Node betweenness has been studied recently by a number of authors, but until now less attention has been paid to edge betweenness. In this paper, we present an exact analytic study of edge betweenness in evolving scale-free and non-scale-free trees. We aim at the probability distribution of edge betweenness under the condition that a local property, the in-degree of the ``younger'' node of a randomly selected edge, is known. En route to the conditional distribution of edge betweenness the exact joint distribution of cluster size and in-degree, and its one dimensional marginal distributions have been presented in the paper as well. From the derived probability distributions the expectation values of different quantities have been calculated. Our results provide an exact solution not only for infinite, but for finite networks as well.
In recent years, the statistical properties of complex networks have been extensively investigated by the physics community [1,2,3,4]. With the increasing computing power of modern computers, analysis of large-scale networks and databases has become possible.
It has been shown that the degree statistics of many natural and artificial networks follow power law. Examples for such networks vary from social interconnections and scientific collaborations [5] to the world-wide web [6] and the Internet [7,8]. These networks are usually referred to as scale-free networks, since the power law distribution indicates that there is no characteristic scale in these systems.
In the early 1960s Erdős and Rényi (ER) introduced random graphs that served as the first mathematical model of complex networks [9]. In their model, the number of nodes is fixed and connections are established randomly, with probability p ER . Although the ER model leads to rich theory, it fails to predict the power law distributions observed in scalefree networks. Barabási and Albert (BA) proposed a more suitable evolving model of these networks [10,11]. The BA model is also based on the random graph theory, but involves two key principles in addition: (a) growth, that is, the size of the network is increasing during development, and (b) preferential attachment, that is, new network elements are connected to higher degree nodes with higher probability. In the BA model every new node connects to the core network with a fixed number of links m.
The study of complex networks usually deals with the structural properties of networks, like degree distribution [12], shortest path distribution [13], degree-degree correlations, clustering [14], etc. For complex networks which involve a transport mechanism betweenness is the matter of importance. Roughly speaking, betweenness is the number of shortest paths passing through a certain network element. For example, in communication networks information flows between remote hosts via intermediate stations and in the Internet data packets are transmitted through routers and cables. The expected traffic flowing through a link or a router is proportional to the particular edge or node betweenness, respectively.
News and rumors spread in social networks, and node betweenness measures the importance or centrality of an individual in society.
Node betweenness has been studied recently by Goh, Kahng, and Kim [15], who argued that it follows power law in scale-free networks, and the exponent δ ≈ 2.2 is indepen-dent from the degree distribution in a certain range. Szabó, Alava, and Kertész [13] used rooted deterministic trees to model scale-free trees, and have found scaling exponent δ t = 2.
The same scaling exponent has been found experimentally by Goh, Kahng, and Kim for scale-free trees. The rigorous proof of the heuristic results of [13] has been provided by Bollobás and Riordan in [16].
Until recently, less attention has been paid to edge betweenness, even though edge betweenness is often essential for estimating the load on links in complex networks. For example, the edge betweenness can measure the “importance” of relationships in social networks, or it can measure the expected amount of data flow on links in computer networks. The probability distribution of edge betweenness gives a rough statistical description of links and it characterizes the network as a whole. Therefore, it is an important tool for an overall description of links in complex networks.
In some cases, a local property of the network is known as well. For instance, if the number of friends of any individual can be counted, then it is reasonable to ask the “importance” of a relationship (i.e. an edge in a social network) under the condition that the number of friends of the related individuals is known. In this case, the conditional probability distribution of edge betweenness provides a much finer description of links than the total distribution.
In this paper we focus on how additional local information could be used to describe links. In particular, we aim at deriving the probability distribution of edge betweenness in evolving scale-free trees, under the condition that the in-degree of the “younger” node of any randomly selected link is known. For the sake of simplicity we consider the in-degree of the “younger” node only. Whether a node is “younger” than another node or not can be defined uniquely in evolving networks, since nodes attach to the network sequentially.
Note that the in-degree is considered instead of total degree for practical reasons only. The construction of the network implies that the in-degree is less than the total degree by one for every “younger” node.
To obtain the desired conditional distribution we calculate the exact joint distribution of cluster size and in-degree for a specific link first. Then, the joint distribution of a randomly selected link is derived, which is comparable with the edge ensemble st
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