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# Tensor Spectral Clustering for Partitioning Higher-order Network Structures.

### Author information

- 1
- Institute for Computational and Mathematical Engineering, Stanford University. Supported by Stanford Graduate Fellowship.
- 2
- Department of Computer Science, Purdue University. Supported by NSF CAREER CCF-1149756 and IIS-1422918.
- 3
- Department of Computer Science, Stanford University. Supported by NSF IIS-1016909, CNS-1010921, IIS-1149837, ARO MURI, DARPA GRAPHS, PayPal, Docomo, Volkswagen, and Yahoo.

### Abstract

Spectral graph theory-based methods represent an important class of tools for studying the structure of networks. 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, and feed-forward loops. Here we propose a *Tensor Spectral Clustering* (TSC) algorithm that allows for modeling higher-order network structures in a graph partitioning framework. Our TSC algorithm allows the user to specify which higher-order network structures (cycles, feed-forward loops, etc.) should be preserved by the network clustering. Higher-order network structures of interest are represented using a tensor, which we then partition by developing a multilinear spectral method. Our framework can be applied to discovering layered flows in networks as well as graph anomaly detection, which we illustrate on synthetic networks. In directed networks, a higher-order structure of particular interest is the directed 3-cycle, which captures feedback loops in networks. We demonstrate that our TSC algorithm produces large partitions that cut fewer directed 3-cycles than standard spectral clustering algorithms.

- PMID:
- 27812399
- PMCID:
- PMC5089081
- DOI:
- 10.1137/1.9781611974010.14