A Local Graph Partitioning Algorithm Using Heat Kernel Pagerank

We give an improved local partitioning algorithm using heat kernel pagerank, a modified version of PageRank. For a subset S with Cheeger ratio (or conductance) h, we show that there are at least a quarter of the vertices in S that can serve as seeds for heat kernel pagerank which lead to local cuts with Cheeger ratio at most O( √ h), improving the previously bound by a factor of p log |S|.

[1]  Jennifer Widom,et al.  Scaling personalized web search , 2003, WWW '03.

[2]  Shang-Hua Teng,et al.  Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems , 2003, STOC '04.

[3]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[4]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[5]  F. Chung.  Random walks and local cuts in graphs , 2007 .

[6]  Santosh S. Vempala,et al.  On clusterings-good, bad and spectral , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[7]  Fan Chung Graham,et al.  Local Graph Partitioning using PageRank Vectors , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[8]  Fan Chung,et al.  The heat kernel as the pagerank of a graph , 2007, Proceedings of the National Academy of Sciences.

[9]  Milena Mihail,et al.  Conductance and convergence of Markov chains-a combinatorial treatment of expanders , 1989, 30th Annual Symposium on Foundations of Computer Science.

[10]  Miklós Simonovits,et al.  The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[11]  Shing-Tung Yau,et al.  Coverings, Heat Kernels and Spanning Trees , 1998, Electron. J. Comb..

[12]  F. Chung,et al.  Weighted graph Laplacians and isoperimetric inequalities , 2000 .

[13]  J. Petersen.  Theorie der algebraischen Gleichungen , 1878, Nature.

[14]  F. Chung,et al.  Complex Graphs and Networks , 2006 .

[15]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[16]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[17]  J. Cheeger.  A lower bound for the smallest eigenvalue of the Laplacian , 1969 .

[18]  Fan Chung Graham,et al.  Detecting Sharp Drops in PageRank and a Simplified Local Partitioning Algorithm , 2007, TAMC.

[19]  Taher H. Haveliwala.  Topic-Sensitive PageRank: A Context-Sensitive Ranking Algorithm for Web Search , 2003, IEEE Trans. Knowl. Data Eng..

[20]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[21]  Miklós Simonovits,et al.  Random Walks in a Convex Body and an Improved Volume Algorithm , 1993, Random Struct. Algorithms.

[22]  Sergey Brin,et al.  The Anatomy of a Large-Scale Hypertextual Web Search Engine , 1998, Comput. Networks.