The theory of percolation on hypergraphs

TL;DR

A message-passing theory of hypergraph percolation is built and it is demonstrated that any fat-tailed cardinality distribution of hyperedges cannot lead to the hyper-resilience phenomenon in hypergraphs in contrast to their factor graphs, where the divergent second moment of a cardinality distribution guarantees zero percolation threshold.

Abstract

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 as been so far only treated adopting factor-graph percolation which describe well higher-order interactions which remain functional even after the removal of one of more of their nodes. This approach, however, fall short to describe situations in which higher-order interactions fail when anyone of their nodes is removed, this latter scenario applying for instance to supply chains, catalytic networks, protein-interaction networks, networks of chemical reactions, etc. Here we show that in these cases the correct process to investigate is hypergraph percolation with is distinct from factor graph percolation. We build a message-passing theory of hypergraph percolation and we investigate its critical behavior using generating function formalism supported by Monte Carlo simulations on random graph and real data. Notably, we show that the node percolation threshold on hypergraphs exceeds node percolation threshold on factor graphs. Furthermore we show that differently from what happens in ordinary graphs, on hypergraphs the node percolation threshold and hyperedge percolation threshold do not coincide, with the node percolation threshold exceeding the hyperedge percolation threshold. These results demonstrate that any fat-tailed cardinality distribution of hyperedges cannot lead to the hyper-resilience phenomenon in hypergraphs in contrast to their factor graphs, where the divergent second moment of a cardinality distribution guarantees zero percolation threshold.

Figures (4)

• Figure 1: Schematic representation of a hypergraph (panel a) and its corresponding factor graph (panel b). The factor graph is a bipartite network of connections between nodes (filled circles) representing the nodes of the hypergraph and factor nodes (filled triangles) representing the hyperedges of the hypergraph with a factor node connected to a node in the factor graph if and only if this node is incident to the corresponding hyperedge in the hypergraph.
• Figure 2: A multiplex hypergraph (panel a) sun2021higherferraz2021phase is a non-uniform (i.e. containing hyperedges with different cardinalities) hypergraph respresented by a multiplex network in which each layer $m$ only accounts for the hyperedge of cadinality $m$ in the hypergraph. In panel a the multiplex hypergraph has two layers $m=2$ and $m=3$. A multiplex hypergraph can be mapped to a multiplex factor graph (panel b) in which each layer is formed by the factor graph corresponding to the hypergraph in corresponding layer of the multiplex hypergraph. When each layer of the hypergraph is drawn from a random hypergraph ensemble, the multiplex hypergraph in general will be correlated. Similarly the corresponding multiplex factor graph will also be correlated.
• Figure 3: Factor graph node percolation and hypergraph node percolation and hyperedge percolation for a factor graph and the corresponding hypergraph with $N=10^4$ nodes and $M=10^4$ factor nodes (hyperedges) having the Poisson degree distribution $P(q)$ with $\langle{q}\rangle=4$ for nodes and $Q(m)=\delta(m,4)$ for factor nodes. The symbols indicate results of the Monte Carlo simulations of the percolation processes and the solid lines indicate results obtained with the corresponding message-passing algorithms. Hyperedge percolation on a hypergraph coincides with factor node percolation on the corresponding factor graph. The percolation threshold for factor graph percolation (and hyperedge percolation) is $p_c=1/12=0.0833\ldots$, Eq. (\ref{['50b']}), while for hypergraph node percolation, it is $p_c=12^{-1/3}=0.437\ldots$, Eq. (\ref{['100']}).
• Figure 4: Factor graph node percolation and hypergraph node percolation and hyperedge percolation for a factor graph and the corresponding hypergraph on the US Senate Commitee hypergraph chodrow2021generativestewart2008congressional. The symbols indicate results of the Monte Carlo simulations of the percolation processes and the solid lines indicate results obtained with the corresponding message-passing algorithms. The dataset has $N=1290$ nodes and $M=341$ hyperedges; the average degree of nodes is $\langle{q}\rangle=9.2$, the average cardinality of the hyperedges is $\langle{m}\rangle=34.8$, and their maximum cardinality is $m_{\max}=82$.