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# K k 0 since the mean number of edges given by goes to

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Unformatted text preview: edges > 0) → 0. p (n ) Assume next that 2/n2 → ∞ as n → ∞. Then, E[number of edges]→ ∞. This does not in general imply that P(number of edges > 0) → 1. Here it follows because the number of edges can be approximated by a Poisson distribution (just like the degree distribution), implying that � e − λ λk � � P(number of edges = 0) = = e −λ . � k! � k =0 Since the mean number of edges, given by λ, goes to inﬁnity as n → ∞, this implies that P(number of edges > 0) → 1. 10 Networks: Lecture 3 Introduction Phase Transitions Hence, the function t (n ) = 1/n2 is a threshold function for the emergence of the ﬁrst link, i.e., When p (n ) << 1/n2 , the network is likely to have no edges in the limit, whereas when p (n ) >> 1/n2 , the network has at least one edge with probability going to 1. How large should p (n ) be to start observing triples in the network? We have E[number of triples] = n3 p 2 , using a similar analysis we can 1 show t (n ) = n3/2 is a threshold function. How large should p (n ) be to start observing a tree with k nodes (and k − 1 arcs)? We have E[number of trees] = nk p k −1 , and the function t (n ) = nk /1 −1 is a threshold function. k 1 The threshold function for observing a cycle with k nodes is t (n ) = n Big trees easier to get than a cycle with arbitrary size! 11 Networks: Lecture 3 Introduction Phase Transitions (Continued) Below the threshold of 1/n, the largest component of the graph includes no more than a factor times log(n ) of the nodes. Above the threshold of 1/n, a giant component emerges, which is the largest component that contains a nontrivial fraction of all nodes, i.e., at least cn for some constant c . The giant component grows in size until the threshold of log(n )/n, at which point the network becomes connected. 12 Networks: Lecture 3 Introduction Phase Transitions (Continued) Figure: A ﬁrst component with more than two nodes: a random network on 50 nodes with p = 0.01. 13 Networks: Lecture 3 Introduction Phas...
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