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The Spacey Random Walk: A Stochastic Process for Higher-Order Data

Austin R. Benson, David F. Gleich, Lek-Heng Lim
2017 SIAM Review  
Here, we present the spacey random walk, a non-Markovian stochastic process whose stationary distribution is given by the tensor eigenvector.  ...  Finally, we provide several applications of the spacey random walk model in population genetics, ranking, and clustering data, and we use the process to analyze taxi trajectory data in New York.  ...  We thank Tao Wu for observing subtleties in our use of the term stable point in the analysis of the two-state case when f (x) ≡ 0 and our convergence analysis of the dynamical system in the limit of Euler's  ... 
doi:10.1137/16m1074023 fatcat:gmhvjrawwbaotntk2blv4pbbga

HeteSpaceyWalk: A Heterogeneous Spacey Random Walk for Heterogeneous Information Network Embedding [article]

Yu He and Yangqiu Song and Jianxin Li and Cheng Ji and Jian Peng and Hao Peng
2019 arXiv   pre-print
In this paper, we systematically formalize the meta-path guided random walk as a higher-order Markov chain process, and present a heterogeneous personalized spacey random walk to efficiently and effectively  ...  Then we propose a generalized scalable framework to leverage the heterogeneous personalized spacey random walk to learn embeddings for multiple types of nodes in an HIN guided by a meta-path, a meta-graph  ...  Thus, this is theoretically a second-order Markovian stochastic process.  ... 
arXiv:1909.03228v1 fatcat:uzpzogehuvhovd2tfsmcf7gutm

Random walks and diffusion on networks

Naoki Masuda, Mason A. Porter, Renaud Lambiotte
2017 Physics reports  
Random walks have been studied for many decades on both regular lattices and (especially in the last couple of decades) on networks with a variety of structures.  ...  They are one of the most fundamental types of stochastic processes; can be used to model numerous phenomena, including diffusion, interactions, and opinions among humans and animals; and can be used to  ...  Introduction Random walks (RWs) are popular models of stochastic processes with a very rich history [1] [2] [3] [4] [5] . 1 The term ''random walk'' was coined by Karl Pearson [6] , and the study of  ... 
doi:10.1016/j.physrep.2017.07.007 fatcat:wvjtv3muo5drhjgbnzzn6j2nsi

General Tensor Spectral Co-clustering for Higher-Order Data [article]

Tao Wu, Austin R. Benson, David F. Gleich
2016 arXiv   pre-print
On synthetic problems with a planted higher-order cluster structure, our method is the only one that can reliably identify the planted structure in all cases.  ...  We develop a new tensor spectral co-clustering method that applies to any non-negative tensor of data.  ...  ARB is supported by a Stanford Graduate Fellowship. We are grateful to our colleagues who helped revise early drafts.  ... 
arXiv:1603.00395v1 fatcat:44o3oryqa5getic3wy7h3us7aa

Recommendation Model Based on a Heterogeneous Personalized Spacey Embedding Method

Qunsheng Ruan, Yiru Zhang, Yuhui Zheng, Yingdong Wang, Qingfeng Wu, Tianqi Ma, Xiling Liu
2021 Symmetry  
The traditional heterogeneous embedding method based on a random walk strategy does not focus on the random walk fundamentally because of higher-order Markov chains.  ...  a meta-path-based heterogenous personalized spacey random walk for recommendation (MPHSRec).  ...  Acknowledgments: The authors would like to show their appreciation for the valuable comments and suggestions from the editors and reviewers.  ... 
doi:10.3390/sym13020290 fatcat:sof2ha4hlrgslfdsavmidkx3tu

Higher-order Network Analysis Takes Off, Fueled by Classical Ideas and New Data [article]

Austin R. Benson, David F. Gleich, Desmond J. Higham
2021 arXiv   pre-print
Higher-order network analysis uses the ideas of hypergraphs, simplicial complexes, multilinear and tensor algebra, and more, to study complex systems.  ...  What's new is that the ideas can be tested and refined on the type of large-scale data arising in today's digital world. This research area therefore is making an impact across many applications.  ...  To address issues arising from the data, each reached for new types of higher-order stochastic processes with more memory, namely higher-order and variable-order Markov chains.  ... 
arXiv:2103.05031v1 fatcat:n4zqmu7fiva4pbiod6p2h5nqzq

Ergodicity coefficients for higher-order stochastic processes [article]

Dario Fasino, Francesco Tudisco
2020 arXiv   pre-print
convergence of lazy higher-order random walks.  ...  The use of higher-order stochastic processes such as nonlinear Markov chains or vertex-reinforced random walks is significantly growing in recent years as they are much better at modeling high dimensional  ...  He would like to thank the department and D.F. for the  ... 
arXiv:1907.04841v3 fatcat:5dk72qd2enaptd53mhdfc6nkm4

Computing tensor Z-eigenvectors with dynamical systems [article]

Austin R. Benson, David F. Gleich
2019 arXiv   pre-print
Our motivation comes from our recent research on spacey random walks, where the long-term dynamics of a stochastic process are governed by a dynamical system that must converge to a Z-eigenvector of a  ...  Here, we apply the ideas more broadly to general tensors and find that our method can compute Z-eigenvectors that algebraic methods like the higher-order power method cannot compute.  ...  DFG is supported by NSF award CCF-1149756, IIS-1422918, IIS-1546488, the NSF Center for Science of Information STC, CCF-0939370, DARPA SIMPLEX, NASA, DOE DE-SC0014543, and the Sloan Foundation.  ... 
arXiv:1805.00903v3 fatcat:attkam4jifd67h4k5kibyp6hgq

Multilinear PageRank

David F. Gleich, Lek-Heng Lim, Yongyang Yu
2015 SIAM Journal on Matrix Analysis and Applications  
The underlying stochastic process is an instance of a vertex-reinforced random walk.  ...  In this paper, we first extend the celebrated PageRank modification to a higher-order Markov chain.  ...  We are extremely grateful to Austin Benson for suggesting an idea that led to the stochastic process as well as some preliminary comments on the manuscript.  ... 
doi:10.1137/140985160 fatcat:3a4jgfwknjgufnng4hsxczc7vu

Tensor Spectral Clustering for Partitioning Higher-order Network Structures [chapter]

Austin R. Benson, David F. Gleich, Jure Leskovec
2015 Proceedings of the 2015 SIAM International Conference on Data Mining  
Spectral methods are based on a first-order Markov chain derived from a random walk on the graph and thus they cannot take advantage of important higher-order network substructures such as triangles, cycles  ...  Here we propose a Tensor Spectral Clustering (TSC) algorithm that allows for modeling higher-order network structures in a graph partitioning framework.  ...  This vector is the stationary distribution of a stochastic process recently termed the spacey random surfer [12] . At any step of the process, a random surfer has just moved from node k to node j.  ... 
doi:10.1137/1.9781611974010.14 pmid:27812399 pmcid:PMC5089081 dblp:conf/sdm/BensonGL15 fatcat:hdbtnc4fhrbrfibdzzasfc43qa

A Local Algorithm for Structure-Preserving Graph Cut

Dawei Zhou, Si Zhang, Mehmet Yigit Yildirim, Scott Alcorn, Hanghang Tong, Hasan Davulcu, Jingrui He
2017 Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining - KDD '17  
In particular, we start with a generic definition of high-order conductance, and define the highorder diffusion core, which is based on a high-order random walk induced by user-specified high-order network  ...  To address this problem, inspired by the family of local graph clustering algorithms for efficiently identifying a low-conductance cut without exploring the entire graph, we propose to generalize the key  ...  Followed by [5] , [33] proposes a tensor spectral co-clustering method by modeling higher-order data with a novel variant of a higher-order Markov chain, i.e., the super-spacey random walk.  ... 
doi:10.1145/3097983.3098015 dblp:conf/kdd/ZhouZYATDH17 fatcat:xnkwecpqpnhmznpzwengybn3yq

Retrospective Higher-Order Markov Processes for User Trails [article]

Tao Wu, David Gleich
2017 arXiv   pre-print
In this paper we propose the retrospective higher-order Markov process (RHOMP) as a low-parameter model for such sequences.  ...  Furthermore, by providing a specific structure to the higher-order chain, RHOMPs improve the model accuracy by efficiently utilizing history states without risks of overfitting the data.  ...  Acknowledgements. is work was supported by NSF IIS-1422918, CAREER award CCF-1149756, Center for Science of Information STC, CCF-093937; DOE award DE-SC0014543; and the DARPA SIMPLEX program.  ... 
arXiv:1704.05982v1 fatcat:6ifkgcuze5epxbcrtpbas3fah4

Extrapolation Methods for fixed-point Multilinear PageRank computations [article]

Stefano Cipolla, Michela Redivo-Zaglia, Francesco Tudisco
2019 arXiv   pre-print
In this work we consider a particular class of nonnegative tensors associated to the multilinear PageRank modification of higher-order Markov chains.  ...  Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher-order power method is one of the most commonly used technique for the computation  ...  It is worth point out that a solution of (4) is a stationary distribution of a stochastic process called "The Spacey Random Surfer" [3] which is an interesting vertexreinforced Markov process that uses  ... 
arXiv:1906.01494v2 fatcat:egww5n3ldzaindwotuoamfkr7y

Network representation learning: A macro and micro view

Xueyi Liu, Jie Tang
2021 AI Open  
Graph is a universe data structure that is widely used to organize data in real-world.  ...  Representation learning can facilitate the design of new algorithms on the graph data. In this survey, we conduct a comprehensive review of current literature on network representation learning.  ...  Acknowledgments The work is supported by the National Key R&D Program of China (2018YFB1402600), NSFC for Distinguished Young Scholar (61825602), NSFC (61672313), NSFC (61836013), and Tsinghua-Bosch Joint  ... 
doi:10.1016/j.aiopen.2021.02.001 fatcat:6ktfheijvjdnfhja5oqobse5b4

Network representation learning: A macro and micro view [article]

Xueyi Liu, Jie Tang
2021 arXiv   pre-print
Graph is a universe data structure that is widely used to organize data in real-world.  ...  Representation learning can facilitate the design of new algorithms on the graph data. In this survey, we conduct a comprehensive review of current literature on network representation learning.  ...  Acknowledgments The work is supported by the National Key R&D Program of China (2018YFB1402600), NSFC for Distinguished Young Scholar (61825602), NSFC (61672313), NSFC (61836013), and Tsinghua-Bosch Joint  ... 
arXiv:2111.10772v1 fatcat:rmmplc4qbzhkxloauzz3micbay
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