Ricerc@Sapienza

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph). Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations.

Given an underlying graph, we consider the following dynamics: Initially, each node locally chooses a value in { - 1, 1}, uniformly at random and independently of other nodes. Then, in each consecutive round, every node updates its local value to the average of the values held by its neighbors, at the same time applying an elementary, local clustering rule that only depends on the current and the previous values held by the node.

Consider the following asynchronous, opportunistic communication model over a graph G: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say ±1 with probability 1/2; then, in each round, the two endpoints of the currently active edge update their values to their average.

Spectral techniques have proved amongst the most effective approaches to graph clustering. However, in general they require explicit computation of the main eigenvectors of a suitable matrix (usually the Laplacian matrix of the graph). Recent work (e.g., Becchetti et al., SODA 2017) suggests that observing the temporal evolution of the power method applied to an initial random vector may, at least in some cases, provide enough information on the space spanned by the first two eigenvectors, so as to allow recovery of a hidden partition without explicit eigenvector computations.

An important feature of complex networks that can help to understand their internal organization is community structure. Recognize such structures in brain networks could be crucial, as the brain functioning is thought to be based on modular organization. Moreover, brain networks are intrinsically multilayer, which essentially means they can vary in time, in frequency or other domains depending on the topology of the network. In the last decades, some multilayer clustering algorithm has been developed with the aim to identify communities in dynamic networks.