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Algorithms to Measure Diversity and Clustering in Social Networks through Dot Product Graphs

  • Matthew Johnson
  • Daniël Paulusma
  • Erik Jan van Leeuwen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 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.

Keywords

Social Network Maximum Clique Structural Result Product Graph Interval Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Matthew Johnson
    • 1
  • Daniël Paulusma
    • 1
  • Erik Jan van Leeuwen
    • 2
  1. 1.School of Engineering and Computer ScienceDurham UniversityEngland
  2. 2.Max-Planck Institut für InformatikSaarbrückenGermany

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