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

Johnson, M. and Paulusma, D. and van Leeuwen, E.J. (2013) 'Algorithms to measure diversity and clustering in social networks through dot product graphs.', in Algorithms and computation : 24th International Symposium, ISAAC 2013, Hong Kong, China, 16-18 December 2013 ; proceedings. Berlin, Heidelberg: Springer, pp. 130-140. Lecture notes in computer science., 8283

Abstract

Social networks are often analyzed through a graph model of the network. 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 a v 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 a u · a v  ≥ t, for some fixed, positive threshold t. The resulting graph is called a d-dot product graph.. We consider two measures for diversity and clustering in social networks by using a d-dot product graph model for the network. Diversity is measured through the size of the largest independent set of the graph, and clustering is measured through the size of the largest clique. We obtain a tight result for the diversity problem, namely that it is polynomial-time solvable for d = 2, but NP-complete 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 consider the situation when two individuals are connected if their preferences are not opposite. This leads to a variant of the standard dot product graph model by taking the threshold t to be zero. We prove in this case that the diversity problem is polynomial-time solvable for any fixed d.

Item Type:Book chapter
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.1007/978-3-642-45030-3_13
Publisher statement:The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-45030-3_13
Date accepted:No date available
Date deposited:15 January 2015
Date of first online publication:2013
Date first made open access:No date available

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