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

ChoiceRank

Identifying Preferences from Node Traffic in Networks

Lucas Maystre, Matthias Grossglauser School of Computer and Communication Sciences, EPFL

ICML — August 8th, 2017

Motivating Example

2

Problem Statement

3

Explain how users
 navigate along edges...

• 0.2

0.6 0.1 0.1

...given network structure and marginal traffic.

• 101

294 51 73 96 127 196 51

Choice Model

4

Underconstrained problem

• λ1

λ2 λ4 λ8 λ5 λ3 λ6 λ7

→ “low-rank” parametrization of pij. Consistent with Luce's choice axiom.


Probability of choosing i over j does not depend on the

• ther alternatives.

pij = λj P

k∈N +

i λk

[Luce 1959]

Prior Work

5

Given:

• directed graph
• model for transitions
• stationary distribution

G = (V, E) π π = πP Find matrix P such that

• if no edge

pij = 0 Inverting a Steady-State [Kumar et al. WSDM 2015] Random-walk framework Our work We merely assume discrete choices

• n a network.

works with:

• finite traffic
• arbitrary network structure

Marginal traffic is a minimally sufficient statistic {(c+

i , c− i ) | i ∈ V }

Marginal Traffic is Sufficient

6

Given network structure + marginal traffic, find “good” parameters λ. `(λ; D) = X

(i,j)∈E

cij  log j − log X

k∈N +

i

k

• =

n

X

i=1

 c−

i log i − c+ i log

X

k∈N +

i

k

• Pretend that we can observe all

transitions D = {cij | (i, j) ∈ E}

pij = λj P

k∈N +

i λk

• c14

c13 ...

X

j∈N −

i

cji X

j∈N +

i

cij

Robust Inference

7

ML estimate is ofuen ill-defined because of graph structure or data sparsity.
 → embed in a Bayesian setting by postulating a prior on λi. Theorem: if α > 1 and β > 0, there is always a unique maximum

n

X

i=1

 c−

i log λi − c+ i log

X

k∈N +

i

λk

• +

n

X

i=1

 (α − 1) log λi − βλi

ChoiceRank Algorithm

8

We maximize the log-posterior using the MM algorithm. [Hunter 2004] λ(t+1)

i

= c−

i

P

j∈N −

i γ(t)

j

, where γ(t)

j

= c+

j

P

k∈N +

j λ(t)

k

λi(t) λi(t+1) One iteration requires two passes over the edges

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

Experimental Results

9

C-Rank Traffic P-Rank Uniform 0.0 0.5 1.0 1.5 2.0 2.5 KL-divergence C-Rank Traffic P-Rank Uniform 0.1 0.2 0.3 0.4 Displacement

English Wikipedia traffic — 2 M nodes, 13 M edges, 1.2 B transitions.
 How well do we recover the transition probabilities?

pij ∝ λj pij ∝ c−

j

pij ∝ PRj pij ∝ 1

Code & Examples

10

github.com/ lucasmaystre/choix

11

ChoiceRank vs. PageRank

ChoiceRank

• Given a network and marginal

traffic, find transition probabilities.

• Assumption: transitions follow

Luce's choice axiom.

• ChoiceRank score corresponds to

a page's utility.

12

PageRank

• Given a network, find steady-

state traffic.

• Assumption: transitions are

uniformly random over neighbors.

• PageRank score corresponds to a

page's popularity.

Issues with ML estimate 1

13

• 1

2 3 4 5 6 7 8

Issues with ML estimate 2

14

1 2 4 3 1 2 3 4 1 2 3 4

c−

2 = 2

c+

4 = 1

c−

4 = 1

c+

2 = 1

c−

1 = 1

c+

1 = 1

c−

3 = 1, c+ 3 = 2

C-Rank Traffic P-Rank Uniform 0.0 0.1 0.2 0.3 KL-divergence C-Rank Traffic P-Rank Uniform 0.20 0.25 0.30 0.35 0.40 0.45 Displacement

NYC Bike Sharing Data

15

Applications beyond clickstream data — e.g., mobility networks.