Lecture 2: Complex Networks

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Lecture 2: Complex Networks Performance Evaluation Lecture 2: Complex Networks Giovanni Neglia INRIA – EPI Maestro 16 December 2013

Configuration model A family of random graphs with given degree distribution

Configuration model A family of random graphs with given degree distribution Uniform random matching of stubs

Configuration model A family of random graphs with given degree distribution Uniform random matching of stubs

Back to Navigation: Random Walks What can we do in networks without a geographical structure? Random walks 1/4 1/4 1/2 1/4 1/4 1/2

Back to Navigation: Random Walks How much time is needed in order to reach a given node?

Random Walks: stationary distribution avg time to come back to node i starting from node i: Avg time to reach node i intuitively ≈Θ(M/ki) i

Another justification Random walk as random edge sampling Prob. to pick an edge (and a direction) leading to a node of degree k is Prob. to arrive to a given node of degree k: Avg. time to arrive to this node 2M/k …equivalent to a RW where at each step we sample a configuration model

Distributed navigation (speed up random walks) Every node knows its neighbors {a,b,c,d} i d a c b

Distributed navigation (speed up random walks) Every node knows its neighbors If a random walk looking for i arrives in a the message is directly forwarded to i {a,b,c,d} i d a c b

Distributed navigation reasoning 1 We discover i when we sample one of the links of i’s neighbors Avg # of these links: Prob. to arrive at one of them: i d a c b

Distributed navigation reasoning 2 Prob that a node of degree k is neighbor of node i given that RW arrives to this node from a node different from i Prob that the next edge brings to a node that is neighbor of node i:

Distributed navigation Avg. Hop# Regular graph with degree d: ER with <k>: Pareto distribution : If α->2…

Distributed navigation Application example: File search in unstructured P2P networks through RWs