Estimating Page Rank on Graph Streams Atish Das

Estimating Page. Rank on Graph Streams Atish Das Sarma (Georgia Tech) Sreenivas Gollapudi, Rina Panigrahy (Microsoft Research)

Page. Rank • Page. Rank – Determine Ranking of nodes in graphs • Typically large graphs - WWW, Social Networks • Run daily by commercial search engines

Page. Rank computation a b u c

Page. Rank Computation a b u c Our Approach: No Matrix-Vector Multiplication!

Our Result Many Random Walk Samples Efficiently. Approximate Page. Rank u

Other results from Random Walks G u We can estimate: Mixing Time Conductance Using Streams

Streaming Input is a “stream” e 1, e 2, e 3, e 4, e 5, e 6, e 7, …. Few Passes 010001011 Frequency moments, quantiles 011101011 0100110111 Graphs: Edges, arbitrary order Small RAM working memory 7

Related Work • Sparsifiers (Benczur-Karger 96, Spielman. Teng 01, Spielman-Srivastava 08) – Given an undirected graph, produces a sparse one – approximately preserves x’Lx – Can be used to compute sparse cuts • Streaming version of BK 96 (Ahn, Guha 09) ~ – Sparse cuts in 1 pass and O(n) space. • Accelarated Page Rank (Mc. Sherry 08) – heuristics 8

Key Idea l u v One walk from u length l efficiently Later extend to Many walks

Single Random Walk - Naive Algo. s One Step with every Pass! Constant Space Passes

Second Naive Algo s Single Pass Sample sufficient edges! If , then sample 2 out-edges from each node. (store order)

Comparison Naive (single walk): l u Our Result: In fact Automatically: walks!

Insight: Merge Short Walks Sample w fraction of nodes (centers) w s a length walks b w w passes - w w Merge and extend short walks! w Two problems: End up at node second time End up at non-sampled node

Stuck Nodes w w s w Again. And again. . . Slow? w w Sample an edge from stuck. w w If new nodes, good in passes!

Stuck nodes Stuck on same Nodes? w w s s s w w w s Must include to set previous seen centers w w Sample s edges from each w s s s progress OR new node!

Summary w w s s s w w w s w s s • Perform short walks from sampled centers • Concatenate walks until stuck • Sample edges from stuck • Make local progress until new node • Local progress = s • New node : center with prob • Amortized progress, every pass

Summary w w s s s w Total number of passes : w w s w w Total Space : w s s

Summary w Set w s s s w Number of passes = w w s w s s Space =

Many Walks Naive Space Bound: w w s s s w We show: w w s w s s Observation: Many short walks not used in Single RW.

Many Random Walks : probability node ’s short walk used in single RW. • If known : save lot of space! • Perform K random walks • Total number of short walks required is about • • Don’t know . But can estimate.

Estimating l u • Run K = (log n) walks of length • Gives a crude estimate of • Sufficient to double K • Continue doubling K • Gives K walks in space • Passes

Distributions samples Space Passes u Distribution:

Mixing Time, Conductance • Undirected graphs: Compare Distribution with Steady State. • Estimating difference: samples. [Batu et. al. ’ 01] – approximate mixing time. • Directed, till distribution “stabilizes”: samples. • Conductance: • Recall space for walks:

Results recap • - Mixing Time for Undirected Graphs : • Quadratic Approximation to Conductance • Page. Rank to accuracy

Open Questions? • Improve passes for random walks. In particular, sub-linear space and constant passes. • Graph Cuts and Graph Sparsification for directed graphs • Better (streaming) algorithms for computing eigenvectors

Thank You!

Summary • • Perform short walks from sampled centers Concatenate walks until stuck Sample edges from stuck Make local progress until new node Local progress = s New node = nodes gives center Amortized, every pass -

Summary • • Perform short walks from sampled centers Concatenate walks until stuck Sample edges from stuck Make local progress until new node Local progress = s New node = nodes gives center Amortized, every pass -

Analysis • • • Total number of passes : Total Space : Set Number of passes = Space =