Perception of similarity: a model for social network dynamics (Q2866982)

In this work the authors study the dynamics of social networks considering the individual behavior of people during the generation of connections with other people. Core of the authors' idea is to create a model for mapping individual perception in the sense of people's interpretation of similarity. Formally, this approach is realized as an agent-based model in which agents (each representing a user) interact and define connections according to a set of rules. Considering the conception of people to evaluate people's popularity, two different metrics are adopted for encoding popularity and similarity. Particularly, the popularity is codified as the number of connections, whereas similarity is codified as the hyperbolic distance among points.NEWLINENEWLINEThe distance \(x\) between two points \((r,\theta)\) and \((r',\theta')\) is computed as: NEWLINE\[NEWLINE\cosh \zeta x=\cosh \zeta r \cdot \cosh \zeta r' - \sinh \zeta r \cdot \sinh \zeta r' \cdot\cos \triangle \theta, \tag{1} NEWLINE\]NEWLINE NEWLINENEWLINEwhere \(\triangle\theta=\pi-|\pi -|\theta-\theta '||\) and \(\zeta>0\) is an individual curvature. The authors formally reflect three main phases of generation and evolution of a network. The first phase, when users try to make connections by evaluating their distances \(d\) with all other users, can be computed using equation (1). It is necessary to perform the following constraint: NEWLINE\[NEWLINE d_{xy} \leq R \cdot \epsilon_x, NEWLINE\]NEWLINE NEWLINENEWLINEwhere \(\epsilon\) stands for the similarity `perspective'. In the second phase, when users generate connections with friends of their friends, the distance could also be computed using (1), but instead, the following relation to evaluate similarity is adopted: NEWLINE\[NEWLINE d_{xy} =\ln(f_x+2)\cdot R\cdot \epsilon_x. NEWLINE\]NEWLINE During the third phase, users can generate connections with people from whom they received a connection request, but this is not considered adequate. Therefore, a preliminary network is formed after the first phase, and then only changes its structure. According to these assumptions the model is created.NEWLINENEWLINEArtificial social networks generated by the proposed model are analyzed considering three different indicators: assortativity, average clustering coefficient, and average degree. There were performed many simulations with a number of agents in the range \([1500,5000]\).NEWLINENEWLINEAssuming that each agent is mapped to a point on a hyperbolic disc, all agents are mapped points which are spread with uniform angular density \(\rho(\theta) =\frac{1}{2\pi} \) and with exponential radial density: \(\rho(r)\approx e^{\alpha r}\) (\(\alpha\in[2,3]\)).NEWLINENEWLINEThe main result of the simulations is that the network structure is deeply affected by \(\alpha\). Also the authors found that the geometrical parameters \(\zeta\) and \(\epsilon\) represent the individual perception and the individual interpretation of similarity, respectively. It should be noted that the coefficient \(\epsilon\) allows users to evaluate the maximum distance to consider someone else alike.NEWLINENEWLINEThe simulations also show that the proposed model reveals the structure of a network, a degree distribution similar to that of E-R graphs, and with small-world behavior (for \(\alpha\geq2.5\)), as evidenced in many real social networks.