c08_rand_graphs_handout

c08_rand_graphs_handout - Network models random graphs...

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1 Properties common to many large-scale networks, independently of their origin and function: 1. The degree and betweenness distribution are decreasing functions, usually power-laws. 2. The distances scale logarithmically with the network size Network models – random graphs N log 3. The clustering coefficient does not seem to depend on the network size As though all these networks were part of the same family/class. k log l k C The average distance and clustering coefficient only depend on the number of nodes and edges in the network. This suggests that general models based only on the number of nodes and edges in the network could be successful in describing the properties of an “expected” (characteristic) network. Random networks Uniformly random network: distributes the edges uniformly among nodes. Probabilistic interpretation: There exists a set (ensemble) of networks with given number of nodes and edges. Select a random member of this set. What are the expected properties of this network? – studied by random graph theory . Ex. 1 Start with 10 isolated nodes. For each pair of nodes, throw with a dice, and connect them if the number on the dice is 1 . Describe the graph you obtained. How many edges are in the graph? Is it connected or not? What is the average degree and the degree distribution? E2 Ex. 2 Now connect node pairs if the number on the dice is 1 or 2. How is the graph different from the previous case? Ex. 2 How many edges do you expect a graph with N nodes would have if each edge is selected with throwing with a dice? Random graph theory Erdös-Rényi algorithm - Publ. Math. Debrecen 6, 290 (1959) • fixed node number N • connecting pairs of nodes with probability p Random graph theory studies the expected properties of graphs with 2 ) 1 N ( N p E = Expected number of edges: N
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2 The properties of random graphs depend on p Properties studied: is the graph connected? does the graph contain a giant connected component? what is the diameter of the graph? does the graph contain cliques (complete subgraphs)? Probabilistic formulation: what is the probability that a graph with N nodes and connection probability p is connected?
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c08_rand_graphs_handout - Network models random graphs...

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