Aug 2026· Chaos· Vol 36 8· 0 citations· 31 references
Medicine
Abstract
The analysis of random walks on networks often relies on global quantities that average over nodes, thereby masking local differences in diffusion speed. This study introduces a vertex-level quantity Hi, defined as the finite-window fitted scaling exponent of the mean squared resistance distance ⟨Ωi2(t)⟩∼Cit2Hi from a given node i. We found nodes with Hi values below 0.5 (echo effect) and above 0.5 (catapult effect). The exponent is computed exactly via matrix powers of the transition matrix. We systematically evaluate Hi on several synthetic network families, generalized Sierpiński graphs, Newman-Watts small-world networks, and a custom grid-path-complete graph, and on two real-world networks (international E-road network and western U.S. power grid). We found nodes with Hi values less than 0.5 (subdiffusive regime) and greater than 0.5 (apparent superdiffusion) in both model networks and real-world networks. Analysis of model networks shows that when a node has an echo effect, its Hi value is less than 0.5, whereas when it has a catapult effect, its Hi value is greater than 0.5. In the two real networks, most nodes are in the subdiffusive regime and the overall heterogeneity of the local diffusion exponents is low, as indicated by Rényi indices of 0.0835 (E-road network) and 0.0555 (power grid). Comparisons with classical centrality measures indicate that Hi provides information not captured by those measures. The local diffusion exponent offers a vertex-level, dynamics-based tool for identifying structural bottlenecks and node roles, complementing global network characterizations.
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.
Functional laws of large numbers (FLLNs) describe the mean-field trajectory of epidemics on networks, but say nothing about the fluctuations around it. These fluctuations are governed by moments of the degree distribution not relevant at the level of the mean. A rigorous functional central limit theorem (FCLT) exists for the susceptible--infected ($SI$) process on configuration-model graphs, but no analogue exists for $SIS$, where recovery reintroduces vertices into the susceptible pool with partially known neighborhoods, breaking the clean neighborhood distribution the $SI$ derivation relies on. We develop a tractable variance approximation for Markovian $SIS$ on configuration-model graphs, combining Gleeson's approximate master equation (AME) framework with a van Kampen system-size expansion in the spirit of the $SI$ FCLT. We derive a closed drift and diffusion matrix for a reduced susceptible/$SI$-edge/$SS$-edge count vector and obtain the time-dependent covariance via the associated Langevin/Lyapunov equation. Validation against Gillespie simulation across Poisson, regular, and power-law networks shows close agreement, with deviations near the epidemic threshold and in strongly heterogeneous networks.
Lucija Nora Farkavs, Sebastian Morel Balbi, Hrvoje vStefanvci'c et al.· 0 citations
Cluster structures are widespread in complex networks, yet conventional network-level measures do not distinguish the accessibility provided inside a cluster from that provided through its external links. This study asks how these two topological contributions can be measured consistently and how rapidly they deteriorate under different node-removal mechanisms. We define internal and external cluster efficiency by combining edge volume, relative cluster size, and a harmonic shortest-path factor based on the mean reciprocal distance rather than the reciprocal of an arithmetic mean distance. The measures are evaluated together with a Cluster Robustness Index (CRI) and Structural Resilience Entropy (SRE) on Barabási–Albert and Lancichinetti–Fortunato–Radicchi networks and five empirical network topologies under six attack strategies. The results show that smaller clusters are generally more vulnerable to attacks on central nodes, whereas larger and less centralized clusters retain more topological efficiency. A source-level audit also shows that replacing the reciprocal of the arithmetic mean distance by the arithmetic mean of reciprocal distances changes most normalized trends only slightly; the clearest quantitative changes occur in the weighted-average efficiency panels. Intra-cluster edge addition increases internal efficiency by 1–5% and CRI-based robustness by up to 11.9%, while cross-cluster edge rewiring improves external efficiency by up to 2.45% and CRI-based robustness by up to 2.33% under the tested targeted attacks. The reported quantities are structural proxies derived from unweighted network topology; they do not represent observed flow, transmission speed, recovery dynamics, or domain-specific functionality. The framework therefore supports cluster-level vulnerability diagnosis and topology-oriented reinforcement without making claims beyond the available network data.
Random walks with long-range jumps can drive superdiffusive transport, replacing ordinary diffusion with an effective long-range kinetic operator. Such superdiffusive kinetics is also central to critical phenomena, notably the self-avoiding walk with long-range jump statistics, or L\'evy-SAW. This work investigates how the critical behavior is affected when the long-range connectivity itself becomes random. We study self-avoiding walks (SAWs) on a one-dimensional long-range random ring graph, where bonds are independently generated with Bernoulli probability $\sim|i-j|^{-(1+\sigma)}$. We term this walk Sparse-SAW. The same random bonds are responsible for both long-range superdiffusive transport and quenched disorder, with both simultaneously controlled by the single parameter $\sigma$, placing the problem beyond the conventional Harris and Weinrib-Halperin frameworks. Through large-scale Monte Carlo simulations and a Gaussian-truncated field theory, we show that Sparse-SAW belongs to the same universality class as the clean superdiffusive L\'evy-SAW. The random bonds generate short-range uncorrelated and long-range correlated mass disorder while simultaneously producing the long-range kinetic operator. Under coarse-graining, the latter dominates, restoring the clean critical behavior. Our study suggests that the full non-Gaussian Bernoulli statistics may lead to disorder physics beyond the conventional theory of quenched disorder, while establishing random graphs as an efficient platform for extracting the critical exponents of the clean superdiffusive L\'evy-SAW universality class.
Adding links generally improves classical transport by increasing the number of available paths. We show that coherent quantum transport can display the opposite behavior. Using continuous-time quantum walks on a percolated hierarchical small-world network, we identify a coherent overconnectivity penalty: root-to-boundary transport is maximized at intermediate bond probability and decreases as the network approaches full connectivity. The effect is quantified by the final-layer limiting probability $\chi_N$ and by the penalty $P_Q=1-\chi_N(p=1)/\max_p\chi_N(p)$, which measures the loss caused by making the architecture fully connected. The optimum results from a competition between shortcut-assisted spreading and interference-induced intra-layer recirculation. Spectral analysis shows that bond dilution creates motif-induced degeneracies and reorganizes the eigenstates connecting the root to the outermost layer. A comparison with dephased and classical transport shows that the non-monotonic landscape is not a purely geometrical percolation effect, but a coherent architecture-dependent phenomenon. These results provide a design principle for coherent transport in disordered photonic and quantum-network architectures.
Miquéias J. Cirino, Marcos César de Oliveira· 0 citations
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.