Higher-order percolation processes on multiplex hypergraphs

Higher-order interactions are increasingly recognized as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraphs as well as simplicial complexes capture the higher-order interactions of complex systems and allow us to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensemble where each layer is characterized by hyperedges of given cardinality. We observe that in presence of the structural cutoff the ensemble of multiplex hypergraphs can be mapped to an ensemble of multiplex bipartite networks. We reveal the relation between higher-order percolation processes in random multiplex hypergraphs, interdependent percolation of multiplex networks, and $K$-core percolation. The structural correlations of the random multiplex hypergraphs are shown to have a significant effect on their percolation properties. The wide range of critical behaviors observed for higher-order percolation processes on multiplex hypergraphs elucidates the mechanisms responsible for the emergence of discontinuous transition and uncovers interesting critical properties which can be applied to the study of epidemic spreading and contagion processes on higher-order networks.

Figure 1

A schematic representation of the multiplex network construction of the hypergraph with given generalized hyperdegree sequences for hyperedges of cardinality $m_1=2$ (layer 1) and $m_2=3$ (layer 2). First a configuration model is used to generate a simple network capturing the two-body interactions of the hypergraph (a). Second, the configuration model of simplicial complexes [4] is used to generate a pure simplicial complex formed exclusively by triangles. Only the information about the three-body interactions is retained (b). Finally, the information of the different layers is aggregated to generate the desired hypergraph including hyperedges of size $m=2$ and $m=3$ (c). This construction can be generalized to an arbitrary number of layers. The factor graph representation of the mulitiplex hypergraph is shown in panels (d), (e), and (f).

Figure 2

An schematic illustration of Eqs. (14) for $\widehat{S}$ and $S$ are shown in panels (a) and (b), respectively. Black circles represent nodes; triangles, squares, and hexagons represent factor nodes (hyperedges) with different cardinality.

Figure 3

The fraction of nodes in the giant component $R$ is shown versus $p^{\left[H\right]}=p$ for random hypergraphs. The hyperdegree distribution $P\left(k\right)$ and distribution of cardinality of hyperedges $\widehat{P}\left(m\right)$ are Poisson distribution, with different expectation $\langle m\rangle$ and $\langle k\rangle$.

Figure 4

A schematic illustration of Eqs. (21) for ${\widehat{S}}_m$ and $S_m$ are shown in panels (a) and (b), respectively. Red circles represent nodes; squares, pentagons, and hexagons represent factor nodes (hyperedges) with different cardinality.

Figure 5

The fraction $R$ of nodes in the giant component for MPCMH (Positive correlations), for the UMH (Uncorrelated) and for MNCMH (Negative correlations) is shown for hyperedge percolation (a) and for node percolation (b). The considered duplex hypergraph has $N={10}^4$ nodes and hyperedges of cardinality $m_1=2$ (layer 1) and $m_2=3$ (layer 2). The generalized hyperdegree distributions are Poisson with $z_2=0.5$ (for layer 1), $z_3=1.5$ (for layer 2).

Figure 6

Schematic representation of the equations for ${\widehat{S}}_m$ and for $S_m$ determining higher-order percolation models defined on multiplex hypergraphs. Panel (a) represents node-interdependent percolation. Panel (b) represents hyperedge-interdependent percolation. Panel (c) represents Node $K$-core percolation. Panel (d) represents hyperedge $K$-core percolation.

Figure 7

The fraction $R$ of active nodes in interdependent node percolation is shown versus $p^{\left[H\right]}$ for a duplex multiplex hypergraph with $p^{\left[N\right]}=1$. The layers of the duplex networks are formed by hyperedges of cardinality $m_1=3$ (layer 1), and $m_2=4$ (layer 2). Both layers have Poisson generalized degree ditribution with $z_3=z_4=2.5$. The inset displays the function $h\left(S\right)$ defined in Eq. (53) calculated at the critical point, i.e., for $p^{\left[H\right]}=p_c^{\left[H\right]}$.

Figure 8

The fraction of active nodes $R$ for interdependent node percolation is plotted versus $p^{\left[H\right]}$ when $p^{\left[N\right]}=1$ (a) and versus $p^{\left[N\right]}$ when $p^{\left[H\right]}=1$ (b) for a MPCMH (Positive correlations) a MNCMH (Negative correlations) and for a UMH (Uncorrelated). The layers of the duplex hypergraph are formed by hyperedges of cardinality $m_1=3$ (layer 1), $m_2=4$ (layer 2), with Poisson layers of average generalized degree $z_3=2.5$, $z_4=2.5$.

Figure 9

The percolation threshold $p_c=p_c^{\left[H\right]}$ of a duplex multiplex hypergraph is plotted versus $r$ for the interdependent node percolation process with partial interdependence. Solid line correspond to the line of continuous critical point, the dashed line corresponds to the line of discontinuous, hybrid transitions. The tricritcal point separating the two lines is obtained for $r=r_T=0.68...$. The inset displays the value $R=R_c$ of the fraction of active nodes at the critical point as a function of $r$ showing that $R_c>0$ for $r>r_T$ indicating that the transition is discontinuous. The layers of the duplex hypergraph are formed by hyperedges of cardinality $m_1=3$ (layer 1), $m_2=4$ (layer 2), with Poisson layers of average generalized degree $z_3=2$, $z_4=2$. Here $p^{\left[N\right]}$ is set equal to one, i.e., $p^{\left[N\right]}=1$.

Figure 10

The critical behavior of the interdependent hyperedge percolation process on a duplex hypergraph is investigated by plotting the function $h\left(S\right)$ defined in Eq. (52) versus $S$ (a, c) and by displaying the the fraction of active nodes $R$ for different values of $p=p^{\left[H\right]}$ (d, f). The duplex hypergraphs have layers with hyperedge cardinalities $m_1=2$, $m_2=3$ (a, d), $m_1=2$, $m_2=10$ (b, e), and $m_1=3$, $m_2=5$ (c, f). Each layer is characterized by Poisson hyperdegree distributions with average degree $z_{m_1}$ (layer 1) and $z_{m_2}$ (layer 2) with $z_{m_1}+z_{m_2}=z=6$. In panel (d) we observe a continuous transitions and a discontinuous transitions occurring for different values of $z_2$. In panel (e) we observe that the model can display, for the same value of $z_3$, two critical points $p_{c1}$ and $p_{c2}$ corresponding to a continuous and discontinuous transition occurring at a nonzero value of the order parameter. In panel (f) we show that all the transitions are discontinuous.

Figure 11

The fraction of active nodes $R$ in the interdependent hyperedge percolation is plotted versus $p^{\left[H\right]}$ when $p^{\left[N\right]}=1$ (a) and versus $p^{\left[N\right]}$ when $p^{\left[H\right]}=1$ (b) for a MPCMH (Positive correlations) a MNCMH (Negative correlations) and for a UMH (Uncorrelated). The layers of the duplex hypergraph are formed by hyperedges of cardinality $m_1=2$ (layer 1), $m_2=3$ (layer 2), with Poisson layers of average generalized degree $z_2=4.8$, $z_3=1.2$ and $z=z_2+z_3=6$.

Figure 12

The fraction $R$ of active nodes for the node $K$-core percolation on duplex hypergraphs with independent Poisson layers is shown versus the probability of retaining a hyperedge $p^{\left[H\right]}=p$. The duplex hypergraph includes $N={10}^4$ nodes and has layers formed by hyperedges of cadinality $m_1=4$ and $m_2=5$ with independent Poisson generalized hyperdegree distributions with average $z_4=z_5=2$. Here $p^{\left[N\right]}$ is fixed to the constant value $p^{\left[N\right]}=1$. The node $K$-core percolation is discontinuous for $K>2$.

Figure 13

The fraction $R$ of active nodes for the hyperedge $K$-core percolation on duplex hypergraphs with independent Poisson layers is shown versus the probability of retaining a hyperedge $p^{\left[H\right]}=p$. The duplex hypergraph includes $N={10}^4$ nodes and has layers formed by hyperedges of cadinality $m_1=4$ and $m_2=5$ with independent Poisson generalized hyperdegree distributions with average $z_4=z_5=2$. Here $p^{\left[N\right]}$ is fixed to the constant value $p^{\left[N\right]}=1$. The transition is discontinuous for $K>2$.