%0 Unpublished work
%T Central loops in random planar graphs
%+ Institut de Physique ThÃ©orique - UMR CNRS 3681 (IPHT)
%A Lion, Benjamin
%A Barthelemy, Marc
%Z Revised and augmented version, 13 pages, 12 figures
%Z t17/060
%8 2017-04-05
%D 2017
%Z 1611.03232
%Z PACS numbers: 89.75.Fb, 89.75.-k, 05.10.Gg and 89.65.Hc
%Z Physics [physics]/Physics [physics]/Physics and Society [physics.soc-ph]
%Z Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]Preprints, Working Papers, ...
%X 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