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.
Mean-field approximations for dynamical processes on networks are widely used, but existing derivations often rely either on moment closures or on idealised assumptions about network structure, leaving the nature of the underlying averaging unclear. Here we present a mathematically principled framework for deriving edge-based mean-field approximations for a broad class of Markov processes on networks using approximate lumping. We consider models in which each vertex is in one of a finite number of vertex states and transitions depend on the number of neighbours in each state. Our approach partitions the full Markov chain state space according to the number of vertices and edges in each possible state, and averages transition rates between partitions. This yields density-dependent population processes that, in the limit of large system size, reduce to a low-dimensional system of ordinary differential equations. We demonstrate the method on single graphs and graph ensembles, such as Erd\H{o}s-R\'enyi random networks, and show that well-known edge-based mean-field approximations arise as special cases of our approach. Our approximate lumping framework clarifies the nature of the averaging underlying mean-field approximations, providing a basis for future work on assessing their accuracy.
G. Timár, Jonathan A. Ward, Péter L. Simon· 0 citations
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
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.
Traffic-induced failures, from packet loss in communication networks to congestion breakdown in transport systems, occur when flows progressively exhaust the edges they traverse. Path percolation models this process by removing edges along sampled origin-destination paths. Existing work assumes locally tree-like networks and deterministic shortest-path routing, leaving unclear how path degeneracy and routing stochasticity affect fragmentation in the clustered networks typical of real systems. We introduce a generalised path-percolation framework where paths are drawn from a temperature-controlled routing ensemble interpolating between geodesic and noisy transport. We argue based on box-covering renormalisation and our numerical experiments that, for any finite routing horizon $C$, the process coarse-grains to ordinary mean-field percolation. Routing details affect non-universal quantities, especially the percolation threshold $p_c$, through the entropy of the load distribution and the capacity of finite clusters to accommodate flow. Load entropy therefore acts as a robustness measure for networks under path-based failures. When the routing horizon is tuned to the mean-field correlation length, $C=N^{1/3}$, within a source-uniform ensemble, the system enters a crossover regime with scaling exponents distinct from shortest-path percolation with infinite budget. In this regime, path elongation becomes decoupled in time from structural fragmentation: the characteristic path length reaches a growing maximum, associated with routing temperature, asymptotically ahead of the collapse of the giant component. These results clarify how microscopic routing organisation shapes macroscopic resilience, and identify path elongation as a measurable precursor of failure in communication and transport infrastructure.
Yunhao Ding, Andreas Münch, R. Lambiotte· 0 citations
The joint asymptotic distribution of any finite collection of network moments in random graphs sampled from a graphon, which includes both the nondegenerate case as well as the degenerate case, provides the higher-order fluctuation theory for subgraph counts in the graphon model.
Anirban Chatterjee, S. Dan, B. Bhattacharya· Annals of Statistics· 0 citations