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A Dynamical System for PageRank with Time-Dependent Teleportation 具有时间相关隐形传态的PageRank动态系统
Q3 Mathematics Pub Date : 2012-11-18 DOI: 10.1080/15427951.2013.814092
D. Gleich, Ryan A. Rossi
Abstract We propose a dynamical system that captures changes to the network centrality of nodes as external interest in those nodes varies. We derive this system by adding time-dependent teleportation to the PageRank score. The result is not a single set of importance scores, but rather a time-dependent set. These can be converted into ranked lists in a variety of ways, for instance, by taking the largest change in the importance score. For an interesting class of dynamic teleportation functions, we derive closed-form solutions for the dynamic PageRank vector. The magnitude of the deviation from a static PageRank vector is given by a PageRank problem with complex-valued teleportation parameters. Moreover, these dynamical systems are easy to evaluate. We demonstrate the utility of dynamic teleportation on both the article graph of Wikipedia, where the external interest information is given by the number of hourly visitors to each page, and the Twitter social network, where external interest is the number of tweets per month. For these problems, we show that using information from the dynamical system helps improve a prediction task and identify trends in the data.
摘要:我们提出了一个动态系统,它可以捕捉到节点的网络中心性随着外部兴趣的变化而变化。我们通过在PageRank分数中添加时间相关的传送来推导这个系统。结果不是一个单一的重要性分数集合,而是一个与时间相关的集合。这些可以通过各种方式转换成排名列表,例如,通过在重要性得分中取最大的变化。对于一类有趣的动态传送函数,我们导出了动态PageRank向量的封闭解。通过一个具有复值隐形传态参数的PageRank问题,给出了与静态PageRank向量偏差的大小。此外,这些动力系统易于评估。我们在维基百科(Wikipedia)的文章图和Twitter社交网络(Twitter social network)上展示了动态传送的效用,前者的外部兴趣信息由每小时访问每个页面的人数给出,后者的外部兴趣是每月的推文数量。对于这些问题,我们表明使用来自动态系统的信息有助于改进预测任务并识别数据中的趋势。
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引用次数: 43
On the Peak-to-Average Power Ratio Reduction Problem for Orthogonal Transmission Schemes 正交传输方案的峰均功率比减小问题
Q3 Mathematics Pub Date : 2012-08-26 DOI: 10.1080/15427951.2012.728100
H. Boche, Brendan Farrell
High peak values of transmission signals in wireless communication systems lead to wasteful energy consumption and out-of-band radiation. However, reducing peak values generally comes at the cost of some other resource. We provide a theoretical contribution toward understanding the relationship between peak value reduction and the resulting cost in information rates. In particular, we address the relationship between peak values and the proportion of transmission signals allocated for information transmission when one is using a strategy known as tone reservation. We show that when tone reservation is used in both OFDM and DS-CDMA systems, if a peak-to-average power ratio criterion is always satisfied, then the proportion of transmission signals that may be allocated for information transmission must tend to zero. We investigate properties of these two systems for sets of both finite and infinite cardinalities. We present properties that OFDM and DS-CDMA share in common as well as ways in which they fundamentally differ.
在无线通信系统中,由于传输信号的峰值过高,导致了能量的浪费和带外辐射。然而,降低峰值通常是以牺牲其他资源为代价的。我们为理解峰值降低和由此产生的信息速率成本之间的关系提供了理论贡献。特别是,我们解决了峰值和分配给信息传输的传输信号比例之间的关系,当一个人使用一种称为音调保留的策略时。研究表明,在OFDM和DS-CDMA两种系统中使用音调保留时,如果始终满足峰均功率比准则,则可分配用于信息传输的传输信号比例必然趋于零。我们研究了这两个系统在有限和无限基数集合下的性质。我们提出了OFDM和DS-CDMA的共同特性,以及它们的根本不同之处。
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引用次数: 12
An Extended Stochastic Model for Quantitative Security Analysis of Networked Systems 网络系统定量安全分析的扩展随机模型
Q3 Mathematics Pub Date : 2012-08-01 DOI: 10.1080/15427951.2012.654480
Maochao Xu, Shouhuai Xu
Abstract Quantitative security analysis of networked computer systems has been an open problem in computer security for decades. Recently, a promising approach was proposed in [Li et al. 11], which, however, made some strong assumptions including the exponential distribution of, and the independence among, the relevant random variables. In this paper, we substantially weaken these assumptions while offering, in addition to the same types of analytical results as in [Li et al. 11], methods for obtaining the desired security quantities in practice. Moreover, we investigate the problem from a higher-level abstraction, which also leads to both analytical results and practical methods for obtaining the desired security quantities. These should represent a significant step toward ultimately solving the problem of quantitative security analysis of networked computer systems.
网络计算机系统的定量安全分析一直是计算机安全领域的一个开放性问题。最近,[Li et al. 11]提出了一种很有前途的方法,然而,该方法做出了一些强有力的假设,包括相关随机变量的指数分布和相互之间的独立性。在本文中,我们大大削弱了这些假设,同时除了提供与[Li et al. 11]中相同类型的分析结果外,还提供了在实践中获得所需安全量的方法。此外,我们从一个更高层次的抽象来研究这个问题,这也导致了获得期望安全量的分析结果和实用方法。这些应该是朝着最终解决联网计算机系统的定量安全分析问题迈出的重要一步。
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引用次数: 35
A Fast Algorithm to Find All High-Degree Vertices in Graphs with a Power-Law Degree Sequence 幂律度序列图中所有高次顶点的快速查找算法
Q3 Mathematics Pub Date : 2012-06-22 DOI: 10.1080/15427951.2013.819210
C. Cooper, T. Radzik, Yiannis Siantos
Abstract We develop a fast method for finding all high-degree vertices of a connected graph with a power-law degree sequence. The method uses a biased random walk, where the bias is a function of the power law c of the degree sequence. Let G(t) be a t-vertex graph, with degree sequence power law c ≥ 3 generated by a generalized preferential attachment process that adds m edges at each step. Let Sa be the set of all vertices of degree at least ta in G(t). We analyze a biased random walk that makes transitions along undirected edges {x, y} with probabilities proportional to (d(x)d(y))b, where d(x) is the degree of vertex x and b > 0 is a constant parameter. With parameter b = (c − 1)(c − 2)/(2c − 3), the random walk discovers the set Sa completely in steps with high probability. The error parameter ε depends on c, a, and m. The cover time of the entire graph G(t) by the biased walk is . Thus the expected time to discover all vertices by the biased walk is not much higher than the Θ(tlog t) cover time of a simple random walk. The standard preferential attachment process generates graphs with power law c = 3. The search parameter b = 2/3 is appropriate for such graphs. We conduct experimental tests on a preferential attachment graph and on a sample of the underlying graph of the World Wide Web with power law c ≈ 3 that support the claimed property.
提出了一种快速求幂律度序列连通图的所有高次顶点的方法。该方法使用有偏随机漫步,其中偏差是度序列幂律c的函数。设G(t)为一个t顶点图,其度序列幂律c≥3是由每步增加m条边的广义优先附加过程生成的。设Sa为G(t)中度数至少为1的所有顶点的集合。我们分析了一个有偏随机漫步,它沿着无向边{x, y}进行过渡,其概率与(d(x)d(y))b成正比,其中d(x)是顶点x的度,b > 0是一个常数参数。当参数b = (c−1)(c−2)/(2c−3)时,随机行走完全分步高概率地发现集合Sa。误差参数ε取决于c, a和m。整个图G(t)的覆盖时间为。因此,通过有偏行走发现所有顶点的期望时间并不比简单随机行走的Θ(tlog t)覆盖时间高多少。标准的优先依恋过程生成的图具有幂律c = 3。对于这样的图,搜索参数b = 2/3是合适的。我们对一个优先依恋图和一个支持所声称属性的幂律c≈3的万维网底层图样本进行了实验测试。
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引用次数: 10
Ranking and Sparsifying a Connection Graph 连接图的排序与稀疏
Q3 Mathematics Pub Date : 2012-06-22 DOI: 10.1080/15427951.2013.800005
F. Graham, Wenbo Zhao, Mark Kempton
Abstract Many problems arising in dealing with high-dimensional data sets involve connection graphs in which each edge is associated with both an edge weight and a d-dimensional linear transformation. We consider vectorized versions of PageRank and effective resistance that can be used as basic tools for organizing and analyzing complex data sets. For example, generalized PageRank and effective resistance can be utilized to derive and modify diffusion distances for vector diffusion maps in data and image processing. Furthermore, the edge-ranking of the connection graphs determined by vectorized PageRank and effective resistance are an essential part of sparsification algorithms that simplify and preserve the global structure of connection graphs. In addition, we examine consistencies in a connection graph, particularly in the applications of recovering low-dimensional data sets and the reduction of noise. In these applications, we analyze the effect of deleting edges with high edge rank.
在处理高维数据集时出现的许多问题涉及到连接图,其中每条边都与边权和d维线性变换相关联。我们考虑了PageRank的矢量化版本和有效阻力,它们可以用作组织和分析复杂数据集的基本工具。例如,在数据和图像处理中,可以利用广义PageRank和有效阻力来推导和修改矢量扩散图的扩散距离。此外,通过向量化PageRank确定连接图的边缘排序和有效阻力是简化和保持连接图全局结构的稀疏化算法的重要组成部分。此外,我们还研究了连接图中的一致性,特别是在恢复低维数据集和降低噪声的应用中。在这些应用中,我们分析了删除高秩边的效果。
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引用次数: 23
Hypergraph Coloring Games and Voter Models 超图着色游戏和选民模型
Q3 Mathematics Pub Date : 2012-06-22 DOI: 10.1080/15427951.2013.833676
F. Graham, Alexander Tsiatas
Abstract We analyze a network coloring game on hypergraphs that can also describe a voter model. Each node represents a voter and is colored according to its preferred candidate (or undecided). Each hyperedge is a subset of voters that can interact and influence one another. In each round of the game, one hyperedge is chosen randomly, and the voters in the hyperedge can change their colors according to some prescribed probability distribution. We analyze this interaction model based on random walks on the associated weighted directed state graph. Under certain “memoryless” restrictions, we can use semigroup spectral methods to explicitly determine the spectrum of the state graph, and the random walk on the state graph converges to its stationary distribution in O(mlog n) steps, where n is the number of voters and m the number of hyperedges. We can then estimate probabilities that events occur within an error bound of ε by simulating the voting game for O(log (1/ε)mlog n) rounds. We also consider a partially memoryless game using the memoryless game for comparison and analysis, which serves as an approximation of the actual interaction dynamics.
我们分析了一个可以描述选民模型的超图上的网络着色游戏。每个节点代表一个选民,并根据其首选候选人(或未决定的候选人)使用颜色。每个超级边缘都是选民的一个子集,可以相互互动和影响。在每一轮博弈中,随机选择一个超级边缘,超级边缘上的选民可以根据规定的概率分布改变自己的颜色。我们在关联的加权有向状态图上分析了基于随机游动的交互模型。在一定的“无记忆”限制下,我们可以使用半群谱方法显式确定状态图的谱,并且状态图上的随机漫步在O(mlog n)步内收敛到其平稳分布,其中n为选民数,m为超边数。然后,我们可以通过模拟O(log (1/ε)mlog n)轮的投票游戏来估计事件在ε误差范围内发生的概率。我们还考虑使用无记忆游戏进行比较和分析的部分无记忆游戏,这是实际交互动态的近似值。
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引用次数: 15
Some Typical Properties of the Spatial Preferred Attachment Model 空间偏好依恋模型的几个典型特征
Q3 Mathematics Pub Date : 2012-06-22 DOI: 10.1080/15427951.2013.796301
C. Cooper, A. Frieze, P. Prałat
Abstract We investigate a stochastic model for complex networks, based on a spatial embedding of the nodes, called the spatial preferred attachment (SPA) model. In the SPA model, nodes have spheres of influence of varying sizes, and a new node may link to a node only if it falls within its region of influence. The spatial embedding of the nodes models the background knowledge or identity of the node, which influences its link environment. In this paper, we focus on the (directed) diameter, small separators, and the (weak) giant component of the model.
摘要研究了一种基于节点空间嵌入的复杂网络随机模型,称为空间偏好依恋(SPA)模型。在SPA模型中,节点具有不同大小的影响范围,新节点只有在其影响范围内才能链接到另一个节点。节点的空间嵌入对节点的背景知识或身份进行建模,从而影响其链接环境。在本文中,我们关注模型的(定向)直径,小分离器和(弱)巨型组件。
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引用次数: 38
Concentration and Stability of Community-Detecting Functions on Random Networks 随机网络上社区检测函数的集中与稳定性
Q3 Mathematics Pub Date : 2012-03-27 DOI: 10.1080/15427951.2012.749437
Weituo Zhang, C. Lim
We propose a general form of community-detecting functions for finding communities—an optimal partition of a random network—and examine the concentration and stability of the function values using the bounded difference martingale method. We derive LDP inequalities for both the general case and several specific community-detecting functions: modularity, graph bipartitioning, and q-Potts community structure. We also discuss the concentration and stability of community-detecting functions on different types of random networks: sparse and nonsparse networks and some examples such as ER and CL networks.
我们提出了一种用于寻找社区的一般形式的社区检测函数-随机网络的最优划分-并使用有界差分鞅方法检验了函数值的集中和稳定性。我们推导了一般情况下的LDP不等式和一些特定的社区检测函数:模块化、图双分区和q-Potts社区结构。讨论了社区检测函数在不同类型的随机网络(稀疏网络和非稀疏网络)上的集中性和稳定性,并举例说明了ER网络和CL网络。
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引用次数: 0
Introduction to the Special Issue on Algorithms and Models for the Web Graph 网络图的算法和模型专题导论
Q3 Mathematics Pub Date : 2012-03-01 DOI: 10.1080/15427951.2012.635562
A. Bonato, Ravi Kumar, D. Sivakumar
This issue of Internet Mathematics includes a selection of papers that were presented at the Seventh Workshop on Algorithms and Models for the Web-Graph, WAW 2010, held at Stanford University in December 2010. The papers in this issue, unlike the conference proceedings of the workshop, do not have page limits and contain full versions of proofs and algorithms. All the articles have been thoroughly reviewed in accordance with the usual high standards of Internet Mathematics. The papers address a number of topics related to complex networks such as network-creation games, applications of PageRank, efficient triangle-counting algorithms, and models for online social networks. The last decade has seen an explosive growth in research on complex networks, ranging from the web graph, to online social networks, to protein–protein interaction networks. Such research has been of great practical importance and has also pushed the frontiers of pure mathematics. One of the goals of the 2010 workshop was to present current research on the theory and applications of complex networks. The papers presented in this special issue should stimulate new and exciting directions in research on complex networks. We would like to thank the authors and reviewers for making the special issue a reality.
本期《互联网数学》收录了2010年12月在斯坦福大学举行的第七届网络图算法和模型研讨会(WAW 2010)上发表的论文。与研讨会的会议论文集不同,本期的论文没有页数限制,并包含证明和算法的完整版本。所有的文章都按照互联网数学通常的高标准进行了彻底的审查。这些论文讨论了许多与复杂网络相关的主题,如网络创建游戏、PageRank的应用、有效的三角形计数算法和在线社交网络模型。过去的十年见证了复杂网络研究的爆炸式增长,从网络图到在线社交网络,再到蛋白质-蛋白质相互作用网络。这样的研究具有重要的现实意义,也推动了纯数学的前沿。2010年研讨会的目标之一是介绍当前复杂网络的理论和应用研究。在这期特刊上发表的论文应该会激发复杂网络研究的新的和令人兴奋的方向。我们要感谢作者和审稿人使特刊成为现实。
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引用次数: 3
Quick Detection of Nodes with Large Degrees 大度节点的快速检测
Q3 Mathematics Pub Date : 2012-02-15 DOI: 10.1080/15427951.2013.798601
K. Avrachenkov, Nelly Litvak, Marina Sokol, Donald F. Towsley
Abstract Our goal is to find top-k lists of nodes with the largest degrees in large complex networks quickly. If the adjacency list of the network is known (not often the case in complex networks), a deterministic algorithm to find the top-k list of nodes with the largest degrees requires an average complexity of , where n is the number of nodes in the network. Even this modest complexity can be very high for large complex networks. We propose to use a random-walk-based method. We show theoretically and by numerical experiments that for large networks, the random-walk method finds good-quality top lists of nodes with high probability and with computational savings of orders of magnitude. We also propose stopping criteria for the random-walk method that requires very little knowledge about the structure of the network.
我们的目标是快速找到大型复杂网络中度最大的节点top-k列表。如果网络的邻接表是已知的(在复杂网络中并不常见),那么找到top-k的度最大的节点列表的确定性算法需要的平均复杂度为,其中n是网络中的节点数。即使这种适度的复杂性对于大型复杂网络来说也可能非常高。我们建议使用基于随机行走的方法。我们通过理论和数值实验表明,对于大型网络,随机行走方法以高概率找到高质量的节点顶级列表,并节省了数量级的计算。我们还提出了随机漫步方法的停止准则,该方法对网络结构的了解很少。
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引用次数: 26
期刊
Internet Mathematics
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