Shortest Path of a Random Graph and its Application

Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within computer science, probability and statistics. A fundamental initial question regarding these models is: when are these random graphs affected by their latent geometry and when are they indistinguishable from simpler models without latent structure, such as the Erdős–Rényi graph $${\mathcal {G}}(n, p)$$ ? We address this question for two of the most well-studied models of random graphs with latent geometry—the random intersection and random geometric graph. Our results are as follows: (a) The random intersection graph is defined by sampling n random sets $$S_1, S_2, \ldots , S_n$$ by including each element of a set of size d in each $$S_i$$ independently with probability $$\delta $$ , and including the edge $$\{i, j\}$$ if $$S_i \cap S_j \ne \emptyset $$ . We prove that the random intersection graph converges in total variation to an Erdős–Rényi graph if $$d = {\tilde{\omega }}(n^3)$$ , and does not if $$d = o(n^3)$$ , for both dense and sparse edge densities p. This resolves an open problem in Fill et al. (Random Struct Algorithms 16(2):156–176, 2000), Rybarczyk (Random Struct Algorithms 38(1–2):205–234, 2011) and Kim et al. (Random Struct Algorithms 52(4):662–679, 2018). The same result was obtained simultaneously and independently by Bubeck et al. (When random intersection graphs lose geometry. Manuscript, 2019). (b) We strengthen the preceding argument to show that the matrix of random intersection sizes $$|S_i \cap S_j|$$ converges in total variation to a symmetric matrix with independent Poisson entries. This yields the first total variation convergence result for $$\tau $$ -random intersection graphs, where the edge $$\{i, j\}$$ is included if $$|S_i \cap S_j| \ge \tau $$ . More precisely, our results imply that, if p is bounded away from 1, then the $$\tau $$ -random intersection graph with edge density p converges to $${\mathcal {G}}(n, p)$$ if $$d = \omega (\tau ^3 n^3)$$ . (c) The random geometric graph on $${\mathbb {S}}^{d - 1}$$ is defined by sampling $$X_1, X_2, \dots , X_n$$ uniformly at random from $${\mathbb {S}}^{d - 1}$$ and including the edge $$\{i, j\}$$ if $$\Vert X_i - X_j \Vert _2 \le \tau $$ . A result of Bubeck et al. (Random Struct Algorithms 49:503–532, 2016) showed that this model converges to $${\mathcal {G}}(n, p)$$ in total variation, where p is chosen so that the models have matching edge densities, as long as $$d = \omega (n^3)$$ . It was conjectured in Bubeck et al. (2016) that this threshold decreases drastically for p small. We make the first progress towards this conjecture by showing convergence if $$d = {\tilde{\omega }}\left( \min \{ pn^3, p^2 n^{7/2} \} \right) $$ . Our proofs are a hybrid of combinatorial arguments, direct couplings and applications of information inequalities. Previous upper bounds on the total variation distance between random graphs with latent geometry and $${\mathcal {G}}(n, p)$$ have typically not been both combinatorial and information-theoretic, while this interplay is essential to the sharpness of our bounds.

Consider a complete graph Kn with edge weights drawn independently from a uniform distribution U(0, 1). The weight of the shortest (minimum‐weight) path P1 between two given vertices is known to be , asymptotically. Define a second‐shortest path P2 to be the shortest path edge‐disjoint from P1, and consider more generally the shortest path Pk edge‐disjoint from all earlier paths. We show that the cost Xk of Pk converges in probability to uniformly for all k ≤ n − 1. We show analogous results when the edge weights are drawn from an exponential distribution. The same results characterize the collectively cheapest k edge‐disjoint paths, that is, a minimum‐cost k‐flow. We also obtain the expectation of Xk conditioned on the existence of Pk.

This paper studies the dynamic single-source shortest paths SSSP in Erdos-Renyi random graphs generated by Gn,i¾?p model. In 2014, Ding and Lin AAIM 2014, LNCS 8546, 197---207 first considered the dynamic SSSP in general digraphs with arbitrary positive weights, and devised a nontrivial local search algorithm named DSPI which takes at most $$On\cdot \max \{1, n\log n / m\}$$Oni¾?max{1,nlogn/m} expected update time to handle a single weight increase, where n is the number of nodes and m is the number of edges in the digraph. DSPI also works on undirected graphs. This paper analyzes the expected update time of DSPI dealing with edge weight increases or edge deletions in Erdos-Renyi a.k.a., Gn,i¾?p random graphs. For weighted Gn,i¾?p random graphs with arbitrary positive edge weights, DSPI takes at most $$OhT_s$$OhTs expected update time to deal with a single edge weight increase as well as $$Opn^2 hT_s$$Opn2hTs total update time, where $$hT_s$$hTs is the height of input SSSP tree $$T_s$$Ts. For Gn,i¾?p random graphs, DSPI takes $$O\ln n$$Olnn expected update time to handle a single edge deletion as well as $$Opn^2 \ln n$$Opn2lnn total update time when $$20\ln n / n \le p \sqrt{2\ln n / n}$$p>2lnn/n. Specifically, DSPI takes the least total update time of $$On\ln n hT_s$$OnlnnhTs for weighted Gn,i¾?p random graphs with $$p = c\ln n / n, c > 1$$p=clnn/n,c>1 as well as $$On^{3/2}\ln n^{1/2}$$On3/2lnn1/2 for Gn,i¾?p random graphs with $$p = c\sqrt{\ln n / n}, c > \sqrt{2}$$p=clnn/n,c>2.

In spite of intensive research, little progress has been made towards fast and work-efficient parallel algorithms for the single source shortest path problem. Our Δ-stepping algorithm, a generalization of Dial's algorithm and the Bellman-Ford algorithm, improves this situation at least in the following "average-case" sense: For random directed graphs with edge probability d/n and uniformly distributed edge weights a PRAM version works in expected time O(log3 n/ log log n) using linear work. The algorithm also allows for efficient adaptation to distributed memory machines. Implementations show that our approach works on real machines. As a side effect, we get a simple linear time sequential algorithm for a large class of not necessarily random directed graphs with random edge weights.

Graph-based approaches for simulating pedestrian dynamics in building models

Preface. 1 Introduction. 1.1 Two examples of combinatorial optimization. 1.2 Why study combinatorial optimization using statistical physics? 1.3 Textbooks. Bibliography. 2 Algorithms. 2.1 Pidgin Algol. 2.2 Iteration and recursion. 2.3 Divide-and-conquer. 2.4 Dynamic programming. 2.5 Backtracking. Bibliography. 3 Introduction to graphs. 3.1 Basic concepts and graph problems. 3.2 Basic graph algorithms. 3.3 Random graphs. Bibliography. 4 Introduction to complexity theory. 4.1 Turing machines. 4.2 Church's thesis. 4.3 Languages. 4.4 The halting problem. 4.5 Class P. 4.6 Class NP. 4.7 Definition of NP-completeness. 4.8 NP-complete problems. 4.9 Worst-case vs. typical-case complexity. Bibliography. 5 Statistical mechanics of the Ising model. 5.1 Phase transitions. 5.2 Some general notes on statistical mechanics. 5.3 The Curie-Weiss model of a ferromagnet. 5.4 The Ising model on a random graph. Bibliography. 6 Algorithms and numerical results for vertex covers. 6.1 Definitions. 6.2 Heuristic algorithms. 6.3 Branch-and-bound algorithm. 6.4 Results: Covering random graphs. 6.5 The leaf-removal algorithm. 6.6 Monte Carlo simulations. 6.7 Backbone. 6.8 Clustering of minimum vertex covers. Bibliography. 7 Statistical mechanics of vertex covers on a random graph. 7.1 Introduction. 7.2 The first-moment bound. 7.3 The hard-core lattice gas. 7.4 Replica approach. Bibliography. 8 The dynamics of vertex-cover algorithms. 8.1 The typical-case solution time of a complete algorithm. 8.2 The dynamics of generalized leaf-removal algorithms. 8.3 Random restart algorithms. Bibliography. 9 Towards new, statistical-mechanics motivated algorithms. 9.1 The cavity graph. 9.2 Warning propagation. 9.3 Belief propagation. 9.4 Survey propagation. 9.5 Numerical experiments on random graphs. Bibliography. 10 The satisfiability problem. 10.1 SAT algorithms. 10.2 Phase transitions in random K-SAT. 10.3 Typical-case dynamics of RandomWalkSAT. 10.4 Message-passing algorithms for SAT. Bibliography. 11 Optimization problems in physics. 11.1 Monte Carlo optimization. 11.2 Hysteric optimization. 11.3 Genetic algorithms. 11.4 Shortest paths and polymers in random media. 11.5 Maximum flows and random-field systems. 11.6 Submodular functions and free energy of Potts model. 11.7 Matchings and spin glasses. Bibliography. Index.

Privacy preserving network publishing has been studied extensively in recent years. Although more works have adopted un-weighted graphs to model network relationships, weighted graph modeling can provide deeper analysis of the degree of relationships. Previous works on weighted graph privacy have concentrated on preserving the shortest path characteristic between pairs of vertices. Two common types of privacy have been proposed. One type of privacy tried to add random noise edge weights to the graph but still maintain the same shortest path. The other privacy, k-shortest path privacy, minimally perturbed edge weights so that there exists k shortest paths. However, the k-shortest path privacy only considers anonymizing same fixed number of shortest paths for all pairs of source and destination vertices. In this work, we present a new concept called [k1, k2]-shortest path privacy to allow different number of shortest paths for different pairs of vertices. A published network graph with [k1, k2]-shortest path privacy has at least k' indistinguishable shortest paths between the source and destination vertices, where k1 l k' l k2. A heuristic algorithm based on modifying only Non-Visited (NV) edges is proposed and experimental results showing the feasibility and characteristics of the proposed approach are presented.

Robustness and Optimization of Complex Networks: Reconstructability, Algorithms and Modeling

In most studies, street networks are considered as undirected graphs while one-way streets and their effect on shortest paths are usually ignored. Here, we first study the empirical effect of one-way streets in about 140 cities in the world. Their presence induces a detour that persists over a wide range of distances and is characterized by a nonuniversal exponent. The effect of one-ways on the pattern of shortest paths is then twofold: they mitigate local traffic in certain areas but create bottlenecks elsewhere. This empirical study leads naturally to considering a mixed graph model of 2d regular lattices with both undirected links and a diluted variable fraction p of randomly directed links which mimics the presence of one-ways in a street network. We study the size of the strongly connected component (SCC) versus p and demonstrate the existence of a threshold p_{c} above which the SCC size is zero. We show numerically that this transition is nontrivial for lattices with degree less than 4 and provide some analytical argument. We compute numerically the critical exponents for this transition and confirm previous results showing that they define a new universality class different from both the directed and standard percolation. Finally, we show that the transition on real-world graphs can be understood with random perturbations of regular lattices. The impact of one-ways on the graph properties was already the subject of a few mathematical studies, and our results show that this problem has also interesting connections with percolation, a classical model in statistical physics.

In recent years, there is growing literature concerning the cascading failure of network characteristics. The object of this paper is to investigate the cascade failures on road traffic network, considering the aeolotropism of road traffic network topology and road congestion dissipation in traffic flow. An improved coupled map lattice (CML) model is proposed. Furthermore, in order to match the congestion dissipation, a recovery mechanism is put forward in this paper. With a real urban road traffic network in Beijing, the cascading failures are tested using different attack strategies, coupling strengths, external perturbations, and attacked road segment numbers. The impacts of different aspects on road traffic network are evaluated based on the simulation results. The findings confirmed the important roles that these characteristics played in the cascading failure propagation and dissipation on road traffic network. We hope these findings are helpful to find out the optimal road network topology and avoid cascading failure on road network.

This study presents a comparative evaluation of Dijkstra’s algorithm, A*, Bellman-Ford, AI-Augmented A* and a neural AI-based model for shortest-path computation across diverse graph topologies, with a focus on time efficiency and memory consumption under standardized experimental conditions. We analyzed grids, random graphs, and scale-free graphs of sizes up to 103,103 nodes, specifically examining 100- and 1000-node grids, 100- and 1000-node random graphs, and 100-node scale-free graphs. The algorithms were benchmarked through repeated runs per condition on a server-class system equipped with an Intel Xeon Gold 6248R processor, NVIDIA Tesla V100 GPU (32 GB), 256 GB RAM, and Ubuntu 20.04. A* consistently outperformed Dijkstra’s algorithm when paired with an informative admissible heuristic, exhibiting faster runtimes by approximately 1.37× to 1.91× across various topologies. In comparison, Bellman-Ford was slower than Dijkstra’s by approximately 1.50× to 1.92×, depending on graph type and size; however, it remained a robust option in scenarios involving negative edge weights or when early-termination conditions reduced practical iterations. The AI model demonstrated the slowest performance across conditions, incurring runtimes that were 2.60× to 3.23× higher than A* and 1.62× to 2.15× higher than Bellman-Ford, offering limited gains as a direct solver. These findings underscore topology-sensitive trade-offs: A* is preferred when a suitable heuristic is available; Dijkstra’s serves as a strong baseline in the absence of heuristics; Bellman-Ford is appropriate for handling negative weights; and current AI approaches are not yet competitive for exact shortest paths but may hold promise as learned heuristics to augment A*. We provide environmental details and comparative results to support reproducibility and facilitate further investigation into hybrid learned-classical strategies.

Comparison of weighted and unweighted network analysis in the case of a pig trade network in Northern Germany

A sampling method based on distributed learning automata for solving stochastic shortest path problem

Because of the increasingly complex spatio-temporal relationships among transportation networks, accurately predicting traffic flow has become a challenging task. Most existing frameworks primarily utilize given spatial adjacency graphs and other complex mechanisms to construct spatio-temporal information of transportation networks. However, the relationships between non-adjacent spatial positions in the road network may affect the effectiveness of these models. Therefore, this paper proposes a new framework called the “spatio-temporal aggregated graph neural network” to enhance the feature relationships in spatio-temporal data. Firstly, a method for generating temporal graphs from spatio-temporal data is introduced to compensate for the spatial graphs’ potential inability to reflect temporal correlations. Next, the correlation coefficient matrix is computed to further enhance the spatio-temporal correlations in the traffic road network. Finally, multiple graph structures are overlayed to comprehensively consider the spatio-temporal correlations of the traffic road network. Experimental results on extensive datasets demonstrate the superiority of this model.

This chapter extends an established model of Brownian agent to investigate the role of social networks in migration and economic aggregation under a multiregion system. The regional conditions are represented by three variables: wage level, tax rate, and welfare. In the model, Brownian agents make their movements by exploiting their social networks to find regional locations with better incomes. We examined the model under four types of network structures: (1) complete network, (2) random network, (3) regular network, and (4) scale-free network. Via computer simulations, the results prove that social networks affect agent migration and economic aggregation and reduce the stability of the whole system. Some findings are summarized as follows: 1. Regardless of the network structure, agents always move to locations with a high welfare service. 2. Agents move frequently when the network has highly connected links (complete and random network) and no obvious economic center appears and remains, whereas in the case of a loosely connected network (regular and scale-free network), agents mainly migrate to regions with a high tax rate for better welfare; the multiregion system finally evolves toward a stable state.