Cookies

We use cookies to ensure that we give you the best experience on our website. By continuing to browse this repository, you give consent for essential cookies to be used. You can read more about our Privacy and Cookie Policy.


Durham Research Online
You are in:

Algorithms for diversity and clustering in social networks through dot product graphs.

Johnson, M. and Paulusma, D. and van Leeuwen, E.J. (2015) 'Algorithms for diversity and clustering in social networks through dot product graphs.', Social networks., 41 . pp. 48-55.

Abstract

In this paper, we investigate a graph-theoretical model of social networks. The dot product model assumes that two individuals are connected in the social network if their attributes or opinions are similar. In the model, a d -dimensional vector View the MathML source represents the extent to which individual v has each of a set of d attributes or opinions. Then two individuals u and v are assumed to be friends, that is, they are connected in the graph model, if and only if View the MathML source, for some fixed, positive threshold t. The resulting graph is called a d-dot product graph. We consider diversity and clustering in social networks by using a d-dot product graph model for the network. Diversity is considered through the size of the largest independent set of the graph, and clustering through the size of the largest clique. We present both positive and negative results on the potential of this model. We obtain a tight result for the diversity problem, namely that it is polynomial-time solvable for d = 2, but NP-hard for d ≥ 3. We show that the clustering problem is polynomial-time solvable for d = 2. To our knowledge, these results are also the first on the computational complexity of combinatorial optimization problems on dot product graphs. We also give new insights into the structure of dot product graphs. We also consider the situation when two individuals u and v are connected if and only if their preferences are not antithetical, that is, if and only if View the MathML source, and the situation when two individuals u and v are connected if and only if their preferences are neither antithetical nor “orthogonal”, that is, if and only if View the MathML source. For these two cases we prove that the diversity problem is polynomial-time solvable for any fixed d and that the clustering problem is polynomial-time solvable for d ≤ 3.

Item Type:Article
Keywords:Social network, d-Dot product graph, Independent set, Clique.
Full text:(AM) Accepted Manuscript
Download PDF
(343Kb)
Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1016/j.socnet.2015.01.001
Publisher statement:NOTICE: this is the author’s version of a work that was accepted for publication in Social Networks. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Social Networks, 41, May 2015, 10.1016/j.socnet.2015.01.001.
Date accepted:No date available
Date deposited:09 January 2015
Date of first online publication:May 2015
Date first made open access:No date available

Save or Share this output

Export:
Export
Look up in GoogleScholar