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- 1. Spacey Random Walks on Higher-Order Markov Chains David F. Gleich! Purdue University! Joint work with Austin Benson, Lek-Heng Lim, supported by " NSF CAREER CCF-1149756 IIS-1422918 SIAM NetSci15 David Gleich · Purdue 1
- 2. 2 Spacey walk ! on Google Images From Film.com
- 3. WARNING!! This talk presents the “forward” explicit derivation (i.e. lots of little steps) rather than the implicit “backwards” derivation (i.e. big intuitive leaps) SIAM NetSci15 David Gleich · Purdue 3
- 4. PageRank:The initial condition My dissertation" Models & Algorithms for PageRank Sensitivity The essence of PageRank! Take any Markov chain P, PageRank " creates a related chain with great “utility” • Unique stationary distribution • Fast convergence • Modeling ﬂexibility (I ↵P)x = (1 ↵)v PageRank beyond the Web arXiv:1407.5107 by Jessica Leber Fast Magazine SIAM NetSci15 David Gleich · Purdue 4
- 5. Be careful about what you discuss after a talk… I gave a talk! at the Univ. of Chicago and visited Lek-heng Lim! He told me about a new idea! in Markov chains analysis and tensor eigenvalues SIAM NetSci15 David Gleich · Purdue 5
- 6. Approximate stationary distributions of higher-order Markov chains A higher order Markov chain! depends on the last few states. These become Markov chains on the product state space." But that’s usually too large for stationary distributions. The approximation! is that we form a rank-1 approximation of that stationary distribution object. Due to Michael Ng and collaborators P(Xt+1 = i | history) = P(Xt+1 = i | Xt = j, Xt 1 = k) P(X = [i, j]) = xi xj SIAM NetSci15 David Gleich · Purdue 6 P(X = [i, j]) = Xi,j
- 7. Why? SIAM NetSci15 David Gleich · Purdue 7 Multidimensional, multi- ceted data from inform- ics and simulations a b m li This propos dimensiona We want to analyze higher-order relationships and multi-way data and … Things like • Enron emails • Regular hypergraphs And there’s three+ indices! So it’s a " higher-order Markov chain
- 8. Approximate stationary distributions of higher-order Markov chains The new problem! of computing an approx. stationary dist. is a tensor eigenvector The new problem’! • existence is guaranteed under mild conditions • uniqueness … • convergence … Due to Michael Ng and collaborators xi = X jk Pijk xj xk or x = Px2 require heroic algebra (and are hard to check) SIAM NetSci15 David Gleich · Purdue 8
- 9. Some small quick notes A stochastic matrix M is a Markov chain A stochastic hypermatrix / tensor / probability P table " is a higher-order Markov chain SIAM NetSci15 David Gleich · Purdue 9 Multidimensional, multi- faceted data from inform- atics and simulations a b m li This propos dimensiona
- 10. PageRank to the rescue! What if we looked at these approx. stat. distributions of a PageRank modiﬁed higher- order chain? Multilinear PageRank! • Formally the Li & Ng approx. stat. dist. of the PageRank modiﬁed higher order Markov chain • Guaranteed existence! • Fast convergence ? • Uniqueness ? x = ↵Px2 + (1 ↵)v Multilinear PageRank" Gleich, Lim, Yu" arXiv:1409.1465 when alpha < 1/order ! when alpha < 1/order ! SIAM NetSci15 David Gleich · Purdue 10
- 11. One nagging question …! Is there a stochastic process that underlies this approximation? SIAM NetSci15 David Gleich · Purdue 11
- 12. Meanwhile … " Spectral clustering of tensors Austin Benson (a colleague) asked" if there were any interesting method to “cluster” tensors. “Recall” spectral clustering on graphs! ! SIAM Data Mining 2015, arXiv:1502.05058 graph ! random walk ! second eigenvector ! sweep cut partition SIAM NetSci15 David Gleich · Purdue 12 MT y = 2y ¯SS min S (S) = min S #(edges cut) min(vol(S), vol( ¯S))
- 13. Meanwhile … " Spectral clustering of tensors Austin Benson (a colleague) asked" if there were any interesting method to “cluster” tensors. “Conjecture” spectral clustering on tensors! ! SIAM Data Mining 2015, arXiv:1502.05058 graph/tensor ! higher-order random walk ! second eigenvector ! sweep cut partition ??????! SIAM NetSci15 David Gleich · Purdue 13
- 14. We tried many • apriori good and • retrospectively bad ideas for the second eigenvector SIAM NetSci15 David Gleich · Purdue 14
- 15. Austin and I were talking one day … ... about the problem of the process. (He was using Multilinear PageRank as the “ﬁrst” eigenvector.) He observed that One of the ﬁve algorithms ! for multilinear PageRank uses a seq. of Markov chains. Is there some way to turn this into a random walk? xk+1 = stat. dist. of Markov chain based on ↵, v, P, and xk SIAM NetSci15 David Gleich · Purdue 15
- 16. EUREKA! SIAM NetSci15 David Gleich · Purdue 16
- 17. The spacey random walk Consider a higher-order Markov chain. If we were perfect, we’d ﬁgure out the stationary distribution of that. But we are spacey! • On arriving at state j, we promptly " “space out” and forget we came from k. • But we still believe we are “higher-order” • So we invent a state k by drawing a random state from our history. P(Xt+1 = i | history) = P(Xt+1 = i | Xt = j, Xt 1 = k) SIAM NetSci15 David Gleich · Purdue 17
- 18. The spacey random walk This is a vertex-reinforced random walk! " e.g. Polya’s urn. Pemantle, 1992; Benaïm, 1997; Pemantle 2007 SIAM NetSci15 David Gleich · Purdue 18 P(Xt+1 = i | Xt = j and the right filtration on history) = X k Pi,j,k Ck (t)/(t + n) Let Ct (k) = (1 + Pt s=1 Ind{Xs = k}) How often we’ve visited state k in the past
- 19. Stationary distributions of vertex reinforced random walks A vertex-reinforced random walk at time t transitions according to a Markov matrix M given the observed frequencies. This has a stationary distribution, iff the dynamical system converges. SIAM NetSci15 David Gleich · Purdue 19 dx dt = ⇡[M(x)] x P(Xt+1 = i | Xt = j and the right filtration on history) = [M(t)]i,j = [M(c(t))]i,j ⇡[M] is a map to the stat. dist. M. Benïam 1997
- 20. The Markov matrix for " Spacey Random Walks A necessary condition for a stationary distribution (otherwise makes no sense) SIAM NetSci15 David Gleich · Purdue 20 Property B. Let P be an order-m, n dimensional probability table. Then P has property B if there is a unique stationary distribution associated with all stochastic combinations of the last m 2 modes. That is, M = P k,`,... P(:, :, k, `, ...) k,`,... defines a Markov chain with a unique Perron root when all s are positive and sum to one. dx dt = ⇡[M(x)] x M = X k P(:, :, k)xk This is the transition probability associated with guessing the last state based on history!
- 21. We have all sorts of cool results on spacey random walks… e.g. Suppose you have a Polya Urn with memory… " Then it always has a stationary distribution! SIAM NetSci15 David Gleich · Purdue 21
- 22. Back to Multilinear PageRank The Multilinear PageRank problem is what we call a spacey random surfer model. • This is a spacey random walk • We add random jumps with probability (1-alpha) It’s also a vertex-reinforced random walk. Thus, it has a stationary probability if converges. SIAM NetSci15 David Gleich · Purdue 22 dx dt = ⇡[M(x)] x M(x) = ↵ P k P(:, :, k)xk + (1 ↵)v Which occurs when alpha < 1/order !
- 23. Some interesting notes about vertex reinforced random walks • The power method is NOT the natural algorithm! It’s to evolve the ODE. • It’s unclear if there are any structural properties that guarantee a stationary distribution (except for something like the Multilinear PageRank equation) • Can be tough to analyze the resulting ODEs • Asymptotically creates a Markov chain! SIAM NetSci15 David Gleich · Purdue 23
- 24. … back to spectral clustering … SIAM NetSci15 David Gleich · Purdue 24
- 25. Meanwhile … " Spectral clustering of tensors Austin Benson (a colleague) asked" if there were any interesting method to “cluster” tensors. “Conjecture” spectral clustering on tensors! ! SIAM Data Mining 2015, arXiv:1502.05058 graph/tensor ! higher-order random walk ! second eigenvector ! sweep cut partition ??????! SIAM NetSci15 David Gleich · Purdue 25
- 26. Meanwhile … " Spectral clustering of tensors Austin Benson (a colleague) asked" if there were any interesting method to “cluster” tensors. “Conjecture” spectral clustering on tensors! ! SIAM Data Mining 2015, arXiv:1502.05058 graph/tensor ! higher-order random walk ! second eigenvector ! sweep cut partition SIAM NetSci15 David Gleich · Purdue 26 M(x)T y = 2y Use the asymptotic Markov matrix!
- 27. Problem current methods only consider edges … and that is not enough for current problems SIAM NetSci15 David Gleich · Purdue 27 In social networks, we want to penalize cutting triangles more than cutting edges. The triangle motif represents stronger social ties.
- 28. Problem current methods only consider edges SIAM NetSci15 David Gleich · Purdue 28 SPT16 HO CLN1 CLN2 SWI4_SWI6 In transcription networks, the ``feedforward loop” motif represents biological function. Thus, we want to look for clusters of this structure.
- 29. An example with a layered ﬂow network SIAM NetSci15 David Gleich · Purdue 29 0 12 3 4 5 6 7 8 9 10 11 § The network “ﬂows” downward § Use directed 3-cycles to model ﬂow i kj i kj i kj i kj 1 1 1 2 § Tensor spectral clustering: {0,1,2,3}, {4,5,6,7}, {8,9,10,11} § Standard spectral: {0,1,2,3,4,5,6,7}, {8,10,11}, {9}
- 30. SIAM NetSci15 David Gleich · Purdue 30 WAW2015 EURANDOM – Eindhoven – Netherlands Workshop on Algorithms and Models for the Web Graph (but it’s grown to be all types of network analysis) December 10-‐11 Winter School on Complex Network and Graph Models December 7-‐8 Submissions Due July 25th!
- 31. Time for Lots of Questions! Manuscripts! Li, Ng. On the limiting probability distribution of a transition probability tensor. Linear & Multilinear Algebra 2013. Gleich. PageRank beyond the Web. (accepted at SIAM Review) Gleich, Lim, Yu. Multilinear PageRank. (under review…) Benson, Gleich, Leskovec. Tensor Spectral Clustering for partitioning higher order network structures. SDM 2015, arXiv:" https://github.com/arbenson/tensor-sc Benson, Gleich, Leskovec. Forthcoming. (Much better method…) Benson, Gleich, Lim. The Spacey Random Walk. In prep. SIAM NetSci15 David Gleich · Purdue 31

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