ChoiceRank Identifying Preferences from Node Tra ff ic in Networks - PowerPoint PPT Presentation

ChoiceRank Identifying Preferences from Node Tra ff ic in Networks Lucas Maystre, Matthias Grossglauser School of Computer and Communication Sciences, EPFL ICML — August 8 th , 2017

Motivating Example 2

Problem Statement Explain how users 
 ...given network structure and navigate along edges ... marginal tra ff ic . 101 294 � � � � 73 96 � � � � 51 0.2 � � 0.6 � � 127 0.1 0.1 � � � � 196 51 3

Choice Model Underconstrained problem λ 1 λ 4 → “low-rank” parametrization of p ij . � � λ j λ 2 p ij = � � P i λ k λ 8 k ∈ N + � λ 3 � λ 5 Consistent with Luce's choice axiom . 
 Probability of choosing i over j does not depend on the other alternatives. � � λ 7 λ 6 [Luce 1959 ] 4

Prior Work Inverting a Steady-State Our work [Kumar et al. WSDM 2015 ] Random-walk framework We merely assume discrete choices on a network. Given: • directed graph G = ( V, E ) • model for transitions works with: • stationary distribution π • finite tra ff ic Find matrix P such that • arbitrary network structure • π = π P • if no edge p ij = 0 5

Marginal Tra ff ic is Su ff icient Given network structure + marginal tra ff ic, find “good” parameters λ . Pretend that we can observe all  � X X ` ( λ ; D ) = log � j − log � k transitions D = { c ij | ( i, j ) ∈ E } c ij ( i,j ) ∈ E k ∈ N + i n  � c 14 X X i log � i − c + c − = i log � � � k c 13 λ j i =1 k ∈ N + p ij = � � i P i λ k k ∈ N + � X X c ji c ij � j ∈ N + j ∈ N − i i ... � � Marginal tra ff ic is a minimally su ff icient statistic { ( c + i ) | i ∈ V } i , c − 6

Robust Inference ML estimate is o fu en ill-defined because of graph structure or data sparsity. 
 → embed in a Bayesian setting by postulating a prior on λ i . n  � X X i log λ i − c + c − i log λ k i =1 k ∈ N + i n  � X + ( α − 1) log λ i − βλ i i =1 Theorem : if α > 1 and β > 0, there is always a unique maximum 7

ChoiceRank Algorithm We maximize the log-posterior using the MM algorithm . [Hunter 2004 ] Scales well to large graphs . Tested on Common Crawl hyperlink graph: 
 • 3.5 B nodes, 128 B edges • Takes 20 min / iteration on a recent machine λ i( t +1) λ i( t ) c + c − λ ( t +1) where γ ( t ) j i One iteration requires two = = , i j i γ ( t ) j λ ( t ) P P passes over the edges k ∈ N + j k j ∈ N − 8

Experimental Results English Wikipedia tra ff ic — 2 M nodes, 13 M edges, 1.2 B transitions. 
 How well do we recover the transition probabilities? 2 . 5 0 . 4 2 . 0 KL-divergence Displacement 1 . 5 0 . 3 1 . 0 0 . 2 0 . 5 0 . 0 0 . 1 C-Rank Traffic P-Rank Uniform C-Rank Traffic P-Rank Uniform p ij ∝ λ j p ij ∝ c − p ij ∝ PR j p ij ∝ 1 j 9

Code & Examples github.com/ lucasmaystre/choix 10

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ChoiceRank vs. PageRank PageRank ChoiceRank • Given a network and marginal • Given a network , find steady- state tra ff ic . tra ff ic , find transition probabilities . • Assumption: transitions are • Assumption: transitions follow uniformly random over neighbors. Luce's choice axiom . • ChoiceRank score corresponds to • PageRank score corresponds to a page's popularity . a page's utility . 12

Issues with ML estimate 1 1 4 � � 2 8 � � 3 � 5 � 6 7 � � 13

Issues with ML estimate 2 3 = 1 , c + 3 = 2 c − 3 3 3 4 4 4 4 = 1 c − c + 4 = 1 2 = 2 c − c + 2 = 1 2 2 2 1 = 1 c − c + 1 = 1 1 1 1 14

NYC Bike Sharing Data Applications beyond clickstream data — e.g., mobility networks . 0 . 45 0 . 3 0 . 40 KL-divergence Displacement 0 . 35 0 . 2 0 . 30 0 . 1 0 . 25 0 . 0 0 . 20 C-Rank Traffic P-Rank Uniform C-Rank Traffic P-Rank Uniform 15