arXiv subjects

Jin-Hua Zhao

Publications and source records attributed to Jin-Hua Zhao.

15 recordsLinked to original sources

arXiv2026

Long-range frustration in minimal vertex cover problem on random graphs

A vertex cover on a graph is a set of vertices in which each edge of the graph is adjacent to at least one vertex in the set. The minimal vertex cover (MVC) problem concerns finding vertex covers with the smallest cardinality, which is a typical computationally hard problem among combinatorial optimization on graphs. Here, we follow the idea of the long-range frustration (LRF) in MVC configurations proposed in [\textsl{Physical Review Letters} \textbf{94} (2005) 217203]. We correct its analytical framework and further extend it from Erdös-Rényi random graphs to general random graphs. We formulate the framework of LRF into a percolation model, and analytically estimate the energy density of MVCs on uncorrelated random graphs only with their degree distributions. We test our framework on some typical random graph models along with other methods, such as a hybrid algorithm of greedy leaf removal (GLR) procedure combined with survey propagation-guided decimation (SPD) algorithm and an analytical theory based on the GLR procedure which ignores LRF effect. We show that, when there is a percolation of LRF effect, the above three predictions of energy density, say $x_{\rm LRF}$, $x_{\rm GLR + SPD}$, and $x_{\rm GLR}$, follow a scenario as $x_{\rm LRF} > x_{\rm GLR+SPD} > x_{\rm GLR}$ in most cases and $x_{\rm GLR+SPD} > x_{\rm LRF} > x_{\rm GLR}$ in the other cases, and $x_{\rm LRF}$ is much closer to $x_{\rm GLR+SPD}$ than $x_{\rm GLR}$ as $|x_{\rm LRF} - x_{\rm GLR+SPD} | < x_{\rm GLR+SPD} - x_{\rm GLR}$. Our results show that LRF is a proper mechanism for the formation of complex energy landscape in the MVC problem and a theoretical framework of LRF helps to characterize its ground-state properties.

cond-mat.stat-mech↗

arXiv2024

Buffon-Laplace Needle Problem as a geometric probabilistic approach to filtration process

Buffon-Laplace Needle Problem considers a needle of a length $l$ randomly dropped on a large plane distributed with vertically parallel lines with distances $a$ and $b$ ($a \geqslant b$), respectively. As a classical problem in stochastic probability, it serves as a mathematical basis of various physical literature, such as the efficiency of a filter and the emergence of clogging in filtration process. Yet its potential application is limited by previous focus on its original form of the `short' needle case of $l < b$ and its analytical difficulty in a general sense. Here, rather than a `short' needle embedded in two-dimensional space, we analytically solve problem versions with needles and spherocylinders of arbitrary length and radius embedded in two- and three-dimensional spaces dropped on a grid with any rectangular shape. We further confirm our analytical theory with Monte Carlo simulation. Our framework here helps to provide a geometric analytical perspective to filtration process, and also extend the analytical power of the needle problem into unexplored parameter regions for physical problems involving stochastic processes.

math.HO↗

arXiv2023

Random node reinforcement and $K$-core structure of complex networks

To enhance robustness of complex networked systems, a simple method is introducing reinforced nodes which always function during failure propagation. A random scheme of node reinforcement can be considered as a benchmark for finding an optimal reinforcement solution. Yet there still lacks a systematic evaluation on how node reinforcement affects network structure at a mesoscopic level upon failures. Here we study this problem through the lens of $K$-cores of networks. Based on an analytical percolation framework, we first show that, on uncorrelated random graphs, with a critical size of reinforced nodes, an abrupt emergence of $K$-cores is smoothed out to a continuous one, and a detailed phase diagram is derived. We then show that, with a cost-benefit analysis on random reinforcement, for proper weight factors in cost functions with constant and increasing marginal costs, a gain function shows a unimodality, thus we can analytically find an optimal reinforcement fraction by locating the maximal gain. In all, our framework offers a gain-oriented analytical perspective to designing robust interconnected systems.

physics.soc-ph↗

arXiv2023

A local algorithm and its percolation analysis of bipartite $z$-matching problem

A $z$-matching on a bipartite graph is a set of edges, among which each vertex of two types of the graph is adjacent to at most $1$ and at most $z$ ($\geqslant 1$) edges, respectively. The $z$-matching problem concerns finding $z$-matchings with the maximum size. Our approach to this combinatorial optimization problem is twofold. From an algorithmic perspective, we adopt a local algorithm as a linear approximate solver to find $z$-matchings on any graph instance, whose basic component is a generalized greedy leaf removal procedure in graph theory. From a theoretical perspective, on uncorrelated random bipartite graphs, we develop a mean-field theory for percolation phenomenon underlying the local algorithm, leading to an analytical estimation of $z$-matching sizes on random graphs. Our analytical theory corrects the prediction by belief propagation algorithm at zero-temperature limit in (Kreačić and Bianconi 2019 \textsl{EPL} \textbf{126} 028001). Besides, our theoretical framework extends a core percolation analysis of $k$-XORSAT problems to a general context of uncorrelated random hypergraphs with arbitrary degree distributions of factor and variable nodes.

physics.soc-ph↗

arXiv2022

A residual-based message passing algorithm for constraint satisfaction problems

Message passing algorithms, whose iterative nature captures well complicated interactions among interconnected variables in complex systems and extracts information from the fixed point of iterated messages, provide a powerful toolkit in tackling hard computational tasks in optimization, inference, and learning problems. In the context of constraint satisfaction problems (CSPs), when a control parameter (such as constraint density) is tuned, multiple threshold phenomena emerge, signaling fundamental structural transitions in their solution space. Finding solutions around these transition points is exceedingly challenging for algorithm design, where message passing algorithms suffer from a large message fluctuation far from convergence. Here we introduce a residual-based updating step into message passing algorithms, in which messages varying large between consecutive steps are given high priority in the updating process. For the specific example of model RB, a typical prototype of random CSPs with growing domains, we show that our algorithm improves the convergence of message updating and increases the success probability in finding solutions around the satisfiability threshold with a low computational cost. Our approach to message passing algorithms should be of value for exploring their power in developing algorithms to find ground-state solutions and understand the detailed structure of solution space of hard optimization problems.

cond-mat.dis-nn↗

arXiv2020

Induced Percolation on Networked Systems

Percolation theory has been widely used to study phase transitions in complex networked systems. It has also successfully explained several macroscopic phenomena across different fields. Yet, the existent theoretical framework for percolation places the focus on the direct interactions among the system's components, while recent empirical observations have shown that indirect interactions are common in many systems like ecological and social networks, among others. Here, we propose a new percolation framework that accounts for indirect interactions, which allows to generalize the current theoretical body and understand the role of the underlying indirect influence of the components of a networked system on its macroscopic behavior. We report a rich phenomenology in which first-order, second-order or hybrid phase transitions are possible depending on whether the links of the substrate network are directed, undirected or a mix, respectively. We also present an analytical framework to characterize the proposed induced percolation, paving the way to further understand network dynamics with indirect interactions.

physics.soc-ph↗

arXiv2019

Two faces of greedy leaf removal procedure on graphs

The greedy leaf removal (GLR) procedure on a graph is an iterative removal of any vertex with degree one (leaf) along with its nearest neighbor (root). Its result has two faces: a residual subgraph as a core, and a set of removed roots. While the emergence of cores on uncorrelated random graphs was solved analytically, a theory for roots is ignored except in the case of Erdös-Rényi random graphs. Here we analytically study roots on random graphs. We further show that, with a simple geometrical interpretation and a concise mean-field theory of the GLR procedure, we reproduce the zero-temperature replica symmetric estimation of relative sizes of both minimal vertex covers and maximum matchings on random graphs with or without cores.

physics.soc-ph↗

arXiv2019

Controllability and maximum matchings of complex networks

Previously, the controllability problem of a linear time-invariant dynamical system was mapped to the maximum matching (MM) problem on the bipartite representation of the underlying directed graph, and the sizes of MMs on random bipartite graphs were calculated analytically with the cavity method at zero temperature limit. Here we present an alternative theory to estimate MM sizes based on the core percolation theory and the perfect matching of cores. Our theory is much more simplified and easily interpreted, and can estimate MM sizes on random graphs with or without symmetry between out- and in-degree distributions. Our result helps to illuminate the fundamental connection between the controllability problem and the underlying structure of complex systems.

physics.soc-ph↗

arXiv2017

Generalized $k$-core pruning process on directed networks

The resilience of a complex interconnected system concerns the size of the macroscopic functioning node clusters after external perturbations based on a random or designed scheme. For a representation of the interconnected systems with directional or asymmetrical interactions among constituents, the directed network is a convenient choice. Yet how the interaction directions affect the network resilience still lacks thorough exploration. Here, we study the resilience of directed networks with a generalized $k$-core pruning process as a simple failure procedure based on both the in- and out-degrees of nodes, in which any node with an in-degree $< k_{in}$ or an out-degree $< k_{ou}$ is removed iteratively. With an explicitly analytical framework, we can predict the relative sizes of residual node clusters on uncorrelated directed random graphs. We show that the discontinuous transitions rise for cases with $k_{in} \geq 2$ or $k_{ou} \geq 2$, and the unidirectional interactions among nodes drive the networks more vulnerable against perturbations based on in- and out-degrees separately.

physics.soc-ph↗

arXiv2016

Feedback arcs and node hierarchy in directed networks

Directed networks such as gene regulation networks and neural networks are connected by arcs (directed links). The nodes in a directed network are often strongly interwound by a huge number of directed cycles, which lead to complex information-processing dynamics in the network and make it highly challenging to infer the intrinsic direction of information flow. In this theoretical paper, based on the principle of minimum-feedback, we explore the node hierarchy of directed networks and distinguish feedforward and feedback arcs. Nearly optimal node hierarchy solutions, which minimize the number of feedback arcs from lower-level nodes to higher-level nodes, are constructed by belief-propagation and simulated-annealing methods. For real-world networks, we quantify the extent of feedback scarcity by comparison with the ensemble of direction-randomized networks and identify the most important feedback arcs. Our methods are also useful for visualizing directed networks.

physics.soc-ph↗

arXiv2016

Optimal Disruption of Complex Networks

The collection of all the strongly connected components in a directed graph, among each cluster of which any node has a path to another node, is a typical example of the intertwining structure and dynamics in complex networks, as its relative size indicates network cohesion and it also composes of all the feedback cycles in the network. Here we consider finding an optimal strategy with minimal effort in removal arcs (for example, deactivation of directed interactions) to fragment all the strongly connected components into tree structure with no effect from feedback mechanism. We map the optimal network disruption problem to the minimal feedback arc set problem, a non-deterministically polynomial hard combinatorial optimization problem in graph theory. We solve the problem with statistical physical methods from spin glass theory, resulting in a simple numerical method to extract sub-optimal disruption arc sets with significantly better results than a local heuristic method and a simulated annealing method both in random and real networks. Our results has various implications in controlling and manipulation of real interacted systems.

physics.soc-ph↗

arXiv2015

The Directed Dominating Set Problem: Generalized Leaf Removal and Belief Propagation

A minimum dominating set for a digraph (directed graph) is a smallest set of vertices such that each vertex either belongs to this set or has at least one parent vertex in this set. We solve this hard combinatorial optimization problem approximately by a local algorithm of generalized leaf removal and by a message-passing algorithm of belief propagation. These algorithms can construct near-optimal dominating sets or even exact minimum dominating sets for random digraphs and also for real-world digraph instances. We further develop a core percolation theory and a replica-symmetric spin glass theory for this problem. Our algorithmic and theoretical results may facilitate applications of dominating sets to various network problems involving directed interactions.

physics.soc-ph↗

arXiv2015

Statistical Mechanics of the Minimum Dominating Set Problem

The minimum dominating set problem has wide applications in network science and related fields. It consists of assembling a node set of global minimum size such that any node of the network is either in this set or is adjacent to at least one node of this set. Although this is a difficult optimization problem in general, we show it can be exactly solved by a generalized leaf-removal process if the network contains no core. If the network has an extensive core, we estimate the size of minimum dominating sets by a mean-field theory and implement a belief-propagation algorithm to obtain near-optimal solutions. Our algorithms also perform well on real-world network instances.

physics.soc-ph↗

arXiv2014

Statistical physics of hard combinatorial optimization: The vertex cover problem

Typical-case computation complexity is a research topic at the boundary of computer science, applied mathematics, and statistical physics. In the last twenty years the replica-symmetry-breaking mean field theory of spin glasses and the associated message-passing algorithms have greatly deepened our understanding of typical-case computation complexity. In this paper we use the vertex cover problem, a basic nondeterministic-polynomial (NP)-complete combinatorial optimization problem of wide application, as an example to introduce the statistical physical methods and algorithms. We do not go into the technical details but emphasize mainly the intuitive physical meanings of the message-passing equations. A nonfamiliar reader shall be able to understand to a large extent the physics behind the mean field approaches and to adjust them in solving other optimization problems.

cond-mat.dis-nn↗

arXiv2013

Inducing Effect on the Percolation Transition in Complex Networks

Percolation theory concerns the emergence of connected clusters that percolate through a networked system. Previous studies ignored the effect that a node outside the percolating cluster may actively induce its inside neighbours to exit the percolating cluster. Here we study this inducing effect on the classical site percolation and K-core percolation, showing that the inducing effect always causes a discontinuous percolation transition. We precisely predict the percolation threshold and core size for uncorrelated random networks with arbitrary degree distributions. For low-dimensional lattices the percolation threshold fluctuates considerably over realizations, yet we can still predict the core size once the percolation occurs. The core sizes of real-world networks can also be well predicted using degree distribution as the only input. Our work therefore provides a theoretical framework for quantitatively understanding discontinuous breakdown phenomena in various complex systems.

cond-mat.stat-mech↗