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10 ople are co-authors if there edge to each of them. derived from the five largest PH, HEP TH, HEP PH, place a time-stamp on each h paper, and for each perission to the arxiv. The the period from April 1992 the graphs (category GR thors, 13,454 papers) and largest graph, with 57,381 pers) and 133,170 edges. It of the other categories also uthors per paper. s we observe similar pheve densification exponents lack of space we present s only for ASTRO PH, the s 1(d) and 2(d) show the me, and a densification exthe effective diameter over rk datasets. Following the ic, we expected the underould detect the differences Effective diameter Effective diameter Full graph Post 95 subgraph Post 95 subgraph, no past Time [years] (a) arxiv citation graph Time [years] (c) Patents Full graph Post 85 subgraph Post 85 subgraph, no past Effective diameter Effective diameter Full graph Post 95 subgraph Post 95 subgraph, no past Time [years] (b) Affiliation network Size of the graph [number of nodes] (d) AS Figure 3: The effective diameter over time. the result of subsequent papers acting as bridges by citing earlier papers from disparate areas. Note that for other Linear fit diameter has the following interpretation: Since all the links out of a node are frozen at the moment it joins the graph, the decreasing distance between pairs of nodes appears to be
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21 Probability mass function of a Zipf random variable; differing α values 0.9 α = α = probability of k k
22 Probability 10 0 mass function of a Zipf random variable; differing α values α =2.0 α = probability of k k
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25 2.0 PDF of a Pareto random variable; differing α values α =2.0 α = probability of x x
26 10 1 PDF of a Pareto random variable; differing α values α =2.0 α = probability of x x
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31 14 The structure and function of complex networks (a) collaborations in mathematics (b) citations (c) World Wide Web (d) Internet (e) power grid (f) protein interactions 1 10 FIG. 6 Cumulative degree distributions for six different networks. The horizontal axis for each panel is vertex degree k (or indegree for the citation and Web networks, which are directed) and the vertical axis is the cumulative probability distribution of degrees, i.e., the fraction of vertices that have degree greater than or equal to k. The networks shown are: (a) the collaboration network of mathematicians [182]; (b) citations between 1981 and 1997 to all papers cataloged by the Institute for Scientific Information [351]; (c) a 300 million vertex subset of the World Wide Web, circa 1999 [74]; (d) the Internet at the level of
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