Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena

Collective phenomena emerge from the interaction of natural or artificial units with a complex organization. The interplay between structural patterns and dynamics might induce functional clusters that, in general, are different from topological ones. In biological systems, like the human brain, the overall functionality is often favored by the interplay between connectivity and synchronization dynamics, with functional clusters that do not coincide with anatomical modules in most cases. In social, sociotechnical, and engineering systems, the quest for consensus favors the emergence of clusters. Despite the unquestionable evidence for mesoscale organization of many complex systems and the heterogeneity of their interconnectivity, a way to predict and identify the emergence of functional modules in collective phenomena continues to elude us. Here, we propose an approach based on random walk dynamics to define the diffusion distance between any pair of units in a networked system. Such a metric allows us to exploit the underlying diffusion geometry to provide a unifying framework for the intimate relationship between metastable synchronization, consensus, and random search dynamics in complex networks, pinpointing the functional mesoscale organization of synthetic and biological systems.

Figure 1

Emergence of functional clusters. (a) A Girvan-Newman benchmark network [20] of $N=128$ oscillators. The mesoscale structure is organized into four clusters. (b) Phases ${\theta }_i$ ($i=1,2,\dots ,N$) of identical oscillators (${\omega }_i=0$) reported on a polar coordinate system with unitary radius for the original system (outer ring) and with smaller radius (inner ring) for one realization of its configuration model (which preserves the connectivity distribution of the original data and removes other correlations). The oscillators have initial phases uniformly distributed in $[0,2\pi ]$ at time $\tau =0$ (left panel). They are free to interact with each other, according to the underlying topology, and drive the systems to collective synchronization at $\tau =20$ (right panel). The original system reaches a metastable state—with intracluster units synchronized to a common phase, with small fluctuations around a reference value—whereas the configuration model—where the mesoscale structure has been destroyed—quickly reaches the global attractor. (c) Opinion-formation dynamics of agents in the DeGroot model of decentralized consensus [21]. Each line represents the evolution of an opinion $x_i(t)$. In the metastable state local consensus is first achieved within clusters (see the inset) and later evolves into a collective opinion.

Figure 2

Identifying functional clusters in diffusion space. (a) A Girvan-Newman benchmark network [20] with four clusters, embedded in the Euclidean space by using multidimensional scaling applied to the diffusion-distance matrix ${\mathbf{\Delta }}_{\tau =1}$. (b) Two units from the same cluster are closer across time ($\tau$) than units from different clusters. (c) Diffusion-distance matrices corresponding to different times: the mesoscale structure becomes more evident as time goes by. (d) The diffusion-distance matrix at time $\tau$ is normalized as ${\mathbf{\Delta }}_{\tau }/{\max }_{ij}\mathbf{(}{\mathrm{\Delta }}_{ij}(\tau )\mathbf{)}$ ($i,j=1,2,\dots ,N$): the normalized distance between units from the same cluster quickly shrinks, while the one between units from different clusters slowly shrinks; this peculiar behavior is used to probe the mesoscale structure at different resolutions. (e) The diffusion-distance matrix built by averaging the matrices up to a certain time ${\tau }_{\max }$ accounts for persistence of mesoscale across time and it is used to unveil its hierarchical organization by means of hierarchical clustering. (f) All resulting hierarchies are screened and the corresponding networks of clusters are built. The network which better represents the mesoscale structure is the one where the average diffusion distance among clusters is maximized. The significance of such a structure can be easily quantified by comparing with the result obtained from a network model preserving the degree distribution of the original data while destroying other correlations (i.e., a configuration model). See the Supplemental Material [30] for further analysis of synthetic networks.