Quantum statistics and networks by asymmetric preferential attachment of nodes -- between bosons and fermions

In this article, we discuss the random graph, Barab\'asi-Albert (BA) model, and lattice networks from a unified view point, with the parameter $\omega$ with values $1,0,-1$ characterizing these networks, respectively. The parameter is related to the preferential attachment of nodes in the networks and has different weights for the incoming and outgoing links. In addition, we discuss the correspondence between quantum statistics and the networks. Positive and negative $\omega$ correspond to Bose and Fermi-like statistics, respectively, and we obtain the distribution that connects the two. When $\omega$ is positive, it is related to 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, nodes have limits in the number of attachments for newly added nodes (outgoing links), which corresponds to Fermi statistics. We also observe the Fermi degeneracy of the network. When $\omega=-1$, a standard Fermion-like network is observed. Fermion networks are realized in the cryptocurrency network "Tangle."