Exactly solvable scale-free network model.

We study a deterministic scale-free network recently proposed by Barabási, Ravasz, and Vicsek. We find that there are two types of nodes: the hub and rim nodes, which form a bipartite structure of the network. We first derive the exact numbers P (k) of nodes with degree k for the hub and rim nodes in each generation of the network, respectively. Using this, we obtain the exact exponents of the distribution function P (k) of nodes with k degree in the asymptotic limit of k-->infinity . We show that the degree distribution for the hub nodes exhibits the scale-free nature, P (k) proportional to k(-gamma) with gamma=ln 3/ln 2=1.584 962 , while the degree distribution for the rim nodes is given by P(k) proportional to e(-gamma'k) with gamma' =ln (3/2) =0.405 465 . Second, we analytically calculate the second-order average degree of nodes, d(-) . Third, we numerically as well as analytically calculate the spectra of the adjacency matrix A for representing topology of the network. We also analytically obtain the exact number of degeneracies at each eigenvalue in the network. The density of states (i.e., the distribution function of eigenvalues) exhibits the fractal nature with respect to the degeneracy. Fourth, we study the mathematical structure of the determinant of the eigenequation for the adjacency matrix. Fifth, we study hidden symmetry, zero modes, and its index theorem in the deterministic scale-free network. Finally, we study the nature of the maximum eigenvalue in the spectrum of the deterministic scale-free network. We will prove several theorems for it, using some mathematical theorems. Thus, we show that most of all important quantities in the network theory can be analytically obtained in the deterministic scale-free network model of Barabási, Ravasz, and Vicsek. Therefore, we may call this network model the exactly solvable scale-free network.