HAL CCSD
Central loops in random planar graphs
Lion, Benjamin
Barthelemy, Marc
Institut de Physique Théorique - UMR CNRS 3681 (IPHT) ; Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
Revised and augmented version, 13 pages, 12 figures
cea-01502150
https://cea.hal.science/cea-01502150
https://cea.hal.science/cea-01502150/document
https://cea.hal.science/cea-01502150/file/1611.03232.pdf
https://cea.hal.science/cea-01502150
2017
ARXIV: 1611.03232
info:eu-repo/semantics/altIdentifier/arxiv/1611.03232
en
PACS numbers: 89.75.Fb, 89.75.-k, 05.10.Gg and 89.65.Hc
[PHYS.PHYS.PHYS-SOC-PH]Physics [physics]/Physics [physics]/Physics and Society [physics.soc-ph]
[PHYS.COND.CM-SM]Physics [physics]/Condensed Matter [cond-mat]/Statistical Mechanics [cond-mat.stat-mech]
info:eu-repo/semantics/preprint
Preprints, Working Papers, ...
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 $w<w_c$ where the threshold of this transition scales as $w_c\sim n/N_b$. In this regime, there is an optimal position of the loop that scales as $\ell_{opt}\sim N_b w/4$. This simple model sheds some light on the organization of these random structures and allows us to discuss the effect of randomness on the centrality of loops. In particular, it suggests that the number and the spatial extension of radial branches are the crucial ingredients that control the existence of central loops.
2017-04-05
info:eu-repo/semantics/OpenAccess