based on a network of couplings inferred by an non-restricted Boltzmann machine, which allows finding a maximum entropy distribution (ME). For N elements, the process starts by removing from the set of N(N − 1)/2 bonds, those with the lowest intensity and calculating the Kullback-Leibler divergence (KL) in each step. The edge removal process stops before there is a drastic increase in the KL divergence. This process was applied to the European market indices over two different periods. The results provide an interesting description of the most significant interactions driving the market and, at the same time, identify markets with higher system importance.