Random walks are ubiquitous in network science, for they are simple yet powerful models of diffusion and information exchange, they also allow to gather information about the network structure finding, e.g., communities, or the deepest node aka the [network median](https://gbertagnolli.github.io/publication/network-depth/). One of the main reasons of their success lies in their mathematics: they are Markov chains on a finite state space with transition probabilities prescribed by the network connectivity. Unfortunately, not all real dynamics satisfy the Markov property and so generalised random walks are needed. In this work, we extensively explore the self-avoiding random walk, and a variation thereof which embeds a resetting mechanism, as an efficient network exploration strategy inspired by the _run-and-tumble_ motion of some bacteria. A significant novelty in this work is provided by the statistical approach we used to describe the evolution of important network features during self-avoiding-type random walks, which are stochastic chains on an evolving state space with structure- and time-dependent transitions.
Many real systems (social groups, the brain, economic and transportation networks) display a particular organization of their hubs (nodes with a lot of connections): they are more densely connected among them than expected by chance. In other words, these rich nodes form clubs, called 'rich clubs'. How do these (structural) rich clubs reflect on dynamical processes taking place on the network? It depends on many aspects: of course, it depends on the dynamics, so here we focus on diffusion dynamics modelled through continuous-time random walks. Then, it also depends on which other types of structures are present in the network, and on their hierarchical relations. In brief: the presence of a strong structural does not imply the existence a strong functional rich club (easier diffusion among rich nodes). The vice versa is also true: functional rich clubs do not translate directly into denser connectivity among hubs. For more detail, and cool plots, see the paper!
Exchanging information is crucial for many real systems and consequently also assessing the how efficiently a system carries on this task. Here we assume that the pairwise communication efficiency is inversely proportional to their distance (metric on the network). Furthermore, we focus on the efficiency that can be quantified through the topology of and the flows on the network.