5 GRAPH GENERATION MODELS 9The preferential attachment model was refined Bollobas and Riordan [8,9] who introduced the scale freemodel to make detailed calculations <strong>of</strong> degree sequence and diameter. The model was generalized by manyauthors, including the web-graph model <strong>of</strong> Cooper and Frieze [15]. The web-graph model is very generaland allows the number <strong>of</strong> edges added at each step to vary, for edges from new vertices to choose their endpoints preferentially or uniformly at random, as well as for insertion <strong>of</strong> edges between existing vertices. Byvarying these parameters, preferential attachment graphs with degree sequences exhibiting power laws c inthe interval (2, ∞) are obtained. Assuming that m edges are added at every step, we refer to this generalized(web-graph) process with power law c as G(c, m, t).In [13], Cooper noted the result that the power law c for preferential attachment graphs and web-graphscan be written explicitly asc = 1 + 1/η, (5.1)where η is the expected proportion <strong>of</strong> edge end points added preferentially. In the Barabási and Albertmodel, η = 1/2, as each new edge chooses an existing neighbor vertex preferentially; thus explaining thepower law <strong>of</strong> 3 for this model.The value η occurs naturally in such models in the expression for the expected degree <strong>of</strong> a vertex. Letd(s, t) denote the degree at step t <strong>of</strong> the vertex v s added at step s. The expected value <strong>of</strong> d(s, t) is given by( ) t ηEd(s, t) ∼ m , (5.2)swhere η is the parameter defined above (see e.g. [17]). Thus, in the preferential attachment model <strong>of</strong> [5],Ed(s, t) ∼ m(t/s) 1/2 .The actual value <strong>of</strong> d(s, t) is not particularly concentrated around Ed(s, t), but the following inequalitiesproved in e.g. [17] and [13], are adequate for our pro<strong>of</strong>s. The inequalities hold With High Probability (whp),for all vertices in G(c, m, t).( ) t η(1−ɛ) ( ) t η≤ d(s, t) ≤ log 2 t, (5.3)sswhere ɛ > 0 is some arbitrarily small positive constant (e.g. ɛ = 0.00001). The upshot <strong>of</strong> this, and ourreason for explaining this to the reader, is that all vertices v added after step s log 2/η+1 t have degreed(v, t) = o ((t/s) η ) whp.Preferential attachment graphs have diameterDiam(G(m, t)) = O(log t) (5.4)whp This was improved for scale free graphs by Bollobas and Riordan, but crude pro<strong>of</strong>s can be made forthe general web-graph model based on expansion properties <strong>of</strong> the graph.The resulting degree distribution from such a graph can be seen in Figure 25.3 Edge Copying modelLike the preferential attachment, the edge copying model [32] produces scale-free graphs. However there isno explicit concept <strong>of</strong> preferential attachment involved.The model works as follows:• Starting with an initial graph at each step a new vertex v arrives• For this vertex v we select a vertex u UAR which will serve as a proxy for v• For each edge (u, w i ) with probability 1 − γ we direct an edge from v to w i• With probability γ we select a vertex v ′ UAR and direct an edge from v to v ′This model has been shown to produce power-laws with a co-efficient <strong>of</strong> 1+ 1has been observed in the resulting graphs.1−γand community structureThe resulting degree distribution from the above generation model can be seen in Figure 3.
5 GRAPH GENERATION MODELS 10Figure 2: Degree distribution <strong>of</strong> the preferential attachment graphFigure 3: Degree distribution <strong>of</strong> the edge copying graph with γ = 0.5
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