Voting to the Link: a Static Network Formation Model
Acta Polytechnica Hungarica
It is well known that the structure of social, organization and economic networks have a huge effect on the behaviour of the underlying system. This structure is often considered as a network, and modelling the formation of these networks is an active research area of complex systems. In this paper we present a simple network generation model: there is a closed population of agents, and the agents are voting to the connections according to their fixed preferences. These preferences are denoted
... rences are denoted by real numbers, and can be considered as simply the desired number of connections or social capital. -207 -R. Pethes et al. Voting to the Link: a Static Network Formation Model random process, and the second is game theory, where strategic models of how networks are formed are developed. The static voting model is in the random network model family. Here we are not going to review the huge area of random networks, but just mention some important milestones. The field of random graphs was first introduced by the famous paper of Erdos and Rényi  . The Erdos-Rényi ER n (p) random graph has n vertices and each pair of vertices is independently connected with probability p. Despite the fact that ER n (p) is the simplest imaginable model of a random network, it has fascinating phase transition when p varies. Looking at many real world networks  such as social network, Internet, transport networks, biological networks, etc., we can see that their degree distribution is power-law, and they have the so called small-world property. Since the ER model has Poisson degree distribution, it is not a good model of real world networks. There are many extensions of this random graph model, for example the inhomogeneous version of the ER model  that we will use later in this paper, the configuration model , the generalized random graph model  , and the most general form of the inhomogeneous random graphs  defined by kernel functions. In this paper we only deal with static models, but we have to mention one network growth model family: the class of preferential attachment models are such that new elements are more likely to attach to elements with high degree compared to elements with small degree. This phenomenon was first published by Barabási and Albert  and their model is called Barabási -Albert (BA) model. Networks generated by the BA model show power-law degree distribution. In this paper we continue and extend the work on static edge voting models  . The structure of this paper is as follows: after an introduction to the necessary notions and definitions, in Section 2 we turn to the Inhomogeneous Random Graphs. In this section we have two main points: the Chernoff-bounds and the fact that we can use the Gauss distribution to approximate the distribution of the degree of a given node in the network. In Section 3 we define the static edge voting model (SEV), introduce the Mixture of Gaussians (MOG) method to estimate the degree distribution of the graph (Subsection 3.1) and discuss how we generate the parameters of the SEV models to be able to execute numerical tests (Subsection 3.2). In Subsections 3.3, 3.4 and 3.5 we define and analyse 3 special cases of the SEV model: the proportional, the Poission and the biased voting model. We will demonstrate the usability of the MOG approximation with the proportional model, and in the following models we will use only this.