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Network (graph) Models

Network (graph) Models. Lecture 6. What is a network (graph) model?. Network models. We will cover the following network models: Erdos–Renyi random graphs Generalized random graphs (with the same degree distribution as the data networks) Small-world networks Scale-free networks

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Network (graph) Models

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  1. Network (graph) Models Lecture 6

  2. What is a network (graph) model?

  3. Network models We will cover the following network models: • Erdos–Renyi random graphs • Generalized random graphs (with the same degree distribution as the data networks) • Small-world networks • Scale-free networks • Hierarchical model • Geometric random graphs

  4. Erdos–Renyi random graphs (ER) • Model a data network G(V,E) with |V|=n and |E|=m • An ER graph that models G is constructed as follows: • It has n nodes • Edges are added between pairs of nodes uniformly at random with the same probability p • Two (equivalent) methods for constructing ER graphs: • Gn,p: pick p so that the resulting model network has m edges • Gn,m: pick randomly m pairs of nodes and add edges between them with probability 1

  5. Erdos–Renyi random graphs (ER) • Number of edges, |E|=m, in Gn,pis: • Average degree is:

  6. Erdos–Renyi random graphs (ER) • Many properties of ER can be proven theoretically (See: Bollobas, "Random Graphs," 2002) • Example: • When m=n/2,suddenly the giant component emerges, i.e.: • One connected component of the network has O(n) nodes • The next largest connected component has O(log(n)) nodes

  7. Probability of drawing a graph of m edges • Look familiar? • The mean value of m • The mean degree of graph is

  8. DEGREE DISTRIBUTION OF A RANDOM GRAPH probability of missing N-1-k edges Select k nodes from N-1 probability of having k edges As the network size increases, the distribution becomes increasingly narrow—we are increasingly confident that the degree of a node is in the vicinity of <k>. Network Science: Random Graphs

  9. DEGREE DISTRIBUTION OF A RANDOM GRAPH For largeNand small k, we can use the following approximations: for Network Science: Random Graphs

  10. POISSON DEGREE DISTRIBUTION For largeNand small k, we arrive to the Poisson distribution: Network Science: Random Graphs

  11. Erdos–Renyi random graphs (ER) • The degree distribution is binomial: • For large n, this can be approximated with Poisson distribution: where z is the average degree • However, many real world networkshave power-law degree distribution

  12. Erdos–Renyi random graphs (ER) • Clustering coefficient, C, of ER is low (for low p) • C=p, since probability p of connecting any two nodes in an ER graph is the same, regardless of whether the nodes are neighbors • However, many real world networkshave high clustering coefficients

  13. Erdos–Renyi random graphs (ER) • Average diameter of ER graphs is small • It is equal to • Real networks alsohave small average diameters • Summary

  14. Generalized random graphs (ER-DD) • Preserve the degree distribution of data (“ER-DD”) • Constructed as follows: • An ER-DD network has n nodes (so does the data) • Edges are added between pairs of nodes using the “stubs method” [configuration model discussed earlier]

  15. Generalized random graphs (ER-DD) • The “stubs method” for constructing ER-DD graphs: • The number of “stubs” (to be filled by edges) is assigned to each node in the model network according to the degree distribution of the real network to be modeled • Edges are created between pairs of nodes with “available” stubs picked at random • After an edge is created, the number of stubs left available at the corresponding “end nodes” of the edges is decreased by one • Multiple edges between the same pair of nodes are not allowed

  16. Generalized random graphs (ER-DD) • Summary • 2 global network properties are matched by ER-DD

  17. Small-world networks (SW) • Watts and Strogatz, 1998 • Created from regular ring lattices by random rewiring of a small percentage of their edges • E.g.

  18. Small-world networks (SW) • SW networks have: • High clustering coefficients – introduced by “ring regularity” • Large average diameters of regular lattices – fixed by randomly re-wiring a small percentage of edges • Summary

  19. Scale-free networks (SF) • Power-law degree distributions: P(k) = k−γ • γ > 0; 2 < γ < 3

  20. Scale-free networks (SF) • Power-law degree distributions: P(k) = k−γ • γ > 0; 2 < γ < 3

  21. Scale-free networks (SF) • Different models exist, e.g.: • A popular one is: • Preferential Attachment Model (SF-BA) (Barabasi-Albert, 1999)

  22. Scale-free networks (SF) • Preferential Attachment Model (SF-BA) • “Growth” model: nodes are added to an existing network • New nodes preferentially attach to existing nodes with probability proportional to the degrees of the existing nodes; e.g.: • This is repeated until the size of SF network matches the size of the data • “Rich getting richer”

  23. Scale-free networks (SF) • Summary

  24. Hierarchical model • Preserves network “modularity” via a fractal-like generation of the network

  25. Hierarchical model • These graphs do not match any biological data and are highly unlikely to be found in data sets

  26. Geometric random graphs • “Uniform” geometric random graphs (GEO) • Take any metric space and, using a uniform random distribution, place nodes within the space • If any nodes are within radius r (calculated via any chosen distance norm for the space), they will be connected • Choose r so that the size of the GEO network matches that of the data • There are many possible metric spaces (e.g., Euclidean space) • There are many possible distance norms (e.g. the Euclidean distance, the Chessboard distance, and the Manhattan/Taxi Driver distance)

  27. Geometric random graphs • “Uniform” geometric random graphs (GEO) • Summary

  28. Stochastic Block Models • M matrix (constant p)  ErdosRenyi model • M matrix (not constant)  ErdosRenyi within community; random bipartite across communities. • Vertices within a community are considered exchangeable (i.e. probabilistically equivalent with respect to their interactions with other vertices)

  29. SBM: Representation and Instantiation

  30. SBM: ASSORTATIVE NETWORKS

  31. SBM: Dissortative Networks

  32. SBM contd. • A number of other composable models can be viewed as Stochastic Block Models • One may compose/create multiple generative models in this fashion • Can handle directed networks (M is not symmetric in this case) • Can potentially model all three properties of real networks one is often interested in!

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