Nonextensive aspects of self-organized scale-free gas-like networks

We explore the possibility to interpret as a “gas” the dynamical self-organized scale-free network recently introduced by Kim et al. (2005). The role of “momentum” of individual nodes is played by the degree of the node, the “configuration space” (metric defining distance between nodes) being determined by the dynamically evolving adjacency matrix. In a constant-size network process, “inelastic” interactions occur between pairs of nodes, which are realized by the merger of a pair of two nodes into one. The resulting node possesses the union of all links of the previously separate nodes. We consider chemostat conditions, i.e., for each merger there will be a newly created node which is then linked to the existing network randomly. We also introduce an interaction “potential” (node-merging probability) which decays with distance dij as $1/d_{ij}^{\,\alpha}\;(\alpha \ge 0)$. We numerically show that this system exhibits nonextensive statistics in the degree distribution, and calculate how the entropic index q depends on α. The particular cases $\alpha=0$ and $\alpha \to \infty$ recover the two models introduced by Kim et al