On the Diversity of Non-Linear Transient Dynamics in Several Types of Complex Networks

TL;DR

This paper investigates the diversity of non-linear transient dynamics in various complex networks using self-avoiding random walks, revealing how diversity varies across network types and nodes, and applying PCA for analysis.

Contribution

It introduces a method to quantify and analyze the diversity of transient dynamics in complex networks, comparing multiple models and applying PCA for dimensionality reduction.

Findings

Diversity tends to increase with average degree across models.

Watts-Strogatz and geographical models show gradual diversity increase.

Principal component analysis effectively characterizes nodes and network subgraphs.

Abstract

Dynamic systems characterized by diversified evolutions are not only more flexible, but also more resilient to attacks, failures and changing conditions. This article addresses the quantification of the diversity of non-linear transient dynamics obtained in undirected and unweighted complex networks as a consequence of self-avoiding random walks. The diversity of walks starting at a specific node $i$ is quantified in terms of a signature composed by the entropies of the node visit probabilities along each of the initial steps. Six theoretical models of complex networks are considered: Erd\H{o}s-R\'enyi, Barab\'asi-Albert, Watts-Strogatz, a geographical model, as well as two recently introduced knitted networks formed by paths. The random walk diversity is explored at the level of network categories and of individual nodes. Because the diversity at successive steps of the walks tends to be correlated, principal component analysis is systematically applied in order to identify the more relevant linear combinations of the diversity entropies and to obtain optimal dimensionality reduction. Several interesting results are reported, including the facts that the transient diversity tends to increase with the average degree for all considered network models and that the Watts and Strogatz and geographical models tend to yield diversity entropies which increase more gradually with the number of steps, contrasting sharply with the steep increases verified for the other four considered models. The principal linear combination of the diversities identified by the principal component analysis method is shown to allow an interesting characterization of individual nodes as well as partitioning of networks into subgraphs of similar diversity.