  The problem of decomposing networks into modules (or clusters) has gained
much attention in recent years, as it can account for a coarse-grained
description of complex systems, often revealing functional subunits of these
systems. A variety of module detection algorithms have been proposed, mostly
oriented towards finding hard partitionings of undirected networks. Despite the
increasing number of fuzzy clustering methods for directed networks, many of
these approaches tend to neglect important directional information. In this
paper, we present a novel random walk based approach for finding fuzzy
partitions of directed, weighted networks, where edge directions play a crucial
role in defining how well nodes in a module are interconnected. We will show
that cycle decomposition of a random walk process connects the notion of
network modules and information transport in a network, leading to a new,
symmetric measure of node communication. walk process, for which we will prove
that although being time-reversible it inherits all necessary information about
directions and modular structure of the original network. Finally, we will use
this measure to introduce a communication graph, for which we will show that
although being undirected it inherits all necessary information about modular
structures from the original network.
