In the name of God. Sharif University of Technology Analysis of Biological Networks CE 558 Spring 2020 Dr. H.R. Rabiee

Homework 3 (Network ) (100 points)

1. Compute centrality of the nodes with the most and least centrality for the following network according to the following measures:

• • Eccentricity • Closeness • Shortest Betweenness

1 2. For a complete (m,n)- compute Katz centrality measure with α = 2mn for each node and determine which nodes have the most centrality. ( (m, n)-bipartite graph consists of two independent partitions with m and n nodes each) 3. (a) For a graph prove that we don’t have a central node. (The centrality of all nodes are the same). Prove it for following centrality measures. • Degree • Eccentricity • Closeness • Shortest Path Betweenness • Katz • PageRank (b) Prove that in a graph that has a full-cycle automorphism, there is no central measure. Prove it for an arbitrary centrality measure. (Note that an appropriate centrality measure only depends on the structure of the graph and not the node labels) (c) for n > 2, find a graph with n nodes that has an automorphism but the centrality of the nodes are not all equal for some measure.

1 4. Prove that for any d-regular graph, PageRank centrality measure approaches to nm CKatz as the number of steps approaches to infinity, where n is the number of nodes, m is the number of steps and CKatz is 1 the Katz centrality measure with α = d . 5. (Fast Algorithm to Calculate Shortest-Path-Betweenness Centrality Measure) Consider an arbitrary undirected graph G = (V,E). Define d(s, t) as the and σs,t as the number of shortest path between any two node s, t in the graph. Also define δs,t(v) as the proportion of paths between s and t that traverse v. (a) Prove that vector v is in the shortest path between s and t if and only if:

d(s, t) = d(s, v) + d(v, t) (1)

1 (b) Define Ps(v) as the set of predecessors of v in the path from s:

Ps(v) = {u ∈ V |{u, v} ∈ E, d(s, v) = d(s, u) + 1} (2)

Prove that: X σs,t = σs,u (3)

u∈Ps(t) (c) Prove that for a fixed node v, the number and length of shortest paths from v to all other nodes of the graph can be computed in O(m) where m = |E|. (d) Define X δs,·(v) = δs,t(v) (4) t∈V Prove that if there is exactly one path from s ∈ V to any t ∈ V we have: X δs,·(v) = (1 + δs,·(w)) (5)

w:v∈Ps(w)

(e) * Prove that: X σs,v δs,·(v) = (1 + δs,·(w)) (6) σs,w w:v∈Ps(w) (f) Prove that Shortest Path Centrality of all nodes of the graph can be computed in O(mn).

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