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 forCyber 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
Outline
1 MotivationWhat is conductance?Why estimate conductances?
2 Notation
3 Prior Work
4 AlgorithmMotivationIdeaPseudocode
5 Theoretical results
6 Simulation ExperimentsCardinal EstimationOrdinal Estimation
7 Discussion
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 2 / 35
Outline
1 MotivationWhat is conductance?Why estimate conductances?
2 Notation
3 Prior Work
4 AlgorithmMotivationIdeaPseudocode
5 Theoretical results
6 Simulation ExperimentsCardinal EstimationOrdinal Estimation
7 Discussion
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 3 / 35
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
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
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 differencebetween them.
Effective conductance is inverse of effective resistance.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 6 / 35
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 differencebetween them.
Effective conductance is inverse of effective resistance.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 6 / 35
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 differencebetween them.
Effective conductance is inverse of effective resistance.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 6 / 35
Outline
1 MotivationWhat is conductance?Why estimate conductances?
2 Notation
3 Prior Work
4 AlgorithmMotivationIdeaPseudocode
5 Theoretical results
6 Simulation ExperimentsCardinal EstimationOrdinal Estimation
7 Discussion
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 7 / 35
Why estimate conductances?
Effective resistance as a robust measure of distance ([1], [2])
Considers all pathsLess sensitive to edge or node insertions and deletions
Sum of effective resistances across all pairs
measure of network robustnessequals network criticality parameter [4]
Edge resistances for graph sparsification [3]
Edges sampled (with replacement) according to their effectiveresistanceApproximately 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
Why estimate conductances?
Effective resistance as a robust measure of distance ([1], [2])
Considers all pathsLess sensitive to edge or node insertions and deletions
Sum of effective resistances across all pairs
measure of network robustnessequals network criticality parameter [4]
Edge resistances for graph sparsification [3]
Edges sampled (with replacement) according to their effectiveresistanceApproximately 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
Why estimate conductances?
Effective resistance as a robust measure of distance ([1], [2])
Considers all pathsLess sensitive to edge or node insertions and deletions
Sum of effective resistances across all pairs
measure of network robustnessequals network criticality parameter [4]
Edge resistances for graph sparsification [3]
Edges sampled (with replacement) according to their effectiveresistanceApproximately 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
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
Prior Work
Nodes of the graph can be embedded in Euclidean space so that theresulting pair-wise distances encode the effective resistances.
The embedding depends on the edge-node adjacency matrix and theLaplacian of the graph.
[3] uses low dimensional random projection to preserve pairwisedistances to estimate resistances. Takes only O(m/ε2) steps, butrequires centralized computation.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 10 / 35
Prior Work
Nodes of the graph can be embedded in Euclidean space so that theresulting pair-wise distances encode the effective resistances.
The embedding depends on the edge-node adjacency matrix and theLaplacian of the graph.
[3] uses low dimensional random projection to preserve pairwisedistances to estimate resistances. Takes only O(m/ε2) steps, butrequires centralized computation.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 10 / 35
Prior Work
Nodes of the graph can be embedded in Euclidean space so that theresulting pair-wise distances encode the effective resistances.
The embedding depends on the edge-node adjacency matrix and theLaplacian of the graph.
[3] uses low dimensional random projection to preserve pairwisedistances to estimate resistances. Takes only O(m/ε2) steps, butrequires centralized computation.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 10 / 35
Outline
1 MotivationWhat is conductance?Why estimate conductances?
2 Notation
3 Prior Work
4 AlgorithmMotivationIdeaPseudocode
5 Theoretical results
6 Simulation ExperimentsCardinal EstimationOrdinal Estimation
7 Discussion
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 11 / 35
AlgorithmMotivation
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
AlgorithmMotivation
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
AlgorithmMotivation
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
AlgorithmMotivation
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
AlgorithmMotivation
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
Outline
1 MotivationWhat is conductance?Why estimate conductances?
2 Notation
3 Prior Work
4 AlgorithmMotivationIdeaPseudocode
5 Theoretical results
6 Simulation ExperimentsCardinal EstimationOrdinal Estimation
7 Discussion
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 13 / 35
AlgorithmIdea
We focus on edge effective conductance estimation as needed forgraph sparsification algorithm in [3].
We use random walks on the graph to estimate effective conductanceand effective resistances.
A random walk on the graph picks, from the current position, one ofthe neighbors with equal probability. Such random walks naturallygive 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
AlgorithmIdea
We focus on edge effective conductance estimation as needed forgraph sparsification algorithm in [3].
We use random walks on the graph to estimate effective conductanceand effective resistances.
A random walk on the graph picks, from the current position, one ofthe neighbors with equal probability. Such random walks naturallygive 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
AlgorithmIdea
We focus on edge effective conductance estimation as needed forgraph sparsification algorithm in [3].
We use random walks on the graph to estimate effective conductanceand effective resistances.
A random walk on the graph picks, from the current position, one ofthe neighbors with equal probability. Such random walks naturallygive 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
AlgorithmIdea
We focus on edge effective conductance estimation as needed forgraph sparsification algorithm in [3].
We use random walks on the graph to estimate effective conductanceand effective resistances.
A random walk on the graph picks, from the current position, one ofthe neighbors with equal probability. Such random walks naturallygive 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
AlgorithmIdea
Let pij denote the probability that a random walk starting at node ivisits 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
AlgorithmIdea
A key fact that underlies our algorithm is:
pij = Gij/di
We estimate this probability by averaging results from several i to ipaths in a random walk.
We will show how this can be done only with local communication foredge conductance estimation.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 16 / 35
AlgorithmIdea
A key fact that underlies our algorithm is:
pij = Gij/di
We estimate this probability by averaging results from several i to ipaths in a random walk.
We will show how this can be done only with local communication foredge conductance estimation.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 16 / 35
AlgorithmIdea
A key fact that underlies our algorithm is:
pij = Gij/di
We estimate this probability by averaging results from several i to ipaths in a random walk.
We will show how this can be done only with local communication foredge conductance estimation.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 16 / 35
AlgorithmVariables
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 jin an instance of a return path from node i to node i of the randomwalk.
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
AlgorithmVariables
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 jin an instance of a return path from node i to node i of the randomwalk.
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
AlgorithmVariables
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 jin an instance of a return path from node i to node i of the randomwalk.
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
AlgorithmVariables
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 jin an instance of a return path from node i to node i of the randomwalk.
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
AlgorithmObservation 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
AlgorithmObservation 2
A visit to node i at time t will be a part of a cycle that originated atnode 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 tonode 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
Outline
1 MotivationWhat is conductance?Why estimate conductances?
2 Notation
3 Prior Work
4 AlgorithmMotivationIdeaPseudocode
5 Theoretical results
6 Simulation ExperimentsCardinal EstimationOrdinal Estimation
7 Discussion
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 20 / 35
AlgorithmPseudocode
Algorithm 1 Visit Before Return
Require: T , G = (V ,E )1: ∀ i ∈ V ,Ni ← 02: ∀ (i , j) ∈ E , pij = pij = 0.3: Sample initial node X1 from the stationary distribution π.4: for t = [1, 2, 3, · · · ,T ] do5: Let i = Xt
6: for all j in ∂i do7: pij ← (pijNi + pij)/(Ni + 1)8: pij ← 09: pji ← 1
10: Ni ← Ni + 111: Jump to a neighbor of the current node as identified by the walk.
12: For every (i , j) ∈ E , output Gij = max(
1, di2 pij +dj2 pji
).
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 21 / 35
Complexity result
Theorem (Performance of VisitBeforeReturn)
Fix an edge (i , j) ∈ E . For any 0 < ε < d2ij/(4m) and 0 < δ < 1/2,
T = O
(Dij ·max{m,Dij tmix} ·
1
ε2log
1
δ
)steps suffice to ensure that the output Gij of the algorithmVisitBeforeReturn satisfies
P(|Gij − Gij | ≥ ε) ≤ δ.
If the algorithm is run for T steps, it requires O(davgT ) computation stepson the average (worst case O(dmaxT ) computations), and usesO(m logT ) space.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 22 / 35
Proof ideas
pij =
∑Nik=1 p
kij
Ni
.
Get concentration for Ni using McDiarmids inequality for Markovchains.
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
Proof ideas
pij =
∑Nik=1 p
kij
Ni
.
Get concentration for Ni using McDiarmids inequality for Markovchains.
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
Proof ideas
pij =
∑Nik=1 p
kij
Ni
.
Get concentration for Ni using McDiarmids inequality for Markovchains.
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
Proof ideas
pij =
∑Nik=1 p
kij
Ni
.
Get concentration for Ni using McDiarmids inequality for Markovchains.
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
Outline
1 MotivationWhat is conductance?Why estimate conductances?
2 Notation
3 Prior Work
4 AlgorithmMotivationIdeaPseudocode
5 Theoretical results
6 Simulation ExperimentsCardinal EstimationOrdinal Estimation
7 Discussion
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 24 / 35
Simulation ExperimentsCardinal Estimation
ε(t) =1
m
∑(i ,j)∈E
∣∣∣Gij(t)− Gij
∣∣∣
0 2 4 6 8 10
x 105
0
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
Simulation ExperimentsCardinal Estimation
0 2 4 6 8 10
x 105
0
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
Outline
1 MotivationWhat is conductance?Why estimate conductances?
2 Notation
3 Prior Work
4 AlgorithmMotivationIdeaPseudocode
5 Theoretical results
6 Simulation ExperimentsCardinal EstimationOrdinal Estimation
7 Discussion
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 27 / 35
Simulation ExperimentsTop-k set estimation
Incremental approximate estimation algorithms can typically recoverordering much faster than exact values.We test performance of our algorithm for recovering top-k edges withhigh conductance by plotting fraction of top-k largest conductanceedges correctly identified at time t.
0 2 4 6 8 10
x 105
0
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
Figure:Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 28 / 35
Discussion
Our algorithm can be naturally distributed: each node storesinformation about its neighbors.
Parallelization can be easily obtained by running multiple randomwalks and avergaing their results.
Our guarantees are weaker as compared to [3]. Particularly, therestriction on ε forces it to be too small. Whether this can beremoved is an open question.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 29 / 35
Discussion
Our algorithm can be naturally distributed: each node storesinformation about its neighbors.
Parallelization can be easily obtained by running multiple randomwalks and avergaing their results.
Our guarantees are weaker as compared to [3]. Particularly, therestriction on ε forces it to be too small. Whether this can beremoved is an open question.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 29 / 35
Discussion
Our algorithm can be naturally distributed: each node storesinformation about its neighbors.
Parallelization can be easily obtained by running multiple randomwalks and avergaing their results.
Our guarantees are weaker as compared to [3]. Particularly, therestriction on ε forces it to be too small. Whether this can beremoved is an open question.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 29 / 35
Discussion
Our algorithm can be naturally distributed: each node storesinformation about its neighbors.
Parallelization can be easily obtained by running multiple randomwalks and avergaing their results.
Our guarantees are weaker as compared to [3]. Particularly, therestriction on ε forces it to be too small. Whether this can beremoved is an open question.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 29 / 35
Conclusion
We have presented a MCMC based scheme to approximate edgeconductances.
Our algorithm is incremental and iterative, can be easily distributed,works with local communication, and uses very little memory andcomputation 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
References
Aykut Firat, Sangit Chatterjee, and Mustafa Yilmaz. “Genetic clustering ofsocial networks using random walks”. In: Computational Statistics & DataAnalysis 51.12 (2007), pp. 6285–6294.
Francois Fouss et al. “Random-walk computation of similarities betweennodes 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 effectiveresistances”. In: SIAM Journal on Computing 40.6 (2011), pp. 1913–1926.
Ali Tizghadam and Alberto Leon-Garcia. “On robust traffic engineering intransport networks”. In: Global Telecommunications Conference, 2008. IEEEGLOBECOM 2008. IEEE. IEEE. 2008, pp. 1–6.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 31 / 35
Thank you for your attention!Questions?
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 32 / 35
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 33 / 35
Appendix A
Estimating resistances from conductances
P
(|Rij − Rij |
Rij≥ ε
)= P
(∣∣∣∣∣ Rij
Rij− 1
∣∣∣∣∣ ≥ ε)
= P
(∣∣∣∣∣Gij
Gij
− 1
∣∣∣∣∣ ≥ ε)
= P(|Gij − Gij | ≥ εGij)
≤ P(|Gij − Gij | ≥ ε),
where the last inequality follows because Gij ≥ 1.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 34 / 35
Appendix B
Our algorithm can be easily adapted for estimating the conductancevalue across any pair of nodes: maintain and update the variable pijand pij .
If effective conductances between far-off nodes is desired, thecommunication is however no longer local.
Ashish Bora, Vivek S. Borkar , Dinesh Garg, Rajesh Sundaresan Edge Conductance Estimation using MCMC 35 / 35