Theory of percolation on hypergraphs.

Hypergraphs capture the higher-order interactions in complex systems and always admit a factor graph representation, consisting of a bipartite network of nodes and hyperedges. As hypegraphs are ubiquitous, investigating hypergraph robustness is a problem of major research interest. In the literature the robustness of hypergraphs so far only has been treated adopting factor-graph percolation, which describes well higher-order interactions which remain functional even after the removal of one of more of their nodes.

Nature of hypergraph k-core percolation problems.

Hypergraphs are higher-order networks that capture the interactions between two or more nodes. Hypergraphs can always be represented by factor graphs, i.e.

Localization of nonbacktracking centrality on dense subgraphs of sparse networks.

The nonbacktracking matrix and the related nonbacktracking centrality (NBC) play a crucial role in models of percolation-type processes on networks, such as nonrecurrent epidemics. Here we study the localization of NBC in infinite sparse networks that contain an arbitrary finite subgraph. Assuming the local tree likeness of the enclosing network, and that branches emanating from the finite subgraph do not intersect at finite distances, we show that the largest eigenvalue of the nonbacktracking matrix of the composite network is equal to the highest of the two largest eigenvalues: that of the finite subgraph and of the enclosing network.

Hidden transition in multiplex networks.

Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, [Formula: see text].

Approximating nonbacktracking centrality and localization phenomena in large networks.

Message-passing theories have proved to be invaluable tools in studying percolation, nonrecurrent epidemics, and similar dynamical processes on real-world networks. At the heart of the message-passing method is the nonbacktracking matrix, whose largest eigenvalue, the corresponding eigenvector, and the closely related nonbacktracking centrality play a central role in determining how the given dynamical model behaves. Here we propose a degree-class-based method to approximate these quantities using a smaller matrix related to the joint degree-degree distribution of neighboring nodes.