Upgrade Report - Department of Informatics - King's College London

5 GRAPH GENERATION MODELS 8Figure 1: Erdós-Rényi Graph G(n, p) graph with n = 50000, p = 0.00015.2 Preferential attachment modelThe preferential attachment model is a graph process used to generate graphs with degree distributionswhich follow a power-law. This generative method was proposed by Barabási and Albert [5] as a generativeprocedure for a model of the WWW. Surveys by Bollobás and Riordan [7] and Drinea, Enachescuand Mitzenmacher [21] give many related generative procedures to obtain graphs with power-law degreesequences.In this model, the graph G(t) = G(m, t) is obtained from G(t−1) by adding a new vertex v t with m edgesbetween v t and G(t − 1). The end points of these edges are chosen preferentially, that is to say proportionalto the existing degree of vertices in G(t − 1). Thus the probability p(x, t) that vertex x ∈ G(t − 1) ischosen as the end point of a given edge is equal to p(x, t) = d(x, t − 1)/(2m(t − 1)), and this choice is madeindependently for each of the m edges added. A model generated in this way has a power law of 3 for thedegree sequence, irrespective of the number of edges m ≥ 1 added at each step.It was empirically shown by Barabási that there are two required elements in order in order to obtain apower-law degree distribution and two models were defined.Model A This model retains growth but all new vertices created are attached UAR on existing edges.Model B This model consists of a graph with a static vertex count which at each epoch (or time frame)an edge is added preferentially.Each of the individual models above fails to create scale-free graphs, and it is shown in [5] that only bya combination of models A and B do we obtain a power-law degree distribution.Preferential attachment graphs have a heavy tailed degree sequence. Thus, although the majority of thevertices have constant degree, a very distinct minority have very large degrees. This particular propertyis the significant defining features of such graphs. A log-log plot of the degree sequence breaks naturallyinto three parts. The lower range (small constant degree) where there may be curvature, as the power lawapproximation is incorrect. The middle range, of large but well represented vertex degrees, which give thecharacteristic straight line log-log plot of the power law coefficient. In the upper tail, where the sequenceis far from concentrated, and the plot is a spiky mess. Due to the structure of the model as well as theextensive analysis that has been done, this is going to be the main reference model for our experiments.