Keywords: Google matrix, PageRank, link analysis, communication network, social networking, ranking algorithm.

In this paper we focus on the mathematical analysis of social network sites (SNSs). Social network analysis is important because it is a useful tool in different areas such as internet social activities, networking evolution, community analysis, etc. We are interested in users of the SNSs, their features and their relations (links). To identify the most important nodes on a network some centrality parameters have been introduced in the literature [1,2]. This paper makes some contributions to the concept of competitivity groups recently defined by the author in [3], where a new model of classification of nodes in an SNS was shown. The main idea consists in using the personalization vector to bias the PageRank [4,5] to users that are important according to some specific features of the SNS, such as number of friends or activity. Competitivity groups are sets of nodes that compete among each other to gain PageRank via the personalization vector. In this paper, the definition of competitivity groups is revised. A new parameter, called the amplification factor, has been introduced and some properties have been shown. In particular, the relation between the personalization vectors and the amplification factor is given. Using a test case it is shown that the amplification factor can be used to classify the nodes of a network in different ways. The concepts introduced allow a given network to be represented in different ways, showing different structures. A comparison between the usual PageRank algorithm (i.e. using the normalized personalization vector of all ones) and the model proposed, based on the definition of competitivity groups- is also shown for the test case. The conclusions obtained with this test case suggest that the concept of competitivity groups can be useful for the managers of SNSs to classify and rank users.

1

R. Albert, A.L. Barabási, "Statistical mechanics of complex networks", Rev. Mod. Phys., 74(1), 2002. doi:10.1103/RevModPhys.74.47

2

M.E.J. Newman, "The structure and function of complex networks", SIAM Rev., 45(2), 167-256, 2003. doi:10.1137/S003614450342480

3

F. Pedroche, "Competitivity groups on Social Network Sites", Mathematical and Computer Modelling, 2010. doi:10.1016/j.mcm.2010.02.031

4

L. Page, S. Brin, R. Motwani, T. Winograd, "The PageRank Citation Ranking: Bringing Order to the Web", Stanford Digital Library Technologies Project, 1999.

5

A.N. Langville, C.D. Meyer. "Google's Pagerank and Beyond: The Science of Search Engine Rankings", Princeton University Press, 2006.

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