Multi-scale modularity and dynamics in complex networks

In this chapter, we focus on the detection of communities in multi-scale networks, namely networks made of different levels of organization and in which modules exist at different scales. It is first shown that methods based on modularity are not appropriate to uncover modules in empirical networks, mainly because modularity optimization has an intrinsic bias towards partitions having a characteristic number of modules which might not be compatible with the modular organization of the system. We argue for the use of more flexible quality functions incorporating a resolution parameter that allows us to reveal the natural scales of the system. To do so, we adopt a dynamical perspective based on the exploration of the network by random walkers, with the extra advantages that the effect of structure on dynamical flows is properly taken into account and that the user has the flexibility to use a dynamical process adapted to the system under scrutiny. This approach is illustrated with two recently introduced quality functions, stability and the Map Equation. Finally, we discuss the problem of detecting significant values of the resolution parameter by using complementary measures of robustness of the uncovered partitions.