percolation theory

This work addresses the degradation of belief propagation (BP) accuracy on loopy networks. We propose embedding non-iterative belief (NIB) message passing into the generalized belief propagation framework to systematically improve accuracy in two fundamental tasks: percolation threshold prediction and sparse matrix spectral estimation. Unlike classical Kikuchi cluster-based methods (e.g., KCN), NIB avoids loop-induced cyclic dependencies, thereby enhancing inference stability and convergence—particularly on high-loop-density graphs. Experiments on Erdős–Rényi (ER), Barabási–Albert (BA), and real-world networks demonstrate over 30% reduction in percolation critical-point estimation error. Moreover, NIB achieves superior reconstruction of sparse matrix eigenvalue distributions, especially near spectral edges. This study extends the applicability of message-passing algorithms to interdisciplinary problems at the interface of statistical physics and spectral graph theory, establishing a new paradigm for efficient approximate inference on complex networks.

This study investigates the connected component structure of stochastic block models (SBMs) and their degree-corrected variants under node and bond percolation, with a focus on the size of the giant component, the distribution of finite components, and percolation thresholds. Leveraging the probability generating function formalism, the work establishes—for the first time—a rigorous analytical framework for percolation on degree-corrected SBMs, yielding closed-form expressions for the giant component size and the mean cluster size. It further uncovers a precise mapping between the generating functions of microcanonical and canonical SBMs, enabling efficient extension of the results to canonical ensembles. Theoretical predictions are shown to be highly accurate and broadly applicable, significantly expanding the scope of percolation theory in complex networks.

This work addresses the computational complexity of Monte Carlo algorithms for estimating percolation thresholds on two-dimensional (and higher-dimensional) lattices, focusing on the efficiency of critical-point identification. Method: We introduce a novel average-case paradigm for threshold characterization and rigorously derive tight asymptotic bounds—optimal, average, and worst-case—for time complexity in one and two dimensions. Our approach integrates percolation cluster connectivity testing, asymptotic analysis, and large-scale Monte Carlo simulation. Contribution/Results: We establish precise quantitative relationships among runtime, iteration cost, and threshold estimation accuracy. Our bounds are provably optimal for 1D and 2D lattices. For dimensions ≥3, while rigorous proofs remain open, extensive empirical evidence strongly indicates that computation time grows dominantly with system size. This study provides the first systematic theoretical complexity framework for percolation simulation, enabling principled algorithm design and scalability analysis.

Whether “counting” (estimating the number and sizes of communities) is computationally easier than “locating” (exactly recovering community assignments) in community detection. Method: We establish the first computational lower bound for the planted–planted hypothesis testing problem, leveraging the low-degree polynomial framework, statistical-to-computational phase transition analysis, and classical hypothesis testing theory. Results: We rigorously prove that community counting and exact community recovery are computationally equivalent—refuting the conjecture that counting is inherently easier than locating. This constitutes the first rigorous evidence of computational intractability for distinguishing between multiple planted distribution models. The result reveals the intrinsic hardness of counting in community structure inference and provides foundational insights into computational limits in statistical learning and graphical models.

This paper addresses the problem of computing $(1+varepsilon)$-approximate shortest $s$-$t$ distances in graphs with dynamically weighted edges, where edge weights are unknown base values multiplied by random factors and querying edge weights is costly. The authors propose an active learning–based querying strategy that adaptively selects edges to query. They establish, for the first time, a theoretical upper bound on query complexity for continuous doubling-dimension graphs, proving that only $left(frac{ ho log n}{varepsilon} ight)^{O(1)}$ edge queries suffice to guarantee approximation accuracy. By integrating classical first-passage percolation theory with metric geometry and probabilistic analysis, they extend the model to settings with correlated edge-weight dependencies. Experiments demonstrate that the proposed strategy significantly improves distance estimation accuracy and outperforms baseline methods in query efficiency. The core contribution lies in theoretically characterizing the minimal information requirement—i.e., the fewest edge queries—for approximating shortest paths under uncertainty, and providing a practical, provably efficient active querying framework.

This paper addresses a fundamental problem in network science: quantitatively assessing whether an empirical network is typical of a given generative model. To this end, we propose a typicity criterion grounded in the theory of regular ensembles, which measures the alignment between a network and a model via the “most typical state”—a deterministic proxy for the ensemble’s dominant configuration. Our key methodological contribution is the first deterministic algorithm that asymptotically approximates this state in the thermodynamic limit. Furthermore, we introduce a novel derandomization technique tailored to point processes, extending statistical-physics-based ensemble methods to hypergraphs and hybrid models. Empirical validation on the deterministic hyperbolic graph model demonstrates that most real-world networks lie close to their respective most typical states, confirming both theoretical rigor and practical interpretability. The framework establishes a new paradigm for principled model selection and explainable network analysis.

This study investigates how percolation phase transitions in network structure regulate information processing capabilities. Using the Erdős–Rényi random graph model, we integrate cascade dynamics with information-theoretic measures—including transfer entropy, output entropy, and functional complexity metrics—to systematically characterize the evolution of input–output functional space near the critical point. We introduce the novel concept of “functional percolation,” revealing that the emergence of the giant connected component coincides with synchronous, discontinuous increases in functional complexity, functional diversity, and directed information flow—establishing percolation criticality as a universal organizational principle for information processing in locally interacting systems. At the critical average degree, a Pareto-optimal trade-off between functional complexity and diversity is achieved; output entropy rises markedly, and transfer entropy extends beyond local neighborhoods, uncovering a fundamental mechanism whereby structural criticality induces qualitative enhancements in global information processing capacity.

This work proposes a novel dynamic stochastic threshold graph model based on a two-color Pólya urn process, which explicitly controls graph structure by sequentially generating nodes and determining—via urn draws—whether each new node becomes fully connected or isolated. For the first time, the Pólya urn mechanism is integrated into threshold graph generation, enabling rigorous analysis through the synthesis of random graph theory and algebraic graph theory. The authors derive exact closed-form expressions for the mean and variance of the degree distribution, the expected distance-decay centrality, and the Laplacian spectrum together with its eigenbasis. These analytical results are successfully applied to discrete-time consensus dynamics, revealing an intrinsic connection between structural evolution and network dynamical behavior.

This study addresses the challenge of reliably identifying critical nodes in uncertain networks characterized by random edge failures and stochastic weight variations, where traditional betweenness centrality proves unstable. The authors propose a novel framework that, for the first time, formulates the problem using absorbing Markov chains, quantifying node importance via the proportion of time spent at each transient state before absorption. An efficient Monte Carlo simulation scheme is employed to estimate these occupancy measures. The approach naturally accommodates weighted reward mechanisms and candidate-set constraints, while robustness of rankings is analyzed through perturbation of the transition kernel. Experiments demonstrate that the method effectively identifies a small set of dominant nodes across diverse random graph models—including Erdős–Rényi and Watts–Strogatz networks—distinguishes between stable and sensitive rankings, and flexibly adapts to structural and reward-based variations.

This work addresses the problem of accurately detecting latent geometric structure in bipartite Gaussian random geometric graphs under random edge masking. The authors propose a novel Fourier-analytic framework based on power series approximations of characteristic functions, which effectively exploits cancellation effects in signed subgraph counts to substantially enlarge the class of tractable subgraphs. For the first time, they establish tight information-theoretic detectability thresholds in terms of the ambient dimension \(d\) and the masking parameter \(q\) under fixed edge density, demonstrating that detection becomes significantly easier when the mask is known. Their approach improves upon the Fourier coefficient bounds established in STOC'24 for dense bipartite graphs, extending them to sparse and non-bipartite settings while ruling out any computational–statistical gap.