We introduce a discrete curvature measure for graphs based on the Cayley-Menger determinant, which we term Cayley-Menger curvature ($ \symbf{\kappa_{ extbf{CM}}} $). For each node, we construct local tetrahedra from its nearest neighbors under the effective resistance metric, compute the Cayley-Menger determinant to embed them in Euclidean space, and extract the circumradius as a proxy for local curvature. Unlike $ \symbf{\kappa_{ extbf{FR}}} $ and $ \symbf{\kappa_{ extbf{OR}}} $, which assign signed curvature values, $ \symbf{\kappa_{ extbf{CM}}} $ is strictly positive by construction and is best interpreted as a measure of local geometric compactness with curvature-like behavior, complementary to-rather than a replacement for-signed discrete curvatures. We validate $ \symbf{\kappa_{ extbf{CM}}} $ on graphs with analytically known geometric properties-lattices, trees, complete graphs, and cycles-and demonstrate that it produces curvature values consistent with expected flat, hyperbolic, and spherical geometries. Comparison with Forman-Ricci ($ \symbf{\kappa_{ extbf{FR}}} $) and Ollivier-Ricci ($ \symbf{\kappa_{ extbf{OR}}} $) curvatures across a battery of benchmark networks reveals that $ \symbf{\kappa_{ extbf{CM}}} $ captures geometric information that is largely independent of these established measures (Spearman $ \symbf{| ho|\leq 0.47} $ in most cases, while $ \symbf{\kappa_{ extbf{FR}}} $ and $ \symbf{\kappa_{ extbf{OR}}} $ correlate at $ \symbf{ ho \gt 0.7} $). As a demonstration of practical utility, we show that $ \symbf{\kappa_{ extbf{CM}}} $ alone outperforms both $ \symbf{\kappa_{ extbf{FR}}} $ and $ \symbf{\kappa_{ extbf{OR}}} $ as a feature for community detection in stochastic block models, achieving an order-of-magnitude improvement in Adjusted Rand Index over $ \symbf{\kappa_{ extbf{FR}}} $ and $ \symbf{\kappa_{ extbf{OR}}} $ used alone, though we note that absolute ARI values remain moderate and the baseline clustering method (Ward linkage) was chosen for simplicity rather than performance; the result demonstrates discriminative power of the $ \symbf{\kappa_{ extbf{CM}}} $ feature, not a claim about state-of-the-art community detection. Scalability benchmarks confirm that $ \symbf{\kappa_{ extbf{CM}}} $ computes faster than $ \symbf{\kappa_{ extbf{OR}}} $ for graphs up to $ \symbf{n\,=\,5000} $ nodes. These results establish the Cayley-Menger curvature as a complementary tool in the discrete curvature toolkit, offering direct geometric interpretability grounded in distance geometry.
Abstract We study a discrete-time, infinite-horizon process of partition formation on a fixed network. We introduce a particular cost function for maintaining connections within a network, allowing the cost of a link between two players to depend on the composition of each player’s neighborhood. Under the cost function, the two players connected by a link may bear asymmetric costs. The stable partitions that arise under the best-response dynamics are identified and analyzed. We characterize stable partitions when players interact on a star, complete, and balanced and complete bipartite networks. Particular attention is paid to the qualitative differences between stable partitions. We also highlight the substantial influence of the initial partition and order in which players are chosen to update on the limiting partitions obtained under best-response dynamics, and we contrast these results with those obtained under the Nash equilibrium concept. Finally, we apply the model to Zachary’s karate club network, providing an explanation for why two specific individuals can be viewed as group leaders. Numerical simulations on Erdős–Rényi random networks further document how stable partitions vary with network density, cost parameters, and initial partition.
Abstract Identifying influential nodes in complex networks is a fundamental problem in network analysis, with wide-ranging applications in epidemic control and information diffusion. Shortest path-based centrality metrics play a crucial role in analyzing such networks. Although these metrics provide valuable insights, they are often limited to single-dimensional representations and are not capable of capturing the complexity of real-world networks characterized by multicriteria features. In addition, many existing approaches rely on fixed weighting schemes to represent multidimensional aspects, which do not necessarily reflect the true importance of different features. In this study, we propose the Bicriteria Entropy-based Shortest Path Centrality (BESP-C) to identify influential nodes for the analysis of multidimensional networks. BESP-C integrates multiple shortest path-based features, including the number of shortest paths, and average shortest paths, within a bicriteria framework in which each edge is associated with two distinct attributes. To effectively balance these heterogeneous features, information entropy is employed to adaptively determine their weights, resulting in a more accurate and data-driven evaluation of node importance. Furthermore, BESP-C adopts a semi-local perspective, enabling it to simultaneously capture both local and nonlocal interactions while maintaining computational efficiency. The effectiveness of the proposed metric is evaluated on a set of real-world networks using the susceptible–infected–recovered model and Kendall’s tau correlation coefficient. The results demonstrate that BESP-C outperforms existing advanced centrality metrics in terms of both performance and scalability in identifying influential nodes.
Abstract Economic complexity analysis has shown that countries diversify through a structured product space in which productive capabilities condition future development paths. This paper extends that approach by studying the geometry and dynamics of the product-space network. We document three empirical regularities. First, multicomponent-aware spectral diagnostics reveal recurrent episodes of structural compression around major global shocks, suggesting periodic critical states in the diversification network. Second, Forman–Ricci curvature is uniformly negative across the benchmark cross-sections examined, with more than $ \boldsymbol{99.9\%} $ of retained edges exhibiting strictly negative curvature. This pattern is stable across aggregation checks and reveals a hyperbolic-like geometry not captured by conventional symmetric relatedness measures. Third, three-year-ahead changes in the Economic Complexity Index (ECI) display a power-law tail in structurally critical states, with exponent $ \boldsymbol{\alpha\approx 2.89} $ and a log-likelihood ratio of 57.3 favoring the power law over a lognormal alternative. We introduce the Sandpile Diversification Index (SDI), a country-year measure of structural fragility combining export concentration, local Forman–Ricci curvature exposure, and a global spectral-network modulator. The SDI is informative beyond ECI (Spearman $ \boldsymbol{\rho=-0.14} $), orthogonal to development level ($ \boldsymbol{\rho=-0.05} $ with log GDP per capita), robust to HS-2 aggregation checks ($ \boldsymbol{\rho = 0.81} $), and stable after excluding extractive economies and microstates ($ \boldsymbol{P\,=\,1.0\mathrm{e}{-58}} $). Cross-sectional regressions show that SDI remains significant after controlling for ECI, log GDP per capita, openness, and R&D intensity ($ \boldsymbol{\beta = 0.113} $, $ \boldsymbol{P\,=\,1.7}\textbf{e}\boldsymbol{-07} $). The SDI therefore provides a second coordinate for economic complexity analysis: ECI measures productive sophistication, while SDI measures structural fragility in country-level capability portfolios.
Abstract We study phase allocation in networks with an assumed cyclic timing frame. Each node chooses where in a cycle of length $ P $ to update; directed edges reward useful phase offsets, and a crowding term penalizes excessive use of the same phase. The model therefore does not explain the origin of the cycle itself. It asks how graph structure shapes the stable allocation of nodes across phases once such a frame is available. We show that the resulting finite timing game converges under asynchronous local improvement moves. Its equilibria are not a single kind of clock: they include diffuse timing, phase clustering, hub- or core-anchored timing, and distributed or defected scaffolds. We analyze these outcomes through circular phase modes, normalized phase entropy, and degree-weighted timing fields. In simulations across synthetic graph families, high-degree momentum terms provide compact diagnostics of the converged phase-entropy organization and improve within-ensemble prediction relative to mean degree alone. Within the controlled graph ensembles studied here, these quantities diagnose phase-regime organization; they should not be read as universal graph-family-independent predictors. Degree-weighted order parameters, interpreted against frozen-random baselines, distinguish learned phase organization from structural concentration artifacts in highly heterogeneous graphs. Additional local-bandit experiments show that sampled decentralized learners can recover similar regime-level outcomes without evaluating all phases at every update.
