Recently, random graphs in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices have attracted much attention. This paper presents a specific realizatio...Recently, random graphs in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices have attracted much attention. This paper presents a specific realization of a class of random network models in which the connection probability between two vertices （i, j） is a specific function of degrees ki and kj. In the framework of the configuration model of random graphsp we find the analytical expressions for the degree correlation and clustering as a function of the variance of the desired degree distribution. The obtained expressions are checked by means of numerical simulations. Possible applications of our model are discussed.展开更多
The first problem considered in this article reads： is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs？ It is proved that ther...The first problem considered in this article reads： is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs？ It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n3/2, where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n3/2.展开更多
We study the number of k-cycles in a random graph G（n, p）. We estimate the probability that a random graph contains more k-cycles than expected. In this case, the usual martingale inequality with bounded difference ...We study the number of k-cycles in a random graph G（n, p）. We estimate the probability that a random graph contains more k-cycles than expected. In this case, the usual martingale inequality with bounded difference is not effective. By construct- ing a variable that approximates to the number of k-cycles in a random graph and using a new and extensive martingale inequality, we get the results in this paper.展开更多
The flooding distance is an important parameter in the design and evaluation of a routing protocol, which is related not only to the delay time in the route discovery, but also to the stability and reliability of the ...The flooding distance is an important parameter in the design and evaluation of a routing protocol, which is related not only to the delay time in the route discovery, but also to the stability and reliability of the route. In this paper, the average flooding distance （AFD） for a mobile ad hoc network （MANET） in a random graph model was given based on the dynamic source routing （DSR） protocol. The influence of spatial reuse on the AFD was also studied. Compared with that in the model without the spatial reuse, the AFD in the model with the spatial reuse has much smaller value, when the connetivity probability between nodes in the network is small and when the number of reused times is large. This means that the route discovery with the spatial reuse is much more effective.展开更多
Inspired by previous work of Diaz, Petit, Serna, and Trevisan (Approximating layout problems on random graphs, Discrete Mathematics, 235, 2001, 245-253), we show that several well-known graph layout problems are appro...Inspired by previous work of Diaz, Petit, Serna, and Trevisan (Approximating layout problems on random graphs, Discrete Mathematics, 235, 2001, 245-253), we show that several well-known graph layout problems are approximable to within a factor arbitrarily close to 1 of the optimal with high probability for random graphs drawn from an Erdös-Renyi distribution with appropriate sparsity conditions using only elementary probabilistic analysis. Moreover, we show that the same results hold for the analogous problems on directed acyclic graphs.展开更多
Exponential random graph models(ERGMs) are flexible probability models allowing edge dependency.However,it is known that to a first-order approximation,many ERGMs behave like Erd?sRényi random graphs,where edges ...Exponential random graph models(ERGMs) are flexible probability models allowing edge dependency.However,it is known that to a first-order approximation,many ERGMs behave like Erd?sRényi random graphs,where edges are independent.In this paper,to distinguish ERGMs from Erd?s-Rényi random graphs,we consider second-order approximations of ERGMs using two-stars and triangles.We prove that the second-order approximation indeed achieves second-order accuracy in the triangle-free case.The new approximation is formally obtained by the Hoeffding decomposition and rigorously justified using Stein's method.展开更多
In this paper, the continuous-time independent edge-Markovian random graph process model is constructed. The authors also define the interval isolated nodes of the random graph process, study the distribution sequence...In this paper, the continuous-time independent edge-Markovian random graph process model is constructed. The authors also define the interval isolated nodes of the random graph process, study the distribution sequence of the number of isolated nodes and the probability of having no isolated nodes when the initial distribution of the random graph process is stationary distribution, derive the lower limit of the probability in which two arbitrary nodes are connected and the random graph is also connected, and prove that the random graph is almost everywhere connected when the number of nodes is sufficiently large.展开更多
It is a common practice to simulate some historical or test systems to validate the efficiency of new methods or concepts. However, there are only a small number of existing power system test cases, and validation and...It is a common practice to simulate some historical or test systems to validate the efficiency of new methods or concepts. However, there are only a small number of existing power system test cases, and validation and evaluation results, obtained using such a limited number of test cases, may not be deemed sufficient or convincing. In order to provide more available test cases, a new random graph generation algorithm, named ‘‘dualstage constructed random graph’’ algorithm, is proposed to effectively model the power grid topology. The algorithm generates a spanning tree to guarantee the connectivity of random graphs and is capable of controlling the number of lines precisely. No matter how much the average degree is,whether sparse or not, random graphs can be quickly formed to satisfy the requirements. An approach is developed to generate random graphs with prescribed numbers of connected components, in order to simulate the power grid topology under fault conditions. Our experimental study on several realistic power grid topologies proves that the proposed algorithm can quickly generate a large number of random graphs with the topology characteristics of real-world power grid.展开更多
Link prediction attempts to estimate the likelihood of the existence of links between nodes based on available brain network information, such as node attributes and observed links. In response to the problem of the p...Link prediction attempts to estimate the likelihood of the existence of links between nodes based on available brain network information, such as node attributes and observed links. In response to the problem of the poor efficiency of general link prediction methods applied to brain networks, this paper proposes a hierarchical random graph model based on maximum likelihood estimation. This algorithm uses brain network data to create a hierarchical random graph model. Then, it samples the space of all possible dendrograms using a Markov-chain Monte Carlo algorithm. Finally, it calculates the average connection probability. It also employs an evaluation index. Comparing link prediction in a brain network with link prediction in three different networks （Treponemapallidum metabolic network, terrorist networks, and grassland species food webs） using the hierarchical random graph model, experimental results show that the algorithm applied to the brain network has the highest prediction accuracy in terms of AUC scores. With the increase of network scale, AUC scores of the brain network reach 0.8 before gradually leveling off. In addition, the results show AUC scores of various algorithms computed in networks of eight different scales in 28 normal people. They show that the HRG algorithm is far better than random prediction and the ACT global index, and slightly inferior to local indexes CN and LP. Although the HRG algorithm does not produce the best results, its forecast effect is obvious, and shows good time complexity.展开更多
In previous papers, the stationary distributions of a class of discrete and continuoustime random graph processes with state space consisting of the simple and directed graphs on Nvenices were studied. In this paper, ...In previous papers, the stationary distributions of a class of discrete and continuoustime random graph processes with state space consisting of the simple and directed graphs on Nvenices were studied. In this paper, the random graph graph process is extended one impotent stepfurther by allowing interaction of edges. Similarly, We obtha the expressions of the stationarydistributions and prove that the process is ergodic under different editions.展开更多
This paper studies distributed convex optimization over a multi-agent system,where each agent owns only a local cost function with convexity and Lipschitz continuous gradients.The goal of the agents is to cooperativel...This paper studies distributed convex optimization over a multi-agent system,where each agent owns only a local cost function with convexity and Lipschitz continuous gradients.The goal of the agents is to cooperatively minimize a sum of the local cost functions.The underlying communication networks are modelled by a sequence of random and balanced digraphs,which are not required to be spatially or temporally independent and have any special distributions.The authors use a distributed gradient-tracking-based optimization algorithm to solve the optimization problem.In the algorithm,each agent makes an estimate of the optimal solution and an estimate of the average of all the local gradients.The values of the estimates are updated based on a combination of a consensus method and a gradient tracking method.The authors prove that the algorithm can achieve convergence to the optimal solution at a geometric rate if the conditional graphs are uniformly strongly connected,the global cost function is strongly convex and the step-sizes don’t exceed some upper bounds.展开更多
Let G ： Gn,p be a binomial random graph with n vertices and edge probability p = p（n）, and f be a nonnegative integer-valued function defined on V（G） such that 0 〈 a ≤ f（x） ≤ b 〈 np- 2√nplogn for every ...Let G ： Gn,p be a binomial random graph with n vertices and edge probability p = p（n）, and f be a nonnegative integer-valued function defined on V（G） such that 0 〈 a ≤ f（x） ≤ b 〈 np- 2√nplogn for every E V（G）. An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h（e） in [0, 1] so that for each vertex x, we have d^hG（x） = f（x）, where dh（x） = ∑ h（e） is the fractional degree xEe ofx inG. Set Eh = {e ： e e E（G） and h（e） ≠ 0}. IfGh isaspanningsubgraphofGsuchthat E（Gh） = Eh, then Gh is called an fractional f-factor of G. In this paper, we prove that for any binomial random graph Gn,p 2 with p 〉 n^-2/3, almost surely Gn,p contains an fractional f-factor.展开更多
We study the number of edges in the inhomogeneous random graph when vertex weights have an infinite mean and show that the number of edges is O(n log n).Central limit theorems for the number of edges are also establis...We study the number of edges in the inhomogeneous random graph when vertex weights have an infinite mean and show that the number of edges is O(n log n).Central limit theorems for the number of edges are also established.展开更多
A vertex-colored path P is rainbow if its internal vertices have distinct colors;whereas P is monochromatic if its internal vertices are colored the same.For a vertex-colored connected graph G,the rainbow vertex-conne...A vertex-colored path P is rainbow if its internal vertices have distinct colors;whereas P is monochromatic if its internal vertices are colored the same.For a vertex-colored connected graph G,the rainbow vertex-connection number rvc(G)is the minimum number of colors used such that there is a rainbow path joining any two vertices of G;whereas the monochromatic vertex-connection number mvc(G)is the maximum number of colors used such that any two vertices of G are connected by a monochromatic path.These two opposite concepts are the vertex-versions of rainbow connection number rc(G)and monochromatic connection number mc(G)respectively.The study on rc(G)and mc(G)of random graphs drew much attention,and there are few results on the rainbow and monochromatic vertex-connection numbers.In this paper,we consider these two vertex-connection numbers of random graphs and establish sharp threshold functions for them,respectively.展开更多
Generalized random graphs are considered where the presence or absence of an edge de- pends on the weights of its nodes. Our main interest is to investigate large deviations for the number of edges per node in such a ...Generalized random graphs are considered where the presence or absence of an edge de- pends on the weights of its nodes. Our main interest is to investigate large deviations for the number of edges per node in such a generalized random graph, where the node weights are deterministic under some regularity conditions, as well as chosen i.i.d, from a finite set with positive components. When the node weights are random variables, obstacles arise because the independence among edges no longer exists, our main tools are some results of large deviations for mixtures. After calculating, our results show that the corresponding rate functions for the deterministic case and the random case are very different.展开更多
Interpreting deep neural networks is of great importance to understand and verify deep models for natural language processing(NLP)tasks.However,most existing approaches only focus on improving the performance of model...Interpreting deep neural networks is of great importance to understand and verify deep models for natural language processing(NLP)tasks.However,most existing approaches only focus on improving the performance of models but ignore their interpretability.In this work,we propose a Randomly Wired Graph Neural Network(RWGNN)by using graph to model the structure of Neural Network,which could solve two major problems(word-boundary ambiguity and polysemy)of ChineseNER.Besides,we develop a pipeline to explain the RWGNNby using Saliency Map and Adversarial Attacks.Experimental results demonstrate that our approach can identify meaningful and reasonable interpretations for hidden states of RWGNN.展开更多
The properties of generalized flip Markov chains on connected regular digraphs are discussed.The 1-Flipper operation on Markov chains for undirected graphs is generalized to that for multi-digraphs.The generalized 1-F...The properties of generalized flip Markov chains on connected regular digraphs are discussed.The 1-Flipper operation on Markov chains for undirected graphs is generalized to that for multi-digraphs.The generalized 1-Flipper operation preserves the regularity and weak connectivity of multi-digraphs.The generalized 1-Flipper operation is proved to be symmetric.Moreover,it is presented that a series of random generalized 1-Flipper operations eventually lead to a uniform probability distribution over all connected d-regular multi-digraphs without loops.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10375025 and 10275027） and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China （Grant No 704035）
文摘Recently, random graphs in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices have attracted much attention. This paper presents a specific realization of a class of random network models in which the connection probability between two vertices （i, j） is a specific function of degrees ki and kj. In the framework of the configuration model of random graphsp we find the analytical expressions for the degree correlation and clustering as a function of the variance of the desired degree distribution. The obtained expressions are checked by means of numerical simulations. Possible applications of our model are discussed.
文摘The first problem considered in this article reads： is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs？ It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n3/2, where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n3/2.
基金Supported by the National Natural Science Foundation of China (10571139)
文摘We study the number of k-cycles in a random graph G（n, p）. We estimate the probability that a random graph contains more k-cycles than expected. In this case, the usual martingale inequality with bounded difference is not effective. By construct- ing a variable that approximates to the number of k-cycles in a random graph and using a new and extensive martingale inequality, we get the results in this paper.
基金supported by the National Natural Science Foundation of China（Grant No.60572126）
文摘The flooding distance is an important parameter in the design and evaluation of a routing protocol, which is related not only to the delay time in the route discovery, but also to the stability and reliability of the route. In this paper, the average flooding distance （AFD） for a mobile ad hoc network （MANET） in a random graph model was given based on the dynamic source routing （DSR） protocol. The influence of spatial reuse on the AFD was also studied. Compared with that in the model without the spatial reuse, the AFD in the model with the spatial reuse has much smaller value, when the connetivity probability between nodes in the network is small and when the number of reused times is large. This means that the route discovery with the spatial reuse is much more effective.
文摘Inspired by previous work of Diaz, Petit, Serna, and Trevisan (Approximating layout problems on random graphs, Discrete Mathematics, 235, 2001, 245-253), we show that several well-known graph layout problems are approximable to within a factor arbitrarily close to 1 of the optimal with high probability for random graphs drawn from an Erdös-Renyi distribution with appropriate sparsity conditions using only elementary probabilistic analysis. Moreover, we show that the same results hold for the analogous problems on directed acyclic graphs.
基金supported by the National Social Science Fund of China (Grant No.20CTQ005)supported by Hong Kong Research Grants Council General Research Fund (Grant Nos.14305821,14304822 and 14303423)a direct grant from the Chinese University of Hong Kong。
文摘Exponential random graph models(ERGMs) are flexible probability models allowing edge dependency.However,it is known that to a first-order approximation,many ERGMs behave like Erd?sRényi random graphs,where edges are independent.In this paper,to distinguish ERGMs from Erd?s-Rényi random graphs,we consider second-order approximations of ERGMs using two-stars and triangles.We prove that the second-order approximation indeed achieves second-order accuracy in the triangle-free case.The new approximation is formally obtained by the Hoeffding decomposition and rigorously justified using Stein's method.
基金supported by the National Natural Science Foundation of China(Nos.60872060,11101265)the Shanghai Natural Science Foundation of China(No.12ZR1421000)the Shanghai Education Commission Innovation Project Fund(Nos.12ZZ193,14YZ152,15ZZ099)
文摘In this paper, the continuous-time independent edge-Markovian random graph process model is constructed. The authors also define the interval isolated nodes of the random graph process, study the distribution sequence of the number of isolated nodes and the probability of having no isolated nodes when the initial distribution of the random graph process is stationary distribution, derive the lower limit of the probability in which two arbitrary nodes are connected and the random graph is also connected, and prove that the random graph is almost everywhere connected when the number of nodes is sufficiently large.
文摘It is a common practice to simulate some historical or test systems to validate the efficiency of new methods or concepts. However, there are only a small number of existing power system test cases, and validation and evaluation results, obtained using such a limited number of test cases, may not be deemed sufficient or convincing. In order to provide more available test cases, a new random graph generation algorithm, named ‘‘dualstage constructed random graph’’ algorithm, is proposed to effectively model the power grid topology. The algorithm generates a spanning tree to guarantee the connectivity of random graphs and is capable of controlling the number of lines precisely. No matter how much the average degree is,whether sparse or not, random graphs can be quickly formed to satisfy the requirements. An approach is developed to generate random graphs with prescribed numbers of connected components, in order to simulate the power grid topology under fault conditions. Our experimental study on several realistic power grid topologies proves that the proposed algorithm can quickly generate a large number of random graphs with the topology characteristics of real-world power grid.
基金financially supported by the National Natural Science Foundation of China (Nos. 61170136, 61373101, 61472270, and 61402318)the Natural Science Foundation of Shanxi (No. 2014021022-5)+1 种基金the Special/Youth Foundation of Taiyuan University of Technology (No. 2012L014)Youth Team Fund of Taiyuan University of Technology (Nos. 2013T047 and 2013T048)
文摘Link prediction attempts to estimate the likelihood of the existence of links between nodes based on available brain network information, such as node attributes and observed links. In response to the problem of the poor efficiency of general link prediction methods applied to brain networks, this paper proposes a hierarchical random graph model based on maximum likelihood estimation. This algorithm uses brain network data to create a hierarchical random graph model. Then, it samples the space of all possible dendrograms using a Markov-chain Monte Carlo algorithm. Finally, it calculates the average connection probability. It also employs an evaluation index. Comparing link prediction in a brain network with link prediction in three different networks （Treponemapallidum metabolic network, terrorist networks, and grassland species food webs） using the hierarchical random graph model, experimental results show that the algorithm applied to the brain network has the highest prediction accuracy in terms of AUC scores. With the increase of network scale, AUC scores of the brain network reach 0.8 before gradually leveling off. In addition, the results show AUC scores of various algorithms computed in networks of eight different scales in 28 normal people. They show that the HRG algorithm is far better than random prediction and the ACT global index, and slightly inferior to local indexes CN and LP. Although the HRG algorithm does not produce the best results, its forecast effect is obvious, and shows good time complexity.
文摘In previous papers, the stationary distributions of a class of discrete and continuoustime random graph processes with state space consisting of the simple and directed graphs on Nvenices were studied. In this paper, the random graph graph process is extended one impotent stepfurther by allowing interaction of edges. Similarly, We obtha the expressions of the stationarydistributions and prove that the process is ergodic under different editions.
基金supported by the Basic Research Project of Shanghai Science and Technology Commission under Grant No.20JC1414000。
文摘This paper studies distributed convex optimization over a multi-agent system,where each agent owns only a local cost function with convexity and Lipschitz continuous gradients.The goal of the agents is to cooperatively minimize a sum of the local cost functions.The underlying communication networks are modelled by a sequence of random and balanced digraphs,which are not required to be spatially or temporally independent and have any special distributions.The authors use a distributed gradient-tracking-based optimization algorithm to solve the optimization problem.In the algorithm,each agent makes an estimate of the optimal solution and an estimate of the average of all the local gradients.The values of the estimates are updated based on a combination of a consensus method and a gradient tracking method.The authors prove that the algorithm can achieve convergence to the optimal solution at a geometric rate if the conditional graphs are uniformly strongly connected,the global cost function is strongly convex and the step-sizes don’t exceed some upper bounds.
基金Supported by NSFSD(No.ZR2013AM001)NSFC(No.11001055),NSFC11371355
文摘Let G ： Gn,p be a binomial random graph with n vertices and edge probability p = p（n）, and f be a nonnegative integer-valued function defined on V（G） such that 0 〈 a ≤ f（x） ≤ b 〈 np- 2√nplogn for every E V（G）. An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h（e） in [0, 1] so that for each vertex x, we have d^hG（x） = f（x）, where dh（x） = ∑ h（e） is the fractional degree xEe ofx inG. Set Eh = {e ： e e E（G） and h（e） ≠ 0}. IfGh isaspanningsubgraphofGsuchthat E（Gh） = Eh, then Gh is called an fractional f-factor of G. In this paper, we prove that for any binomial random graph Gn,p 2 with p 〉 n^-2/3, almost surely Gn,p contains an fractional f-factor.
基金supported by National Natural Science Foundation of China(Grant No.11671373)。
文摘We study the number of edges in the inhomogeneous random graph when vertex weights have an infinite mean and show that the number of edges is O(n log n).Central limit theorems for the number of edges are also established.
基金supported by the National Natural Science Foundation of China(Nos.11901196)Natural Science Foundation of Anhui Province(Nos.JZ2020AKZR0295)by the Scholarship Promotion Program of Hefei University of Technology(Nos.JZ2019HGTA0038)。
文摘A vertex-colored path P is rainbow if its internal vertices have distinct colors;whereas P is monochromatic if its internal vertices are colored the same.For a vertex-colored connected graph G,the rainbow vertex-connection number rvc(G)is the minimum number of colors used such that there is a rainbow path joining any two vertices of G;whereas the monochromatic vertex-connection number mvc(G)is the maximum number of colors used such that any two vertices of G are connected by a monochromatic path.These two opposite concepts are the vertex-versions of rainbow connection number rc(G)and monochromatic connection number mc(G)respectively.The study on rc(G)and mc(G)of random graphs drew much attention,and there are few results on the rainbow and monochromatic vertex-connection numbers.In this paper,we consider these two vertex-connection numbers of random graphs and establish sharp threshold functions for them,respectively.
基金Supported by National Natural Science Foundation of China(Grant Nos.11371169 and 11671168)
文摘Generalized random graphs are considered where the presence or absence of an edge de- pends on the weights of its nodes. Our main interest is to investigate large deviations for the number of edges per node in such a generalized random graph, where the node weights are deterministic under some regularity conditions, as well as chosen i.i.d, from a finite set with positive components. When the node weights are random variables, obstacles arise because the independence among edges no longer exists, our main tools are some results of large deviations for mixtures. After calculating, our results show that the corresponding rate functions for the deterministic case and the random case are very different.
基金supported by the National Science Foundation of China(NSFC)underGrants 61876217 and 62176175the Innovative Team of Jiangsu Province under Grant XYDXX-086Jiangsu Postgraduate Research and Innovation Plan(KYCX20_2762).
文摘Interpreting deep neural networks is of great importance to understand and verify deep models for natural language processing(NLP)tasks.However,most existing approaches only focus on improving the performance of models but ignore their interpretability.In this work,we propose a Randomly Wired Graph Neural Network(RWGNN)by using graph to model the structure of Neural Network,which could solve two major problems(word-boundary ambiguity and polysemy)of ChineseNER.Besides,we develop a pipeline to explain the RWGNNby using Saliency Map and Adversarial Attacks.Experimental results demonstrate that our approach can identify meaningful and reasonable interpretations for hidden states of RWGNN.
基金National Natural Science Foundation of China(No.11671258)。
文摘The properties of generalized flip Markov chains on connected regular digraphs are discussed.The 1-Flipper operation on Markov chains for undirected graphs is generalized to that for multi-digraphs.The generalized 1-Flipper operation preserves the regularity and weak connectivity of multi-digraphs.The generalized 1-Flipper operation is proved to be symmetric.Moreover,it is presented that a series of random generalized 1-Flipper operations eventually lead to a uniform probability distribution over all connected d-regular multi-digraphs without loops.
