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High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks

In this work we analyze basic properties of Random Apollonian Networks \cite{zhang,zhou}, a popular stochastic model which generates planar graphs with power law properties. Specifically, let $k$ be a constant and $Δ_1 \geq Δ_2 \geq .. \geq Δ_k$ be the degrees of the $k$ highest degree vertices. We prove that at time $t$, for any function $f$ with $f(t) \rightarrow +\infty$ as $t \rightarrow +\infty$, $\frac{t^{1/2}}{f(t)} \leq Δ_1 \leq f(t)t^{1/2}$ and for $i=2,...,k=O(1)$, $\frac{t^{1/2}}{f(t)} \leq Δ_i \leq Δ_{i-1} - \frac{t^{1/2}}{f(t)}$ with high probability (\whp). Then, we show that the $k$ largest eigenvalues of the adjacency matrix of this graph satisfy $λ_k = (1\pm o(1))Δ_k^{1/2}$ \whp. Furthermore, we prove a refined upper bound on the asymptotic growth of the diameter, i.e., that \whp the diameter $d(G_t)$ at time $t$ satisfies $d(G_t) \leq ρ\log{t}$ where $\frac{1}ρ=η$ is the unique solution greater than 1 of the equation $η- 1 - \logη = \log{3}$. Finally, we investigate other properties of the model.