Unravelling the size distribution of social groups with information theory in complex networks

Abstract

The minimization of Fisher's information (MFI) approach of Frieden et al. [Phys. Rev. E 60 48 (1999)] is applied to the study of size distributions in social groups on the basis of a recently established analogy between scale invariant systems and classical gases [arXiv:0908.0504]. Going beyond the ideal gas scenario is seen to be tantamount to simulating the interactions taking place in a network's competitive cluster growth process. We find a scaling rule that allows to classify the final cluster-size distributions using only one parameter that we call the competitiveness. Empirical city-size distributions and electoral results can be thus reproduced and classified according to this competitiveness, which also allows to correctly predict well-established assessments such as the "six-degrees of separation", which is shown here to be a direct consequence of the maximum number of stable social relationships that one person can maintain, known as Dunbar's number. Finally, we show that scaled city-size distributions of large countries follow the same universal distribution. PACS. 89.70.Cf Entropy and other measures of information -05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems -89.75.Da Systems obeying scaling laws -89.75.-k Complex systems

Fisher information and the thermodynamics of scale-invariant systems

“…This process just renders the inverse function of the ensuing cumulative distribution, normalized to the number of elements. We call r the rank that ranges from 1 to N. For large N, the density distribution (24) correspond to an exponential rank-size distribution This behaviour, which corresponds to the class of universality γ = 0 in (1), is that empirically found by Costa Filho et al [8] in the distribution of votes in the Brazilian electoral results. We have found such a behaviour [22]).…”

“…We exhibit in figure 4 the raw empirical data as compared with the distribution obtained from the transformation k ′ = c/k (u ′ = −u + ln c), where c = 3.3 × 10 6 . The main part of the density distribution reaches the bulk density obeying (24), whereas Zipf's law emerges at the edges, which could be understood as constituting the surface of the system, since they explain how the density (exponentially) falls from its bulk-value to zero in u-space when the system is exposed to an infinitely empty volume. This effect is clearly visible in figure 5, where the empirical density (27).…”

The workings of the maximum entropy principle in collective human behaviour

“…Inspired by opinion dynamics models [18,22,23], we described in a previous work [24] a numerical process that reproduces the shapes of the empirical city-population distributions. The model is based on a competitive cluster growth process inside a scale-free ideal network (a scale-free network in which degree distribution is described as an SFIG).…”

Privacy dynamics

“…As the number of people in a typical user's social network, we used "Dunbar's number" which is the cognitive limit to the number of people with whom one can maintain stable social relationships [11]. In our experiments, we use the commonly used value, which is 150 [19]. On the other hand, the parameters we used to generate social identity groups are the number of social identity groups in a typical user's social network, social identity group size and finally pattern of a typical social network.…”