R. Albert and A. Barab asi. Topology of evolving networks: Local events and universality. Physical Review Letters, 85:5234--5237, 2000.  Home/Search   Document Not in Database   Summary   APS   Related Articles   Check

.... topology generators; earlier approaches to router level topology generation incorporated transit networks that were essentially random graphs [12] With the discovery of power laws in the degree distribution of Internet graphs [17] there is renewed interest in topology generation [20] 22] [5], 25] 4] Our work can provide input to, or validate, router level generators that emerge from this activity. Our paper represents a first look at the topological and geographic structure of ASs. It combines the results of nearly four million traceroutes from six different vantage points (all ....

R. Albert and A.-L. Barabasi. Topology of Evolving Networks: Local Events and Universality . Physical Review Letters, 85:5234--5237, 2000.

....generators were unsuitable models for the Internet. Subsequently, there have been an increasing number of proposals for topology generators that are designed primarily to match the Internet s degree distribution and do not attempt to model the Internet s hierarchical structure; for example, see [23, 27, 2, 30, 1, 8]. These degree based topology generators embody the implicit assumption that it is more important to match a certain local property the degree distribution than to capture the large scale hierarchical structure of the Internet. The rapid adoption of these degree based generators suggests that ....

....2 Related Work We have already mentioned several important areas of related work: the Waxman, Transit Stub and Tiers topology generators, and Faloutsos et al. s observations of power law degree distributions in the Internet. We have also mentioned in passing several new degree based generators [23, 27, 2, 30, 1]. They all attempt to generate networks with power law degree distributions, but differ in the way in which nodes are connected. We describe some of these generators in slightly more detail in Appendix D. Perhaps closest in spirit to the work presented in this paper is the pioneering exploration ....

ALBERT, R., AND BARABASI, A.-L. Topology of Evolving Networks: Local Events and Universality . Physical Review Letters 85 (2000), 5234--5237.

....encoder and a decoder, we can use an indicator variable for each server pair and each session and obtain the same IP structure. A. A simulation study We generated networks based on the newly discovered power log law [13] Our generator is based on the algorithm suggested by Albert and Barabasi [14]. In all the generated network we picked the parameter to be # # # ###############. For each instance we generated ### sessions that are randomly generated according to two models, where # is the number of nodes in the network. In the uniform model, both session end points were selected uniformly ....

Reka Albert and Albert-Laszlo Barabasi, "Topology of evolving networks: local events and universality," Physical Review Letters, vol. 85, no. 24, pp. 5234--5237, 11 Dec. 2000.

....has motivated the development of new topology generators. However, much of the efforts to date has been either abstract (e.g. generate a graph with a given vertex degree distribution [1] 21] or based on some exogenously imposed mechanisms (e.g. a presumed preferential type connectivity rule [2], 24] While the resulting models and generators are generally successful in reproducing and matching the powerlaw type node degree distributions of measured AS graphs, their relevance to networking is seriously hampered by their generic nature they are mostly designed to model all types of ....

....to critical phase transition [6] our approach suggests a simple recipe for separating sound from specious claims and theories: use domain knowledge and check against appropriate measurements. For example, when comparing the scale free models for Internet growth at the AS level introduced in [8] [2] with our multivariate HOT model, we find that the former is void of any domain knowledge and can be easily refuted using available measurements about the network s historical evolution [12] 32] In contrast, the latter not only thrives on domain knowledge and incorporates it explicitly into the ....

R. Albert and A.-L. Barab asi. Topology of Evolving Networks: Local Events and Universality. Physical Review Letters, 85(24):5234--5237, 2000.

....of the Internet. While the exact nature of the Internet topology is in debate [4] it was found that many realistic networks posses a power law, or scale free degree distribution [11] These results were also verified by [12, 14, 5, 8] who conducted further investigations. Albert and Barabasi [2, 1] suggested a dynamic graph generation model that generates such networks. One of their main findings was the self similarity characteristic of such networks. Interestingly, empirical findings on partial views obtained similar results, which may lead to the assumption that due to the self ....

....characteristics of the depth rings around the root node of shortest path trees. All of our findings were also validated on real Internet data. 5.1 Topology and Tree Generation Our method for producing trees is the following. First, we generate power law topologies based on the Notre Dame model [1]. The model specifies 4 parameters: a 0 , a, p and q . Where a 0 is the initial number of detached nodes, and a is the initial connectivity of a node. When a link is added, one of its end points is chosen randomly, and the other with probability that is proportional to the nodes degree. This ....

[Article contains additional citation context not shown here]

R. Albert and A.-L. Barabasi. Topology of evolving networks: local events and universality. Physical Review Letters, 85(24):5234--5237, 11 Dec. 2000.

....and has motivated the development of new topology generators. However, much of the e#orts to date has been either abstract (e.g. generate a graph with a given vertex degree distribution [1, 20] or based on some exogenously imposed mechanisms (e.g. a presumed preferential type connectivity rule [2, 23]) While the resulting models and generators are generally successful in reproducing and matching the power law type node degree distributions of measured AS graphs, their relevance to networking is seriously hampered by their generic nature they are mostly designed to model all types of ....

....to critical phase transition [5] our approach suggests a simple recipe for separating sound from specious claims and theories: use domain knowledge and check against appropriate measurements. For example, when comparing the scale free models for Internet growth at the AS level introduced in [7, 2] with our multivariate HOT model, we find that the former is void of any domain knowledge and can be easily refuted using available measurements about the network s historical evolution [12, 30] In contrast, the latter not only thrives on domain knowledge and incorporates it explicitly into the ....

R. Albert and A.-L. Barabasi. Topology of Evolving Networks: Local Events and Universality. Physical Review Letters, 85(24), 2000.

....power law distributions about the Internet, leading to the creation of new Internet topology generators. Tangmunarunkit et al. divide network topology generators into two categories [38] Structural and Degree Based network generators. Other recently proposed generators are [1] 14] 39] 40] [41], 42] The major difference between these two categories is that the former explicitly injects hierarchical strcuture into the network, while the later generates graphs with power law degree distributions without any consideration of network hierarchy. Tangmunarunkit et al. argue that even though ....

.... of nodes and edges increases quadratically and we can predict the number of nodes in the near future with the equations given in Figure 9(a) and 9(b) Average degrees of the Internet topologies are shown in Figure 9(c) In most of the time step based Internet topology generators including [1] [41], 42] the number of links added at each time step is fixed. However, the average degree of the Internet increased linearly until the end of 1999 but suddenly decreased from early 2000 even though the number of nodes was increasing. This implies that the approaches of timestep and fixed number of ....

[Article contains additional citation context not shown here]

R. Albert and A. Barab asi, "Topology of evolving networks: local events and universality," Tech. Rep., LANL ArXiv, 2000.

....constructed trees from a graph based on true routing paths in the Internet, also showed a high frequency of relay nodes in the tree graphs [17] A. Topology and Tree Generation Our method for producing trees is the following. First, we generate power law topologies based on the Notre Dame model [18]. The model specifies 4 parameters: a 0 , a, p and q . Where a 0 is the initial number of detached nodes, and a is the initial connectivity of a node. When a link is added, one of its end points is chosen randomly, and the other with probability that is proportional to the nodes degree. This ....

....the fact that new links often attach to popular (high degree) nodes. The growth model is the following: with probability p, a new links are added to the topology. With probability q, a links are rewired, and with probability p q a new node with a links is added. Note that The notations in [18] are m 0 , m, p and q. a, p and q determine the average degree of the nodes. We created a vast range of topologies, but concentrated on several parameter combinations that can be roughly described as very sparse (VS) Internet like sparse (IS) and less sparse (LS) Table I summarizes the main ....

R. Albert and A.-L. Barab asi, "Topology of evolving networks: local events and universality," Physical Review Letters, vol. 85, no. 24, pp. 5234--5237, 11 Dec. 2000.

.... Krapivsky, John Byers, Mark Crovella, David Finkel, Sid Redner ABSTRACT Recent studies have noted that vertex degree in the autonomous system (AS) graph exhibits a highly variable distribution [15, 22] The most prominent explanatory model for this phenomenon is the Barab asi Albert (B A) model [5, 2]. A central feature of the B A model is preferential connectivity meaning that the likelihood a new node in a growing graph will connect to an existing node is proportional to the existing node s degree. In this paper we ask whether a more general explanation than the B A model, and absent the ....

.... papers have proposed mechanisms (more complicated than purely random connection) that may give rise to powerlaw degree distributions in graphs [5, 20, 19] The most prominent model attempting to explain the emergence of power law degree distributions is the Barab asi Albert model (or B A model) [5, 2]. In fact, it has been considered in a number of papers as a model for AS graphs [3, 7, 27, 24, 32] The B A model assumes the network is formed through incre1 mental addition of nodes. In the simplest form of the model, a new node forms a connection to an existing node with probability ....

R. Albert and A. Barab asi. Topology of evolving networks: local events and universality. Phys. Rev. Lett., 85:5234--5237, 2000.

.... Krapivsky, John Byers, Mark Crovella, David Finkel, Sid Redner ABSTRACT Recent studies have noted that vertex degree in the autonomous system (AS) graph exhibits a highly variable distribution [15, 22] The most prominent explanatory model for this phenomenon is the Barab asi Albert (B A) model [5, 2]. A central feature of the B A model is preferential connectivity meaning that the likelihood a new node in a growing graph will connect to an existing node is proportional to the existing node s degree. In this paper we ask whether a more general explanation than the B A model, and absent the ....

.... papers have proposed mechanisms (more complicated than purely random connection) that may give rise to powerlaw degree distributions in graphs [5, 20, 19] The most prominent model attempting to explain the emergence of power law degree distributions is the Barab asi Albert model (or B A model) [5, 2]. In fact, it has been considered in a number of papers as a model for AS graphs [3, 7, 27, 24, 32] The B A model assumes the network is formed through incre1 mental addition of nodes. In the simplest form of the model, a new node forms a connection to an existing node with probability ....

R. Albert and A. Barab asi. Topology of evolving networks: local events and universality. Phys. Rev. Lett., 85:5234--5237, 2000.

....paper is organized as follows. The BBS simulation, initial condition, friction, and an example embedding in the Euclidean plane are discussed in next section. In Sec. III, BBS is compared to four other methods, using simulated graphs created according to both Waxman [12] and Barab asi Albert (BA) [13] methods. In Sec. IV we apply BBS in two practical applications, topology aggregation and Internet distance maps, comparing it with previous results from [2] and [4] 5] respectively. II. BIG BANG SIMULATION A. The Model The vertices of the graph, the network nodes, are modeled as a set of ....

....this clique. This method had the lowest distance distortion among the aggregation methods considered in [2] and our comparison was done using the C code used there. B. Environment Details The network graphs in our comparison were created according to both Waxman [12] and Barab asi Albert (BA) [13] methods. Only a subset of the graph nodes, was selected to be embedded, according to the characteristics of the two applications. For IDMaps [4] the subset was selected using two Tracer placement methods, TRANSIT or LAN (stub) which select the Tracers among the highest or lowest degree nodes, ....

R. Albert and A.-L. Barab asi, "Topology of evolving networks: local events and universality," Physical Review Letters, pp. 5234-- 5237, 11 Dec. 2000.

....are derived from extensive research we performed on the structure and characteristics of multicast trees in the Internet. Recently, Faluotsos et al. 8] found power laws that characterize the Internet structure, mainly in the AS granularity, but also at the router level. Albert and Barabasi [2, 1] suggested a dynamic graph generation model that generates such networks and thus aided in the understanding of the evolvement of the Internet. One of their main findings was the self similarity characteristic of such networks. Additional understanding of the Internet structure and the relation ....

....in the fields of multicast routing protocols, reliability and congestion control, who can benefit from a realistic environment for their conclusion evaluations. To investigate the structure of multicast trees derived from a power law obeying topology, we built a topology generator based on [1]. We then produced multicast trees from the generated topologies, and investigated their characteristics. In section 2 we detail some of our relevant findings. Among them is the observation that nodes with degree higher than 5 tend to be rare in the resulting trees. These high degree nodes can ....

[Article contains additional citation context not shown here]

R. Albert and A.-L. Barabasi. Topology of evolving networks: local events and universality. Physical Review Letters, 85(24):5234--5237, 11 Dec. 2000.

....generators were unsuitable models for the Internet. Subsequently, there have been an increasing number of proposals for topology generators that are designed primarily to match the Internet s degree distribution and do not attempt to model the Internet s hierarchical structure; for example, see [23, 28, 2, 31, 1, 8]. These degree based topology generators embody the implicit assumption that it is more important to match a certain local property the degree distribution than to capture the large scale hierarchical structure of the Internet. The rapid adoption of these degree based generators suggests that ....

....2. RELATED WORK We have already mentioned several important areas of related work: the Waxman, Transit Stub and Tiers topology generators, and Faloutsos et al. s observations of power law degree distributions in the Internet. We have also mentioned in passing several new degree based generators [23, 28, 2, 31, 1]. They all attempt to generate networks with power law degree distributions, but differ in the way in which nodes are connected. We describe some of these generators in slightly more detail in [42] Perhaps closest in spirit to the work presented in this paper is the pioneering exploration of ....

R. Albert and A.-L. Barabasi. Topology of Evolving Networks: Local Events and Universality . Physical Review Letters, 85:5234--5237, 2000.

....having extremely large numbers of neighbors (e.g. popular actors and popular Web pages in the examples above) The degree distributions were often observed to follow approximately a power law. The BA Model: Power law networks are particularly emphasized by the work of Barabasi and Albert [6] [4] who explored a promising class of models that yield strict power law degree distributions. In their model (the BA model) three generic mechanisms are defined: 1) Incremental growth, which follows from the observation that networks develop by adding new vertices or new connections. 2) ....

R. Albert and A. Barabasi. Topology of evolving networks: Local events and universality. Physical Review Letters, 85:5234--5237, 2000.

....having extremely large numbers of neighbors (e.g. popular actors and popular Web pages in the examples above) The degree distributions were often observed to follow approximately a power law. The BA Model: Power law networks are particularly emphasized by the work of Barabasi and Albert [6] [4] who explored a promising class of models that yield strict power law degree distributions. In their model (the BA model) three generic mechanisms are defined: 1) Incremental growth, which follows from the observation that networks develop by adding new vertices or new connections. 2) ....

R. Albert and A. Barabasi. Topology of evolving networks: Local events and universality. Physical Review Letters, 85:5234--5237, 2000.

....Our conclusions are mixed. Although they (mostly) do a reasonable job at capturing the power law exponent, they do less well in capturing the clustering phenomena exhibited by the Internet topology. Based on these results we propose a variation of the recent incremental topology generator of [6] that is more successful at matching the power law exponent and the clustering behavior of the Internet. Last, we comment on the small world behavior of the Internet topology. I. INTRODUCTION Recent work has shown that the node degree in the World Wide Web (WWW) induced graph, and the AS level ....

.... and the AS level topology of the Internet exhibit power laws [3] 7] 14] This work has stimulated considerable activity aimed at understanding the implications on web and network design [4] 8] 18] 19] and on the development of algorithms for generating graphs exhibiting such power laws [5] [6] [10] 20] Past research has shown that the network topology can have a significant effect on the performance of network protocols. For example, the AS level topology of the Internet has a major impact on the convergence of the inter domain routing protocol BGP [16] The growth in the ASlevel ....

[Article contains additional citation context not shown here]

R. Albert and A. Barabasi. "Topology of Evolving Network: Local Events and Universality." Physica Review Letters, 85:5234-5237,2000

.... especially at the level of autonomous systems (ASs) For example, these activities include (1) analyzing and modeling measurements to infer the Internet s AS connectivity graph to describe its properties [2] 2) explaining the origins and causes of some of the observed surprising features [3] [4], 3) building topology generators that produce graph structures that match those of the measured AS connectivity graphs [5] 6] 7] 4) investigating the problem of routing path inflation [8] 9] 5) studying the effectiveness of proposed algorithms for detection prevention of attacks on ....

....Clearly, this latter distinction has direct implications for the generation of Internet like graphs or for the more challenging question of explaining the origins and causes of the highly variable vertex degrees in the Internet context. To illustrate, the work by Barabasi and Albert [3] [4] takes the quantitative power law observations at face value and provides a suite of results, including constructions that attempt to explain the causes that lead to power law vertex degree distributions. The applicability of these results and constructions to the Internet has been claimed in [4] ....

[Article contains additional citation context not shown here]

R. Albert and A.-L. Barabasi, "Topology of evolving networks: Local events and universality," Physical Review Letters, vol. 85, pp. 5234--5237, 2000.

....encoder and a decoder, we can use an indicator variable for each server pair and each session and obtain the same IP structure. 4.1. A simulation study We generated networks based on the newly discovered power log law [10] Our generator is based on the algorithm suggested by Albert and Barabasi [1]. In all the generated network we picked the parameter to be ,71 ,1 ( DR G1j, For each instance we generated E sessions that are randomly generated according to two models, where is the number of nodes in the network. In the uniform model, both session end points were selected ....

R. Albert and A.-L. Barabasi. Topology of evolving networks: local events and universality. Physical Review Letters, 85(24):5234--5237, 11 Dec. 2000.

....Our conclusions are mixed. Although they (mostly) do a reasonable job at capturing the power law exponent, they do less well in capturing the clustering phenomena exhibited by the Internet topology. Based on these results we propose a variation of the recent incremental topology generator of [6] that is more successful at matching the power law exponent and the clustering behavior of the Internet. Last, we comment on the small world behavior of the Internet topology. I. INTRODUCTION Recent work has shown that the node degree in the World Wide Web (WWW) induced graph, and the ASlevel ....

.... and the ASlevel topology of the Internet exhibit power laws [3] 7] 12] This work has stimulated considerable activity aimed at understanding the implications on web and network design [4] 8] 16] 17] and on the development of algorithms for generating graphs exhibiting such power laws [5] [6] [10] 15] 18] Past research has shown that the network topology can have a significant effect on the performance of network protocols. For example, the AS level topology of the Internet has a major impact on the convergence of the inter domain routing protocol BGP [14] The growth in the ....

[Article contains additional citation context not shown here]

R. Albert and A. Barabasi. "Topology of Evolving Network: Local Events and Universality." Physica Review Letters, 85:5234-5237,2000

....mechanisms that cause massive graphs such as Web linkage [18] telephone call [19] or bibliographic citation [20] graphs to exhibit phenomena similar to the power law vertex degree distribution observed in the AS map constructed from the Oregon data set. Such works include the papers [21] 2] [22], 23] 24] 20] Of these works, 2] 23] have attracted the most attention in the networking community as their authors propose a very appealing construction of network topologies. This construction was later used to form the basis for the claimed error intolerance and attack vulnerability of ....

....BA model was proposed as a general model for the evolution of scale free networks. We emphasize that our results do not question the applicability of the model to scale free networks in general, but only to the AS topology. A. The Barabasi Albert (BA) Model The Barabasi Albert (BA) model [2] [22] consists of three generic mechanisms that drive the evolution of graph structures over time to produce graphs with power law vertex degree: 1. Incremental growth. Incremental growth follows from the observation that most networks develop over time by adding new nodes and new links to the ....

[Article contains additional citation context not shown here]

R. Albert and A.-L. Barabasi, "Topology of Evolving Networks: Local Events and Universality ," Physical Review Letters, vol. 85, pp. 5234-- 5237, 2000.

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R. Albert and A. Barab asi. Topology of evolving networks: Local events and universality. Physical Review Letters, 85:5234--5237, 2000.

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R. Albert and A. Barab asi. Topology of evolving networks: Local events and universality. Physical Review Letters, 85:5234--5237, 2000.

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Reka Albert and Albert-Laszlo Barabasi. Topology of evolving networks: Local events and universality. Physical Review Letters, pages 5234--5237, 2000.

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ALBERT, R., AND BARAB ASI, A.-L. Topology of evolving networks: Local events and universality. Physical Review Letters (December 11 2000), 5234--5237.

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R. Albert and A. Barabasi. topology of evolving network: local events and universality. physical review letters, 85(24):5234--5237, Dec. 2000.

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R. Albert and A. Barabasi. Topology of Evolving Network: Local Events and Universality. Physica Review Letters, 85:52345237, 2000.

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ALBERT, R., AND BARABASI, A.-L. Topology of evolving networks: local events and universality. Physical Review Letters 85(24) (2000), 5234--5237.

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R. Albert, A.L. Barabasi, Topology of evolving networks: local events and universality, Phys. Rev. Lett. 85 (2000) 5234.

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R. Albert and A.-L. Barab asi, "Topology of evolving networks: local events and universality," Physical Review Letters, pp. 5234-- 5237, Dec. 2000.

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R. Albert and A. Barabasi, "Topology of evolving networks: local events and universality," Physical Review Letters, vol. 85, no. 24, pp. 5234-- 5237, 11 Dec. 2000.

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Albert, R., and Barabasi, A.-L. Topology of Evolving Networks: Local Events and Universality. Physical Review Letters 85(24) (2000), 5234-5237.

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R. Albert and A. Barabasi, "Topology of evolving networks: local events and universality," Tech. Rep., LANL ArXiv, 2000.

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Albert, R. and Barabasi, A.-L., Topology of evolving networks: Local events and universality, Phys. Rev. Lett. 85, 5234--5237 (2000).

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Reka Albert and Albert-Laszlo Barabasi. Topology of evolving networks: local events and universality. Physcal Review Letters, 85:5234, 2000.

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Reka Albert and Albert-Laszlo Barabasi, \Topology of evolving networks: local events and universality, " Physical Review Letters, pp. 5234-5237, 11 Dec. 2000.

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Albert, R., and Barabasi, A.-L. Topology of evolving networks: local events and universality. Physical Review Letters 85(24) (2000), 5234-5237.

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R. Albert, and A.-L. Barabsi, "Topology of Evolving Networks: Local Events and Universality," Physical review letters, vol.85, 2000, p.5234.

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R. Albert and A.-L. Barabasi. Topology of evolving networks: local events and universality. Physical Review Letters, 85(24):5234--5237, 11 Dec. 2000.

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R. Albert and A.-L. Barab asi, "Topology of evolving networks: local events and universality," Physical Review Letters, pp. 5234-- 5237, Dec. 2000.

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Reka Albert and Albert-Laszlo Barabasi, "Topology of evolving networks: local events and universality," Physical Review Letters, vol. 85, no. 24, pp. 5234--5237, 11 Dec. 2000.

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R. Albert and A.-L. Barabasi. Topology of evolving networks: local events and universality. Physical Review Letters, 85(24):5234--5237, 11 Dec. 2000.

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R eka Albert and Albert-L aszl o Barab asi, "Topology of evolving networks: local events and universality," Physical Review Letters, pp. 5234--5237, 11 Dec. 2000.

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Rka Albert and Albert-Lszl Barabsi, "Topology of Evolving Networks: Local Events and Universality" Phys. Rev. Lett. Vol.85,Iss.24,pp.5234-5237

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R. Albert and A. Barab asi. Topology of evolving networks: local events and universality. Phys. Rev. Lett., 85:5234--5237, 2000.

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ALBERT, R., AND BARABASI, A.-L. Topology of Evolving Networks: Local Events and Universality . Physical Review Letters 85 (2000), 5234--5237.

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R. Albert and A. Barab asi. Topology of evolving networks: local events and universality. Phys. Rev. Lett., 85:5234--5237, 2000.

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R. Albert and A.-L. Barabasi. Topology of Evolving Networks: Local Events and Universality . Physical Review Letters, 85:5234--5237, 2000.

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