Weighted network motifs as random walk patterns

Over the last two decades, network theory has shown to be a fruitful paradigm in understanding the organization and functioning of real-world complex systems. One technique helpful to this endeavor is identifying functionally influential subgraphs, shedding light on underlying evolutionary processes. Such overrepresented subgraphs, motifs, have received much attention in simple networks, where edges are either on or off. However, for weighted networks, motif analysis is still undeveloped. Here, we proposed a novel methodology - based on a random walker taking a fixed maximum number of steps - to study weighted motifs of limited size. We introduce a sink node to balance the network and allow the detection of configurations within an a priori fixed number of steps for the random walker. We applied this approach to different real networks and selected a specific null model based on maximum-entropy to test the significance of weighted motifs occurrence. We found that identified similarities enable the classifications of systems according to functioning mechanisms associated with specific configurations: economic networks exhibit close patterns while differentiating from ecological systems without any a priori assumption.