### Citations

4667 | The Anatomy of a Large-scale Hypertextual Web Search Engine
- Brin, Page
- 1998
(Show Context)
Citation Context ...he number of copies of the directed edge uv in the graph (which is zero if there is no edge from u to v), and out-deg(u) denotes the out-degree of u. PageRank is used as a ranking mechanism in Google =-=[25]-=-. More details and applications can be found in [91]. It will be convenient to have an ‘algorithmic’ definition for PageRank, namely, a way to sample from the PageRank distribution. This is achieved b... |

3928 | Emergence of scaling in random networks
- Barabási, Albert
- 1999
(Show Context)
Citation Context ...atter, which had been analyzed before. The difficulty of analyzing evolving models over non-evolving ones arises perhaps from the dependencies between edges in the former models. Barabási and Albert =-=[9]-=- in 1999 introduced one of the first evolving models for realworld networks, sometimes known as the preferential attachment model. Their model can be roughly described as follows (see [20] for a forma... |

3274 | An introduction to probability theory and its applications vol I - Feller - 1968 |

2142 | Statistical mechanics of complex networks
- Albert, AL
(Show Context)
Citation Context ...k. The diameter of the giant component of this graph was found to be 41, and the average distance between reachable pairs was found to be around 4.74. For other examples, see, e.g., Tables 1 and 2 in =-=[4]-=-, Table 8.1 in [105] or Table 4 in [95]. Due to the ever growing interest in social networks, the Webgraph, biological networks, etc., in recent years a great deal of research has been built around st... |

880 |
Epidemic algorithms for replicated database maintenance
- Demers, Greene, et al.
- 1987
(Show Context)
Citation Context ...eas in human communities. A well studied rumour spreading protocol is the (synchronous) push&pull protocol, introduced by Demers, Greene, Hauser, Irish, Larson, Shenker, Sturgis, Swinehart, and Terry =-=[45]-=- and popularized by Karp, Schindelhauer, Shenker, and Vöcking [82]. Suppose that one node in a network is aware of a piece of information, the ‘rumour’, and wants to spread it to all nodes quickly. T... |

532 | Randomized gossip algorithms
- Boyd, Ghosh, et al.
- 2006
(Show Context)
Citation Context ... See [45] for details. Other than the aforementioned applications, rumour spreading protocols have successfully been applied in various contexts such as resource discovery [75], distributed averaging =-=[24]-=-, data aggregation [83], and the spread of computer viruses [12]. A discussion of these and other applications can be found in Keshav [84]. In the second part of the thesis we study two randomized rum... |

468 | Gossip-based computation of aggregate information
- Kempe, Dobra, et al.
(Show Context)
Citation Context ...Other than the aforementioned applications, rumour spreading protocols have successfully been applied in various contexts such as resource discovery [75], distributed averaging [24], data aggregation =-=[83]-=-, and the spread of computer viruses [12]. A discussion of these and other applications can be found in Keshav [84]. In the second part of the thesis we study two randomized rumour spreading protocols... |

380 | Markov chains and mixing times - Levin, Peres, et al. - 2009 |

330 |
A survey of gossiping and broadcasting in communication networks
- Hedetniemi, Hedetniemi, et al.
- 1988
(Show Context)
Citation Context ...uting in such networks, of which we mention just two. The first is in broadcasting algorithms: a single processor wants to broadcast a piece of information to all other processors in the network (see =-=[76]-=- for a survey). There are at least four advantages to the push&pull protocol: it puts much less load on the edges than naive flooding, it is simple (each node makes 8 a simple local decision in each r... |

304 | Graph structure in the web
- Broder, Kumar, et al.
- 2000
(Show Context)
Citation Context ... web pages,1 and there is an edge joining two vertices if there is a hyperlink in the first page pointing to the second page. Broder, Kumar, Maghoul, Raghavan, Rajagopalan, Stata, Tomkins, and Wiener =-=[26]-=- in 1999 crawled about 200 million web pages and found that the expected shortest-directed-path distance between two random web pages (when a path exists at all) is 16.18; this figure is 6.83 in the c... |

289 | Stochastic models for the web graph.
- Kumar, Raghavan, et al.
- 2000
(Show Context)
Citation Context ...m graph models, and demonstrate it by proving logarithmic upper bounds for the diameters of a variety of models, including the following well known ones: the forest fire model [92], the copying model =-=[89]-=-, the PageRank-based selection model [108], the Aiello-Chung-Lu models [2], the generalized linear preference model [28], directed scalefree graphs [17], the Cooper-Frieze model [38], and random k-tre... |

286 | The average distance in random graphs with given expected degrees
- Chung, Lu
(Show Context)
Citation Context ...cording to a power law distribution with exponent β and study the resulting ‘power law random graph.’ We call this model the Chung-Lu model with exponent β. Let y = ∑ w2k/ ∑ wk. In 2001, Chung and Lu =-=[33]-=- introduced this model and proved that for β > 3, a.a.s. the average diameter is Θ(log n/ log y) and the diameter is Θ(log n), whereas for 2 < β < 3, a.a.s. the average diameter is O(log log n) and th... |

286 | Randomized rumor spreading
- Karp, Schindelhauer, et al.
- 2000
(Show Context)
Citation Context ...is the (synchronous) push&pull protocol, introduced by Demers, Greene, Hauser, Irish, Larson, Shenker, Sturgis, Swinehart, and Terry [45] and popularized by Karp, Schindelhauer, Shenker, and Vöcking =-=[82]-=-. Suppose that one node in a network is aware of a piece of information, the ‘rumour’, and wants to spread it to all nodes quickly. The protocol proceeds in rounds. In each round, every informed node ... |

266 | Graph evolution: Densification and shrinking diameters
- Leskovec, Kleinberg, et al.
(Show Context)
Citation Context ...cted ones. For directed graphs, since there is no guarantee that there is always a directed path between two given vertices, there is a question of how to define the diameter. We take the approach of =-=[92]-=- and ignore the edge directions when calculating the diameter. In other words, we define the diameter of directed graph as that of its underlying undirected graph. In this thesis we work with (weakly)... |

256 | On Distinguishing between Internet power law topology generators," - Bu, Towsley - 2002 |

231 |
Concentration, Probabilistic methods for algorithmic discrete mathematics
- McDiarmid
- 1998
(Show Context)
Citation Context ...r tail Chernoff bounds, respectively. See McDiarmid [97, Theorem 2.3] for proofs. P [X ≤ (1− ε)E [X]] ≤ exp(−ε2E [X] /2) . (2.2) P [X ≥ (1 + ε)E [X]] ≤ exp ( − ε 2E [X] 2 + 2ε/3 ) . (2.3) As noted in =-=[97]-=-, these inequalities are true also if X is a sum of arbitrary but independent indicator random variables. 27 Stirling’s approximation. We will use the following bounds for n!, known as Stirling’s appr... |

208 | Deeper inside pagerank
- Langville, Meyer
- 2003
(Show Context)
Citation Context ...raph (which is zero if there is no edge from u to v), and out-deg(u) denotes the out-degree of u. PageRank is used as a ranking mechanism in Google [25]. More details and applications can be found in =-=[91]-=-. It will be convenient to have an ‘algorithmic’ definition for PageRank, namely, a way to sample from the PageRank distribution. This is achieved by the following proposition. Proposition 2.25. If we... |

207 |
Urn Models and Their Applications,
- Johnson, Kotz
- 1977
(Show Context)
Citation Context ...an networks and random k-trees (Chapters 4 and 6), which are evolving random graphs with reinforcement. In this section we give some definitions and review some relevant results. See Johnson and Kotz =-=[81]-=- for a general introduction, and Mahmoud [96] for a more recent survey. Definition 2.7 (Eggenberger-Pólya urn). Start with w0 white and b0 black balls in the urn. In every step a ball is drawn from t... |

200 |
Organization of growing random networks
- Krapivsky, Redner
- 2001
(Show Context)
Citation Context ...ls we studied in Chapter 3: in every step a new vertex arrives and is joined to one or more vertices of the existing graph. In the preferential attachment model and most of its variations (see, e.g., =-=[9, 51, 54, 88]-=-) the probability that the new vertex attaches to an old vertex v, called the attraction of v, is proportional to a deterministic function of the degree of v. In other variations (see, e.g., [14, 61])... |

180 | The spectra of random graphs with given expected degrees
- Chung, Lu, et al.
(Show Context)
Citation Context ...erwald [69] proved the same upper bound (up to constant factors) for the spread time on Chung-Lu graphs. It is known that a.a.s. Barabási-Albert graphs and Chung-Lu graphs have conductance Ω(1) (see =-=[36, 103]-=-). So it is not surprising that rumours spread fast on these graphs. All the above results assumed a synchronized model, i.e. all nodes take action simultaneously at discrete time steps. In many appli... |

179 | The phase transition in inhomogeneous random graphs. Random Structures Algorithms. 2007; 31(1):3–122. doi: 10.1002/rsa.20168
- Bollobás, Janson, et al.
(Show Context)
Citation Context ...e graph.) An important non-evolving model that has been studied extensively is the random graph model with given expected degrees. (A more general version was studied by Bollobás, Janson and Riordan =-=[18]-=- under the name ‘inhomogenous random graphs.’) Given positive numbers w1, w2, . . . , wn satisfying ∑ wk ≥ max{w2k}, we build a random graph G = G(w) as follows. The vertices are numbered 1 to n, and ... |

171 |
The Diameter of Scale-Free Random Graphs
- Bollobás, Riordan
(Show Context)
Citation Context ...i and Albert [9] in 1999 introduced one of the first evolving models for realworld networks, sometimes known as the preferential attachment model. Their model can be roughly described as follows (see =-=[20]-=- for a formal definition). We start with a fixed small graph, and in each time-step a new vertex appears and is joined to a fixed number of old vertices, where the probability of joining to each old v... |

132 | Graph mining: Laws, generators, and algorithms - Chakrabarti, Faloutsos |

117 | A general model of web graphs.
- Cooper, Frieze
- 2003
(Show Context)
Citation Context ...the copying model [89], the PageRank-based selection model [108], the Aiello-Chung-Lu models [2], the generalized linear preference model [28], directed scalefree graphs [17], the Cooper-Frieze model =-=[38]-=-, and random k-trees [71]. This means that in each of these models, with probability close to 1, for every pair (u, v) of vertices there exists a very short (u, v)-path, a path connecting u and v whos... |

115 | Competition and multiscaling in evolving networks
- Bianconi, Barabási
(Show Context)
Citation Context ..., 54, 88]) the probability that the new vertex attaches to an old vertex v, called the attraction of v, is proportional to a deterministic function of the degree of v. In other variations (see, e.g., =-=[14, 61]-=-) the attraction also depends on the so-called ‘fitness’ of v, which is a random variable generated independently for each vertex and does not depend on the structure of the graph. For analyzing such ... |

110 | Complex networks and decentralized search algorithms
- Kleinberg
- 2006
(Show Context)
Citation Context ... this thesis however, we are not concerned about the clustering coefficients of graphs. Also, we do not discuss the algorithmic aspects of the small-world phenomenon, which are discussed by Kleinberg =-=[85]-=-. All models we consider are probabilistic: due to the complexity of real-world graphs, probabilistic modelling seems inevitable. Consequently, in the analysis we mostly use probabilistic tools (Chapt... |

105 | Random evolution in massive graphs.
- Aiello, Chung, et al.
- 2002
(Show Context)
Citation Context ...the diameters of a variety of models, including the following well known ones: the forest fire model [92], the copying model [89], the PageRank-based selection model [108], the Aiello-Chung-Lu models =-=[2]-=-, the generalized linear preference model [28], directed scalefree graphs [17], the Cooper-Frieze model [38], and random k-trees [71]. This means that in each of these models, with probability close t... |

104 |
Treewidth, Computations and Approximations.
- Kloks
- 1994
(Show Context)
Citation Context ...m k-tree is its treewidth. Roughly speaking, the treewidth of a graph measures its similarity to a tree: the treewidth of a tree is 1, and if the treewidth of a graph is small, it is ‘tree-like’ (see =-=[86]-=- for a comprehensive survey). Gao [72] proved that many random graph models, including Erdős-Rényi random graphs with expected degree ω(log n) and preferential attachment graphs with out-degree grea... |

98 | Probability and measure. Wiley Series in Probability and Mathematical Statistics - Billingsley - 1995 |

96 | Resource discovery in distributed networks.
- Harchol-balter, Leighton, et al.
- 1999
(Show Context)
Citation Context ...nverge to the same contents. See [45] for details. Other than the aforementioned applications, rumour spreading protocols have successfully been applied in various contexts such as resource discovery =-=[75]-=-, distributed averaging [24], data aggregation [83], and the spread of computer viruses [12]. A discussion of these and other applications can be found in Keshav [84]. In the second part of the thesis... |

85 | On certain connectivity properties of the internet topology
- Mihail, Papadimitriou, et al.
(Show Context)
Citation Context ...erwald [69] proved the same upper bound (up to constant factors) for the spread time on Chung-Lu graphs. It is known that a.a.s. Barabási-Albert graphs and Chung-Lu graphs have conductance Ω(1) (see =-=[36, 103]-=-). So it is not surprising that rumours spread fast on these graphs. All the above results assumed a synchronized model, i.e. all nodes take action simultaneously at discrete time steps. In many appli... |

74 | Directed scale-free graphs
- Bollobás, Borgs, et al.
- 2003
(Show Context)
Citation Context ...: the forest fire model [92], the copying model [89], the PageRank-based selection model [108], the Aiello-Chung-Lu models [2], the generalized linear preference model [28], directed scalefree graphs =-=[17]-=-, the Cooper-Frieze model [38], and random k-trees [71]. This means that in each of these models, with probability close to 1, for every pair (u, v) of vertices there exists a very short (u, v)-path, ... |

66 | On the spread of viruses on the internet.
- Berger, Borgs, et al.
- 2005
(Show Context)
Citation Context ...s, rumour spreading protocols have successfully been applied in various contexts such as resource discovery [75], distributed averaging [24], data aggregation [83], and the spread of computer viruses =-=[12]-=-. A discussion of these and other applications can be found in Keshav [84]. In the second part of the thesis we study two randomized rumour spreading protocols, namely the synchronous push&pull protoc... |

60 |
Random Graph Dynamics. Cambridge
- Durrett
- 2007
(Show Context)
Citation Context ... social networks, the Webgraph, biological networks, etc., in recent years a great deal of research has been built around studying mathematical properties of real world networks (see, e.g., the books =-=[22, 30, 35, 56]-=-). Another fascinating observation on many real-world graphs is that their degree sequences are heavy-tailed and almost obey a power law. Namely, for each positive integer k, the fraction of vertices ... |

60 | A Geometric Preferential Attachment Model of Networks
- Flaxman, Frieze, et al.
(Show Context)
Citation Context ...Other evolving models whose diameters have been studied in the literature include the Fabrikant-Koutsoupias-Papadimitriou model [11], protean graphs [112], the geometric preferential attachment model =-=[66, 94]-=-, and the spatial preferred attachment model [40]. See [18, Section 14] and [117] for collections of results on diameters of non-evolving models. 1.1.2 Chapter 3 There are dozens of research papers in... |

58 |
Uber die statistik verketteter vorgange. Zeitschrift fur Angewandte Mathematik und Mechanik,
- Eggenberger, Polya
- 1923
(Show Context)
Citation Context ... Urn(w0, b0, s, n) denote the number of white balls right after n draws. In Chapter 7 we will consider more complicated urns, see Definition 7.9. The following result is due to Eggenberger and Pólya =-=[58]-=- (see, e.g., Mahmoud [96, Theorem 5.1.2]). Theorem 2.8 (limiting distribution in Eggenberger-Pólya urns [58]). For any α ∈ [0, 1] we have lim n→∞ P [ Urn(a, b, s, n) sn < α ] =P [ Beta ( w s , b s ) ... |

56 | Four degrees of separation.
- Backstrom, Boldi, et al.
- 2011
(Show Context)
Citation Context ...ected-path distance between two random web pages (when a path exists at all) is 16.18; this figure is 6.83 in the corresponding underlying undirected graph. Backstrom, Boldi, Rosa, Ugander, and Vigna =-=[8]-=- studied the Facebook graph in May 2011, which had about 721 million vertices. The vertices of this graph are people, and two of them are joined by an edge if they are friends on Facebook. The diamete... |

53 | Limit theorems for triangular urn schemes. Probability Theory and Related Fields,
- Janson
- 2006
(Show Context)
Citation Context ...ur in certain triangular urn models. In our proofs, we exploit these connections and apply some recent results on triangular urn models from the work of Flajolet, Dumas and Puyhaubert [65] and Janson =-=[79]-=-. Our second main result is the following theorem, which gives a polynomial lower bound for the spread time. Theorem 7.5. Let k ≥ 2 be fixed and let f(n) = o(log log n) be an arbitrary function going ... |

48 | Some exactly solvable models of urn process theory, Discrete Mathematics and Computer Science,
- Flajolet, Dumas, et al.
- 2006
(Show Context)
Citation Context ... of a given colour in certain triangular urn models. In our proofs, we exploit these connections and apply some recent results on triangular urn models from the work of Flajolet, Dumas and Puyhaubert =-=[65]-=- and Janson [79]. Our second main result is the following theorem, which gives a polynomial lower bound for the spread time. Theorem 7.5. Let k ≥ 2 be fixed and let f(n) = o(log log n) be an arbitrary... |

42 | A survey of models of the web graph. - Bonato, Laurier - 2004 |

40 |
Models of first-passage percolation.
- Howard
- 2004
(Show Context)
Citation Context ...del has so far received less attention. Rumour spreading protocols in this model turn out to be closely related to Richardson’s model for the spread of a disease [55] and to first-passage percolation =-=[77]-=- with edges having i.i.d. exponential weights. The main difference is that in rumour spreading protocols each vertex contacts one neighbour at a time. So, for instance in the push protocol, the net co... |

38 |
Randomized broadcast in networks. Random Structures and Algorithms
- Feige, Peleg, et al.
- 1990
(Show Context)
Citation Context ... independent of the size of network: it does not grow more complex as the network grows) and robust (the protocol tolerates random node/link failures without the use of error recovery mechanisms, see =-=[62]-=-). A second application comes from the maintenance of databases replicated at many sites, e.g., yellow pages, name servers, or server directories. There are updates injected at various nodes, and thes... |

33 | Coupling online and offline analyses for random power law graphs
- Chung, Lu
(Show Context)
Citation Context ...ph [30], or [29, Table III], or the table in [21, p. 162]: each cited table contains a summary of known results on the diameter and other properties of several real-world network models. Chung and Lu =-=[34]-=- defined an evolving (online) and a non-evolving (offline) model. They state that ‘The online model is obviously much harder to analyze than the offline model’, and hence analyze the former by couplin... |

32 | Tight bounds for rumor spreading in graphs of a given conductance.
- Giakkoupis
- 2011
(Show Context)
Citation Context ...he graph. Let Φ(G) and α(G) denote the conductance and the vertex expansion of a graph G, respectively (see Section 7.2 for the definitions). After a series of results by various scholars, Giakkoupis =-=[73, 74]-=- showed the spread time is O (min{Φ(G)−1 · log n, α(G)−1 · log ∆(G) · log n}). This protocol has recently been used to model news propagation in social networks. Doerr, Fouz, and Friedrich [48] proved... |

30 |
Complex graphs and networks, volume 107
- Chung, Lu
- 2006
(Show Context)
Citation Context ... social networks, the Webgraph, biological networks, etc., in recent years a great deal of research has been built around studying mathematical properties of real world networks (see, e.g., the books =-=[22, 30, 35, 56]-=-). Another fascinating observation on many real-world graphs is that their degree sequences are heavy-tailed and almost obey a power law. Namely, for each positive integer k, the fraction of vertices ... |

30 | Large deviations, volume 14 of Fields Institute Monographs - Hollander - 2000 |

29 | Social networks spread rumors in sublogarithmic time.
- Doerr, Fouz, et al.
- 2011
(Show Context)
Citation Context ...is [73, 74] showed the spread time is O (min{Φ(G)−1 · log n, α(G)−1 · log ∆(G) · log n}). This protocol has recently been used to model news propagation in social networks. Doerr, Fouz, and Friedrich =-=[48]-=- proved an upper bound of O(log n) for the spread time on Barabási-Albert graphs, and Fountoulakis, Panagiotou, and Sauerwald [69] proved the same upper bound (up to constant factors) for the spread ... |

24 | On the runtime and robustness of randomized broadcasting - Elsässer, Sauerwald |

22 | Large deviations for the weighted height of an extended class of trees. Algorithmica 46: 271–297
- Broutin, Devroye
- 2006
(Show Context)
Citation Context .... We remark that although the models studied in this chapter are quite different from RANs, surprisingly the same engine is used for proving these results, namely the technique of Broutin and Devroye =-=[27]-=-. Our two applications of this technique demonstrates its flexibility. 1.2 Randomized rumour spreading Randomized rumour spreading is an important primitive for information dissemination in networks a... |

21 |
Adversarial deletion in a scale-free random graph process.
- Flaxman, Frieze, et al.
- 2007
(Show Context)
Citation Context ...roof technique we introduced in Chapter 3 could be a fundamental step in building this theory. One can try to further develop this technique to cover other network models, e.g. growth-deletion models =-=[34, 42]-=-, accelerated network growth models [52], and spatial models [80]. In this thesis we only considered growth models: vertices and edges are never deleted from the graph. In most real-world networks, ho... |

20 |
L.R.: Apollonian networks: simultaneously scale-free, small world Euclidean, space filling, and with matching graphs Phys
- Andrade, Herrmann, et al.
(Show Context)
Citation Context ...gure 4.1 for an illustration. The term ‘Apollonian network’ refers to a deterministic version of this process, formed by subdividing all triangles the same number of times, which was first studied in =-=[6, 53]-=-. Andrade, Herrmann, Andrade, and Silva [6] studied power laws in the degree sequences of these networks. Random Apollonian networks were defined by Zhou, Yan and Wang [120] 1This chapter is based on ... |

20 | A Preferential Attachment Model with Random Initial Degrees. Arkiv for Matematik, 47:41–72
- Deijfen, Esker, et al.
- 2009
(Show Context)
Citation Context ...ally imply logarithmic upper bounds for average diameters of these models. We also prove polylogarithmic upper bounds for the diameter of the preferential attachment model with random initial degrees =-=[44]-=- in the case that the initial degrees’ distribution has an exponential decay. Prior to this work no sublinear upper bound was known even for the average diameter of any of these models. (This claim ca... |

20 | Balls and bins models with feedback. In:
- Drinea, Frieze, et al.
- 2002
(Show Context)
Citation Context ...ls we studied in Chapter 3: in every step a new vertex arrives and is joined to one or more vertices of the existing graph. In the preferential attachment model and most of its variations (see, e.g., =-=[9, 51, 54, 88]-=-) the probability that the new vertex attaches to an old vertex v, called the attraction of v, is proportional to a deterministic function of the degree of v. In other variations (see, e.g., [14, 61])... |

19 | A random-surfer web-graph model”,
- Blum, Rwebangira
- 2006
(Show Context)
Citation Context ...el for the Webgraph, in which the main idea is that a new web page prefers to link to web pages that have higher PageRanks (see Section 2.6 for the definition of PageRank). Blum, Chan, and Rwebangira =-=[16]-=- introduced a random-surfer model for the Webgraph, in which the links of a new web page are chosen by doing independent random walks that start from random web pages and whose lengths 7 are geometric... |

19 | Reliable broadcasting in random networks and the effect of density
- Fountoulakis, Huber, et al.
(Show Context)
Citation Context ....a.s. the spread time of the push protocol is Ω(log n) and O(∆(G)·(diam(G)+log n)). This protocol has been studied on many graph classes such as complete graphs [62, 110], Erdős-Rényi random graphs =-=[62, 67, 106]-=-, random regular graphs [10, 68], and hypercube graphs [62]. For most of these classes it turns out that a.a.s. the spread time is Θ(diam(G) + log n), which does not depend on the maximum degree. Inte... |

18 | Accelerated growth on networks
- Dorogovtsev, Mendes
- 2003
(Show Context)
Citation Context ...uld be a fundamental step in building this theory. One can try to further develop this technique to cover other network models, e.g. growth-deletion models [34, 42], accelerated network growth models =-=[52]-=-, and spatial models [80]. In this thesis we only considered growth models: vertices and edges are never deleted from the graph. In most real-world networks, however, deletions exist. Growth-deletion ... |

18 |
Rumor spreading on random regular graphs and expanders.
- Fountoulakis, Panagiotou
- 2010
(Show Context)
Citation Context ...tocol is Ω(log n) and O(∆(G)·(diam(G)+log n)). This protocol has been studied on many graph classes such as complete graphs [62, 110], Erdős-Rényi random graphs [62, 67, 106], random regular graphs =-=[10, 68]-=-, and hypercube graphs [62]. For most of these classes it turns out that a.a.s. the spread time is Θ(diam(G) + log n), which does not depend on the maximum degree. Interesting connections between the ... |

18 | Ultra-fast rumor spreading in social networks
- Fountoulakis, Panagiotou, et al.
- 2012
(Show Context)
Citation Context ...odel news propagation in social networks. Doerr, Fouz, and Friedrich [48] proved an upper bound of O(log n) for the spread time on Barabási-Albert graphs, and Fountoulakis, Panagiotou, and Sauerwald =-=[69]-=- proved the same upper bound (up to constant factors) for the spread time on Chung-Lu graphs. It is known that a.a.s. Barabási-Albert graphs and Chung-Lu graphs have conductance Ω(1) (see [36, 103]).... |

18 | One, two and three times log n/n for paths in a complete graph with random weights.
- Janson
- 1999
(Show Context)
Citation Context ...re chosen. In this sense, Fill and Pemantle [64] and Bollobás and Kohayakawa [19] showed that a.a.s. the spread time of the asynchronous push&pull protocol is Θ(log n) on the hypercube graph. Janson =-=[78]-=- and Amini, Draief and Lelarge [5] showed the same results (up to constant factors) for the complete graph and for random regular graphs, respectively. These bounds match the same order of magnitude a... |

17 | Urn models and connections to random trees: A review,
- Mahmoud
- 2003
(Show Context)
Citation Context ...d 6), which are evolving random graphs with reinforcement. In this section we give some definitions and review some relevant results. See Johnson and Kotz [81] for a general introduction, and Mahmoud =-=[96]-=- for a more recent survey. Definition 2.7 (Eggenberger-Pólya urn). Start with w0 white and b0 black balls in the urn. In every step a ball is drawn from the urn uniformly at random, the ball is retur... |

16 | Diameters in preferential attachment models.
- Dommers, Hofstad, et al.
- 2010
(Show Context)
Citation Context ... where d ∈ N is fixed. If d = 1 and δ ≥ 0, Pittel [111] showed the diameter is Θ(log n). If d > 1 and δ ∈ (−d, 0), the diameter is Θ(log log n) as proved by Dommers, van der Hofstad, and Hooghiemstra =-=[51, 117]-=-. If d > 1 and δ = 0, the diameter is Θ(log n/ log log n), see Bollobás and Riordan [20]. Finally, if d > 1 and δ > 0, the diameter is Θ(log n) [51, 117]. Remark 3.21. Chung and Lu [34] studied a var... |

16 | Further inequalities for the gamma function - Laforgia - 1984 |

15 |
C.P.: Self-similar disk packings as model spatial scale-free networks Phys
- Doye, Massen
(Show Context)
Citation Context ...gure 4.1 for an illustration. The term ‘Apollonian network’ refers to a deterministic version of this process, formed by subdividing all triangles the same number of times, which was first studied in =-=[6, 53]-=-. Andrade, Herrmann, Andrade, and Silva [6] studied power laws in the degree sequences of these networks. Random Apollonian networks were defined by Zhou, Yan and Wang [120] 1This chapter is based on ... |

14 | Some families of increasing planar maps
- Albenque, Marckert
- 2008
(Show Context)
Citation Context ...o the vertices on the face. After n − 3 steps, we obtain a (random) triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). See Figure 1.1 for an illustration. In =-=[3]-=- it was shown that a.a.s. the average diameter of a RAN is asymptotic to η1 log n, where η1 = 6/11 ≈ 0.545. Frieze and Tsourakakis [70] showed that the diameter of a RAN is a.a.s. at most η2 log n, wh... |

14 | Percolation, first-passage percolation and covering times for Richardson’s model on the n-cube
- Fill, Pemantle
- 1993
(Show Context)
Citation Context ..., the asynchronous push&pull protocol, Richardson’s model, and first-passage percolation are essentially the same process, assuming appropriate parameters are chosen. In this sense, Fill and Pemantle =-=[64]-=- and Bollobás and Kohayakawa [19] showed that a.a.s. the spread time of the asynchronous push&pull protocol is Θ(log n) on the hypercube graph. Janson [78] and Amini, Draief and Lelarge [5] showed th... |

14 | The degree distribution of random k-trees,
- Gao
- 2009
(Show Context)
Citation Context ...networks tend to have large clustering coefficients (see, e.g., [118, Table 1]). 11 Figure 1.2: the construction of an instance of a random 2-tree with 13 vertices (last few steps are not shown.) Gao =-=[71]-=- proved that a.a.s. the degree sequence of a random k-tree asymptotically follows a power law distribution with exponent 2 + 1 k−1 . In Chapter 3 we show that a.a.s. its diameter is O(log n). We also ... |

12 | Efficient randomised broadcasting in random regular networks with applications in peer-to-peer systems
- Berenbrink, Elsässer, et al.
- 2008
(Show Context)
Citation Context ...tocol is Ω(log n) and O(∆(G)·(diam(G)+log n)). This protocol has been studied on many graph classes such as complete graphs [62, 110], Erdős-Rényi random graphs [62, 67, 106], random regular graphs =-=[10, 68]-=-, and hypercube graphs [62]. For most of these classes it turns out that a.a.s. the spread time is Θ(diam(G) + log n), which does not depend on the maximum degree. Interesting connections between the ... |

12 | Long cycles in 3-connected graphs - Chen, Yu |

12 |
New kid on the block: Exploring the Google+ social graph.
- Magno, Comarela, et al.
- 2012
(Show Context)
Citation Context ... of this graph was found to be 41, and the average distance between reachable pairs was found to be around 4.74. For other examples, see, e.g., Tables 1 and 2 in [4], Table 8.1 in [105] or Table 4 in =-=[95]-=-. Due to the ever growing interest in social networks, the Webgraph, biological networks, etc., in recent years a great deal of research has been built around studying mathematical properties of real ... |

11 |
On a characteristic property of Polya’s urn.
- Athreya
- 1969
(Show Context)
Citation Context ...rate a random variable distributed as Urn(w, b, s, n). Theorem 2.8 can be generalized to get the joint distribution for the proportion of balls of each colour: the following theorem is due to Athreya =-=[7]-=- (see also [81, page 378]). Theorem 2.10 (limiting joint distribution in multicolour Eggenberger-Pólya urns [7]). Consider an urn with k different colours. Suppose that initially there are ci > 0 bal... |

11 | Treewidth of Erdös-Rényi random graphs, random intersection graphs, and scale-free random graphs - Gao |

10 | Universal techniques to analyze preferential attachment trees: Global and Local analysis. Available online at http://www.unc.edu/~bhamidi/preferent.pdf,
- Bhamidi
- 2007
(Show Context)
Citation Context ... When the new vertex is joined to exactly one vertex in the existing graph (so the resulting evolving graph is always a tree), a general technique based on branching processes is developed by Bhamidi =-=[13]-=-, using which he proved the diameter of a variety of preferential attachment trees is a.a.s Θ(log n). (In this chapter n always denotes the number of vertices in the graph.) An important non-evolving ... |

9 | R.: Scale Free Properties of random k-trees
- Cooper, Uehara
- 2010
(Show Context)
Citation Context ...lique in a graph is a complete subgraph of size k.) See Figure 1.2 for an illustration with k = 2. We remark that this process is different from the random k-tree process defined by Cooper and Uehara =-=[43]-=- which was further studied in [41]. As in the preferential attachment scheme, the random k-tree process enjoys a ‘the rich get richer’ effect. Think of the number of k-cliques containing any vertex v ... |

8 | Cover Time and Broadcast Time,” in
- Elsasser, Sauerwald
- 2009
(Show Context)
Citation Context ...iam(G) + log n), which does not depend on the maximum degree. Interesting connections between the spread time and the cover time/mixing time of the simple random walk on the graph have been proved in =-=[59, 115]-=-. Fountoulakis et al. [69] studied the asynchronous push&pull protocol on Chung-Lu random graphs with exponent between 2 and 3. For these graphs, they showed that a.a.s. after some constant time, n − ... |

8 |
Growing random networks with fitness.
- Ergun, Rodgers
- 2002
(Show Context)
Citation Context ..., 54, 88]) the probability that the new vertex attaches to an old vertex v, called the attraction of v, is proportional to a deterministic function of the degree of v. In other variations (see, e.g., =-=[14, 61]-=-) the attraction also depends on the so-called ‘fitness’ of v, which is a random variable generated independently for each vertex and does not depend on the structure of the graph. For analyzing such ... |

8 | Spatial models for virtual networks
- Janssen
(Show Context)
Citation Context ...des lots of new ones, and will help in proving many of the forthcoming network models are small-world. We hope this theory will be developed further to cover other network models, e.g. spatial models =-=[80]-=-, as well. The results of this chapter shed light on why the small-world phenomenon is observed in so many real-world graphs. At their core, our arguments are based on the fact that in all considered ... |

7 | On Richardson’s Model on the Hypercube.
- Bollobas, Thomason
- 1997
(Show Context)
Citation Context ...col, Richardson’s model, and first-passage percolation are essentially the same process, assuming appropriate parameters are chosen. In this sense, Fill and Pemantle [64] and Bollobás and Kohayakawa =-=[19]-=- showed that a.a.s. the spread time of the asynchronous push&pull protocol is Θ(log n) on the hypercube graph. Janson [78] and Amini, Draief and Lelarge [5] showed the same results (up to constant fac... |

7 |
Tight bounds for rumor spreading with vertex expansion
- Giakkoupis
- 2014
(Show Context)
Citation Context ...he graph. Let Φ(G) and α(G) denote the conductance and the vertex expansion of a graph G, respectively (see Section 7.2 for the definitions). After a series of results by various scholars, Giakkoupis =-=[73, 74]-=- showed the spread time is O (min{Φ(G)−1 · log n, α(G)−1 · log ∆(G) · log n}). This protocol has recently been used to model news propagation in social networks. Doerr, Fouz, and Friedrich [48] proved... |

7 | Efficient and decentralized computation of approximate global state
- Keshav
- 2006
(Show Context)
Citation Context ...ntexts such as resource discovery [75], distributed averaging [24], data aggregation [83], and the spread of computer viruses [12]. A discussion of these and other applications can be found in Keshav =-=[84]-=-. In the second part of the thesis we study two randomized rumour spreading protocols, namely the synchronous push&pull protocol and the asynchronous push&pull protocol. We investigate their spread ti... |

6 | Flooding in weighted sparse random graphs,
- Amini, Draief, et al.
- 2013
(Show Context)
Citation Context ...Pemantle [64] and Bollobás and Kohayakawa [19] showed that a.a.s. the spread time of the asynchronous push&pull protocol is Θ(log n) on the hypercube graph. Janson [78] and Amini, Draief and Lelarge =-=[5]-=- showed the same results (up to constant factors) for the complete graph and for random regular graphs, respectively. These bounds match the same order of magnitude as in the synchronized case. Doerr,... |

6 |
PageRank and the random surfer model”,
- Chebolu, Melsted
- 2008
(Show Context)
Citation Context ...he Webgraph, in which the links of a new web page are chosen by doing independent random walks that start from random web pages and whose lengths 7 are geometric random variables. Chebolu and Melsted =-=[31]-=- showed that under certain conditions, the previous two models are equivalent. See Chapter 5 for details. The diameter of the Barabási -Albert model was analyzed by Bollobás and Riordan [20]. Previo... |

6 | Some typical properties of the Spatial Preferred Attachment model
- Cooper, Frieze, et al.
(Show Context)
Citation Context ...ed in the literature include the Fabrikant-Koutsoupias-Papadimitriou model [11], protean graphs [112], the geometric preferential attachment model [66, 94], and the spatial preferred attachment model =-=[40]-=-. See [18, Section 14] and [117] for collections of results on diameters of non-evolving models. 1.1.2 Chapter 3 There are dozens of research papers in which models for complex networks have been defi... |

6 | Asynchronous rumor spreading in preferential attachment graphs
- Doerr, Fouz, et al.
- 2012
(Show Context)
Citation Context ...push&pull protocol on Chung-Lu random graphs with exponent between 2 and 3. For these graphs, they showed that a.a.s. after some constant time, n − o(n) nodes are informed. Doerr, Fouz, and Friedrich =-=[49]-=- showed that for the preferential attachment graph (the non-tree case), a.a.s. all but o(n) vertices receive the rumour in time O (√ log n ) , but to inform all vertices a.a.s., Θ(log n) time is neces... |

5 | The height of random k-trees and related branching processes. arXiv, 1309.4342v2 [math.CO
- Cooper, Frieze
- 2013
(Show Context)
Citation Context ...graph of size k.) See Figure 1.2 for an illustration with k = 2. We remark that this process is different from the random k-tree process defined by Cooper and Uehara [43] which was further studied in =-=[41]-=-. As in the preferential attachment scheme, the random k-tree process enjoys a ‘the rich get richer’ effect. Think of the number of k-cliques containing any vertex v as the ‘wealth’ of v (note that th... |

4 |
Degree distribution of the FKP network model. Theoret
- Berger, Bollobás, et al.
(Show Context)
Citation Context ...a.s. the average diameter is O(log log n) and the diameter is Θ(log n). Other evolving models whose diameters have been studied in the literature include the Fabrikant-Koutsoupias-Papadimitriou model =-=[11]-=-, protean graphs [112], the geometric preferential attachment model [66, 94], and the spatial preferred attachment model [40]. See [18, Section 14] and [117] for collections of results on diameters of... |

4 | Depth properties of scaled attachment random recursive trees. Random Structures Algorithms
- Devroye, Fawzi, et al.
(Show Context)
Citation Context ... the weighted height of the auxiliary tree is at most η log n. We work with the tree in the rest of the proof. Let us consider an alternative way to grow the tree, used by Devroye, Fawzi, and Fraiman =-=[47]-=-, which results in the same distribution. Let U1, U2, . . . be i.i.d. uniform random variables in (0, 1). Then for each new vertex vs, we attach it to the vertex vbsUsc, which is indeed a vertex unifo... |

4 | On the longest paths and the diameter in random apollonian networks
- Ebrahimzadeh, Farczadi, et al.
- 2013
(Show Context)
Citation Context ...ltaneously by three groups who presented it obliviously at the same conference (Random Structures and Algorithms, Poland, August 2013)! The three proofs are quite different. Our proof has appeared in =-=[57]-=- (see [41, Theorem 2] and [87, Theorem 2.2] for the other proofs). We also study the length of longest simple paths in RANs. Let Lm be a random variable denoting the number of vertices in a longest pa... |

4 | Degrees and distances in random and evolving Apollonian networks http://arxiv.org/pdf/1310.3864v1.pdf - Kolossváry, Komjáthy, et al. |

4 |
The small-community phenomenon in networks
- Li, Peng
(Show Context)
Citation Context ...Other evolving models whose diameters have been studied in the literature include the Fabrikant-Koutsoupias-Papadimitriou model [11], protean graphs [112], the geometric preferential attachment model =-=[66, 94]-=-, and the spatial preferred attachment model [40]. See [18, Section 14] and [117] for collections of results on diameters of non-evolving models. 1.1.2 Chapter 3 There are dozens of research papers in... |

3 |
A course on the web graph, volume 89
- Bonato
- 2008
(Show Context)
Citation Context ... social networks, the Webgraph, biological networks, etc., in recent years a great deal of research has been built around studying mathematical properties of real world networks (see, e.g., the books =-=[22, 30, 35, 56]-=-). Another fascinating observation on many real-world graphs is that their degree sequences are heavy-tailed and almost obey a power law. Namely, for each positive integer k, the fraction of vertices ... |

3 | Experimental analysis of rumor spreading in social networks
- Doerr, Fouz, et al.
- 2012
(Show Context)
Citation Context ...results (up to constant factors) for the complete graph and for random regular graphs, respectively. These bounds match the same order of magnitude as in the synchronized case. Doerr, Fouz, Friedrich =-=[50]-=- experimentally compared the spread time in the two time models. They state that ‘Our experiments show that the asynchronous model is faster on all graph classes [considered here].’ However, a general... |

3 |
and C.E Tsourakakis. Some properties of random apollonian networks
- Frieze
- 2013
(Show Context)
Citation Context ... Apollonian Network (RAN). See Figure 1.1 for an illustration. In [3] it was shown that a.a.s. the average diameter of a RAN is asymptotic to η1 log n, where η1 = 6/11 ≈ 0.545. Frieze and Tsourakakis =-=[70]-=- showed that the diameter of a RAN is a.a.s. at most η2 log n, where η2 ≈ 7.081 is the unique solution greater than 1 6 Figure 1.1: the construction of an instance of a RAN with seven vertices of exp ... |

2 | Long paths in random Apollonian networks. arXiv, 1403.1472v1 [math.PR
- Cooper, Frieze
- 2014
(Show Context)
Citation Context ...oting the number of vertices in a longest path in a RAN with m faces (and (m+ 5)/2 vertices). In [57] we showed there exists a fixed δ > 0 such that P [Lm < m/(logm)δ]→ 1. Recently, Cooper and Frieze =-=[39]-=- improved this by showing that for every constant c < 2/3, we have P [Lm ≤ m exp(− logcm)] → 1, and conjectured there exists a fixed δ < 1 such that P [Lm ≤ mδ] → 1. In Chapter 4 we confirm this conje... |

2 | Its a Small World for Random Surfers
- Mehrabian, Wormald
- 2014
(Show Context)
Citation Context ... n)/p. In Theorem 5.8 we show the same conclusion holds for the PageRank-based selection Webgraph model. These results were proved in collaboration with Wormald and appear in the submitted manuscript =-=[101]-=-, an extended abstract of which has been published [102]. 7. In Theorems 5.3 and 5.4, we determine the a.a.s. asymptotic value of the height and diameter of a random surfer tree with parameter p in th... |

1 |
On the push&pull protocol for rumour spreading. arXiv, 1411.0948 [cs.DC
- Acan, Collevecchio, et al.
- 2014
(Show Context)
Citation Context ...e show these bounds are asymptotically best possible, up to the constant factors. These results were proved in collaboration with Acan, Collevecchio, and Wormald, and appear in the submitted preprint =-=[1]-=-. 14 9. In Theorem 6.4 we show the following hold for any connected graph G: (1− 1/n) wasts(G) ≤ gsts(G) ≤ ewasts(G) log n , wasts(G) = O(n), and gsts(G) = O(n log n) . Here, wasts and gsts denote the... |

1 | Complex networks
- Bonato, Chung
- 2013
(Show Context)
Citation Context ...andom graphs do not satisfy this property, scholars have defined lots of models recently, aiming at capturing the aforementioned and other properties of real-world graphs (see, e.g., Bonato and Chung =-=[23]-=- or Chakrabarti and Faloutsos [30, Part II] and the references therein). Lots of mathematical models have been defined so far, yet very few rigorously analyzed. The diameter of an undirected graph is ... |

1 |
Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees. arXiv, 1404.2425 [math.PR
- Collevecchio, Mehrabian, et al.
- 2014
(Show Context)
Citation Context ...ove that a.a.s. a random Apollonian network does not contain a path of length n0.99999996. This result was proved in collaboration with Collevecchio and Wormald, and appears in the submitted preprint =-=[37]-=-. 6. In Theorem 5.2 we show that a.a.s. the diameter of the random-surfer Webgraph model with parameters p and d is at most 8ep(log n)/p. In Theorem 5.8 we show the same conclusion holds for the PageR... |

1 |
Stochastic growth models: recent results and open problems. In Mathematical approaches to problems in resource management and epidemiology
- Durrett
- 1987
(Show Context)
Citation Context ...ng problem in the asynchronous time model has so far received less attention. Rumour spreading protocols in this model turn out to be closely related to Richardson’s model for the spread of a disease =-=[55]-=- and to first-passage percolation [77] with edges having i.i.d. exponential weights. The main difference is that in rumour spreading protocols each vertex contacts one neighbour at a time. So, for ins... |

1 | Justifying the small-world phenomenon via random recursive trees. arXiv, 1410.6397 [cs.DM
- Mehrabian
- 2014
(Show Context)
Citation Context ...nd 3.27), the generalized linear preference model (Theorem 3.22), directed scale-free graphs (Theorem 3.32), and the Cooper-Frieze model (Theorem 3.34). These results appear in the submitted preprint =-=[98]-=-. 2. In Theorem 3.24 we prove for the preferential attachment model with random initial degrees in the case that the initial degrees’ distribution has an exponential decay, that, a.a.s. the diameter i... |

1 | Randomized rumor spreading in poorly connected small-world networks
- Mehrabian, Pourmiri
- 2014
(Show Context)
Citation Context ... rumour. In Theorem 7.6 we show the same conclusion holds in a random k-Apollonian network with k ≥ 3. These results were proved in collaboration with Pourmiri, and appear in the submitted manuscript =-=[99]-=-, an extended abstract of which has been published [100]. 12. In Theorem 7.5 we show that a.a.s. the spread time of the synchronous push&pull protocol on a random k-tree with k ≥ 2 is at least n1/(5k)... |