This work shows that a dual-threshold bootstrap percolation model on random hypergraphs separates a connected active backbone from large-scale endogenous activation, providing a basis for predicting cascade risk and designing targeted node- and group-level interventions in complex systems.
Abstract
Collective behaviors such as infrastructure failures and social adoption propagate through groups governed by fixed quorum requirements rather than by pairwise contacts. However, it remains unclear how absolute group- and individual-level thresholds jointly shape cascades in higher-order networks. Here we show that a dual-threshold bootstrap percolation model on random hypergraphs separates a connected active backbone from large-scale endogenous activation. A seed-driven giant component emerges continuously at a structural percolation threshold, whereas macroscopic amplification ignites only at a higher dynamical tipping point, creating a metastable safety margin in which communication is possible without systemic outbreak. This decoupling reflects an asymmetric division of labor: the group quorum
M
acts as a source-side filter, while the individual barrier
K
serves as a receiver-side gatekeeper. On homogeneous substrates, the structural onset is protected to leading order against
K
, but heterogeneous connectivity erodes this protection. Our framework provides a basis for predicting cascade risk and designing targeted node- and group-level interventions in complex systems.
A statistical filter that benchmarks node-level bipartite clustering against degree-preserving randomizations to classify nodes as geometric (signal) or degree constrained noise is introduced, offering a simple, scalable way to disentangle structure from noise in bipartite networks.
L. Ramirez, Roya Aliakbarisani, M. Serrano et al.· Nature Communications· 0 citations
Predicting the percolation threshold of highly clustered networks from local statistics remains difficult, because short loops break the independence assumption underlying tree-like message passing. Existing remedies address loopy connectivity either through prescribed local motifs in random-graph ensembles or through a single network's realized topology, leaving an ensemble-level treatment of arbitrary connectivity patterns absent. Here, we develop a loopy message-passing framework for random clustered graph ensembles based on generalized-edge statistics, which characterize overlap patterns among the neighborhoods of different nodes. This yields a progressively refined approximation scheme based on neighborhoods of increasing size around each node. The low-order approximations recover previous equations for random network ensembles, and the new result that yields refined threshold prediction is developed by the second-order approximation. We show that the effectiveness of this framework depends not only on short-cycle density but also on the internal consistency of generalized edges. To diagnose this effectiveness, we introduce the generalized-edge closure coefficient (GECC) to quantify this consistency. Because GECC is computed entirely from local statistics and does not rely on any percolation calculation, it serves as an a priori diagnostic for the reliability of the approximation. Using synthetic and real networks, the threshold is evaluated via the second-order and lower-order approximations. Comparisons with Monte Carlo simulations show that GECC captures key structural features that strongly affect the percolation threshold. These results establish ensemble-based loopy message passing as an efficient route for predicting the percolation threshold in large clustered networks.
The study of epidemic spreading in complex networks is fundamental to understanding diffusion processes across natural and social systems. While traditional graphs capture only pairwise interactions, many real-world processes involve higher-order group interactions that can be naturally represented by hypergraphs. In this work, we propose a community-based preferential attachment hypergraph model with tunable modularity and a heavy-tailed degree distribution, reproducing key structural properties in real systems. Based on this model, we develop a hypergraph-based SAIR framework to describe epidemic dynamics with asymptomatic transmission. A mean-field approximation is derived and compared with classical mean-field, heterogeneous mean-field, and Monte Carlo simulations, demonstrating improved predictive accuracy for community hypergraphs. The results show that epidemic spreading is regulated by community structure, transmission probability, and initial conditions, giving rise to localized, heterogeneous, and global diffusion regimes. By introducing a cross-community hyperedge index, we reveal that community structure suppresses spreading primarily through the reduction in inter-community transmission pathways. These factors collectively determine the spreading radius, propagation speed, and epidemic peak. We further evaluate behavioral, hyperedge-based, and node-based intervention strategies. Overall, this study provides a quantitative framework for analyzing epidemic spreading and control on community-structured hypergraphs, with potential relevance to studies of information diffusion and risk propagation.
Communication allows large language model agents to pool evidence, but it also creates paths along which an erroneous claim can spread. We formulate a correction-aware network model that tracks susceptible, exposed, infectious, and corrected agents and derive its early-invasion condition for heterogeneous communication networks. We then couple this propagation model to an analytic majority-vote benchmark in which a clean-task reliability target imposes a minimum connectivity requirement. Under fixed exposure per communication edge, reliability and error control impose opposing graph constraints. We characterize when their intersection is empty and when it contains an intermediate connectivity range, and identify regular graphs that attain the smallest invasion factor within the reliable graph class when such graphs exist. Under a fixed sender budget, the homogeneous first-order threshold is independent of network density, showing that the communication-budget convention determines whether added edges increase early propagation risk. Finite-network simulations on 21,000 trajectories illustrate these directional predictions. A controlled grok-4.3 experiment then evaluates three six-node topologies on 36 new closed-world tasks, with a balanced 12-task subset continued to full cascades. Mean first-generation offspring increased from 0.667 to 1.333 and 1.667 as degree increased from 2 to 4 and 5, while the adoption fraction among exposed neighbours remained 0.333. Mean non-seed erroneous adoption in the full-cascade subset was 0.200, 0.333, and 0.333. Together, these results provide a tractable basis for selecting connectivity under explicit reliability and propagation constraints.
We study how physical mobility, algorithmic exposure, and repulsive social influence interact in a stochastic multiplex model of opinion dynamics. Agents diffuse in physical space while a directed digital network rewires under a conserved attention budget, so digital exposure displaces rather than supplements local interaction. With purely assimilative bounded-confidence influence, opinion-blind long-range exposure reduces locality-induced fragmentation whereas homophilic recommendation preserves echo chambers. When a contested repulsive response to sufficiently distant opinions is activated, this ordering reverses at the reference parameters: a neutral platform reaches the maximal polarization permitted by the bounded opinion space, controversy-seeking curation drives faster initial separation but slows sharply near the boundary, and homophilic curation delays radicalization by suppressing cross-bloc exposure. In a late-stage symmetric two-bloc reduction, any curation kernel maps to a state-dependent cross-bloc exposure profile $p(y)$ and an exact quadrature for the radicalization time. Pointwise-ordered profiles inherit a global kinetic ordering; crossing profiles yield target- and horizon-dependent rankings. For similarity-driven curation the quadrature has a closed form involving the exponential integral. Simulations, finite-size scans to $N=1600$, structural controls, and a well-mixed particle comparison support the mechanism. Heavy-tailed influence strengths are not required for the inversion; in the well-mixed heavy-tail regime they additionally produce a non-self-averaging stable-weighted asymptotic description. Finally, opinion-independent Brownian mobility produces no detectable geographic opinion structure in the explored regime, whereas opinion-dependent drift produces spatial domains through a P\'eclet-controlled crossover near $\chi\ell/D \sim 1$.
Complex contagion models, in which adoption requires reinforcement from multiple neighbors, have been extensively studied in the monotone (no-recovery) setting, but the phase diagram of threshold models with SIS-like recovery on networks remains unmapped. We study a stochastic Watts-threshold SIS model on Erdos-Renyi and Barabasi-Albert networks and reconstruct its extinction-persistence phase boundary in the joint parameter space of transmission rate $\beta$, adoption threshold $\theta$, and infectious duration $d$. Using adaptive Delaunay-based sampling and weighted logistic regression on over 180,000 Monte Carlo trials, we find that: (i) the boundary is well described by a six-parameter interaction model whose structure is invariant across both topologies; (ii) the transition is sharp, with the 10-90\% extinction-probability band spanning only $\Delta\theta \approx 0.005$-$0.008$; and (iii) the adoption threshold is the dominant parameter governing epidemic feasibility, with transmission rate and infectious duration playing secondary and asymmetric roles. The characterization provides a quantitative reference for the complex-contagion analogue of the classical SIS epidemic threshold.