Cross-order induced behaviors in contagion dynamics on higher-order networks

TL;DR

The novel phenomenon of cross-order induced behaviors is uncovered, where behavioral signatures emerge at interaction orders where no direct mechanism is present, and offer new insights into the relationship between the network structure and observed dynamics of higher-order systems.

Abstract

Recent studies have shown that novel collective behaviors emerge in complex systems due to higher-order interactions. However, the way in which the structural correlations of these interactions shape such behaviors remains a significant gap in current research. To address this, we use signatures of higher-order behaviors (HOBs) to identify the underlying dynamical rules, or higher-order mechanisms (HOMs). In this work, we compare several HOB measures derived from information theory. Utilizing a simplicial SIS contagion model, we demonstrate that simpler, computationally efficient measures can serve as robust indicators of HOMs. We uncover the novel phenomenon of cross-order induced behaviors, where behavioral signatures emerge at interaction orders where no direct mechanism is present. Crucially, these cross-order HOBs are not simply induced by structural correlations -- such as nestedness and hyperedge overlap -- but they appear in the neighborhood of any HOM. Among the information-theoretic measures we tested, synergy is the most reliable indicator of the true order where the underlying mechanism is at play. These findings offer new insights into the relationship between the network structure and observed dynamics of higher-order systems.

Figures (4)

• Figure 1: Higher order behaviors in 2-simpleces. Statistical distance $\delta$ between 3-clique and 2-simplex distributions for different lower and higher order information measures: (a) sum of pairwise transfer entropies $\mathcal{T}(\vb{X})$; (b) average dynamical O-information $d\Omega(\vb{X})$; (c-d) unconditioned $\hat{\mathcal{S}}$ and conditioned $\mathcal{S}$ average synergy; (e-f) unconditioned $\hat{\mathcal{I}}$ and conditioned $\mathcal{I}$ average mutual information. Phase space is composed of the pairwise and 2-simplex rescaled infectivities $(\lambda_1,\lambda_2)$ in the simplicial SIS model. Lines in panels a and b correspond to the mean-field critical infectivity due to pairwise (dashed) and higher-order (dash-dotted) mechanisms.
• Figure 2: Behaviors at higher orders. Statistical distance $\delta$ for the average dynamical O-information $d\Omega$ (left column, a-b) and average synergy $\mathcal{S}$ (right column, c-d) distributions of $3$-hyperedges (top row, a and c) and $5$-hyperedges (bottom row, b and d) relative to their respective $(\ell+1)$-cliques. Distributions are based on separate simulations on random simplicial complexes with maximum order $L\in\{3,5\}$ and all mechanisms of intermediate order disabled, i.e. $\lambda_\ell=0$ for $2 \leq \ell < L$, effectively leaving only pairwise and order $L$ edges.
• Figure 3: Higher-order behaviors in mixed-order scenario. Statistical distance $\delta$ between clique and simplex distributions of average synergy $\mathcal{S}$ for 2-simpleces (left, a-b) and 3-simpleces (right, c-d) depending on the strength of the other's mechanism in a random simplicial complex with maximum order $L=3$. Panels a and c show the HOBs at order $\ell$ when the $\ell$-th order mechanisms are disabled (equivalent to removing $\ell$-hyperedges). Panels b and d show the HOBs when both mechanisms are active.
• Figure 4: Induced higher-order behaviours in nested hyperedges. Statistical distance $\delta$ between HOB measure distributions for nested groups and random nodes, on a random hypergraph of maximum order $L=\ell_M=2$. Rows {a,b,c,d} correspond to different orders $\ell_B\in\{2,3,4,5\}$ for the nested groups, respectively. In this case, row a corresponds to the actual order of the nested hyperedges, i.e. $\ell_B=\ell_M=2$. Columns correspond to different measures of HOB: average dynamical O-information $d\Omega$ (column 1), average synergy $\mathcal{S}$ (column 2), average mutual information $\mathcal{I}$ (column 3) and sum of pairwise transfer entropies $\mathcal{T}$ (column 4).