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Central loops in random planar graphs

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Abstract

Random planar graphs appear in a variety of context and it is important for many different applications to be able to characterize their structure. Local quantities fail to give interesting information and it seems that path-related measures are able to convey relevant information about the organization of these structures. In particular, nodes with a large betweenness centrality (BC) display non-trivial patterns, such as central loops. We first discuss empirical results for different random planar graphs and we then propose a toy model which allows us to discuss the condition for the emergence of non-trivial patterns such as central loops. This toy model is made of a star network with $N_b$ branches of size $n$ and links of weight $1$, superimposed to a loop at distance $\ell$ from the center and with links of weight $w$. We estimate for this model the BC at the center and on the loop and we show that the loop can be more central than the origin if $w
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cea-01502150 , version 1 (05-04-2017)

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Benjamin Lion, Marc Barthelemy. Central loops in random planar graphs. 2017. ⟨cea-01502150⟩
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