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By Alspach B., Xu M.Y.

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3 for an instance of a power law graph. 3. A power law graph with 400 vertices. with the high degree vertices clustered toward the centre of the figure. 1), then the relationship is expressed as log(Nk,G) N log(t) - ,Q log(k). Hence, in the log-log plot, we obtain a straight line with slope In both real-world networks and graphs generated by theoretical models, the power law may only fit for a certain range of degrees, with discrepancies 2. The Web Graph 24 for small or large degree vertices. 4, where a straight line is quite evident for lower degrees.

Fix p E (0, 1), and let n be a positive integer. s. G E 9p(q, A) is n-e. c. Proof. Fix G E 9p (q, A), and fix disjoint sets of vertices X and Y in G, with I X UYI = n. Let U = X UY. We prove that for large q, with probability 1 there is a vertex z correctly joined to X and Y. To accomplish this, we construct a set PU of points, disjoint from U, such that with probability 1, z is in PU. We set s = [qb] , where b is chosen so that 0 < b < 2. Fix a point v of A. The projection from v onto l,, is the map 7rv : A\{v} -j f,, taking a point x to the intersection of vx with 2, Hence, is the slope of the line vx.

5) Determine if the following (undirected) networks are power law graphs, and heuristically justify your answer. (a) The highway graph. Vertices are cities, while edges correspond to roads between them. (b) The airport graph. Vertices are airports, and edges are airline routes. (c) The musician graph. Vertices are musicians, and edges correspond to having played together in a concert or recording. (d) The word graph. Vertices are words in the English language, and two words are joined if they appear in the same sentence.

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