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# RobinsonBCslides - Array Based Betweenness Centrality Eric...

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Array Based Betweenness Centrality Eric Robinson Northeastern University MIT Lincoln Labs Jeremy Kepner MIT Lincoln Labs

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Vertex Betweenness Centrality Which Vertices are Important?
Loss Of Communication Slower Communication Vertex Betweenness Centrality Which Vertices are Important?

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Vertex Betweenness Centrality How do we Measure Importance? Number of Shortest Paths Through Node 0 0 0 50 32 45 8 8 9 9 1
Vertex Betweenness Centrality Traditional Algorithm 0 0 0 50 32 45 8 8 9 9 1 = V v t s st st B v v C σ ) ( ) (

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Traditional Algorithm Theoretical Time and Space Time: O(N 3 ) Storage: O(N 2 ) 0 0 0 50 32 45 8 8 9 9 1
Vertex Betweenness Centrality Updating Algorithm For each starting node: Once you know: The depth of each node in the BFS, D The centrality updates for nodes at depth d, u The shortest path counts from the root, s Can determine centrality of nodes at depth d-1: For each node, v, at depth d-1, it's update is the sum: u v = ∀ v ,w ∈ E ,w D d , 1 u d ∗ s v / s w

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Updating Algorithm An Example Find Single Source Shortest Path Counts 7 8 6 1 5 4 9 2 3 10 11 1 O(N + M)
Updating Algorithm An Example Find Single Source Shortest Path Counts 7 8 6 1 5 4 9 2 3 10 11 1 1 1 O(N + M)

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Updating Algorithm An Example Find Single Source Shortest Path Counts 7 8 6 1 5 4 9 2 3 10 11 1 1 1 2 O(N + M)
Updating Algorithm An Example Find Single Source Shortest Path Counts 7 8 6 1 5 4 9 2 3 10 11 1 1 1 2 2 2 O(N + M)

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Updating Algorithm An Example Find Single Source Shortest Path Counts 7 8 6 1 5 4 9 2 3 10 11 1 1 1 2 2 2 2 2 O(N + M)
Updating Algorithm An Example Find Single Source Shortest Path Counts 7 8 6 1 5 4 9 2 3 10 11 1 1 1 2 2 2 2 2 2 2 2 O(N + M)

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## This note was uploaded on 12/27/2011 for the course CMPSC 240A taught by Professor Gilbert during the Fall '09 term at UCSB.

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RobinsonBCslides - Array Based Betweenness Centrality Eric...

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