Power-law distributions from additive preferential redistributions

Abstract

We introduce a nongrowth model that generates the power-law distribution with the Zipf exponent. There are elements, each of which is characterized by a quantity, and at each time step these quantities are redistributed through binary random interactions with a simple additive preferential rule, while the sum of quantities is conserved. The situation described by this model is similar to those of closed -particle systems when conservative two-body collisions are only allowed. We obtain stationary distributions of these quantities both analytically and numerically while varying parameters of the model, and find that the model exhibits the scaling behavior for some parameter ranges. Unlike well-known growth models, this alternative mechanism generates the power-law distribution when the growth is not expected and the dynamics of the system is based on interactions between elements. This model can be applied to some examples such as personal wealths, city sizes, and the generation of scale-free networks when only rewiring is allowed.

Network rewiring dynamics with convergence towards a star network

“…If either of these conditions are true, then the rewiring procedure is aborted [16]. The steady-state scaling of this model is related to parameter β in equation (2.1) and the mean degreek of N. Ree shows [15] that this relationship results in either a steady-state exponential or power-law graph. For a non-degenerate, undirected graph the algorithm converges to a power-law when α c (β) ≈k, where α c (β) is an empirically derived formula (see [15]).…”

“…The steady-state scaling of this model is related to parameter β in equation (2.1) and the mean degreek of N. Ree shows [15] that this relationship results in either a steady-state exponential or power-law graph. For a non-degenerate, undirected graph the algorithm converges to a power-law when α c (β) ≈k, where α c (β) is an empirically derived formula (see [15]). However, the setting of β was quite sensitive to the initial regular network configuration, using the algorithm somewhat difficult to tune for different network configurations.…”

“…The algorithm presented here is based on the work of Ree [15,16], and so a brief review of the Ree algorithm is appropriate. Given a graph G with N nodes (vertices) and E edges, the Ree algorithm commences by selecting two random nodes i and j with degrees k i and k j .…”

“…Variations of this model have been used to study the effect of network structure on processes such as genetic drift [12] and military communication [13], although this model often produced disconnected networks, and the convergence properties were not well defined. Other models for fixed network architectures that have stationary scale-free distributions under a range of conditions include Park & Lai [14], Ree [15,16] and Xie et al [17]. The field of statistical mechanics has also been used to examine network construction [18,19], using the metaphor of temperature and total energy to determine the phase of the network.…”

Dynamics of three-agent games

“…A reasonable approach to model the interactions is to involve a finite number of agents at a time and via the rules decide a winner which gains one unit of the attribute used to compare the agents. The interaction may represent, a competitive game for wealth [5]- [8], trophies in sports [9], [10] , opinion dynamics [11]- [13], idea or rumor propagation [14]- [16]. One can also contemplate the emergence of social hierarchies from such models [17]- [21].…”