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Collective phenomena emerge from the interaction of natural or artificialunits with a complex organization. The interplay between structural patternsand dynamics might induce functional clusters that, in general, are differentfrom topological ones. In biological systems, like the human brain, the overallfunctionality is often favored by the interplay between connectivity andsynchronization dynamics, with functional clusters that do not coincide withanatomical modules in most cases. In social, socio-technical and engineeringsystems, the quest for consensus favors the emergence of clusters.  Despite the unquestionable evidence for mesoscale organization of manycomplex systems and the heterogeneity of their inter-connectivity, a way topredict and identify the emergence of functional modules in collectivephenomena continues to elude us. Here, we propose an approach based on randomwalk dynamics to define the diffusion distance between any pair of units in anetworked system. Such a metric allows to exploit the underlying diffusiongeometry to provide a unifying framework for the intimate relationship betweenmetastable synchronization, consensus and random search dynamics in complexnetworks, pinpointing the functional mesoscale organization of synthetic andbiological systems.

Diffusion geometry unravels the emergence of functional clusters in collective phenomena