Pub Date : 2015-10-27DOI: 10.1080/15427951.2016.1177802
E. Bergamini, Henning Meyerhenke
Abstract Betweenness is a well-known centrality measure that ranks the nodes of a network according to their participation in shortest paths. Because exact computations are prohibitive in large networks, several approximation algorithms have been proposed. Besides that, recent years have seen the publication of dynamic algorithms for efficient recomputation of betweenness in networks that change over time. In this article, we propose the first betweenness centrality approximation algorithms with a provable guarantee on the maximum approximation error for dynamic networks. Several new intermediate algorithmic results contribute to the respective approximation algorithms: (i) new upper bounds on the vertex diameter, (ii) the first fully dynamic algorithm for updating an approximation of the vertex diameter in undirected graphs, and (iii) an algorithm with lower time complexity for updating single-source shortest paths in unweighted graphs after a batch of edge actions. Using approximation, our algorithms are the first to make in-memory computation of betweenness in dynamic networks with millions of edges feasible. Our experiments show that our algorithms can achieve substantial speedups compared to recomputation, up to several orders of magnitude. Moreover, the approximation accuracy is usually significantly better than the theoretical guarantee in terms of absolute error. More importantly, for reasonably small approximation error thresholds, the rank of nodes is well preserved, in particular for nodes with high betweenness.
{"title":"Approximating Betweenness Centrality in Fully Dynamic Networks","authors":"E. Bergamini, Henning Meyerhenke","doi":"10.1080/15427951.2016.1177802","DOIUrl":"https://doi.org/10.1080/15427951.2016.1177802","url":null,"abstract":"Abstract Betweenness is a well-known centrality measure that ranks the nodes of a network according to their participation in shortest paths. Because exact computations are prohibitive in large networks, several approximation algorithms have been proposed. Besides that, recent years have seen the publication of dynamic algorithms for efficient recomputation of betweenness in networks that change over time. In this article, we propose the first betweenness centrality approximation algorithms with a provable guarantee on the maximum approximation error for dynamic networks. Several new intermediate algorithmic results contribute to the respective approximation algorithms: (i) new upper bounds on the vertex diameter, (ii) the first fully dynamic algorithm for updating an approximation of the vertex diameter in undirected graphs, and (iii) an algorithm with lower time complexity for updating single-source shortest paths in unweighted graphs after a batch of edge actions. Using approximation, our algorithms are the first to make in-memory computation of betweenness in dynamic networks with millions of edges feasible. Our experiments show that our algorithms can achieve substantial speedups compared to recomputation, up to several orders of magnitude. Moreover, the approximation accuracy is usually significantly better than the theoretical guarantee in terms of absolute error. More importantly, for reasonably small approximation error thresholds, the rank of nodes is well preserved, in particular for nodes with high betweenness.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 1","pages":"281 - 314"},"PeriodicalIF":0.0,"publicationDate":"2015-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2016.1177802","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948094","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 : 2015-09-03DOI: 10.1080/15427951.2015.1069522
A. Bonato, P. Prałat
This issue of Internet Mathematics includes a selection of papers that were presented at the Eleventh Workshop on Algorithms and Models for the Web-Graph (WAW 2013) held at Harvard University in December 2013. The workshop was co-located with the 9th Conference on Web and Internet Economics (WINE 2013). All the articles have been thoroughly reviewed in accordance with the usual high standards of Internet Mathematics. The World Wide Web has become part of our everyday life, and information retrieval and data mining on the Web are now of enormous practical interest. The algorithms supporting these activities combine the view of the Web as a text repository and as a graph, induced in various ways by links among pages, hosts and users. The aim of WAW 2013 was to further the understanding of graphs that arise from the Web and various user activities on the Web, and stimulate the development of high-performance algorithms and applications that exploit these graphs. The workshop included talks from researchers working on graph-theoretic and algorithmic aspects of complex networks such as on-line social networks. We would like to thank the authors and reviewers for making this special issue a reality.
{"title":"Special Issue on Algorithms and Models for the Web-graph","authors":"A. Bonato, P. Prałat","doi":"10.1080/15427951.2015.1069522","DOIUrl":"https://doi.org/10.1080/15427951.2015.1069522","url":null,"abstract":"This issue of Internet Mathematics includes a selection of papers that were presented at the Eleventh Workshop on Algorithms and Models for the Web-Graph (WAW 2013) held at Harvard University in December 2013. The workshop was co-located with the 9th Conference on Web and Internet Economics (WINE 2013). All the articles have been thoroughly reviewed in accordance with the usual high standards of Internet Mathematics. The World Wide Web has become part of our everyday life, and information retrieval and data mining on the Web are now of enormous practical interest. The algorithms supporting these activities combine the view of the Web as a text repository and as a graph, induced in various ways by links among pages, hosts and users. The aim of WAW 2013 was to further the understanding of graphs that arise from the Web and various user activities on the Web, and stimulate the development of high-performance algorithms and applications that exploit these graphs. The workshop included talks from researchers working on graph-theoretic and algorithmic aspects of complex networks such as on-line social networks. We would like to thank the authors and reviewers for making this special issue a reality.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"307 - 307"},"PeriodicalIF":0.0,"publicationDate":"2015-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1069522","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948242","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 : 2015-08-03DOI: 10.1080/15427951.2015.1022626
R. La, Maya Kabkab
We introduce a new random graph model. In our model, n, n ≥ 2, vertices choose a subset of potential edges by considering the (estimated) benefits or utilities of the edges. More precisely, each vertex selects k, k ≥ 1, incident edges it wishes to set up, and an undirected edge between two vertices is present in the graph if and only if both of the end vertices choose the edge. First, we examine the scaling law of the smallest k needed for graph connectivity with increasing n and prove that it is Θ(log (n)). Second, we study the diameter of the random graph and demonstrate that, under certain conditions on k, the diameter is close to log (n)/log (log (n)) with high probability. In addition, as a byproduct of our findings, we show that, for all sufficiently large n, if k > β⋆log (n), where β⋆ ≈ 2.4626, there exists a connected Erds–Rnyi random graph that is embedded in our random graph, with high probability.
{"title":"A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter","authors":"R. La, Maya Kabkab","doi":"10.1080/15427951.2015.1022626","DOIUrl":"https://doi.org/10.1080/15427951.2015.1022626","url":null,"abstract":"We introduce a new random graph model. In our model, n, n ≥ 2, vertices choose a subset of potential edges by considering the (estimated) benefits or utilities of the edges. More precisely, each vertex selects k, k ≥ 1, incident edges it wishes to set up, and an undirected edge between two vertices is present in the graph if and only if both of the end vertices choose the edge. First, we examine the scaling law of the smallest k needed for graph connectivity with increasing n and prove that it is Θ(log (n)). Second, we study the diameter of the random graph and demonstrate that, under certain conditions on k, the diameter is close to log (n)/log (log (n)) with high probability. In addition, as a byproduct of our findings, we show that, for all sufficiently large n, if k > β⋆log (n), where β⋆ ≈ 2.4626, there exists a connected Erds–Rnyi random graph that is embedded in our random graph, with high probability.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"528 - 554"},"PeriodicalIF":0.0,"publicationDate":"2015-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1022626","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948137","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 : 2015-07-23DOI: 10.1080/15427951.2015.1103339
A. Bonato, J. Janssen, Elham Roshanbin
ABSTRACT We introduce a new graph parameter called the burning number, inspired by contact processes on graphs such as graph bootstrap percolation, and graph searching paradigms such as Firefighter. The burning number measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We provide a number of properties of the burning number, including characterizations and bounds. The burning number is computed for several graph classes, and is derived for the graphs generated by the Iterated Local Transitivity model for social networks.
{"title":"How to Burn a Graph","authors":"A. Bonato, J. Janssen, Elham Roshanbin","doi":"10.1080/15427951.2015.1103339","DOIUrl":"https://doi.org/10.1080/15427951.2015.1103339","url":null,"abstract":"ABSTRACT We introduce a new graph parameter called the burning number, inspired by contact processes on graphs such as graph bootstrap percolation, and graph searching paradigms such as Firefighter. The burning number measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We provide a number of properties of the burning number, including characterizations and bounds. The burning number is computed for several graph classes, and is derived for the graphs generated by the Iterated Local Transitivity model for social networks.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 1","pages":"100 - 85"},"PeriodicalIF":0.0,"publicationDate":"2015-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1103339","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947921","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 : 2015-05-21DOI: 10.1080/15427951.2015.1110543
J. Janssen, P. Prałat, Rory Wilson
ABSTRACT The spatial preferential attachment (SPA) is a model for complex networks. In the SPA model, nodes are embedded in a metric space, and each node has a sphere of influence whose size increases if the node gains an in-link, and otherwise decreases with time. In this work, we study the behavior of the SPA model when the distribution of the nodes is nonuniform. Specifically, the space is divided into dense and sparse regions, where it is assumed that the dense regions correspond to coherent communities. We prove precise theoretical results with regard to the degree of a node, the number of common neighbors, and the average out-degree in a region. Moreover, we show how these theoretically derived results about the graph properties of the model can be used to formulate a reliable estimator for the distance between certain pairs of nodes, and to estimate the density of the region containing a given node.
{"title":"Nonuniform Distribution of Nodes in the Spatial Preferential Attachment Model","authors":"J. Janssen, P. Prałat, Rory Wilson","doi":"10.1080/15427951.2015.1110543","DOIUrl":"https://doi.org/10.1080/15427951.2015.1110543","url":null,"abstract":"ABSTRACT The spatial preferential attachment (SPA) is a model for complex networks. In the SPA model, nodes are embedded in a metric space, and each node has a sphere of influence whose size increases if the node gains an in-link, and otherwise decreases with time. In this work, we study the behavior of the SPA model when the distribution of the nodes is nonuniform. Specifically, the space is divided into dense and sparse regions, where it is assumed that the dense regions correspond to coherent communities. We prove precise theoretical results with regard to the degree of a node, the number of common neighbors, and the average out-degree in a region. Moreover, we show how these theoretically derived results about the graph properties of the model can be used to formulate a reliable estimator for the distance between certain pairs of nodes, and to estimate the density of the region containing a given node.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 1","pages":"121 - 144"},"PeriodicalIF":0.0,"publicationDate":"2015-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1110543","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947982","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 : 2015-05-04DOI: 10.1080/15427951.2014.950875
Thang N. Dinh, M. Thai
Abstract Many networks, including the Internet, social networks, and biological relations, are found to be naturally divided into communities of densely connected nodes, known as community structure. Since Newman’s suggestion of using modularity as a measure to qualify the goodness of community structures, many efficient methods to maximize modularity have been proposed but without optimality guarantees. In this work we study exact and theoretically near-optimal algorithms for maximizing modularity. In the first part, we investigate the complexity and approximability of the problem on tree graphs. Somewhat surprisingly, the problem is still NP-complete on trees. We then provide a polynomial time algorithm for uniform-weighted trees and a pseudopolynomial time algorithm and a PTAS for trees with arbitrary weights. In the second part, we present a family of compact linear programming formulations for the problem in general graphs. These formulations exploit the graph connectivity structure and reduce substantially the number of constraints, thus, they vastly improve the running time for solving linear programming and integer programming. As a result, networks of thousands of vertices can be solved in minutes, whereas the current largest instance solved with mathematical programming has fewer than 250 vertices.
{"title":"Toward Optimal Community Detection: From Trees to General Weighted Networks","authors":"Thang N. Dinh, M. Thai","doi":"10.1080/15427951.2014.950875","DOIUrl":"https://doi.org/10.1080/15427951.2014.950875","url":null,"abstract":"Abstract Many networks, including the Internet, social networks, and biological relations, are found to be naturally divided into communities of densely connected nodes, known as community structure. Since Newman’s suggestion of using modularity as a measure to qualify the goodness of community structures, many efficient methods to maximize modularity have been proposed but without optimality guarantees. In this work we study exact and theoretically near-optimal algorithms for maximizing modularity. In the first part, we investigate the complexity and approximability of the problem on tree graphs. Somewhat surprisingly, the problem is still NP-complete on trees. We then provide a polynomial time algorithm for uniform-weighted trees and a pseudopolynomial time algorithm and a PTAS for trees with arbitrary weights. In the second part, we present a family of compact linear programming formulations for the problem in general graphs. These formulations exploit the graph connectivity structure and reduce substantially the number of constraints, thus, they vastly improve the running time for solving linear programming and integer programming. As a result, networks of thousands of vertices can be solved in minutes, whereas the current largest instance solved with mathematical programming has fewer than 250 vertices.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"181 - 200"},"PeriodicalIF":0.0,"publicationDate":"2015-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.950875","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947756","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 : 2015-03-04DOI: 10.1080/15427951.2014.912994
A. E. Maftouhi, Ararat Harutyunyan, Y. Manoussakis
In this article, we strengthen some results in our previous work on balance in random signed graphs and study the weak balance. We show that many of the phenomena observed for the balance of random signed graphs extend to weak balance.
{"title":"Weak Balance in Random Signed Graphs","authors":"A. E. Maftouhi, Ararat Harutyunyan, Y. Manoussakis","doi":"10.1080/15427951.2014.912994","DOIUrl":"https://doi.org/10.1080/15427951.2014.912994","url":null,"abstract":"In this article, we strengthen some results in our previous work on balance in random signed graphs and study the weak balance. We show that many of the phenomena observed for the balance of random signed graphs extend to weak balance.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"143 - 154"},"PeriodicalIF":0.0,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.912994","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947619","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 : 2015-03-04DOI: 10.1080/15427951.2013.862884
Yanhua Li, Wei Chen, Yajun Wang, Zhi-Li Zhang
Online social networks (OSNs) are becoming increasingly popular and are generating great interest in the study of the influence diffusion and influence maximization with applications to online viral marketing. Existing studies focus on social networks with only friendship relations, whereas the foe or enemy relations that commonly exist in many OSNs, e.g., Epinions and Slashdot, are completely ignored. In this study, we make the first attempt to investigate the influence diffusion and influence maximization in OSNs with both friend and foe relations, which are modeled using positive and negative edges on signed networks. In particular, we extend the classic voter model to signed networks and analyze the dynamics of influence diffusion of two opposite opinions. We first provide systematic characterization of both short-term and long-term dynamics of influence diffusion in this model and illustrate that the steady state behaviors of the dynamics depend on three types of graph structures, which we refer to as balanced graphs, anti balanced graphs, and strictly unbalanced graphs. We then apply our results to solve the influence maximization problem and develop efficient algorithms to select initial seeds of one opinion that maximize either its short-term influence coverage or long-term steady state influence coverage. Extensive simulation results on both synthetic and real-world networks, such as Epinions and Slashdot, confirm our theoretical analysis on influence diffusion dynamics, and demonstrate the efficacy of our influence maximization algorithm over other heuristic algorithms.
{"title":"Voter Model on Signed Social Networks","authors":"Yanhua Li, Wei Chen, Yajun Wang, Zhi-Li Zhang","doi":"10.1080/15427951.2013.862884","DOIUrl":"https://doi.org/10.1080/15427951.2013.862884","url":null,"abstract":"Online social networks (OSNs) are becoming increasingly popular and are generating great interest in the study of the influence diffusion and influence maximization with applications to online viral marketing. Existing studies focus on social networks with only friendship relations, whereas the foe or enemy relations that commonly exist in many OSNs, e.g., Epinions and Slashdot, are completely ignored. In this study, we make the first attempt to investigate the influence diffusion and influence maximization in OSNs with both friend and foe relations, which are modeled using positive and negative edges on signed networks. In particular, we extend the classic voter model to signed networks and analyze the dynamics of influence diffusion of two opposite opinions. We first provide systematic characterization of both short-term and long-term dynamics of influence diffusion in this model and illustrate that the steady state behaviors of the dynamics depend on three types of graph structures, which we refer to as balanced graphs, anti balanced graphs, and strictly unbalanced graphs. We then apply our results to solve the influence maximization problem and develop efficient algorithms to select initial seeds of one opinion that maximize either its short-term influence coverage or long-term steady state influence coverage. Extensive simulation results on both synthetic and real-world networks, such as Epinions and Slashdot, confirm our theoretical analysis on influence diffusion dynamics, and demonstrate the efficacy of our influence maximization algorithm over other heuristic algorithms.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"133 - 93"},"PeriodicalIF":0.0,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2013.862884","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947356","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 : 2015-03-04DOI: 10.1080/15427951.2014.884513
Shi Li, G. Tucci
In this article we define the notion of (p, δ)–Gromov hyperbolic space where we relax the Gromov slimness condition to allow that not all, but a positive fraction of all triangles, are δ–slim. Furthermore, we study their traffic congestion under geodesic routing. We also construct a constant degree family of expanders with congestion Θ(n2) in contrast to random regular graphs that have congestion O(nlog3(n)).
{"title":"Traffic Congestion in Expanders and (p,δ)–Hyperbolic Spaces","authors":"Shi Li, G. Tucci","doi":"10.1080/15427951.2014.884513","DOIUrl":"https://doi.org/10.1080/15427951.2014.884513","url":null,"abstract":"In this article we define the notion of (p, δ)–Gromov hyperbolic space where we relax the Gromov slimness condition to allow that not all, but a positive fraction of all triangles, are δ–slim. Furthermore, we study their traffic congestion under geodesic routing. We also construct a constant degree family of expanders with congestion Θ(n2) in contrast to random regular graphs that have congestion O(nlog3(n)).","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"134 - 142"},"PeriodicalIF":0.0,"publicationDate":"2015-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.884513","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947553","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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