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Presented by: Yuchen Bian

4.16.2015

MRWC: Clustering based on Multiple Random Walks Chain

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1. Introduction and Motivation----Background

2. Multiple Random Walks Chain (MRWC)----Intuition----Definitions

3. Experiments

4. Conclusion

5. Future Work

Content

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1. Introduction and Motivation----Background

2. Multiple Random Walks Chain (MRWC)----Intuition----Definitions

3. Experiments

4. Conclusion

5. Future Work

Content

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1. Introduction and Motivation

Random Walk Model:

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t=0

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t=1

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t=2

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t=3

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xt+1(i) = ∑j(Probability of being at node j)*Pr(j->i) =∑jxt(j)*P(j,i)

xt+1 = PTxt

Long time after…

xt+1 ≈ xt

xt = PTxt

Converge to a stationary distribution π no matter what the initial distribution is.

For each πi

πi=d(i)/2m

1. Introduction and Motivation

Random Walk Model:

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1. Introduction and Motivation

Random Walk Model:

πi=d(i)/2m

Query node: 8

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xt = PTxt

ei is a vector in which only the i-th (query node) element is 1, otherwise 0.------Restart

c 0≤c<1

1. Introduction and Motivation

Random Walk with Restart Model:

xt = (1-c)PTxt+cei

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1. Introduction and Motivation

Query node bias: sharp peak

Query node: 8

Random Walk with Restart Model:

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1. Introduction and Motivation

For large graph, convergence needs more time. Query node: 8

Local clustering:

Find cluster before convergence, even the RW will not reach some nodes.

In fact, a RW might be restricted in the cluster with high probability, HOWEVER, it is also hard to travel back if RW pass through boundary

Targets: restricted in the cluster which contains the query nodes.

What if the query node(s) send out a series of RWs, not a single RW, hopefully, this RWs group is harder than single RW to travel through boundary.

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1. Introduction and Motivation----Background

2. Multiple Random Walks Chain (MRWC)----Intuition----Definitions

3. Experiments

4. Conclusion

5. Future Work

Content

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Intuition:

2. Multiple Random Walks Chain (MRWC)

From each query node, send a series of RWs to explore the graph, all RWs walk one by one,

but the next vertex the current RW will explore is not only follow its own “thought” but also decided by other RWs.

Then all RWs constructs a RWs group and this group is harder than a single RW to travel through the boundary.

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Definitions:

2. Multiple Random Walks Chain (MRWC)

Definition 1: [Multi-Random Walks Chain (MRWC)]

A MRWC is that for each query vertex, from time point 0, sending k random walks at following time point 1, 2,..., k. At time point , assume that the k random walks stand at vertices .

At each time point, there is only one random walk which follows the order to and recursively searching through the graph, and at the same time, other random walks have effects on the next vertex the current random walk will go to.

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Definitions:

2. Multiple Random Walks Chain (MRWC)

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Definitions:

2. Multiple Random Walks Chain (MRWC)

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1. Introduction and Motivation----Background

2. Multiple Random Walks Chain (MRWC)----Intuition----Definitions

3. Experiments

4. Conclusion

5. Future Work

Content

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3. Experiments

Computation and Egs:

A3, P3

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Naïve Method:Iteratively computation

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5253 55

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5657

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A2, P2

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3. Experiments

Fig 1. Basic RW Fig 2. RWR

Fig 3. MRWC (k=2) Fig 4. MRWC (k=3)

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3. Experiments

Not converge! Why?Dynamically choosing the other RWs’ position with largest probability

MRWC (k=2)RWs’ position for each iterationW1-B*, W2-Rs

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3. Experiments

Fig 1. RWR

Fig 3. MRWC (k=3)

Fig 2. MRWC (k=2)

RWs’ position for each iteration (k=2)

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1. Introduction and Motivation----Background

2. Multiple Random Walks Chain (MRWC)----Intuition----Definitions

3. Experiments

4. Conclusion

5. Future Work

Content

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4. Conclusion

• Motivation: restrict into the target cluster• Advantages:

• Increase the number of features• Sharpen the boundary: harder to pass through than single RW• Group activity not single activity (sharp peak)

• Disadvantages:• Convergence issue• Naïve method

Evaluation to MRWC:

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1. Introduction and Motivation----Background

2. Multiple Random Walks Chain (MRWC)----Intuition----Definitions

3. Experiments

4. Conclusion

5. Future Work

Content

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5. Future Work

• Model: • formal and general model

• Mathematical Analysis:• Convergence?• How to sharpen the boundary?

• Algorithm:• Efficient computation or approximation• Compare with other methods

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Yuchen Bian

Thank you! Q & A