1 Presented by: Yuchen Bian MRWC: Clustering based on Multiple Random Walks Chain

2 1. Introduction and Motivation ----Background 2. Multiple Random Walks Chain (MRWC) ----Intuition ----Definitions 3. Experiments 4. Conclusion 5. Future Work Content

3 1. Introduction and Motivation ----Background 2. Multiple Random Walks Chain (MRWC) ----Intuition ----Definitions 3. Experiments 4. Conclusion 5. Future Work Content

4 1. Introduction and Motivation Random Walk Model: a b c 1 1/2 1 t=0 a b c 1 1/2 1 t=1 a b c 1 1/2 1 t=2 a b c 1 1/2 1 t=3

5 5 x t+1 (i) = ∑ j (Probability of being at node j)*Pr(j->i) =∑ j x t (j)*P(j,i) x t+1 = P T x t Long time after… x t+1 ≈ x t x t = P T x t Converge to a stationary distribution π no matter what the initial distribution is. For each π i π i =d(i)/2m 1. Introduction and Motivation Random Walk Model:

6 1. Introduction and Motivation Random Walk Model: π i =d(i)/2m Query node: 8

7 7 x t = P T x t e i is a vector in which only the i-th (query node) element is 1, otherwise Restart c0≤c<1 1. Introduction and Motivation Random Walk with Restart Model: x t = (1-c)P T x t +ce i

8 1. Introduction and Motivation Query node bias: sharp peak Query node: 8 Random Walk with Restart Model:

9 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.

10 1. Introduction and Motivation ----Background 2. Multiple Random Walks Chain (MRWC) ----Intuition ----Definitions 3. Experiments 4. Conclusion 5. Future Work Content

11 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.

12 Definitions: 2. Multiple Random Walks Chain (MRWC)

13 Definitions: 2. Multiple Random Walks Chain (MRWC)

14 Definitions: 2. Multiple Random Walks Chain (MRWC)

15 1. Introduction and Motivation ----Background 2. Multiple Random Walks Chain (MRWC) ----Intuition ----Definitions 3. Experiments 4. Conclusion 5. Future Work Content

16 3. Experiments Computation and Egs: A3, P Naïve Method: Iteratively computation

17 A2, P2

18 3. Experiments Fig 1. Basic RWFig 2. RWR Fig 3. MRWC (k=2) Fig 4. MRWC (k=3)

19 3. Experiments MRWC (k=2) RWs’ position for each iteration W1-B*, W2-Rs

20 3. Experiments Fig 1. RWR Fig 3. MRWC (k=3) Fig 2. MRWC (k=2) RWs’ position for each iteration (k=2)

21 1. Introduction and Motivation ----Background 2. Multiple Random Walks Chain (MRWC) ----Intuition ----Definitions 3. Experiments 4. Conclusion 5. Future Work Content

22 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:

23 1. Introduction and Motivation ----Background 2. Multiple Random Walks Chain (MRWC) ----Intuition ----Definitions 3. Experiments 4. Conclusion 5. Future Work Content

24 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

25 Yuchen Bian Thank you! Q & A