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Comparison of Random Walk based techniques for estimating network - - PowerPoint PPT Presentation

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Comparison of Random Walk based techniques for estimating network averages Jithin K. Sreedharan INRIA, France Arun Kadavankandy Konstantin Avrachenkov Vivek S. Borkar INRIA, France INRIA, France IIT Bombay, India CSoNet 2016, August 2

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SLIDE 1

Comparison of Random Walk based techniques for estimating network averages

Jithin K. Sreedharan

INRIA, France

Vivek S. Borkar

IIT Bombay, India

Konstantin Avrachenkov

INRIA, France

CSoNet 2016, August 2

Arun Kadavankandy

INRIA, France

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SLIDE 2

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 2

  • Estimation in Online Social Network

(OSN)

  • Example:

What proportion of a population supports a given political party? How young a given social network is?

Motivation

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SLIDE 3

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 2

  • Estimation in Online Social Network

(OSN)

  • Example:

What proportion of a population supports a given political party? How young a given social network is?

Motivation

Easy to answer if the graph is fully known beforehand What if the network is not known?

  • Can only crawl network
  • Few queries
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SLIDE 4

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 3

Problem definition

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SLIDE 5

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 3

Problem definition

Let

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SLIDE 6

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 3

Problem definition

  • Undirected graph

Let

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SLIDE 7

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 3

Problem definition

  • Undirected graph
  • Nodes have labels

Let

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SLIDE 8

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 3

Problem definition

  • Undirected graph
  • Nodes have labels
  • Large graph

Let

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SLIDE 9

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 3

Problem definition

  • Undirected graph
  • Nodes have labels
  • Large graph

Let Estimate

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SLIDE 10

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 3

Problem definition

  • Undirected graph
  • Nodes have labels
  • Large graph

Let Estimate

  • Graph is unknown
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SLIDE 11

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 3

Problem definition

  • Undirected graph
  • Nodes have labels
  • Large graph

Let Estimate

  • Graph is unknown
  • Only local information available

Seed nodes and their neighbor IDs Query (visit) a neighbor Visited nodes and their neighbor IDs

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SLIDE 12

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

Random Walk based estimation

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SLIDE 13

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

Random Walk based estimation

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SLIDE 14

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

Random Walk based estimation

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SLIDE 15

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 15

Random Walk based estimation

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SLIDE 16

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

Random Walk based estimation

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SLIDE 17

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

Random Walk based estimation

Random walk has unique stationary distribution if graph 𝐻 is connected and non- bipartite

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SLIDE 18

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

Random Walk based estimation

Random walk has unique stationary distribution if graph 𝐻 is connected and non- bipartite

  • Goal:

Estimate

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SLIDE 19

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

Random Walk based estimation

Random walk has unique stationary distribution if graph 𝐻 is connected and non- bipartite

  • Goal:

Estimate

  • How: Ergodic theorem
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SLIDE 20

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 20

Random Walk based estimation

Random walk has unique stationary distribution if graph 𝐻 is connected and non- bipartite

  • Goal:

Estimate

  • How: Ergodic theorem

For any initial distribution,

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SLIDE 21

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 21

Random Walk based estimation

Random walk has unique stationary distribution if graph 𝐻 is connected and non- bipartite

  • Goal:

Estimate

  • How: Ergodic theorem

For any initial distribution,

How to make ?

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SLIDE 22

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 4 Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 22

Random Walk based estimation

Random walk has unique stationary distribution if graph 𝐻 is connected and non- bipartite

  • Goal:

Estimate

  • How: Ergodic theorem

For any initial distribution,

How to make ? How to compare different random walks?

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SLIDE 23

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 5

Respondent Driven Sampling

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 5

Respondent Driven Sampling

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 5

Respondent Driven Sampling

  • With re-weighting the function 𝑔
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SLIDE 26

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 5

Respondent Driven Sampling

  • With re-weighting the function 𝑔

Requires knowledge of no. of nodes and no. of edges

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SLIDE 27

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 5

Respondent Driven Sampling

  • With re-weighting the function 𝑔

Requires knowledge of no. of nodes and no. of edges Estimator:

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SLIDE 28

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 5

Respondent Driven Sampling

  • With re-weighting the function 𝑔

Requires knowledge of no. of nodes and no. of edges Estimator:

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SLIDE 29

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 5

Respondent Driven Sampling

  • With re-weighting the function 𝑔

Requires knowledge of no. of nodes and no. of edges Estimator:

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SLIDE 30

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 5

Respondent Driven Sampling

  • With re-weighting the function 𝑔

Requires knowledge of no. of nodes and no. of edges Estimator:

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SLIDE 31

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 6

Metropolis Hastings Sampling

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SLIDE 32

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 6

Metropolis Hastings Sampling

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 6

Metropolis Hastings Sampling

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SLIDE 34

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 6

Metropolis Hastings Sampling

If head appears: move to 𝑘

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SLIDE 35

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 6

Metropolis Hastings Sampling

If head appears: move to 𝑘 If tail appears: stays at 𝑌𝑢

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SLIDE 36

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 6

Metropolis Hastings Sampling

If head appears: move to 𝑘 For any initial distribution, If tail appears: stays at 𝑌𝑢

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SLIDE 37

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 7

Reinforcement Learning technique

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SLIDE 38

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 7

Reinforcement Learning technique

  • Graph not necessarily connected or

has included connected components of interest

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SLIDE 39

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 7

Reinforcement Learning technique

  • Graph not necessarily connected or

has included connected components of interest

  • Few seed nodes

Seed nodes

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SLIDE 40

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

Idea of tours

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r

Idea of tours

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

Sample

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

Sample

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

Sample

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

Properties of tours: Sample

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

Properties of tours:

  • Tours are independent

Sample

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SLIDE 48

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

Properties of tours:

  • Tours are independent
  • Fully distributed crawler

Sample

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SLIDE 49

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

Properties of tours:

  • Tours are independent
  • Fully distributed crawler

implementation Sample

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SLIDE 50

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 8

b a c d e f

g

i h k l m n p

q

r b a d e f i h l m n p r

Idea of tours

Properties of tours:

  • Tours are independent
  • Fully distributed crawler

implementation

  • Larger super node size,

shorter the tours Sample

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SLIDE 51

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

Reinforcement Learning technique (contd.)

Stochastic Approximation Algorithm

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SLIDE 52

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

Reinforcement Learning technique (contd.)

Seed set

For each node 𝑗 in Stochastic Approximation Algorithm

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SLIDE 53

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

Reinforcement Learning technique (contd.)

Seed set

For each node 𝑗 in

Function sum inside a tour Cost function

Stochastic Approximation Algorithm

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SLIDE 54

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

Reinforcement Learning technique (contd.)

Seed set

For each node 𝑗 in

sample 1 sample 2 ……. sample 𝑙

Function sum inside a tour Cost function

Stochastic Approximation Algorithm

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SLIDE 55

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

Reinforcement Learning technique (contd.)

Seed set

For each node 𝑗 in

sample 1 sample 2 ……. sample 𝑙

Function sum inside a tour Cost function

Stochastic Approximation Algorithm

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SLIDE 56

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 9

Reinforcement Learning technique (contd.)

Seed set

For each node 𝑗 in

Function sum inside a tour Cost function

Stochastic Approximation Algorithm

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SLIDE 57

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

Which Random Walk method to select ?

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SLIDE 58

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

Which Random Walk method to select ?

  • Mixing time

Not a good criterion here due to burn-in period.

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SLIDE 59

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

Which Random Walk method to select ?

  • Mixing time

Not a good criterion here due to burn-in period.

X1 X2 ……. Xk : accepted sample : rejected sample Burn-in period Approximate stationary regime

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SLIDE 60

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

Which Random Walk method to select ?

  • Mixing time

Not a good criterion here due to burn-in period.

X1 X2 ……. Xk : accepted sample : rejected sample Burn-in period Approximate stationary regime

Reinforcement Learning technique does not require burn-in period

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SLIDE 61

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 10

Which Random Walk method to select ?

  • Mixing time

Not a good criterion here due to burn-in period.

  • Efficiency of the estimator:

How many samples are needed to achieve certain accuracy

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SLIDE 62

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 11

Asymptotic Variance

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SLIDE 63

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 11

Asymptotic Variance

Asymptotic variance of the estimator

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SLIDE 64

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 11

Asymptotic Variance

Asymptotic variance of the estimator Also from Central Limit Theorem equivalent

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

Asymptotic Variance (contd.)

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SLIDE 66

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

Asymptotic Variance (contd.)

  • For Metropolis-Hastings Sampling,

where

Fundamental matrix of Markov chain

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SLIDE 67

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

Asymptotic Variance (contd.)

  • For Respondent Driven Sampling,
  • For Metropolis-Hastings Sampling,

where

Fundamental matrix of Markov chain

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SLIDE 68

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 12

Asymptotic Variance (contd.)

  • For Reinforcement Learning based sampling,
  • For Respondent Driven Sampling,
  • For Metropolis-Hastings Sampling,

where

Fundamental matrix of Markov chain

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SLIDE 69

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

Numerical Studies

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SLIDE 70

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

Numerical Studies

Normalized Root Mean Square Error (NRMSE) vs Budget B

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SLIDE 71

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

Numerical Studies

Normalized Root Mean Square Error (NRMSE) vs Budget B

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SLIDE 72

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

Numerical Studies

Normalized Root Mean Square Error (NRMSE) vs Budget B Why MSE ?

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SLIDE 73

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

Numerical Studies

Normalized Root Mean Square Error (NRMSE) vs Budget B Why MSE ?

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SLIDE 74

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 13

Numerical Studies

Normalized Root Mean Square Error (NRMSE) vs Budget B Budget B: number of allowed samples Why MSE ?

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SLIDE 75

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

Les Misérables network

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SLIDE 76

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

Les Misérables network

Number of nodes: 77, number of edges: 254.

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SLIDE 77

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

Les Misérables network

Number of nodes: 77, number of edges: 254.

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SLIDE 78

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 14

Les Misérables network

Number of nodes: 77, number of edges: 254.

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 15

Les Misérables network contd.

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 15

Les Misérables network contd.

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SLIDE 81

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 15

Les Misérables network contd.

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

Les Misérables network contd.

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

Les Misérables network contd.

Study of asymptotic variance

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

Les Misérables network contd.

Study of asymptotic variance

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Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

Les Misérables network contd.

Study of asymptotic variance

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SLIDE 86

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 16

Les Misérables network contd.

Study of asymptotic variance

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SLIDE 87

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

Friendster network

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SLIDE 88

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

Friendster network

Number of nodes ~ 65K number of edges ~ 1.25M

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SLIDE 89

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

Friendster network

Number of nodes ~ 65K number of edges ~ 1.25M

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SLIDE 90

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 17

Friendster network

Number of nodes ~ 65K number of edges ~ 1.25M

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SLIDE 91

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

Friendster network contd.

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SLIDE 92

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

Friendster network contd.

Stability of sample paths:

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SLIDE 93

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

Friendster network contd.

Stability of sample paths: single path example

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SLIDE 94

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

Friendster network contd.

Stability of sample paths: Varying super-node size single path example

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SLIDE 95

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 18

Friendster network contd.

Stability of sample paths: Varying super-node size Varying step size single path example

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SLIDE 96

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

Conclusions

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SLIDE 97

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

Conclusions

  • Rand Walk based estimators of
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SLIDE 98

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

Conclusions

  • Rand Walk based estimators of
  • Numerical and theoretical study of Mean Square Error & Asymptotic Variance of

 Metropolis-Hastings sampling  Respondent Driven sampling (RDS)  New Reinforcement Learning based sampling (RL)

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SLIDE 99

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

Conclusions

  • Rand Walk based estimators of
  • Numerical and theoretical study of Mean Square Error & Asymptotic Variance of

 Metropolis-Hastings sampling  Respondent Driven sampling (RDS)  New Reinforcement Learning based sampling (RL)

  • Reinforcement Learning technique:

 Tackles disconnected graph  A cross between deterministic iteration and MCMC  Can control the stability of the algorithm with step sizes

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SLIDE 100

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 19

Conclusions

  • Rand Walk based estimators of
  • Numerical and theoretical study of Mean Square Error & Asymptotic Variance of

 Metropolis-Hastings sampling  Respondent Driven sampling (RDS)  New Reinforcement Learning based sampling (RL)

  • Reinforcement Learning technique:

 Tackles disconnected graph  A cross between deterministic iteration and MCMC  Can control the stability of the algorithm with step sizes

  • RDS works better. RL technique comparable, yet more stable and no burn-in !
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SLIDE 101

Jithin K. Sreedharan (jithin.sreedharan@inria.fr) 20

Thank you!

http://bit.do/Jithin