Jun 2026· Nature Communications· Vol 17· 0 citations· 66 references
Medicine
TL;DR
A unified statistical physics framework based on the random cluster model is introduced that encompasses classical notions of connectivity and defines a continuous family of new connectivity measures, offering a powerful tool to analyze structure and dynamics in complex networks.
Abstract
Connectivity is a fundamental concept in network science, characterizing how interactions propagate through indirect pathways. While numerous connectivity metrics exist, such as shortest paths, effective resistance and minimum cut, each highlighting distinct structural features, their relationships remain largely fragmented. Here we show that these classical notions arise as limiting cases of a unified statistical-physics framework based on the random cluster (RC) model, which interprets connectivity as a principled synthesis of series and parallel transmission. By tuning its parameters, the RC model not only recovers classical connectivity measures but also extrapolates into unexplored regimes, leading to emergent notions of connectivity which yield practical tools for network learning tasks. In particular, RC connectivity naturally encodes the kinetics of growing paths, enhancing learning performance in dynamical settings such as epidemic spreading and neurodynamics. By linking structural, dynamical, and learning-based perspectives, RC connectivity establishes a general and interpretable foundation for the analysis of networked systems. Network connectivity is traditionally described by distinct measures like shortest paths or effective resistance. Here, authors introduce a unified statistical physics framework based on the random cluster model that encompasses these classical notions and defines a continuous family of new connectivity measures, offering a powerful tool to analyze structure and dynamics in complex networks.
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
In this work, we explore the concept of equilibrium measures within the framework of Schr¨odinger random walks on networks. Building on previous work, we demonstrate how these equilibrium measures can be leveraged to compute key network parameters such as the Mean First Passage Time (MFPT) and Kemeny’s constant. By expressing these parameters in terms of generalized inverses of the associated M-matrix, we provide new insights and efficient computational tools for network analysis. The results are particularly applicable to both star and path networks, where we offer explicit formulations for these fundamental quantities. Our findings highlight the importance of equilibrium measures as a powerful tool in the study of complex networks.
Á. Carmona, A. Encinas, M. J. Jiménez et al.· The Electronic Journal of Li...· 0 citations
Network control theory applied to structural connectomes typically ranks brain regions as candidate driver nodes by their structural connectivity strength, and evaluates performance through scalar control energy. We test whether this framing captures the most relevant information about how driver-node selection shapes brain network control. We introduce an alternative criterion based on the persistent topological cycles in which each node participates---a measure of mesoscale integration that captures features beyond local connectivity---and compare it to standard degree-based selection across 70 human structural connectomes at three parcellation scales. Topology- and degree-informed driver sets achieve nearly identical scalar control energy, differing by approximately 0.2%. The geometry of the controllable subspace, however, differs substantially: topology-informed sets distribute controllability across more dimensions of state space and produce better-conditioned controllability matrices. This geometric advantage is preserved when high-degree hub nodes are removed, and it carries a functional signature: because the two criteria place driver nodes in different cortical territory, each most efficiently reaches a different class of target state. The choice of node-ranking criterion therefore shapes which brain-state transitions are energetically favored even when average control cost is unchanged. The results reveal a dissociation between control cost and control geometry, and demonstrate that persistent topology captures information about brain network control that scalar energy summaries miss.
Carter Sale, Marco Coraggio, Mengsen Zhang et al.· 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.
Graph-theoretical analysis offers a principled framework for quantifying emergent dynamics in neuronal cultures. Here, we present an analytical pipeline for inferring network-level properties of in vitro cortical cultures from multichannel electrophysiological recordings. The approach builds on a recently proposed Reservoir Computing (RC) framework (Auslender et al., 2025), which enables direct extraction of an Intrinsic Connectivity Map (ICM) from neural activity. We interpret the ICM as an effective adjacency matrix and apply graph-theoretic centrality measures to quantify node- and edge-level contributions to the culture's collective dynamics. We systematically evaluate both local and global graph metrics and examine their relationships with experimentally measured activity features, including firing rates and network-level descriptors. To validate the inference procedure, we also simulate the experimental environment, enabling controlled benchmarking of the RC-derived connectivity against a known ground-truth adjacency matrix and assessment of model performance as a function of graph structure. Our results demonstrate statistically robust associations, of varying strength, between graph-theoretic measures and experimentally observed activity patterns. These findings additionally support the validity of the RC-based connectivity inference and establish a scalable, data-driven framework for functional network characterization in neuronal culture systems.
I. Auslender, G. Letti, Yasaman Heydari et al.· 0 citations
We derive a one-dimensional reduction for nonlinear dynamics on simplicial complexes containing both pairwise and triangular (higher-order) interactions. The effective state is defined using a mixed weight determined by the pairwise and triangular degrees of each node. The resulting reduced equation retains two structural coefficients, associated separately with the pairwise and higher-order coupling channels. A fluctuation expansion identifies the closure assumptions underlying the reduction and shows how deviations of individual node states from the effective state contribute to the approximation error. We numerically validate the proposed framework on Gene-regulatory dynamics, the double-well system, and SIS spreading. The states of the reduced model are compared with full-network simulations through coupling-parameter sweeps, steady-state branch calculations, and progressive node-removal experiments on synthetic and real-world networks. The reduced model successfully reproduces the principal transitions and steady-state branches in all three dynamical systems considered. Agreement is strongest for relatively homogeneous networks and deteriorates when structural heterogeneity produces a broader distribution of node states. The closure diagnostics account for this loss of accuracy and indicate when a single effective state is no longer sufficient. The reduction therefore provides a tractable description of resilience in systems with coexisting pairwise and higher-order interactions.
Amit Tiwari, C. Hens, Prosenjit Kundu· 0 citations