## Abstract

Finding a new mathematical representation for graphs, which allows direct comparison between different graph structures, is an open-ended research direction. Having such a representation is the first prerequisite for a variety of machine learning algorithms like classification, clustering, etc., over graph datasets. In this paper, we propose a symmetric positive semidefinite matrix with the (i, j)-th entry equal to the covariance between normalized vectors A^{i}e and A^{j}e (e being vector of all ones) as a representation for a graph with adjacency matrix A. We show that the proposed matrix representation encodes the spectrum of the underlying adjacency matrix and it also contains information about the counts of small sub-structures present in the graph such as triangles and small paths. In addition, we show that this matrix is a 'graph invariant'. All these properties make the proposed matrix a suitable object for representing graphs. The representation, being a covariance matrix in a fixed dimensional metric space, gives a mathematical embedding for graphs. This naturally leads to a measure of similarity on graph objects. We define similarity between two given graphs as a Bhattacharya similarity measure between their corresponding covariance matrix representations. As shown in our experimental study on the task of social network classification, such a similarity measure outperforms other widely used state-of-theart methodologies. Our proposed method is also computationally efficient. The computation of both the matrix representation and the similarity value can be performed in operations linear in the number of edges. This makes our method scalable in practice. We believe our theoretical and empirical results provide evidence for studying truncated power iterations, of the adjacency matrix, to characterize social networks.

Original language | English (US) |
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Title of host publication | ASONAM 2014 - Proceedings of the 2014 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining |

Editors | Martin Ester, Guandong Xu, Xindong Wu, Xindong Wu |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 62-71 |

Number of pages | 10 |

ISBN (Electronic) | 9781479958771 |

DOIs | |

State | Published - Oct 10 2014 |

Event | 2014 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2014 - Beijing, China Duration: Aug 17 2014 → Aug 20 2014 |

### Publication series

Name | ASONAM 2014 - Proceedings of the 2014 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining |
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### Other

Other | 2014 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ASONAM 2014 |
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Country | China |

City | Beijing |

Period | 8/17/14 → 8/20/14 |

## All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Computer Science Applications