Małgorzata J. Krawczyk, Krzysztof Kułakowski, Janusz A. Hołyst

A model algorithm is proposed to study subsequent hierarchical partitions of complex networks describing social structures. The partitions are supposed to appear as actions of rivaling leaders [1] corresponding to nodes with large degrees. The condition of a partition is that the distance between two leaders is at least three links. This ensures that the layer of nearest neighbours of each leader remains attached to him. The process of cutting links starts from a selection of the shortest path between the leaders. If there is more than one path, we concentrate on one of them. If the length of the path is exactly three, there is only one link in the middle to be cut. If the shortest path consists of more than three links, the cutting can be performed in two ways; either we select the link with the lowest number (variant A) or the ink with the highest number (variant B). As a rule, numerically calculated size distribution of fragments of scale-free Albert-Barabasi networks reveals one large fragment which contains the original leader (hub of the network), and a number of small fragments with opponents that are described by two Weibull distributions [2].

Numerical simulations and mean-field theory reveal that size of the larger fragment scales as the square root of the initial network size. The algorithm is applied to the data on political blogs in U.S. (L. Adamic and N. Glance [3]). The obtained fragments are clearly polarized; either they belong to Democrats, or to the GOP.

[1] K. Kacperski and J.A. Holyst, Phase transitions as a persisten feature of groups with leadersin models of opinion formation, Physica A: Statistical Mechanics and its Applications. 2000; 287(34):631–643.

[2] Malgorzata J. Krawczyk, Krzysztof Kulakowski and Janusz A. Holyst, Consequtive partitions of social networks between revealing leaders, arXiv:1611.05604.

[3] L.A. Adamic and N. Glance, The Political Blogosphere and the 2004 U.S. Election: Divided They Blog. In: Proceedings of the 3rd InternationalWorkshop on Link Discovery. LinkKDD ’05. New York, NY, USA: ACM; 2005. p. 36–43.