Random walks in directed modular networks

TL;DR

This paper studies random walks in directed modular networks, revealing how community efficiency depends on ingoing and outgoing connections, with implications for understanding directed systems like cortical networks.

Contribution

It introduces an analysis of random walk efficiency in directed networks, highlighting the role of in/out degree balance and providing analytical expressions for different network models.

Findings

Efficiency depends mainly on ingoing/outgoing connection balance.

Internal node degree has limited impact on efficiency.

Results applicable to cortical and communication networks.

Abstract

Because diffusion typically involves symmetric interactions, scant attention has been focused on studying asymmetric cases. However, important networked systems underlain by diffusion (e.g. cortical networks and WWW) are inherently directed. In the case of undirected diffusion, it can be shown that the steady-state probability of the random walk dynamics is fully correlated with the degree, which no longer holds for directed networks. We investigate the relationship between such probability and the inward node degree, which we call efficiency, in modular networks. Our findings show that the efficiency of a given community depends mostly on the balance between its ingoing and outgoing connections. In addition, we derive analytical expressions to show that the internal degree of the nodes do not play a crucial role in their efficiency, when considering the Erd\H{o}s-R\'enyi and Barab\'asi-Albert models. The results are illustrated with respect to the macaque cortical network, providing subsidies for improving transportation and communication systems.