Pub Date : 2013-09-02DOI: 10.1080/15427951.2015.1067848
Elisabetta Candellero, N. Fountoulakis
Abstract Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks, or social networks. In this article, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was proposed recently by Krioukov and colleagues as a mathematical model of complex networks, under the fundamental assumption that hyperbolic geometry underlies the structure of these networks. We give a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show how the global clustering coefficient can be tuned by these parameters and we give an explicit formula for this function.
{"title":"Clustering and the Hyperbolic Geometry of Complex Networks","authors":"Elisabetta Candellero, N. Fountoulakis","doi":"10.1080/15427951.2015.1067848","DOIUrl":"https://doi.org/10.1080/15427951.2015.1067848","url":null,"abstract":"Abstract Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks, or social networks. In this article, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was proposed recently by Krioukov and colleagues as a mathematical model of complex networks, under the fundamental assumption that hyperbolic geometry underlies the structure of these networks. We give a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show how the global clustering coefficient can be tuned by these parameters and we give an explicit formula for this function.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1067848","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948208","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 : 2013-08-09DOI: 10.1080/15427951.2013.865686
P. Boldi, S. Vigna
Abstract Given a social network, which of its nodes are more central? This question has been asked many times in sociology, psychology, and computer science, and a whole plethora of centrality measures (a.k.a. centrality indices, or rankings) were proposed to account for the importance of the nodes of a network. In this study, we try to provide a mathematically sound survey of the most important classic centrality measures known from the literature and propose an axiomatic approach to establish whether they are actually doing what they have been designed to do. Our axioms suggest some simple, basic properties that a centrality measure should exhibit. Surprisingly, only a new simple measure based on distances, harmonic centrality, turns out to satisfy all axioms; essentially, harmonic centrality is a correction to Bavelas’s classic closeness centrality [Bavelas 50] designed to take unreachable nodes into account in a natural way. As a sanity check, we examine in turn each measure under the lens of information retrieval, leveraging state-of-the-art knowledge in the discipline to measure the effectiveness of the various indices in locating webpages that are relevant to a query. Although there are some examples of such comparisons in the literature, here, for the first time, we also take into consideration centrality measures based on distances, such as closeness, in an information-retrieval setting. The results closely match the data we gathered using our axiomatic approach. Our results suggest that centrality measures based on distances, which in recent years have been neglected in information retrieval in favor of spectral centrality measures, do provide high-quality signals; moreover, harmonic centrality pops up as an excellent general-purpose centrality index for arbitrary directed graphs.
{"title":"Axioms for Centrality","authors":"P. Boldi, S. Vigna","doi":"10.1080/15427951.2013.865686","DOIUrl":"https://doi.org/10.1080/15427951.2013.865686","url":null,"abstract":"Abstract Given a social network, which of its nodes are more central? This question has been asked many times in sociology, psychology, and computer science, and a whole plethora of centrality measures (a.k.a. centrality indices, or rankings) were proposed to account for the importance of the nodes of a network. In this study, we try to provide a mathematically sound survey of the most important classic centrality measures known from the literature and propose an axiomatic approach to establish whether they are actually doing what they have been designed to do. Our axioms suggest some simple, basic properties that a centrality measure should exhibit. Surprisingly, only a new simple measure based on distances, harmonic centrality, turns out to satisfy all axioms; essentially, harmonic centrality is a correction to Bavelas’s classic closeness centrality [Bavelas 50] designed to take unreachable nodes into account in a natural way. As a sanity check, we examine in turn each measure under the lens of information retrieval, leveraging state-of-the-art knowledge in the discipline to measure the effectiveness of the various indices in locating webpages that are relevant to a query. Although there are some examples of such comparisons in the literature, here, for the first time, we also take into consideration centrality measures based on distances, such as closeness, in an information-retrieval setting. The results closely match the data we gathered using our axiomatic approach. Our results suggest that centrality measures based on distances, which in recent years have been neglected in information retrieval in favor of spectral centrality measures, do provide high-quality signals; moreover, harmonic centrality pops up as an excellent general-purpose centrality index for arbitrary directed graphs.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2013.865686","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947488","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 : 2013-06-09DOI: 10.1080/15427951.2013.763556
E. Yeh
The past few decades have seen tremendous growth in the development and adoption of wireless networking technology. This growth has been accompanied by a flowering of research in the area, which in addition to being practically important, is also theoretically significant for advancing the frontiers of mathematics. This issue of Internet Mathematics includes a selection of invited papers from leading researchers in the area of wireless networks. These papers illustrate the state of the art in research within the field. The topics addressed include the optimal control of wireless networks, stochastic routing in wireless sensor networks, coverage in wireless networks with secrecy constraints, the capacity of large-scale underwater networks, and the interplay between capacity, peak power value, and energy consumption in wireless communication. All the articles have been thoroughly reviewed in accordance with the usual high standards of Internet Mathematics. It is hoped that the papers presented in this special issue will not only lead to new directions in research in wireless networks, but also stimulate increased interactions between researchers in wireless networks and those in other networkrelated research areas. I would like to thank the authors and reviewers for making this special issue a reality, and Professor Fan Chung Graham for initiating this special issue project.
{"title":"Special Issue on Wireless Networks","authors":"E. Yeh","doi":"10.1080/15427951.2013.763556","DOIUrl":"https://doi.org/10.1080/15427951.2013.763556","url":null,"abstract":"The past few decades have seen tremendous growth in the development and adoption of wireless networking technology. This growth has been accompanied by a flowering of research in the area, which in addition to being practically important, is also theoretically significant for advancing the frontiers of mathematics. This issue of Internet Mathematics includes a selection of invited papers from leading researchers in the area of wireless networks. These papers illustrate the state of the art in research within the field. The topics addressed include the optimal control of wireless networks, stochastic routing in wireless sensor networks, coverage in wireless networks with secrecy constraints, the capacity of large-scale underwater networks, and the interplay between capacity, peak power value, and energy consumption in wireless communication. All the articles have been thoroughly reviewed in accordance with the usual high standards of Internet Mathematics. It is hoped that the papers presented in this special issue will not only lead to new directions in research in wireless networks, but also stimulate increased interactions between researchers in wireless networks and those in other networkrelated research areas. I would like to thank the authors and reviewers for making this special issue a reality, and Professor Fan Chung Graham for initiating this special issue project.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2013.763556","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946910","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 : 2013-03-26DOI: 10.1080/15427951.2012.727375
A. Kupavskii, L. Ostroumova, D. Shabanov, P. Tetali
In this article we consider a well-known generalization of the Barabási and Albert preferential attachment model—the Buckley–Osthus model. Buckley and Osthus proved that in this model, the degree sequence has a power law distribution. As a natural (and arguably more interesting) next step, we study the second degrees of vertices. Roughly speaking, the second degree of a vertex is the number of vertices at distance two from the given vertex. The distribution of second degrees is of interest because it is a good approximation of PageRank, where the importance of a vertex is measured by taking into account the popularity of its neighbors. We prove that the second degrees also obey a power law. More precisely, we estimate the expectation of the number of vertices with the second degree greater than or equal to k and prove the concentration of this random variable around its expectation using the now-famous Talagrand's concentration inequality over product spaces. As far as we know, this is the only application of Talagrand's inequality to random web graphs where the (preferential attachment) edges are not defined over a product distribution, making the application nontrivial and requiring a certain degree of novelty.
{"title":"The Distribution of Second Degrees in the Buckley–Osthus Random Graph Model","authors":"A. Kupavskii, L. Ostroumova, D. Shabanov, P. Tetali","doi":"10.1080/15427951.2012.727375","DOIUrl":"https://doi.org/10.1080/15427951.2012.727375","url":null,"abstract":"In this article we consider a well-known generalization of the Barabási and Albert preferential attachment model—the Buckley–Osthus model. Buckley and Osthus proved that in this model, the degree sequence has a power law distribution. As a natural (and arguably more interesting) next step, we study the second degrees of vertices. Roughly speaking, the second degree of a vertex is the number of vertices at distance two from the given vertex. The distribution of second degrees is of interest because it is a good approximation of PageRank, where the importance of a vertex is measured by taking into account the popularity of its neighbors. We prove that the second degrees also obey a power law. More precisely, we estimate the expectation of the number of vertices with the second degree greater than or equal to k and prove the concentration of this random variable around its expectation using the now-famous Talagrand's concentration inequality over product spaces. As far as we know, this is the only application of Talagrand's inequality to random web graphs where the (preferential attachment) edges are not defined over a product distribution, making the application nontrivial and requiring a certain degree of novelty.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2012.727375","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947201","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 : 2013-03-26DOI: 10.1080/15427951.2012.727772
A. Fanelli, M. Flammini, L. Moscardelli
We consider the network-design game introduced by Anshelevich et al. in which n source–destination pairs must be connected by n respective players equally sharing the cost of the used links. It is well known that the price of anarchy for this class of games may be as large as n. One approach for reducing this bound is that of resorting to the Stackelberg model, in which for a subset of at most ⌊αn⌋ coordinated players, with 0⩽α⩽1, communication paths inducing better equilibria are fixed. In this paper we show the effectiveness of Stackelberg strategies by providing optimal and nearly optimal bounds on the performance achievable by such Stackelberg strategies. In particular, in contrast to previous works, we are also able to provide Stackelberg strategies computable in polynomial time and lowering the price of anarchy from n to . Most of the results are extended to the case in which each player aims at connecting k>2 nodes of the network.
{"title":"Stackelberg Strategies for Network Design Games","authors":"A. Fanelli, M. Flammini, L. Moscardelli","doi":"10.1080/15427951.2012.727772","DOIUrl":"https://doi.org/10.1080/15427951.2012.727772","url":null,"abstract":"We consider the network-design game introduced by Anshelevich et al. in which n source–destination pairs must be connected by n respective players equally sharing the cost of the used links. It is well known that the price of anarchy for this class of games may be as large as n. One approach for reducing this bound is that of resorting to the Stackelberg model, in which for a subset of at most ⌊αn⌋ coordinated players, with 0⩽α⩽1, communication paths inducing better equilibria are fixed. In this paper we show the effectiveness of Stackelberg strategies by providing optimal and nearly optimal bounds on the performance achievable by such Stackelberg strategies. In particular, in contrast to previous works, we are also able to provide Stackelberg strategies computable in polynomial time and lowering the price of anarchy from n to . Most of the results are extended to the case in which each player aims at connecting k>2 nodes of the network.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2012.727772","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947211","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 : 2013-03-06DOI: 10.1080/15427951.2014.979380
Samuel D. Johnson, R. D’Souza
Abstract This article addresses the matter of inequality in network formation games. We employ a quantity that we are calling the Nash Inequality Ratio (NIR), defined as the maximal ratio between the highest and lowest costs incurred to individual agents in a Nash equilibrium strategy, to characterize the extent to which inequality is possible in equilibrium. We give tight upper bounds on the NIR for the network formation games of Fabrikant et al. [14] and Ehsani et al. [13]. With respect to the relationship between equality and social efficiency, we show that, contrary to common expectations, efficiency does not necessarily come at the expense of increased inequality.
{"title":"Inequality and Network Formation Games","authors":"Samuel D. Johnson, R. D’Souza","doi":"10.1080/15427951.2014.979380","DOIUrl":"https://doi.org/10.1080/15427951.2014.979380","url":null,"abstract":"Abstract This article addresses the matter of inequality in network formation games. We employ a quantity that we are calling the Nash Inequality Ratio (NIR), defined as the maximal ratio between the highest and lowest costs incurred to individual agents in a Nash equilibrium strategy, to characterize the extent to which inequality is possible in equilibrium. We give tight upper bounds on the NIR for the network formation games of Fabrikant et al. [14] and Ehsani et al. [13]. With respect to the relationship between equality and social efficiency, we show that, contrary to common expectations, efficiency does not necessarily come at the expense of increased inequality.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.979380","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947565","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 : 2013-01-23DOI: 10.1080/15427951.2014.982310
M. Bloznelis, M. Karonski
Vertices of an affiliation network are linked to features and two vertices are declared adjacent whenever they share a common feature. We introduce a random intersection graph process aimed at modeling sparse evolving affiliation networks. We establish the asymptotic degree distribution and provide explicit asymptotic formulas for assortativity and clustering coefficients and show how these edge dependence characteristics vary over time.
{"title":"Random Intersection Graph Process","authors":"M. Bloznelis, M. Karonski","doi":"10.1080/15427951.2014.982310","DOIUrl":"https://doi.org/10.1080/15427951.2014.982310","url":null,"abstract":"Vertices of an affiliation network are linked to features and two vertices are declared adjacent whenever they share a common feature. We introduce a random intersection graph process aimed at modeling sparse evolving affiliation networks. We establish the asymptotic degree distribution and provide explicit asymptotic formulas for assortativity and clustering coefficients and show how these edge dependence characteristics vary over time.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.982310","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947576","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 : 2013-01-01DOI: 10.1080/15427951.2012.678140
A. Frieze, P. Horn, P. Prałat
The 8th Workshop on Algorithms and Models for the Web Graph (WAW 2011) took place at Emory University in Atlanta, Georgia, May 27–29, 2011. This is an annual meeting, which is traditionally co-located with another, related, conference. WAW 2011 was co-located with the 15th International Conference on Random Structures and Algorithms (RSA 2011). Co-location of the workshop and conference provided opportunities for researchers in two different but interrelated areas to interact and to exchange research ideas. It was an effective venue for the dissemination of new results and for fostering research collaboration. 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 the workshop was to further the understanding of graphs that arise from the Web and various user activities on the Web, and to stimulate the development of high-performance algorithms and applications that exploit these graphs. The workshop gathered researchers who are working on graph-theoretic and algorithmic aspects of related complex networks, including citation networks, social networks, biological networks, molecular networks, and other networks arising from the Internet. This issue of Internet Mathematics includes a selection of papers that were presented at the workshop. The papers in this issue, unlike the conference proceedings of the workshop, do not have page limits and contain full versions of
{"title":"Special Issue on Algorithms and Models for the Web Graph","authors":"A. Frieze, P. Horn, P. Prałat","doi":"10.1080/15427951.2012.678140","DOIUrl":"https://doi.org/10.1080/15427951.2012.678140","url":null,"abstract":"The 8th Workshop on Algorithms and Models for the Web Graph (WAW 2011) took place at Emory University in Atlanta, Georgia, May 27–29, 2011. This is an annual meeting, which is traditionally co-located with another, related, conference. WAW 2011 was co-located with the 15th International Conference on Random Structures and Algorithms (RSA 2011). Co-location of the workshop and conference provided opportunities for researchers in two different but interrelated areas to interact and to exchange research ideas. It was an effective venue for the dissemination of new results and for fostering research collaboration. 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 the workshop was to further the understanding of graphs that arise from the Web and various user activities on the Web, and to stimulate the development of high-performance algorithms and applications that exploit these graphs. The workshop gathered researchers who are working on graph-theoretic and algorithmic aspects of related complex networks, including citation networks, social networks, biological networks, molecular networks, and other networks arising from the Internet. This issue of Internet Mathematics includes a selection of papers that were presented at the workshop. The papers in this issue, unlike the conference proceedings of the workshop, do not have page limits and contain full versions of","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2012.678140","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946872","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 : 2013-01-01DOI: 10.1080/15427951.2012.678194
F. Graham, Alexander Tsiatas, Wensong Xu
Motivated by numerous models of representing trust and distrust within a network ranking system, we examine a quantitative vertex ranking with consideration of the influence of a subset of nodes. We propose and analyze a general ranking metric, called Dirichlet PageRank, which gives a ranking of vertices in a subset S of nodes subject to some specified conditions on the vertex boundary of S. In addition to the usual Dirichlet boundary condition (which disregards the influence of nodes outside of S), we consider general boundary conditions allowing the presence of negative (distrustful) nodes or edges. We give an efficient approximation algorithm for computing Dirichlet PageRank vectors. Furthermore, we give several algorithms for solving various trust-based ranking problems using Dirichlet PageRank with general boundary conditions.
{"title":"Dirichlet PageRank and Ranking Algorithms Based on Trust and Distrust","authors":"F. Graham, Alexander Tsiatas, Wensong Xu","doi":"10.1080/15427951.2012.678194","DOIUrl":"https://doi.org/10.1080/15427951.2012.678194","url":null,"abstract":"Motivated by numerous models of representing trust and distrust within a network ranking system, we examine a quantitative vertex ranking with consideration of the influence of a subset of nodes. We propose and analyze a general ranking metric, called Dirichlet PageRank, which gives a ranking of vertices in a subset S of nodes subject to some specified conditions on the vertex boundary of S. In addition to the usual Dirichlet boundary condition (which disregards the influence of nodes outside of S), we consider general boundary conditions allowing the presence of negative (distrustful) nodes or edges. We give an efficient approximation algorithm for computing Dirichlet PageRank vectors. Furthermore, we give several algorithms for solving various trust-based ranking problems using Dirichlet PageRank with general boundary conditions.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2012.678194","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59946994","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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