sb3-hw15-2016 - JHU 580.429 SB3 HW15 Networks 1 Networks...

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JHU 580.429 SB3 HW15: Networks. 1. Networks and degree distributions. Consider a network with N total vertices and E total undirected, unweighted edges. (a) What is the probability f that an edge connects two vertices connected at random? (b) What is the average number of neighbors J per protein? (c) The probability p n that a protein has exactly n neighbors follows a binomial distribu- tion. Provide this distribution. (d) Show that in the limit that f 0, N , and J a finite constant, that the binomial probability distribution for p n approaches a Poisson distribution. Provide the Poisson distribution. (e) Based on the Poisson distribution, for what value of J do we expect on average only one vertex in the entire network to have no neighbors? 2. Network motifs. Again consider a network with N total vertices and E total edges. The edge probability f and the average degree J can be calculated from N and E and can also be used in your answers. You can assume the thermodynamic limit that f 0, N , and fN J . (a) What is the probability that three vertices, selected at random, have all edges (a 3- clique)? How many 3-cliques exist in the network? (b) How many edges are possible for a group of k vertices? Given a set of k vertices, what is the probability that all these edges exist, giving a k -clique? How many k -cliques are expected in a random network?
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