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Proximity in Graphs by Using Random Walks
Many of the slides are borrowed from Dr. Hanghang Tong’ talk slides and Dr. Jure Leskovec’s lecture notes
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Proximity on Graph What is Prox between A and B
‘how close is Smith to Johnson’?
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Proximity on Graphs: Why?
Link prediction Ranking Management Image caption Neighborhooh Formulation Conn. subgraph Pattern match Collaborative Filtering Many more…
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Link Prediction How to predict the existence of the link?
Proximity [Liben-Nowell ]
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Center-Piece Subgraph(Ceps)
Given Q query nodes Find Center-piece ( ) Input of Ceps Q Query nodes Budget b K softand coefficient App. Social Network Law Inforcement Gene Network …
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Example of CEPS
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CEPS: Overview Individual Score Calculation Combine Individual Scores
Measure importance wrt individual query Combine Individual Scores Measure importance wrt query set “Extract” Alg. … the connection subgraphs
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Issue: `degree-1 node’ effect [Faloutsos+] [Koren+]
Esc_Prob(a->b)=1 Esc_Prob(a->b)=1 no influence for degree-1 nodes (E, F)! known as ‘pizza delivery guy’ problem in undirected graph
![Issue: `degree-1 node’ effect [Faloutsos+] [Koren+]](http://slideplayer.com/slide/17266248/100/images/17/Issue%3A+%60degree-1+node%E2%80%99+effect+%5BFaloutsos%2B%5D+%5BKoren%2B%5D.jpg "Esc_Prob(a->b)=1. Esc_Prob(a->b)=1. no influence for degree-1 nodes (E, F)! known as ‘pizza delivery guy’ problem in undirected graph.")
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RWR: Individual Score Calculation
Goal Individual importance score r(i,j) = ri,j For each node j wrt each query i How to Random walk with restart Steady State Prob.
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An Illustrating Example
5 Prob (RW will finally stay at j) 11 12 4 Starting from 1 Randomly to neighbor Some p to return to 1 10 3 13 6 2 7 1 9 8
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Individual Score Calculation
Q1 Q2 Q3 Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Node 9 Node 10 Node 11 Node 12 Node 13
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Individual Score Calculation
Q1 Q2 Q3 Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Node 9 Node 10 Node 11 Node 12 Node 13
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Variant: escape probability
Define Random Walk (RW) on the graph Esc_Prob(AB) Prob (starting at A, reaches B before returning to A) the remaining graph A B Esc_Prob = Pr (smile before cry)
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AND: Combine Scores Q: How to combine scores? A: Multiply
…= prob. 3 random particles coincide on node j
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K_SoftAnd: Combine Scores
Generalization – SoftAND: We want nodes close to k of Q (k<Q) query nodes. Q: How to do that?
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K_SoftAnd: Combine Scores
Generalization – softAND: We want nodes close to k of Q (k<Q) query nodes. Q: How to do that? A: Prob(at least k-out-of-Q will meet each other at j)
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K_SoftAnd: Relaxation of AND
Disconnected Communities Noise Asking AND query? No Answer!
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AND query vs. K_SoftAnd query
x 1e-4 2_SoftAnd Query And Query
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1_SoftAnd query = OR query
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Measuring Importance Individual Scores Combining Scores
Q1 Q2 Q3 Node 1 Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Node 8 Node 9 Node 10 Node 11 Node 12 Node 13 0.4505 0.0710 0.2267 0.1010 OR 0.0103 0.0019 0.0024 0.0046 Random walk with restart K_SoftAnd Steady State Prob And 2_SoftAnd Meeting Prob
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“Extract” Alg. Goal How to…”Extract” Alg. Maximize total scores and
1 2 3 5 4 6 7 8 9 10 11 12 13 14 15 16 Goal Maximize total scores and ‘Appropriate’ Connections How to…”Extract” Alg. Dynamic Programming Greedy Alg. Pickup promising node Find ‘best’ path 2 10 9 6 8 13 11 4 5 7 12 3 1
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Case Study: AND query
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database Statistic 2_SoftAnd query
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