Quantum statistics and networks: Between bosons and fermions.

In this article, we discuss several types of networks: random graph, Barabasi-Albert (BA) model, and lattice networks, from the unified view point. The parameter $\omega$ with values $1,0, -1$ corresponds to these networks, respectively. The parameter is related to the preferential attachment of nodes in the networks and has different weights for the incoming and out-going links. In other viewpoints, we discuss the correspondence between quantum statistics and the networks. Positive (negative) $\omega$ corresponds to Bose (Fermi)-like statistics. We can obtain the distribution that connects Boson and Fermion. When $\omega$ is positive, $\omega$ is the threshold of Bose-Einstein condensation (BEC). As $\omega$ decreases, the area of the BEC phase is narrowed and disappears in the limit $\omega=0$. When $\omega$ is negative, there exists the maximum occupied number which corresponds to that the node has the maximum number of links. We can observe the Fermi degeneracy of the network. When $\omega=-1$, a standard Fermion-like network is observed. In fact Fermion network is realized in the criptcurrency network, "Tangle".