Pub Date : 2014-04-03DOI: 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.
{"title":"Multicommodity Allocation for Dynamic Demands Using PageRank Vectors","authors":"F. Graham, P. Horn, Jacob Hughes","doi":"10.1080/15427951.2013.833148","DOIUrl":"https://doi.org/10.1080/15427951.2013.833148","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2013.833148","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-02-07DOI: 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.
{"title":"Long Paths in Random Apollonian Networks","authors":"C. Cooper, A. Frieze","doi":"10.1080/15427951.2014.925524","DOIUrl":"https://doi.org/10.1080/15427951.2014.925524","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.925524","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-02-06DOI: 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.
{"title":"The Analysis of Kademlia for Random IDs","authors":"Xing Shi Cai, L. Devroye","doi":"10.1080/15427951.2015.1051674","DOIUrl":"https://doi.org/10.1080/15427951.2015.1051674","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2014-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1051674","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-12-14DOI: 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.
{"title":"A Faster Algorithm to Update Betweenness Centrality After Node Alteration","authors":"Keshav Goel, R. Singh, S. Iyengar, Sukrit Gupta","doi":"10.1080/15427951.2014.982311","DOIUrl":"https://doi.org/10.1080/15427951.2014.982311","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.982311","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-12-14DOI: 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.
{"title":"A Local Clustering Algorithm for Connection Graphs","authors":"F. Graham, Mark Kempton","doi":"10.1080/15427951.2014.968295","DOIUrl":"https://doi.org/10.1080/15427951.2014.968295","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.968295","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-12-14DOI: 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.
{"title":"Fast Low-Cost Estimation of Network Properties Using Random Walks","authors":"C. Cooper, T. Radzik, Yiannis Siantos","doi":"10.1080/15427951.2016.1164100","DOIUrl":"https://doi.org/10.1080/15427951.2016.1164100","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2016.1164100","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-10-30DOI: 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.
{"title":"Tree Nash Equilibria in the Network Creation Game","authors":"A. Mamageishvili, Matús Mihalák, D. Müller","doi":"10.1080/15427951.2015.1016248","DOIUrl":"https://doi.org/10.1080/15427951.2015.1016248","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1016248","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-10-24DOI: 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.
{"title":"Degree-Degree Dependencies in Directed Networks with Heavy-Tailed Degrees","authors":"P. V. D. Hoorn, N. Litvak","doi":"10.1080/15427951.2014.927038","DOIUrl":"https://doi.org/10.1080/15427951.2014.927038","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.927038","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-10-12DOI: 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.
{"title":"Sublinear Column-wise Actions of the Matrix Exponential on Social Networks","authors":"D. Gleich, Kyle Kloster","doi":"10.1080/15427951.2014.971203","DOIUrl":"https://doi.org/10.1080/15427951.2014.971203","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.971203","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-10-02DOI: 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.
{"title":"Containing Viral Spread on Sparse Random Graphs: Bounds, Algorithms, and Experiments","authors":"M. Bradonjic, Michael Molloy, Guanhua Yan","doi":"10.1080/15427951.2013.798600","DOIUrl":"https://doi.org/10.1080/15427951.2013.798600","url":null,"abstract":"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.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2013.798600","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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