Structural Properties of the Asymmetric Barabási-Albert Model in the Lattice Limit

Abstract

The Asymmetric BA model extends the Barabási-Albert scale-free network model by introducing a parameter $\omega$. As $\omega$ varies, the model transitions through different network structures: an extended lattice at $\omega =-1$, a random graph at $\omega =0$, and the original scale-free network at $\omega =1$. We derive the exact degree distribution for $\omega =-r/(r+k)$, where $k\in \{0,1,\cdots \}$, and develop a perturbative expansion around these values of $\omega$. Additionally, we show that for $\omega =-1+\epsilon$, the clustering coefficient scales as $\ln t/\sqrt{\epsilon }t$ and approaches zero as $t\to \infty$, confirming the absence of small-world properties.

20 pages,9 figures