Pub Date : 2015-03-02DOI: 10.1080/15427951.2014.982312
Mickey Brautbar, M. Draief, S. Khanna
Over the last decade we have witnessed the rapid proliferation of online networks and Internet activity. Although such activity is generally considered a blessing, it also brings with it a large increase in risk of computer malware—malignant software that actively spreads from one computer to another. To date, the majority of existing models of malware spread use stochastic behavior, when the set of neighbors infected from the current set of infected nodes is chosen obliviously. In this work, we initiate the study of planned-infection strategies that can decide intelligently which neighbors of infected nodes to infect next in order to maximize their spread, while maintaining a “signature” similar to the oblivious stochastic infection strategy in order not to be discovered. We first establish that computing optimal and near-optimal planned strategies is computationally hard. We then identify necessary and sufficient conditions in terms of network structure and edge infection probabilities such that the planned process can infect polynomially more nodes than the stochastic process while maintaining a similar “signature” as the oblivious stochastic infection strategy. Among our results is a surprising connection between an additional structural quantity of interest in a network, the network toughness, and planned infections. Based on the network toughness, we characterize networks where existence of planned strategies that are pandemic (infect all nodes) is guaranteed, as well as efficiently computable.
{"title":"On the Power of Planned Infections in Networks","authors":"Mickey Brautbar, M. Draief, S. Khanna","doi":"10.1080/15427951.2014.982312","DOIUrl":"https://doi.org/10.1080/15427951.2014.982312","url":null,"abstract":"Over the last decade we have witnessed the rapid proliferation of online networks and Internet activity. Although such activity is generally considered a blessing, it also brings with it a large increase in risk of computer malware—malignant software that actively spreads from one computer to another. To date, the majority of existing models of malware spread use stochastic behavior, when the set of neighbors infected from the current set of infected nodes is chosen obliviously. In this work, we initiate the study of planned-infection strategies that can decide intelligently which neighbors of infected nodes to infect next in order to maximize their spread, while maintaining a “signature” similar to the oblivious stochastic infection strategy in order not to be discovered. We first establish that computing optimal and near-optimal planned strategies is computationally hard. We then identify necessary and sufficient conditions in terms of network structure and edge infection probabilities such that the planned process can infect polynomially more nodes than the stochastic process while maintaining a similar “signature” as the oblivious stochastic infection strategy. Among our results is a surprising connection between an additional structural quantity of interest in a network, the network toughness, and planned infections. Based on the network toughness, we characterize networks where existence of planned strategies that are pandemic (infect all nodes) is guaranteed, as well as efficiently computable.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"319 - 332"},"PeriodicalIF":0.0,"publicationDate":"2015-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.982312","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948119","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 : 2015-03-01DOI: 10.1080/15427951.2016.1177801
O. Michail
Abstract A temporal graph is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence G1, G2…, Gl of static graphs over the same (static) set of nodes V. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension is added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community.
{"title":"An Introduction to Temporal Graphs: An Algorithmic Perspective*","authors":"O. Michail","doi":"10.1080/15427951.2016.1177801","DOIUrl":"https://doi.org/10.1080/15427951.2016.1177801","url":null,"abstract":"Abstract A temporal graph is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence G1, G2…, Gl of static graphs over the same (static) set of nodes V. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension is added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 1","pages":"239 - 280"},"PeriodicalIF":0.0,"publicationDate":"2015-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2016.1177801","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948080","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 : 2015-02-23DOI: 10.1080/15427951.2015.1018019
Aron Laszka, A. Gueye
Abstract Technological networks (e.g., telephone and sensor networks, Internet) have provided modern society with increased efficiency, but have also exposed us to the risks posed by their vulnerability to attacks. Mitigating these risks involves designing robust network topologies in situations where resources are economically constrained. In this study, we consider the vulnerability of network topologies from an economic viewpoint and propose security metrics, which are necessary for assessing the efficiency of our solutions. We define the vulnerability of a network as the potential loss in connectivity due to the actions of a strategic adversary. To derive vulnerability metrics, we revisit our recently introduced network blocking game models, which provide a framework for quantifying network topology vulnerability in adversarial environments. We assume that the network operator takes both security and economic goals into consideration. To model these goals, we generalize previous models by introducing usage costs and budget constraints for the operator. We study two natural constraint formulations, the maximum and the expected cost constraints, and derive the feasible vulnerability/cost region. Because the proposed metrics are based on game-theoretic models, computing them can be challenging. To elucidate these challenges, we provide a thorough complexity analysis for solving the proposed games.
{"title":"Network Topology Vulnerability/Cost Trade-Off: Model, Application, and Computational Complexity","authors":"Aron Laszka, A. Gueye","doi":"10.1080/15427951.2015.1018019","DOIUrl":"https://doi.org/10.1080/15427951.2015.1018019","url":null,"abstract":"Abstract Technological networks (e.g., telephone and sensor networks, Internet) have provided modern society with increased efficiency, but have also exposed us to the risks posed by their vulnerability to attacks. Mitigating these risks involves designing robust network topologies in situations where resources are economically constrained. In this study, we consider the vulnerability of network topologies from an economic viewpoint and propose security metrics, which are necessary for assessing the efficiency of our solutions. We define the vulnerability of a network as the potential loss in connectivity due to the actions of a strategic adversary. To derive vulnerability metrics, we revisit our recently introduced network blocking game models, which provide a framework for quantifying network topology vulnerability in adversarial environments. We assume that the network operator takes both security and economic goals into consideration. To model these goals, we generalize previous models by introducing usage costs and budget constraints for the operator. We study two natural constraint formulations, the maximum and the expected cost constraints, and derive the feasible vulnerability/cost region. Because the proposed metrics are based on game-theoretic models, computing them can be challenging. To elucidate these challenges, we provide a thorough complexity analysis for solving the proposed games.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"588 - 626"},"PeriodicalIF":0.0,"publicationDate":"2015-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1018019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948129","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 : 2015-01-28DOI: 10.1080/15427951.2015.1009522
F. Graham, O. Simpson
We present an efficient algorithm for solving local linear systems with a boundary condition using the Green’s function of a connected induced subgraph related to the system. We introduce the method of using the Dirichlet heat kernel pagerank1 vector to approximate local solutions to linear systems in the graph Laplacian, satisfying given boundary conditions over a particular subset of vertices. With an efficient algorithm for approximating Dirichlet heat kernel pagerank, our Local Linear Solver algorithm computes an approximate local solution with multiplicative and additive error ε by performing O(ε−5s3log (s3ε−1)log n) random walk steps, where n is the number of vertices in the full graph, and s is the size of the local system on the induced subgraph.
{"title":"Solving Local Linear Systems with Boundary Conditions Using Heat Kernel Pagerank","authors":"F. Graham, O. Simpson","doi":"10.1080/15427951.2015.1009522","DOIUrl":"https://doi.org/10.1080/15427951.2015.1009522","url":null,"abstract":"We present an efficient algorithm for solving local linear systems with a boundary condition using the Green’s function of a connected induced subgraph related to the system. We introduce the method of using the Dirichlet heat kernel pagerank1 vector to approximate local solutions to linear systems in the graph Laplacian, satisfying given boundary conditions over a particular subset of vertices. With an efficient algorithm for approximating Dirichlet heat kernel pagerank, our Local Linear Solver algorithm computes an approximate local solution with multiplicative and additive error ε by performing O(ε−5s3log (s3ε−1)log n) random walk steps, where n is the number of vertices in the full graph, and s is the size of the local system on the induced subgraph.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"449 - 471"},"PeriodicalIF":0.0,"publicationDate":"2015-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1009522","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948223","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 : 2015-01-02DOI: 10.1080/15427951.2014.902407
Maochao Xu, Gaofeng Da, Shouhuai Xu
Studying models of cyber epidemics over arbitrary complex networks can deepen our understanding of cyber security from a whole-system perspective. In this work, we initiate the investigation of cyber epidemic models that accommodate the dependences between the cyber attack events. Due to the notorious difficulty in dealing with such dependences, essentially all existing cyber epidemic models have disregarded them. Specifically, we introduce the idea of copulas into cyber epidemic models for accommodating the dependences between the cyber attack events. We investigate the epidemic equilibrium thresholds as well as the bounds for both equilibrium and nonequilibrium infection probabilities. We further characterize the side effects of disregarding the due dependences between the cyber attack events by showing that the results thereof are unnecessarily restrictive or even incorrect.
{"title":"Cyber Epidemic Models with Dependences","authors":"Maochao Xu, Gaofeng Da, Shouhuai Xu","doi":"10.1080/15427951.2014.902407","DOIUrl":"https://doi.org/10.1080/15427951.2014.902407","url":null,"abstract":"Studying models of cyber epidemics over arbitrary complex networks can deepen our understanding of cyber security from a whole-system perspective. In this work, we initiate the investigation of cyber epidemic models that accommodate the dependences between the cyber attack events. Due to the notorious difficulty in dealing with such dependences, essentially all existing cyber epidemic models have disregarded them. Specifically, we introduce the idea of copulas into cyber epidemic models for accommodating the dependences between the cyber attack events. We investigate the epidemic equilibrium thresholds as well as the bounds for both equilibrium and nonequilibrium infection probabilities. We further characterize the side effects of disregarding the due dependences between the cyber attack events by showing that the results thereof are unnecessarily restrictive or even incorrect.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"62 - 92"},"PeriodicalIF":0.0,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.902407","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947563","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 : 2015-01-02DOI: 10.1080/15427951.2013.830583
Shouhuai Xu, Wenlian Lu, Hualun Li
The concept of active cyber defense has appeared in the literature in recent years. However, there are no mathematical models for characterizing the effectiveness of active cyber defense. In this paper, we fill the void by proposing a novel Markov process model that is native to the interaction between cyber attack and active cyber defense. Unfortunately, the native Markov process model cannot be tackled by techniques of which we are aware. We therefore simplify, via mean-field approximation, the Markov process model as a dynamical system model that is amenable to analysis. This allows us to derive a set of valuable analytic results that characterize the effectiveness of four types of active cyber defense dynamics. Simulations show that the analytic results are intrinsic to the native Markov process model, and therefore justify the validity of the dynamical system model. We also discuss side effects of the mean-field approximation and their implications.
{"title":"A Stochastic Model of Active Cyber Defense Dynamics","authors":"Shouhuai Xu, Wenlian Lu, Hualun Li","doi":"10.1080/15427951.2013.830583","DOIUrl":"https://doi.org/10.1080/15427951.2013.830583","url":null,"abstract":"The concept of active cyber defense has appeared in the literature in recent years. However, there are no mathematical models for characterizing the effectiveness of active cyber defense. In this paper, we fill the void by proposing a novel Markov process model that is native to the interaction between cyber attack and active cyber defense. Unfortunately, the native Markov process model cannot be tackled by techniques of which we are aware. We therefore simplify, via mean-field approximation, the Markov process model as a dynamical system model that is amenable to analysis. This allows us to derive a set of valuable analytic results that characterize the effectiveness of four types of active cyber defense dynamics. Simulations show that the analytic results are intrinsic to the native Markov process model, and therefore justify the validity of the dynamical system model. We also discuss side effects of the mean-field approximation and their implications.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"23 - 61"},"PeriodicalIF":0.0,"publicationDate":"2015-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2013.830583","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947679","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 : 2014-12-17DOI: 10.1080/15427951.2016.1198281
Silvio Lattanzi, S. Leonardi
Abstract The clustering coefficient of an unweighted network has been extensively used to quantify how tightly connected is the neighbor around a node and it has been widely adopted for assessing the quality of nodes in a social network. The computation of the clustering coefficient is challenging since it requires to count the number of triangles in the graph. Several recent works proposed efficient sampling, streaming and MapReduce algorithms that allow to overcome this computational bottleneck. As a matter of fact, the intensity of the interaction between nodes, that is usually represented with weights on the edges of the graph, is also an important measure of the statistical cohesiveness of a network. Recently various notions of weighted clustering coefficient have been proposed but all those techniques are hard to implement on large-scale graphs. In this work we show how standard sampling techniques can be used to obtain efficient estimators for the most commonly used measures of weighted clustering coefficient. Furthermore we also propose a novel graph-theoretic notion of clustering coefficient in weighted networks.
{"title":"Efficient computation of the Weighted Clustering Coefficient","authors":"Silvio Lattanzi, S. Leonardi","doi":"10.1080/15427951.2016.1198281","DOIUrl":"https://doi.org/10.1080/15427951.2016.1198281","url":null,"abstract":"Abstract The clustering coefficient of an unweighted network has been extensively used to quantify how tightly connected is the neighbor around a node and it has been widely adopted for assessing the quality of nodes in a social network. The computation of the clustering coefficient is challenging since it requires to count the number of triangles in the graph. Several recent works proposed efficient sampling, streaming and MapReduce algorithms that allow to overcome this computational bottleneck. As a matter of fact, the intensity of the interaction between nodes, that is usually represented with weights on the edges of the graph, is also an important measure of the statistical cohesiveness of a network. Recently various notions of weighted clustering coefficient have been proposed but all those techniques are hard to implement on large-scale graphs. In this work we show how standard sampling techniques can be used to obtain efficient estimators for the most commonly used measures of weighted clustering coefficient. Furthermore we also propose a novel graph-theoretic notion of clustering coefficient in weighted networks.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 1","pages":"381 - 401"},"PeriodicalIF":0.0,"publicationDate":"2014-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2016.1198281","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948615","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 : 2014-12-08DOI: 10.1080/15427951.2016.1197167
L. Bulteau, Vincent Froese, Nimrod Talmon
Abstract We study competitive diffusion games on graphs introduced by Alon et al. [1] to model the spread of influence in social networks. Extending results of Roshanbin [8] for two players, we investigate the existence of pure Nash equilibriafor at least three players on different classes of graphs including paths, cycles, grid graphs and hypercubes; as a main contribution, we answer an open question proving that there is no Nash equilibriumfor three players on m × n grids with min {m, n} ≥ 5. Further, extending results of Etesami and Basar [3] for two players, we prove the existence of pure Nash equilibriafor four players on every d-dimensional hypercube.
{"title":"Multi-Player Diffusion Games on Graph Classes","authors":"L. Bulteau, Vincent Froese, Nimrod Talmon","doi":"10.1080/15427951.2016.1197167","DOIUrl":"https://doi.org/10.1080/15427951.2016.1197167","url":null,"abstract":"Abstract We study competitive diffusion games on graphs introduced by Alon et al. [1] to model the spread of influence in social networks. Extending results of Roshanbin [8] for two players, we investigate the existence of pure Nash equilibriafor at least three players on different classes of graphs including paths, cycles, grid graphs and hypercubes; as a main contribution, we answer an open question proving that there is no Nash equilibriumfor three players on m × n grids with min {m, n} ≥ 5. Further, extending results of Etesami and Basar [3] for two players, we prove the existence of pure Nash equilibriafor four players on every d-dimensional hypercube.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"112 1","pages":"363 - 380"},"PeriodicalIF":0.0,"publicationDate":"2014-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2016.1197167","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948544","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 : 2014-11-26DOI: 10.1080/15427951.2014.986778
B. Pattabiraman, Md. Mostofa Ali Patwary, A. Gebremedhin, W. Liao, A. Choudhary
The maximum clique problem is a well-known NP-hard problem with applications in data mining, network analysis, information retrieval, and many other areas related to the World Wide Web. There exist several algorithms for the problem, with acceptable runtimes for certain classes of graphs, but many of them are infeasible for massive graphs. We present a new exact algorithm that employs novel pruning techniques and is able to find maximum cliques in very large, sparse graphs quickly. Extensive experiments on different kinds of synthetic and real-world graphs show that our new algorithm can be orders of magnitude faster than existing algorithms. We also present a heuristic that runs orders of magnitude faster than the exact algorithm while providing optimal or near-optimal solutions. We illustrate a simple application of the algorithms in developing methods for detection of overlapping communities in networks.
{"title":"Fast Algorithms for the Maximum Clique Problem on Massive Graphs with Applications to Overlapping Community Detection","authors":"B. Pattabiraman, Md. Mostofa Ali Patwary, A. Gebremedhin, W. Liao, A. Choudhary","doi":"10.1080/15427951.2014.986778","DOIUrl":"https://doi.org/10.1080/15427951.2014.986778","url":null,"abstract":"The maximum clique problem is a well-known NP-hard problem with applications in data mining, network analysis, information retrieval, and many other areas related to the World Wide Web. There exist several algorithms for the problem, with acceptable runtimes for certain classes of graphs, but many of them are infeasible for massive graphs. We present a new exact algorithm that employs novel pruning techniques and is able to find maximum cliques in very large, sparse graphs quickly. Extensive experiments on different kinds of synthetic and real-world graphs show that our new algorithm can be orders of magnitude faster than existing algorithms. We also present a heuristic that runs orders of magnitude faster than the exact algorithm while providing optimal or near-optimal solutions. We illustrate a simple application of the algorithms in developing methods for detection of overlapping communities in networks.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"421 - 448"},"PeriodicalIF":0.0,"publicationDate":"2014-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.986778","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948176","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 : 2014-11-19DOI: 10.1080/15427951.2014.958250
Jérôme Kunegis
Abstract In this article, we extend several algebraic graph analysis methods to bipartite networks. In various areas of science, engineering, and commerce, many types of information can be represented as networks, and thus, the discipline of network analysis plays an important role in these domains. A powerful and widespread class of network analysis methods is based on algebraic graph theory, i.e., representing graphs as square adjacency matrices. However, many networks are of a very specific form that clashes with that representation: they are bipartite. That is, they consist of two node types, with each edge connecting a node of one type with a node of the other type. Examples of bipartite networks (also called two-mode networks) are persons and the social groups they belong to, musical artists and the musical genres they play, and text documents and the words they contain. In fact, any type of feature that can be represented by a categorical variable can be interpreted as a bipartite network. Although bipartite networks are widespread, most literature in the area of network analysis focuses on unipartite networks, i.e., those networks with only a single type of node. The purpose of this article is to extend a selection of important algebraic network analysis methods to bipartite networks, showing that many methods from algebraic graph theory can be applied to bipartite networks, with only minor modifications. We show methods for clustering, visualization, and link prediction. Additionally, we introduce new algebraic methods for measuring the bipartivity in near-bipartite graphs.
{"title":"Exploiting The Structure of Bipartite Graphs for Algebraic and Spectral Graph Theory Applications","authors":"Jérôme Kunegis","doi":"10.1080/15427951.2014.958250","DOIUrl":"https://doi.org/10.1080/15427951.2014.958250","url":null,"abstract":"Abstract In this article, we extend several algebraic graph analysis methods to bipartite networks. In various areas of science, engineering, and commerce, many types of information can be represented as networks, and thus, the discipline of network analysis plays an important role in these domains. A powerful and widespread class of network analysis methods is based on algebraic graph theory, i.e., representing graphs as square adjacency matrices. However, many networks are of a very specific form that clashes with that representation: they are bipartite. That is, they consist of two node types, with each edge connecting a node of one type with a node of the other type. Examples of bipartite networks (also called two-mode networks) are persons and the social groups they belong to, musical artists and the musical genres they play, and text documents and the words they contain. In fact, any type of feature that can be represented by a categorical variable can be interpreted as a bipartite network. Although bipartite networks are widespread, most literature in the area of network analysis focuses on unipartite networks, i.e., those networks with only a single type of node. The purpose of this article is to extend a selection of important algebraic network analysis methods to bipartite networks, showing that many methods from algebraic graph theory can be applied to bipartite networks, with only minor modifications. We show methods for clustering, visualization, and link prediction. Additionally, we introduce new algebraic methods for measuring the bipartivity in near-bipartite graphs.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"11 1","pages":"201 - 321"},"PeriodicalIF":0.0,"publicationDate":"2014-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2014.958250","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947809","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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