Abstract
Motivated by the problem of graph structure compression under realistic source models, we study the symmetry behavior of preferential and uniform attachment graphs. These are two dynamic models of network growth in which new nodes attach to a constant number $m$ of existing ones according to some attachment scheme. We prove symmetry results for $m=1$ and $2$, and we conjecture that for $m\geq 3$, both models yield asymmetry with high probability. We provide new empirical evidence in terms of graph defect. We also prove that vertex defects in the uniform attachment model grow at most logarithmically with graph size, then use this to prove a weak asymmetry result for all values of $m$ in the uniform attachment model. Finally, we introduce a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and we show that the change introduces symmetry for all values of $m$.
Highlights
Study of the asymptotic behavior of the symmetries of random graphs, originally motivated by combinatorial problems, has relatively recently found a new application in theChoi and Szpankowski (2012) studied this problem in the setting of Erdos-Renyi graphs
We prove that vertex defects in the uniform attachment model grow at most logarithmically with graph size, use this to prove a weak asymmetry result for all values of m in the uniform attachment model
We introduce a natural variation of the two models that incorporates preference of new nodes for nodes of a similar age, and we show that the change introduces symmetry for all values of m
Summary
Study of the asymptotic behavior of the symmetries of random graphs, originally motivated by combinatorial problems, has relatively recently found a new application in the. The primary problem appears to be difficult, so we study a closely related model in which attachment is uniform, in the hope that the proof techniques used there may be generalized In both uniform and preferential attachment models, we prove that when each new vertex chooses only one previous vertex as a neighbor, there is symmetry with high probability, and when each new vertex makes two choices, there is a positive probability of symmetry. The authors precisely characterized the range for which Erdos-Renyi graphs are asymmetric by proving concentration results for random variables defined in terms of vertex defect.
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