High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks
Abstract
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 $\Delta_1 \geq \Delta_2 \geq .. \geq \Delta_k$ 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 $t \rightarrow +\infty$, $\frac{t^{1/2}}{f(t)} \leq \Delta_1 \leq f(t)t^{1/2}$ and for $i=2,...,k=O(1)$, $\frac{t^{1/2}}{f(t)} \leq \Delta_i \leq \Delta_{i-1} - \frac{t^{1/2}}{f(t)}$ with high probability (\whp). Then, we show that the $k$ largest eigenvalues of the adjacency matrix of this graph satisfy $\lambda_k = (1\pm o(1))\Delta_k^{1/2}$ \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 $d(G_t) \leq \rho \log{t}$ where $\frac{1}{\rho}=\eta$ is the unique solution greater than 1 of the equation $\eta - 1 - \log{\eta} = \log{3}$. Finally, we investigate other properties of the model.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2011
- DOI:
- 10.48550/arXiv.1104.5259
- arXiv:
- arXiv:1104.5259
- Bibcode:
- 2011arXiv1104.5259F
- Keywords:
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- Computer Science - Social and Information Networks;
- Computer Science - Discrete Mathematics;
- Mathematics - Combinatorics;
- Physics - Physics and Society;
- 68R10;
- 68R05;
- 68P05
- E-Print:
- (1) 18 pages, 6 figures (2) Updates in 2nd version: added references, corrected typos and simplifications. For more details check http://www.math.cmu.edu/~ctsourak/apolarxiv.txt