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Lecture 17 Feb25 CS4850

# Lecture 17 Feb25 CS4850 - r< 2 always short paths but no...

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CS 4850 LECTURE 17 - FEBRUARY 25, 2009 SCRIBE: YU-KANG CHENG (YC362) GROWING GRAPH WITH PREFERENTIAL ATTACHMENT At time t = 0 no vertices At each unit of time, add one vertex With probability δ add edge connecting new vertex to an existing vertex, selecting the existing vertex with probability proportional to degree of the existing vertex. Let d i ( t ) be degree of i th vertex at time t . d dt d i ( t ) = δ d i ( t ) t j =1 d j ( t ) = δ d i ( t ) 2 δt = d i ( t ) 2 t Solution to differential equation is d i ( t ) = at 1 2 at 1 2 i = δ a = δ t 1 2 i d i ( t ) = δ r t t i FIGURE 1 d = δ r t t i d 2 = δ 2 t i t t i = δ 2 d 2 t Pr ( vertex degree > d ) = 1 - δ 2 d 2 pdf = ∂d ± 1 - δ 2 d 2 ² = 2 δ 2 d 2 1

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FIGURE 2. n x n graph with 3 long edges added SMALL WORLD GRAPHS Find shortest path using path using only local information. Assume an n x n graph, with each vertex connected to 4 neighbors (less on edges), and add one edge to a further vertex. Selection of the further vertex is proportional to distance. Pr [( u,v )] d - r ( u,v ) w 6 = v d - r ( u,w ) r = 0 edge ends uniformly distributed
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Unformatted text preview: r < 2 always short paths, but no efﬁcient algorithm to ﬁnd them probability of encountering edge that gets you closer too low r = 2 ∃ alogrithm (ln n ) 2 take edge that gets you closest to destination each time r > 2 there may or may not be a short path, but no algorithm exists to ﬁnd them probability of encountering long edge too low r = ∞ no long edges Proof of algorithm: s start point, t end point FIGURE 3 A j ∈ (2 j , 2 j +1 ] at most log 2 n phases, E ( time ) = ln n Time of algorithm is (ln n ) 2 Lemma: For r = 2 ∃ constant c such that the probability that a long distance edge from u goes to v is at least c d-2 ( u,v ) ln n . Proof: Pr [( u,v )] ∝ d-r ( u,v ) P w 6 = v d-r ( u,w ) To Be Continued . .....
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Lecture 17 Feb25 CS4850 - r< 2 always short paths but no...

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