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Multicommodity Allocation for Dynamic Demands Using PageRank Vectors 基于PageRank向量的动态需求多商品分配
Q3 Mathematics Pub Date : 2014-04-03 DOI: 10.1080/15427951.2013.833148
F. Graham, P. Horn, Jacob Hughes
Abstract We consider a variant of the contact process concerning multicommodity allocation. In this process, the demands for several types of commodities are initially given at some specified vertices, and then the demands spread interactively in the contact graph. To allocate supplies in such a dynamic setting, we use a modified version of PageRank vectors, called Kronecker PageRank, to identify vertices for shipping supplies. We analyze both the situation that the demand distribution evolves mostly in clusters around the initial vertices and the case that the demands spread to the whole network. We establish sharp upper bounds for the probability that the demands are satisfied as a function of PageRank vectors.
摘要本文考虑了多商品分配中接触过程的一种变体。在这一过程中,首先在一些特定的点上给出几种商品的需求,然后在接触图中交互传播。为了在这样的动态环境中分配物资,我们使用了一种修改版本的PageRank向量,称为Kronecker PageRank,来识别运送物资的顶点。我们分析了需求分布主要在初始点周围的集群中演变的情况和需求向整个网络扩散的情况。作为PageRank向量的函数,我们建立了满足需求的概率的上界。
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引用次数: 1
Long Paths in Random Apollonian Networks 随机阿波罗网络中的长路径
Q3 Mathematics Pub Date : 2014-02-07 DOI: 10.1080/15427951.2014.925524
C. Cooper, A. Frieze
We consider the length L(n) of the longest path in a randomly generated Apollonian Network (ApN) . We show that with high probability for any constant c < 2/3.
我们考虑随机生成的阿波罗网络(ApN)中最长路径的长度L(n)。我们证明了对任意常数c < 2/3都有很高的概率。
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引用次数: 4
The Analysis of Kademlia for Random IDs 随机id的Kademlia分析
Q3 Mathematics Pub Date : 2014-02-06 DOI: 10.1080/15427951.2015.1051674
Xing Shi Cai, L. Devroye
Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uinm. Kademlia is thede factostandard searching algorithm for P2P (peer-to-peer) networks on the Internet. In our earlier work, we introduced two slightly different models for Kademlia and studied how many steps it takes to search for a target node by using Kademlia’s searching algorithm. The first model, in which nodes of the network are labeled with deterministic IDs, was discussed in that article. In the second, the Random ID Model, in which nodes are labeled with random IDs, was only briefly mentioned. Refined results with detailed proofs for this model are given in this article. Our analysis shows that, with high probability, it takes about clog n steps to locate any node, where n is the total number of nodes in the network and c is a constant that does not depend on n.
文章中一个或多个人物的彩色版本可以在www.tandfonline.com/uinm网站上找到。Kademlia是Internet上P2P(点对点)网络的基本标准搜索算法。在我们早期的工作中,我们为Kademlia引入了两个略有不同的模型,并研究了使用Kademlia的搜索算法搜索目标节点需要多少步。本文讨论了第一种模型,其中网络节点用确定性id标记。在第二篇文章中,只简要地提到了随机ID模型,其中节点用随机ID标记。本文给出了该模型的改进结果,并给出了详细的证明。我们的分析表明,在高概率下,定位任何节点大约需要clog n步,其中n是网络中节点的总数,c是一个不依赖于n的常数。
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引用次数: 6
A Faster Algorithm to Update Betweenness Centrality After Node Alteration 一种节点改变后间性中心性更新的快速算法
Q3 Mathematics Pub Date : 2013-12-14 DOI: 10.1080/15427951.2014.982311
Keshav Goel, R. Singh, S. Iyengar, Sukrit Gupta
Betweenness centrality is widely used as a centrality measure, with applications across several disciplines. It is a measure that quantifies the importance of a vertex based on the vertex’s occurrence on shortest paths in a graph. This is a global measure, and in order to find the betweenness centrality of a node, one is supposed to have complete information about the graph. Most of the algorithms that are used to find betweenness centrality assume the constancy of the graph and are not efficient for dynamic networks. We propose a technique to update betweenness centrality of a graph when nodes are added or deleted. Observed experimentally, for real graphs, our algorithm speeds up the calculation of betweenness centrality from 7 to 412 times in comparison to the currently best-known techniques.
中间中心性作为一种中心性度量被广泛使用,其应用跨越多个学科。它是一种基于顶点在图中最短路径上出现的次数来量化顶点重要性的度量。这是一个全局度量,为了找到一个节点的中间性中心性,一个人应该有关于图的完整信息。大多数用于寻找中间性中心性的算法都假设了图的恒定性,对于动态网络来说效率不高。提出了一种在增加或删除节点时更新图的中间性中心性的方法。通过实验观察,对于真实的图,与目前最著名的技术相比,我们的算法将中间性中心性的计算速度从7倍提高到412倍。
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引用次数: 40
A Local Clustering Algorithm for Connection Graphs 连接图的局部聚类算法
Q3 Mathematics Pub Date : 2013-12-14 DOI: 10.1080/15427951.2014.968295
F. Graham, Mark Kempton
We give a clustering algorithm for connection graphs, that is, weighted graphs in which each edge is associated with a d-dimensional rotation. The problem of interest is to identify subsets of small Cheeger ratio that have a high level of consistency, i.e., that have a small edge boundary and for which the rotations along any distinct paths joining two vertices are the same or within some small error factor. We use PageRank vectors as well as tools related to the Cheeger constant to give a clustering algorithm that runs in nearly linear time.
我们给出了连接图的聚类算法,即每条边与一个d维旋转相关联的加权图。感兴趣的问题是识别具有高度一致性的小Cheeger比率子集,即具有较小的边缘边界,并且沿着连接两个顶点的任何不同路径的旋转是相同的或在一些小误差因子内。我们使用PageRank向量以及与Cheeger常数相关的工具来给出一个在接近线性时间内运行的聚类算法。
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引用次数: 11
Fast Low-Cost Estimation of Network Properties Using Random Walks 基于随机漫步的网络属性快速低成本估计
Q3 Mathematics Pub Date : 2013-12-14 DOI: 10.1080/15427951.2016.1164100
C. Cooper, T. Radzik, Yiannis Siantos
Abstract We study the use of random walks as an efficient method to estimate global properties of large connected undirected graphs. Typical examples of the properties of interest include the number of edges, vertices, and triangles, and more generally, the number of small fixed subgraphs. We consider two methods based on first returns of random walks: (1) the cycle formula of regenerative processes and (2) weighted random walks with edge weights defined by the property under investigation. We review the theoretical foundations for these methods and indicate how they can be adapted for the general nonintrusive investigation of large online networks. The expected value and variance of the time of the first return of a random walk decrease with increasing vertex weight, so for a given time budget, returns to high-weight vertices should give the best property estimates. We present theoretical and experimental results on the rate of convergence of the estimates as a function of the number of returns of a random walk to a given start vertex. We made experiments to estimate the number of vertices, edges, and triangles for two test graphs.
摘要研究了随机游走作为估计大连通无向图全局性质的一种有效方法。我们感兴趣的属性的典型示例包括边、顶点和三角形的数量,以及更一般的小固定子图的数量。我们考虑了两种基于随机行走的首次返回的方法:(1)再生过程的循环公式和(2)由所研究的性质定义边权的加权随机行走。我们回顾了这些方法的理论基础,并指出它们如何适用于大型在线网络的一般非侵入性调查。随机漫步第一次返回时间的期望值和方差随着顶点权重的增加而减小,因此对于给定的时间预算,返回高权重顶点应该给出最好的属性估计。我们提出的理论和实验结果的收敛速度估计作为一个函数的随机数漫步的返回到一个给定的开始顶点。我们做了一些实验来估计两个测试图的顶点、边和三角形的数量。
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引用次数: 16
Tree Nash Equilibria in the Network Creation Game 网络创造博弈中的树纳什均衡
Q3 Mathematics Pub Date : 2013-10-30 DOI: 10.1080/15427951.2015.1016248
A. Mamageishvili, Matús Mihalák, D. Müller
In the network creation game with n vertices, every vertex (player) creates an (adjacent) edge and decides to which other vertices the created edge should go. Each created edge costs a fixed amount α > 0. Each player aims to have a good connection with the rest of the vertices and, at the same time, to pay as little as possible. Formally, the cost of each player in the resulting (created) graph is defined as α times the number of edges created by the player plus the sum of the distances to all other vertices. It has been conjectured that for α ≥ n, every Nash equilibrium of this game is a tree and has been confirmed for every α ≥ 273 · n. We improve on this bound and show that this is true for every α ≥ 65 · n. We also show that our approach cannot be used to show the desired bound, but we conjecture that a slightly worse bound α ≥ 1.3 · n can be achieved. Toward this conjecture, we show that if a Nash equilibrium has a cycle of length at most 10, then indeed α < 1.3 · n. We investigate our approach for a coalitional variant of a Nash equilibrium, which coalitions of two players cannot collectively improve, and show that if α ≥ 41 · n, then every such Nash equilibrium is a tree.
在有n个顶点的网络创建游戏中,每个顶点(玩家)创建一个(相邻的)边,并决定创建的边应该去哪些其他顶点。每条创建的边花费固定的α > 0。每个玩家的目标都是与其他顶点保持良好的联系,同时尽可能少花钱。正式地说,在生成的(创建的)图中,每个玩家的成本被定义为α乘以玩家创建的边数加上到所有其他顶点的距离之和。我们已经推测,当α≥n时,这个博弈的每个纳什均衡都是一棵树,并且对于每个α≥273·n都得到了证实。我们改进了这个边界,并证明对于每个α≥65·n都是如此。我们还表明,我们的方法不能用来表示期望的边界,但我们推测可以实现一个稍差的边界α≥1.3·n。针对这一猜想,我们证明了如果纳什均衡的循环长度最多为10,那么α确实< 1.3·n。我们研究了纳什均衡的联盟变体的方法,其中两个参与者的联盟不能共同改进,并证明了如果α≥41·n,那么每个这样的纳什均衡都是树。
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引用次数: 34
Degree-Degree Dependencies in Directed Networks with Heavy-Tailed Degrees 重尾度有向网络中的度-度依赖关系
Q3 Mathematics Pub Date : 2013-10-24 DOI: 10.1080/15427951.2014.927038
P. V. D. Hoorn, N. Litvak
In network theory, Pearson’s correlation coefficients are most commonly used to measure the degree assortativity of a network. We investigate the behavior of these coefficients in the setting of directed networks with heavy-tailed degree sequences. We prove that for graphs where the in- and out-degree sequences satisfy a power law with realistic parameters, Pearson’s correlation coefficients converge to a nonnegative number in the infinite network size limit. We propose alternative measures for degree-degree dependencies in directed networks based on Spearman’s rho and Kendall’s tau. Using examples and calculations on the Wikipedia graphs for nine different languages, we show why these rank correlation measures are more suited for measuring degree assortativity in directed graphs with heavy-tailed degrees.
在网络理论中,皮尔逊相关系数最常用于衡量网络的分类程度。我们研究了这些系数在重尾度序列有向网络设置中的行为。证明了对于进出度序列满足幂律的图,在无限网络规模极限下,Pearson相关系数收敛于一个非负数。我们提出了基于Spearman 's rho和Kendall 's tau的有向网络中度-度依赖的替代度量。通过对九种不同语言的维基百科图的示例和计算,我们展示了为什么这些排名相关度量更适合于测量具有重尾度的有向图中的度的协调性。
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引用次数: 18
Sublinear Column-wise Actions of the Matrix Exponential on Social Networks 社会网络上矩阵指数的亚线性列行为
Q3 Mathematics Pub Date : 2013-10-12 DOI: 10.1080/15427951.2014.971203
D. Gleich, Kyle Kloster
We consider stochastic transition matrices from large social and information networks. For these matrices, we describe and evaluate three fast methods to estimate one column of the matrix exponential. The methods are designed to exploit the properties inherent in social networks, such as a power-law degree distribution. Using only this property, we prove that one of our three algorithms has a sublinear runtime. We present further experimental evidence showing that all three of them run quickly on social networks with billions of edges, and they accurately identify the largest elements of the column.
我们考虑来自大型社会和信息网络的随机转移矩阵。对于这些矩阵,我们描述并评价了三种快速估计矩阵指数一列的方法。这些方法旨在利用社会网络固有的属性,如幂律度分布。仅使用这个性质,我们就证明了三种算法中的一种具有次线性运行时。我们提供了进一步的实验证据,表明这三种方法都能在拥有数十亿条边的社交网络上快速运行,并且它们能准确地识别出列中最大的元素。
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引用次数: 13
Containing Viral Spread on Sparse Random Graphs: Bounds, Algorithms, and Experiments 在稀疏随机图上包含病毒传播:界限、算法和实验
Q3 Mathematics Pub Date : 2013-10-02 DOI: 10.1080/15427951.2013.798600
M. Bradonjic, Michael Molloy, Guanhua Yan
Viral spread on large graphs has many real-life applications such as malware propagation in computer networks and rumor (or misinformation) spread in Twitter-like online social networks. Although viral spread on large graphs has been intensively analyzed on classical models such as Susceptible–Infectious–Recovered, there still exits a deficit of effective methods in practice to contain epidemic spread once it passes a critical threshold. Against this backdrop, we explore methods of containing viral spread in large networks with the focus on sparse random networks. The viral containment strategy is to partition a large network into small components and then to ensure that all messages delivered across different components are free of infection. With such a defense mechanism in place, an epidemic spread starting from any node is limited to only those nodes belonging to the same component as the initial infection node. We establish both lower and upper bounds on the costs of inspecting intercomponent messages. We further propose heuristic-based approaches to partitioning large input graphs into small components. Finally, we study the performance of our proposed algorithms under different network topologies and different edge-weight models.
在大图形上的病毒式传播有许多现实应用,例如计算机网络中的恶意软件传播和类似twitter的在线社交网络中的谣言(或错误信息)传播。尽管在诸如易感-感染-恢复等经典模型上对大图上的病毒传播进行了深入的分析,但在实践中仍然缺乏有效的方法来控制一旦超过临界阈值的流行病传播。在此背景下,我们探索了在大型网络中控制病毒传播的方法,重点是稀疏随机网络。病毒遏制策略是将大型网络划分为小组件,然后确保跨不同组件传递的所有消息都不受感染。有了这样的防御机制,从任何节点开始的流行病传播仅限于与初始感染节点属于同一组件的节点。我们建立了检查组件间消息成本的下界和上界。我们进一步提出了基于启发式的方法来将大的输入图划分为小的组件。最后,我们研究了算法在不同网络拓扑和不同边权模型下的性能。
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引用次数: 2
期刊
Internet Mathematics
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