  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 benchmark 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.
