Edge Conductance Estimation using MCMC Ashish Bora 1 Vivek S. Borkar - - PowerPoint PPT Presentation

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Edge Conductance Estimation using MCMC Ashish Bora 1 Vivek S. Borkar - - PowerPoint PPT Presentation

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Edge Conductance Estimation using MCMC Ashish Bora 1 Vivek S. Borkar 2 Dinesh Garg 3 Rajesh Sundaresan 4 1 Department of Computer Science, University of Texas at Austin 2 Department of Electrical Engineering, IIT Bombay, Mumbai, India 3 IBM India

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Edge Conductance Estimation using MCMC

Ashish Bora1 Vivek S. Borkar2 Dinesh Garg3 Rajesh Sundaresan4

1Department of Computer Science, University of Texas at Austin 2Department of Electrical Engineering, IIT Bombay, Mumbai, India 3IBM India Research Lab, Bengaluru, India 4Department of Electrical Communication Engineering and the Robert Bosch Centre for

Cyber Physical Systems, Indian Institute of Science, Bengaluru, India.

Allerton, 2016

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 1 / 35

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Outline

1

Motivation What is conductance? Why estimate conductances?

2

Notation

3

Prior Work

4

Algorithm Motivation Idea Pseudocode

5

Theoretical results

6

Simulation Experiments Cardinal Estimation Ordinal Estimation

7

Discussion

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 2 / 35

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Outline

1

Motivation What is conductance? Why estimate conductances?

2

Notation

3

Prior Work

4

Algorithm Motivation Idea Pseudocode

5

Theoretical results

6

Simulation Experiments Cardinal Estimation Ordinal Estimation

7

Discussion

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 3 / 35

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What is conductance?

Analogy

Given a graph G = (V , E), imagine each edge as a unit resistor.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 4 / 35

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What is conductance?

Definition

i j Pick any two nodes i, j ∈ V .

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 5 / 35

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What is conductance?

Definition

i j Pick any two nodes i, j ∈ V . Inject unit current at i and extract it at j.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 6 / 35

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What is conductance?

Definition

i j Pick any two nodes i, j ∈ V . Inject unit current at i and extract it at j. Effective resistance between i and j is the potential difference between them.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 6 / 35

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What is conductance?

Definition

i j Pick any two nodes i, j ∈ V . Inject unit current at i and extract it at j. Effective resistance between i and j is the potential difference between them. Effective conductance is inverse of effective resistance.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 6 / 35

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Outline

1

Motivation What is conductance? Why estimate conductances?

2

Notation

3

Prior Work

4

Algorithm Motivation Idea Pseudocode

5

Theoretical results

6

Simulation Experiments Cardinal Estimation Ordinal Estimation

7

Discussion

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 7 / 35

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Why estimate conductances?

Effective resistance as a robust measure of distance ([1], [2])

Considers all paths Less sensitive to edge or node insertions and deletions

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 8 / 35

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Why estimate conductances?

Effective resistance as a robust measure of distance ([1], [2])

Considers all paths Less sensitive to edge or node insertions and deletions

Sum of effective resistances across all pairs

measure of network robustness equals network criticality parameter [4]

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 8 / 35

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Why estimate conductances?

Effective resistance as a robust measure of distance ([1], [2])

Considers all paths Less sensitive to edge or node insertions and deletions

Sum of effective resistances across all pairs

measure of network robustness equals network criticality parameter [4]

Edge resistances for graph sparsification [3]

Edges sampled (with replacement) according to their effective resistance Approximately preseves quadratic form of Graph Laplacian (i.e. x⊤Lx)

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 8 / 35

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Notation

G = (V , E) is an undirected, unweighted, connected, finite graph. m = |E|, n = |V | ∂i = {j | (i, j) ∈ E} di = |∂i| dmax = maxi∈V di, dmin = mini∈V di dij = min{di, dj}, Dij = max{di, dj}. πi = di/2m, stationary distribution of simple random walk on G davg =

i∈V diπi

Gij = the effective conductance between i and j Rij = the effective resistance between i and j

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 9 / 35

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Prior Work

Nodes of the graph can be embedded in Euclidean space so that the resulting pair-wise distances encode the effective resistances.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 10 / 35

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Prior Work

Nodes of the graph can be embedded in Euclidean space so that the resulting pair-wise distances encode the effective resistances. The embedding depends on the edge-node adjacency matrix and the Laplacian of the graph.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 10 / 35

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Prior Work

Nodes of the graph can be embedded in Euclidean space so that the resulting pair-wise distances encode the effective resistances. The embedding depends on the edge-node adjacency matrix and the Laplacian of the graph. [3] uses low dimensional random projection to preserve pairwise distances to estimate resistances. Takes only O(m/ǫ2) steps, but requires centralized computation.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 10 / 35

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Outline

1

Motivation What is conductance? Why estimate conductances?

2

Notation

3

Prior Work

4

Algorithm Motivation Idea Pseudocode

5

Theoretical results

6

Simulation Experiments Cardinal Estimation Ordinal Estimation

7

Discussion

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 11 / 35

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Algorithm

Motivation

For large dynamic graphs, we would like algorithms that Are distributed and use minimal local communication

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 12 / 35

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Algorithm

Motivation

For large dynamic graphs, we would like algorithms that Are distributed and use minimal local communication Have low memory footprint

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 12 / 35

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Algorithm

Motivation

For large dynamic graphs, we would like algorithms that Are distributed and use minimal local communication Have low memory footprint Use very few computations per step

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 12 / 35

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Algorithm

Motivation

For large dynamic graphs, we would like algorithms that Are distributed and use minimal local communication Have low memory footprint Use very few computations per step Are easily parallelizable

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 12 / 35

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Algorithm

Motivation

For large dynamic graphs, we would like algorithms that Are distributed and use minimal local communication Have low memory footprint Use very few computations per step Are easily parallelizable Are incremental and adaptive

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 12 / 35

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Outline

1

Motivation What is conductance? Why estimate conductances?

2

Notation

3

Prior Work

4

Algorithm Motivation Idea Pseudocode

5

Theoretical results

6

Simulation Experiments Cardinal Estimation Ordinal Estimation

7

Discussion

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 13 / 35

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Algorithm

Idea

We focus on edge effective conductance estimation as needed for graph sparsification algorithm in [3].

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 14 / 35

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Algorithm

Idea

We focus on edge effective conductance estimation as needed for graph sparsification algorithm in [3]. We use random walks on the graph to estimate effective conductance and effective resistances.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 14 / 35

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Algorithm

Idea

We focus on edge effective conductance estimation as needed for graph sparsification algorithm in [3]. We use random walks on the graph to estimate effective conductance and effective resistances. A random walk on the graph picks, from the current position, one of the neighbors with equal probability. Such random walks naturally give us many of the desired properties.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 14 / 35

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Algorithm

Idea

We focus on edge effective conductance estimation as needed for graph sparsification algorithm in [3]. We use random walks on the graph to estimate effective conductance and effective resistances. A random walk on the graph picks, from the current position, one of the neighbors with equal probability. Such random walks naturally give us many of the desired properties. We assume positive recurrence of the associated Markov Chain.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 14 / 35

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Algorithm

Idea

Let pij denote the probability that a random walk starting at node i visits node j before returning to node i. i j i j

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 15 / 35

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Algorithm

Idea

A key fact that underlies our algorithm is: pij = Gij/di

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 16 / 35

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Algorithm

Idea

A key fact that underlies our algorithm is: pij = Gij/di We estimate this probability by averaging results from several i to i paths in a random walk.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 16 / 35

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Algorithm

Idea

A key fact that underlies our algorithm is: pij = Gij/di We estimate this probability by averaging results from several i to i paths in a random walk. We will show how this can be done only with local communication for edge conductance estimation.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 16 / 35

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Algorithm

Variables

Let’s introduce some variables that will be used in the algorithm

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 17 / 35

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Algorithm

Variables

Let’s introduce some variables that will be used in the algorithm

  • pij is boolean. It denotes the the success or failure of visiting node j

in an instance of a return path from node i to node i of the random walk.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 17 / 35

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Algorithm

Variables

Let’s introduce some variables that will be used in the algorithm

  • pij is boolean. It denotes the the success or failure of visiting node j

in an instance of a return path from node i to node i of the random walk. Ni is the number of times node i was visited.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 17 / 35

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Algorithm

Variables

Let’s introduce some variables that will be used in the algorithm

  • pij is boolean. It denotes the the success or failure of visiting node j

in an instance of a return path from node i to node i of the random walk. Ni is the number of times node i was visited.

  • pij is a running estimate of pij.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 17 / 35

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Algorithm

Observation 1

In the long run, every visit to node i marks the end of a return path. Thus, on visiting node i, we can update pij using pij. i j

  • pij

i previous visit current visit

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 18 / 35

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Algorithm

Observation 2

A visit to node i at time t will be a part of a cycle that originated at node j prior to time t and a subsequent return to node j after time t (with probability 1, because of positive recurrence). Thus a visit to node i can be used to update pji. j

  • pji ← 1

j previous visit current future visit

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 19 / 35

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Outline

1

Motivation What is conductance? Why estimate conductances?

2

Notation

3

Prior Work

4

Algorithm Motivation Idea Pseudocode

5

Theoretical results

6

Simulation Experiments Cardinal Estimation Ordinal Estimation

7

Discussion

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 20 / 35

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Algorithm

Pseudocode

Algorithm 1 Visit Before Return Require: T, G = (V , E)

1: ∀ i ∈ V , Ni ← 0 2: ∀ (i, j) ∈ E,

pij = pij = 0.

3: Sample initial node X1 from the stationary distribution π. 4: for t = [1, 2, 3, · · · , T] do 5:

Let i = Xt

6:

for all j in ∂i do

7:

  • pij ← (

pijNi + pij)/(Ni + 1)

8:

  • pij ← 0

9:

  • pji ← 1

10:

Ni ← Ni + 1

11:

Jump to a neighbor of the current node as identified by the walk.

12: For every (i, j) ∈ E, output

Gij = max

  • 1, di

2

pij + dj

2

pji

  • .

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 21 / 35

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Complexity result

Theorem (Performance of VisitBeforeReturn)

Fix an edge (i, j) ∈ E. For any 0 < ǫ < d2

ij/(4m) and 0 < δ < 1/2,

T = O

  • Dij · max{m, Dijtmix} · 1

ǫ2 log 1 δ

  • steps suffice to ensure that the output

Gij of the algorithm VisitBeforeReturn satisfies P(| Gij − Gij| ≥ ǫ) ≤ δ. If the algorithm is run for T steps, it requires O(davgT) computation steps

  • n the average (worst case O(dmaxT) computations), and uses

O(m log T) space.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 22 / 35

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Proof ideas

  • pij =

Ni

k=1

pk

ij

Ni .

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 23 / 35

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Proof ideas

  • pij =

Ni

k=1

pk

ij

Ni . Get concentration for Ni using McDiarmids inequality for Markov chains.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 23 / 35

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Proof ideas

  • pij =

Ni

k=1

pk

ij

Ni . Get concentration for Ni using McDiarmids inequality for Markov chains. Get concentration for pij for a fixed Ni using Hoeffding’s inequality.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 23 / 35

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Proof ideas

  • pij =

Ni

k=1

pk

ij

Ni . Get concentration for Ni using McDiarmids inequality for Markov chains. Get concentration for pij for a fixed Ni using Hoeffding’s inequality. Combine the two.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 23 / 35

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Outline

1

Motivation What is conductance? Why estimate conductances?

2

Notation

3

Prior Work

4

Algorithm Motivation Idea Pseudocode

5

Theoretical results

6

Simulation Experiments Cardinal Estimation Ordinal Estimation

7

Discussion

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 24 / 35

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Simulation Experiments

Cardinal Estimation

ǫ(t) = 1 m

  • (i,j)∈E
  • Gij(t) − Gij
  • 2

4 6 8 10 x 10

5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Number of Steps t ǫ(t) Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 25 / 35

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Simulation Experiments

Cardinal Estimation

2 4 6 8 10 x 10

5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of Steps t ǫ(t)

  • t/mdavg

Example 1 Example 2 Example 3 Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 26 / 35

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Outline

1

Motivation What is conductance? Why estimate conductances?

2

Notation

3

Prior Work

4

Algorithm Motivation Idea Pseudocode

5

Theoretical results

6

Simulation Experiments Cardinal Estimation Ordinal Estimation

7

Discussion

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 27 / 35

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Simulation Experiments

Top-k set estimation

Incremental approximate estimation algorithms can typically recover

  • rdering much faster than exact values.

We test performance of our algorithm for recovering top-k edges with high conductance by plotting fraction of top-k largest conductance edges correctly identified at time t.

2 4 6 8 10 x 10

5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of Steps t fk(t) k =100 k =400 k =700 k =1000 increasing k

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 28 / 35

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Discussion

Our algorithm can be naturally distributed: each node stores information about its neighbors.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 29 / 35

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Discussion

Our algorithm can be naturally distributed: each node stores information about its neighbors. Parallelization can be easily obtained by running multiple random walks and avergaing their results.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 29 / 35

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Discussion

Our algorithm can be naturally distributed: each node stores information about its neighbors. Parallelization can be easily obtained by running multiple random walks and avergaing their results. Our guarantees are weaker as compared to [3]. Particularly, the restriction on ǫ forces it to be too small. Whether this can be removed is an open question.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 29 / 35

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Discussion

Our algorithm can be naturally distributed: each node stores information about its neighbors. Parallelization can be easily obtained by running multiple random walks and avergaing their results. Our guarantees are weaker as compared to [3]. Particularly, the restriction on ǫ forces it to be too small. Whether this can be removed is an open question.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 29 / 35

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Conclusion

We have presented a MCMC based scheme to approximate edge conductances. Our algorithm is incremental and iterative, can be easily distributed, works with local communication, and uses very little memory and computation per step. We provide theoretical guarantees on the performance. Simulation experiments support our theoretical results.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 30 / 35

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References

Aykut Firat, Sangit Chatterjee, and Mustafa Yilmaz. “Genetic clustering of social networks using random walks”. In: Computational Statistics & Data Analysis 51.12 (2007), pp. 6285–6294. Francois Fouss et al. “Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation”. In: Knowledge and Data Engineering, IEEE transactions on 19.3 (2007),

  • pp. 355–369.

Daniel A Spielman and Nikhil Srivastava. “Graph sparsification by effective resistances”. In: SIAM Journal on Computing 40.6 (2011), pp. 1913–1926. Ali Tizghadam and Alberto Leon-Garcia. “On robust traffic engineering in transport networks”. In: Global Telecommunications Conference, 2008. IEEE GLOBECOM 2008. IEEE. IEEE. 2008, pp. 1–6.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 31 / 35

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Thank you for your attention! Questions?

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 32 / 35

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Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 33 / 35

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Appendix A

Estimating resistances from conductances P

  • |

Rij − Rij| Rij ≥ ǫ

  • =

P

  • Rij

Rij − 1

  • ≥ ǫ
  • =

P

  • Gij
  • Gij

− 1

  • ≥ ǫ
  • =

P(|Gij − Gij| ≥ ǫ Gij) ≤ P(| Gij − Gij| ≥ ǫ), where the last inequality follows because Gij ≥ 1.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 34 / 35

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Appendix B

Our algorithm can be easily adapted for estimating the conductance value across any pair of nodes: maintain and update the variable pij and pij. If effective conductances between far-off nodes is desired, the communication is however no longer local.

Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 35 / 35