Geometric origin of scaling in large traffic networks.

Large scale traffic networks are an indispensable part of contemporary human mobility and international trade. Networks of airport travel and cargo ship movements are invaluable for the understanding of human mobility patterns [R. Guimera et al., Proc. Natl. Acad. Sci. U.S.A. 102, 7794 (2005))], epidemic spreading [V. Colizza et al., Proc. Natl. Acad. Sci. U.S.A. 103, 2015 (2006)], global trade [International Maritime Organization, http://www.imo.org/], and spread of invasive species [G. M. Ruiz et al., Nature (London) 408, 49 (2000)]. Different studies [M. Barthelemy, Phys. Rept. 499, 1 (2011)] point to the universal character of some of the exponents measured in such networks. Here we show that exponents which relate (i) the strength of nodes to their degree and (ii) weights of links to degrees of nodes that they connect have a geometric origin. We present a simple robust model which exhibits the observed power laws and relates exponents to the dimensionality of 2D space in which traffic networks are embedded. We show that the relation between weight strength and degree is s(k)k(3/2), the relation between distance strength and degree is s(d)(k)k(3/2), and the relation between weight of link and degrees of linked nodes is w(ij)~(k(i)k(j))(1/2) on the plane 2D surface. We further analyze the influence of spherical geometry, relevant for the whole planet, on exact values of these exponents. Our model predicts that these exponents should be found in future studies of port networks and it imposes constraints on more refined models of port networks.