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Applications of Relative Importance. Why is relative importance interesting? Web Social Networks Citation Graphs Biological Data Graphs become too complex for manual analysis. Existing Techniques. Web PageRank (Google) Social Networks ‘Centrality’
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Applications of Relative Importance • Why is relative importance interesting? • Web • Social Networks • Citation Graphs • Biological Data • Graphs become too complex for manual analysis
Existing Techniques • Web • PageRank (Google) • Social Networks • ‘Centrality’ • All focus on global measures of node importance – we’re interested in importance relative to a set of root nodes R
Use Existing Techniques? • Use global algorithm on the subgraph surrounding root nodes? • No preferential treatment of root nodes – just ranking surrounding nodes.
Organization: Relative importance Algorithms • Notation • Problem Formulation • General Framework • Algorithms
Notation • Digraph • G = (V, E) • Edges • Ordered pair of nodes (u, v) • Graphs are directed, unweighted, simple • Walks from u to v • a.k.a. • A walk is a path with no repeated nodes
Notation • k-short paths • P(u,v) – set of paths between u and v • – set of distinct out-going edges from u • Similarly, we have
Problem Formulation • Given G and r and t, where , compute the “importance” of t w.r.t. root node r:
Problem Formulation • Given G and node , rank all vertices in T(G), T V, w.r.t. r.
Problem Formulation • Given G, a set of nodes T(G) to rank, and a set of root nodes R(G) where R V, rank all vertices in T w.r.t. R.This is similar to the last case, except that we compute rather than Average importance:
Problem Formulation (3 cont’d.) • Rather than average each node’s importance score, we could define • This requires ‘important’ nodes to have a high importance score among all nodes in R
Problem Formulation • Given G, rank all nodes where R=T=V.
General Framework:Weighted Paths • Nodes are related according to the paths that connect them • The longer the path, the less importance: is a scalar coefficient,P(r,t) is a set of paths from r to t, pi is the ith path in P. Importance decays exponentially
How to choose P(r,t)? • Path examples Shortest paths from R to T: {R-C-T. R-D-T} which fail to capture much of Connectivity from R to T. a. b.
Shortest Path • e.g.: Transport cargo from r to t • Shortest path doesn’t always give a good approximation of importance. • E.g: the web (graph b)
k-Short Paths • Paths of length K • Idea: there might often be longer paths than the shortest ones that are important to take into account • Fixes problem of longer, important paths in Shortest Paths • e.g.: graph b., 3-short • Problem: capacity constraints • e.g.: network topology
k-Short Node-Disjoint Paths • No nodes and no edges are repeated • Implicitly enforces capacity constraints • Motivated by ‘mass flow’ where importance can ‘flow’ along paths • e.g.: graph b. • Breadth-first with some heuristic, with some K and some
Markov Chains & Relative Importance • Graph viewed as a stochastic process • Explanation of Markov Chains • Token traversing Chain… • Obviously good for modeling the web
Markov Chains & Relative Importance • Markov Centrality • Mean First Passage Time : expected number of steps until first arrival at node t starting at node r : probability that the chain first returns to state t in exactly n steps
Markov Chains & Relative Importance • Bias toward ‘central nodes’ • COMPLEX!! • Time: O(|V|3) (inversion of |V|x|V| transition matrix) • Space: O(|V2|)
Markov Chains & Relative Importance • PageRank • Uses backlinks to assign importance to web pages
Markov Chains & Relative Importance • PageRank • Less complex Converges logarithmically • 322 million links processed in 52 iterations
Markov Chains & Relative Importance • Retrofit PageRank such that all nodes in R have a uniform bias at the start • ‘Surfer’ begins at a root node, traverses graph, returning to root set R with probability at each time-step • I(t|R) = probability that surfer visits t during a walk
Experiments (Simulated Data) • More complex • in and out degrees changed • Shortest path lengths between nodes changed (e.g.: A-B) • Analysis which follows, R={A,F}
Experiments (Simulated Data) • HITSPa A .252 F .241 G .128 C .110 E .099 H .052 D .032 J .025 I .032 B .024 • HITSPh F .225 A .186 D .162 B .119 E .090 I .067 H .061 J .050 G .028 C .008
Experiments (Simulated Data) • MarkovC J .180 C .133 G .130 H .129 E .111 I .101 F .069 D .051 A .047 B .044 • KSMarkov H .146 G .142 E .142 J .140 C .120 I .098 F .087 D .061 A .034 B .024
Experiments (9/11 Terrorist Network) • 63 nodes (terrorists) • 308 edges (interactions)
Conclusion • Provides a first-step to addressing ‘relative-importance’ • Scaling for algorithms such as Markov Chaining can be an issue • Using different algorithms and comparing results can reveal interesting information • …Paper Analysis…
References • White, Smyth. Algorithms for Estimating Relative Importance in Networks. SIGKDD ’03. • Page, Brin, Motwani, Winograd. The PageRank Citation Ranking: Bringing Order to the Web. Stanford University, Computer Science Department Technical Report. • Wikipedia on Markov Chains • http://en.wikipedia.org/wiki/Markov_chain • http://en.wikipedia.org/wiki/Examples_of_Markov_chains