{"title":"Graph search via star sampling with and without replacement","authors":"S. Weber, Jonathan Stokes","doi":"10.24166/IM.04.2019","DOIUrl":"https://doi.org/10.24166/IM.04.2019","url":null,"abstract":"Star sampling (SS) is a random sampling procedure on a graph wherein each sample consists of a randomly selected vertex (the star center) and its (one-hop) neighbors (the star points). We consider the use of SS to find any member of a target set of vertices in a graph, where the figure of merit (cost) is either the expected number of samples (unit cost) or the expected number of star centers plus star points (linear cost) until a vertex in the target set is encountered, either as a star center or as a star point. We analyze these two performance measures on three related star sampling paradigms: SS with replacement (SSR), SS without center replacement (SSC), and SS without star replacement (SSS). Exact and approximate expressions are derived for the expected unit and linear costs of SSR, SSC, and SSS on Erdős-Renyi (ER) random graphs. The approximations are seen to be accurate. SSC/SSS are notably better than SSR under unit cost for low-density ER graphs, while SSS is notably better than SSR/SSC under linear cost for low- to moderate-density ER graphs. Simulations on twelve \"real-world\" graphs shows the cost approximations to be of variable quality: the SSR and SSC approximations are uniformly accurate, while the SSS approximation, derived for an ER graph, is of variable accuracy.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"1 1","pages":"18171"},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45139825","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}
{"title":"Preferential Placement for Community Structure Formation","authors":"A. Dorodnykh, L. Ostroumova, E. Samosvat","doi":"10.1007/978-3-319-67810-8_6","DOIUrl":"https://doi.org/10.1007/978-3-319-67810-8_6","url":null,"abstract":"","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80295424","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}
{"title":"A Multi-type Preferential Attachment Tree","authors":"Sebastian Rosengren","doi":"10.24166/im.05.2018","DOIUrl":"https://doi.org/10.24166/im.05.2018","url":null,"abstract":"A multi-type preferential attachment tree is introduced, and studied using general multi-type branching processes. For the $p$-type case we derive a framework for studying the tree where a type $i$ vertex generates new type $j$ vertices with rate $w_{ij}(n_1,n_2,ldots, n_p)$ where $n_k$ is the number of type $k$ vertices previously generated by the type $i$ vertex, and $w_{ij}$ is a non-negative function from $mathbb{N}^p$ to $mathbb{R}$. The framework is then used to derive results for trees with more specific attachment rates. \u0000In the case with linear preferential attachment---where type $i$ vertices generate new type $j$ vertices with rate $w_{ij}(n_1,n_2,ldots, n_p)=gamma_{ij}(n_1+n_2+dots +n_p)+beta_{ij}$, where $gamma_{ij}$ and $beta_{ij}$ are positive constants---we show that under mild regularity conditions on the parameters ${gamma_{ij}}, {beta_{ij}}$ the asymptotic degree distribution of a vertex is a power law distribution. The asymptotic composition of the vertex population is also studied.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"2018 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49220828","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}
{"title":"A Theory of Network Security: Principles of Natural Selection and Combinatorics","authors":"Angsheng Li, Yicheng Pan","doi":"10.1080/15427951.2015.1098755","DOIUrl":"https://doi.org/10.1080/15427951.2015.1098755","url":null,"abstract":"We propose the definition of security of networks against the cascading failure models of deliberate attacks. We propose a model of networks by the natural selection of homophyly/kinship, randomness and preferential attachment, referred to as security model. We show that the networks generated by the security model are provably secure against any attacks of sizes poly(log n) under the cascading failure models, for which the principles of natural selection and the combinatorial principles of the networks of the security model, including a power law, a self-organizing principle, a small diameter property, a local navigation law, a degree priority principle, an inclusion-exclusion principle, and an infection priority tree principle etc, are the underlying principles. Furthermore, we show that the networks generated by the security model have an expander core. This property ensures that the networks of the security model satisfy the requirement of global communications in engineering. Based on our theory, we propose a security protocol for computer networks. Our theory demonstrates that security of networks can be achieved by a merging of natural selection and combinatorial principles, and that both natural selection principle and combinatorial principles are essential to security of networks.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"16 1","pages":"145 - 204"},"PeriodicalIF":0.0,"publicationDate":"2016-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1098755","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947864","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}
{"title":"Special Issue on Algorithms and Models for the Web-Graph","authors":"A. Bonato, P. Prałat","doi":"10.1080/15427951.2016.1140994","DOIUrl":"https://doi.org/10.1080/15427951.2016.1140994","url":null,"abstract":"The present issue of Internet Mathematics includes a selection of papers that were presented at the Twelfth Workshop on Algorithms and Models for the Web-Graph (WAW 2014) held at the Academy of Mathematics and Systems Science in Beijing, China in December 2014. The workshop was co-located with the Tenth Conference on Web and Internet Economics (WINE 2014). 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 2014 was to further the understanding of graphs that arise from the Web and complex networks, 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":"84 1","pages":"1 - 1"},"PeriodicalIF":0.0,"publicationDate":"2016-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2016.1140994","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948026","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}
{"title":"Firefighting as a Strategic Game","authors":"Carme Àlvarez, M. Blesa, Hendrik Molter","doi":"10.1080/15427951.2015.1110542","DOIUrl":"https://doi.org/10.1080/15427951.2015.1110542","url":null,"abstract":"ABSTRACT The Firefighter Problem was proposed in 1995 as a deterministic discrete-time model for the spread and containment of a fire. The problem is defined on an undirected finite graph G = (V, E), where fire breaks out initially at f nodes. In each subsequent time-step, two actions occur: a certain number b of firefighters are placed on nonburning nodes, permanently protecting them from the fire, then the fire spreads to all nondefended neighbors of the nodes on fire. Because the graph is finite, at some point each node is either on fire or saved, and thus the fire cannot spread further. One of the objectives for the problem is to place the firefighters in such a way that the number of saved nodes is maximized. The applications of the Firefighter Problem reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks. Most research on the problem considers the case in which the fire starts in a single place (i.e., f = 1), and in which the budget of available firefighters per time-step is one (i.e., b = 1). So does the work in this study. This configuration already leads to hard problems and, even in this case, the problem is known to be NP-hard. In this work, we study the problem from a game-theoretical perspective. We introduce a strategic game model for the Firefighter Problem to tackle its complexity from a different angle. We refer to it as the Firefighter Game. Such a game-based context seems very appropriate when applied to large networks where entities may act and make decisions based on their own interests, without global coordination. At every time-step of the game, a player decides whether to place a new firefighter in a nonburning node of the graph. If so, he must decide where to place it. By placing it, the player is indirectly deciding which nodes to protect at that time-step. We define different utility functions in order to model selfish and nonselfish scenarios, which lead to equivalent games. We show that the Price of Anarchy (PoA) is linear for a particular family of graphs, but it is at most two for trees. We also analyze the quality of the equilibria when coalitions among players are allowed. It turns out that it is possible to compute an equilibrium in polynomial time, even for constant-size coalitions. This yields to a polynomial time approximation algorithm for the problem and its approximation ratio equals the PoA of the corresponding game. We show that for some specific topologies, the PoA is constant when constant-size coalitions are considered.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 1","pages":"101 - 120"},"PeriodicalIF":0.0,"publicationDate":"2016-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1110542","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947970","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}
{"title":"Global Clustering Coefficient in Scale-Free Weighted and Unweighted Networks","authors":"Liudmila Ostroumova Prokhorenkova","doi":"10.1080/15427951.2015.1092482","DOIUrl":"https://doi.org/10.1080/15427951.2015.1092482","url":null,"abstract":"Abstract In this article, we present a detailed analysis of the global clustering coefficient in scale-free graphs. Many observed real-world networks of diverse nature have a power-law degree distribution. Moreover, the observed degree distribution usually has an infinite variance. Therefore, we are especially interested in such degree distributions. In addition, we analyze the clustering coefficient for both weighted and unweighted graphs. There are two well-known definitions of the clustering coefficient of a graph: the global and the average local clustering coefficients. There are several models proposed in the literature for which the average local clustering coefficient tends to a positive constant as a graph grows. However, there are no models of scale-free networks with an infinite variance of the degree distribution and with an asymptotically constant global clustering coefficient. Models with constant global clustering and finite variance were also proposed. Therefore, in this work we focus only on the most interesting case: we analyze the global clustering coefficient for graphs with an infinite variance of the degree distribution. For unweighted graphs, we prove that the global clustering coefficient tends to zero with high probability and we also estimate the largest possible clustering coefficient for such graphs. On the contrary, for weighted graphs, the constant global clustering coefficient can be obtained even for the case of an infinite variance of the degree distribution.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 1","pages":"54 - 67"},"PeriodicalIF":0.0,"publicationDate":"2016-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1092482","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59948260","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}
{"title":"LiveRank: How to Refresh Old Datasets","authors":"The Dang Huynh, F. Mathieu, L. Viennot","doi":"10.1080/15427951.2015.1098756","DOIUrl":"https://doi.org/10.1080/15427951.2015.1098756","url":null,"abstract":"ABSTRACT This article considers the problem of refreshing a dataset. More precisely, given a collection of nodes gathered at some time (webpages, users from an online social network) along with some structure (hyperlinks, social relationships), we want to identify a significant fraction of the nodes that still exist at present time. The liveness of an old node can be tested through an online query at present time. We call LiveRank a ranking of the old pages so that active nodes are more likely to appear first. The quality of a LiveRank is measured by the number of queries necessary to identify a given fraction of the active nodes when using the LiveRank order. We study different scenarios from a static setting where the LiveRank is computed before any query is made, to dynamic settings where the LiveRank can be updated as queries are processed. Our results show that building on the PageRank can lead to efficient LiveRanks, for web graphs as well as for online social networks.","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 1","pages":"68 - 84"},"PeriodicalIF":0.0,"publicationDate":"2016-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2015.1098756","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"59947875","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}
Internet MathematicsPub Date : 2016-01-01Epub Date: 2016-03-24DOI: 10.1080/15427951.2016.1164768
Brian Cloteaux
{"title":"Fast Sequential Creation of Random Realizations of Degree Sequences.","authors":"Brian Cloteaux","doi":"10.1080/15427951.2016.1164768","DOIUrl":"https://doi.org/10.1080/15427951.2016.1164768","url":null,"abstract":"<p><p>We examine the problem of creating random realizations of very large degree sequences. Although fast in practice, the Markov chain Monte Carlo (MCMC) method for selecting a realization has limited usefulness for creating large graphs because of memory constraints. Instead, we focus on sequential importance sampling (SIS) schemes for random graph creation. A difficulty with SIS schemes is assuring that they terminate in a reasonable amount of time. We introduce a new sampling method by which we guarantee termination while achieving speed comparable to the MCMC method.</p>","PeriodicalId":38105,"journal":{"name":"Internet Mathematics","volume":"12 3","pages":"205-219"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/15427951.2016.1164768","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"34349458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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