1. 1 CIS 4930/6930 – Recent Advances in Bioinformatics Spring 2014 Network models Tamer Kahveci

2. 2 Graphs Useful for describing networks. G = (V, E) with –V = set of nodes –E = set of edges Topological models –Directed/Undirected –Weighted/Unweighted –Deterministic/Probabilistic (G = (V, E, P)) Concepts –Degree (indegree/outdegree), path

3. 3 Topological properties Degree distribution, P(k) of G=(V, E) –Deg(k) = number of nodes in G with degree = k. –P(k) = Deg(k)/|V| = Probability that a random node in G has degree = k. H.Pylori Todor et al. TCBB. 10:4. 2013 3 2 2 1

4. 4 Topological properties Neighbors of node v, N(v) = set of nodes adjacent to v. Clustering coefficient of node v, C v shows the connectivity of N(v). Slightly different denominator for directed vs undirected graph C v = # edges among N(v) Max # edges possible among N(v) C(k) = Average clustering coefficients for all nodes with k edges. Networks clustering coefficient = average clustering coefficients of all nodes in G = (∑ C v ) / |V| 2/6

5. 5 Centrality of a node Centrality of a node v in graph G = (V, E) indicates relative importance of v in G with respect to the rest of the nodes in G. Lets denote it with f(v | G) or simply f(v). Many centrality measures exists –Degree centrality How popular am I? f Deg (v) = Deg(v) –Closeness centrality –Betweenness centrality

6. 6 Closeness Centrality How close am I to everyone else? Given G = (V, E) Dist(u,v) = shortest path length from u to v in G f Close (u) = ∑ v in G Dist(u, v) Alternative (for disconnected networks) –f Close (u) = ∑ v in V-{u} 1/ Dist(u, v) –1/inf = 0 How do I find shortest path? –Floyd-Warshall algorithm –Johnson’s algorithm 1 1 2 3

7. 7 Betweenness Centrality How many pairs of nodes use me on the cheapest route to communicate? g st = number of shortest path between s & and t. g st (v) = number of shortest path between s & and t that contains v. f Between (v) = (∑ s,t g st (v)/ g st ) / (number of s,t pairs in V- {v}).

8. Floyd-Warshall: shortest path 8 for k = 1 to n do // use node k on path for i = 1 to n do // origin i for j = 1 to n do // destination j if (d[i,k] + d[k,j]) < d[i,j]) { d[i,j] = d[i,k] + d[k,j] // shorter path length visit[i,j] = k // new path goes through k } Given G = (V, E, w) Distance(i, j, 0) = w(i, j) Distance(i, j, k+1) = min{Distance(i, j, k), Distance(i, k+1, k) + Distance(k+1, j, k)} j i k+1 V’ = {1, 2, …, k}

9. 9 Key network models Erdos-Renyi Small world Scale free

10. 10 Erdos-Renyi Totally uniformly random distribution of edges Construction –Given two parameters (n = # of nodes, p = probability of an edge existence) –For all pairs of node (u,v) Create an edge (u,v) with probability p.

11. 11 Small World (Watts-Strogatz) Everyone tends to be close to each other. As the number of nodes (N) in the network grows, the distance between two random nodes grows with the logarithm of N. Construction –Given three parameters: N = # of nodes. K = average degree p = rewiring probability –Construct a ring lattice Connect each ith node to nodes {i-1, i-2, …, i- k/2} and {i+1, i+2, …, i+k/2} with an edge –For each node u For each edge (u, v) –Randomly pick a node v’ = V-{u} –Replace (u, v) with (u, v’) with probability p …

12. 12 Scale-Free A lot of poor work for a few super rich Probability that a node has degree k drops exponentially with k. –P(k) ~ k - Construction (preferential attachment – or rich gets richer) –Given two parameters (n = # of nodes, k = average degree) –Build a small network (e.g. two nodes and one edge) –Repeat Insert a new node v Insert k edges from v to existing nodes. Existing node u gets an edge with probability p u = Deg(u)/ ∑ i Deg(i) –Until we have n nodes

13. 13 Hierarchical Similar to fractals Scale-free networks with high clustering. Construction –Create an initial network (seed) with t peripheral nodes –Create t copies of this network and connect each of them to the central node. Fractal

14. Probabilistic 14 a b c 0.30.6 a b c a b c a b c a b c (1-0.6) x (1-0.3) = 0.28 0.18 0.28 0.120.42 0.28 + 0.12 + 0.42 + 0.18 = 1 G = (V, E, P) P: E -> (0, 1]