Geographical effects on the path length and the robustness in complex networks

Abstract

The short paths between any two nodes and the robustness of connectivity are advanced properties of scale-free (SF) networks; however, they may be affected by geographical constraints in realistic situations. We consider geographical networks with the SF structure based on planar triangulation for online routings, and suggest scaling relations between the average distance or number of hops on the optimal paths and the network size. We also show that the tolerance to random failures and attacks on hubs is weakened in geographical networks, and that even then it is possible for the extremely vulnerable ones to be improved by adding with the local exchange of links.

Improvement of the robustness on geographical networks by adding shortcuts

“…Note that the added shortcuts contribute to create some higher-order cycles which consists of a long path and the overhead bridge in the majority of triangular cycles. The original degree distributions without shortcuts follow a power-law with the exponent nearly 3 in RA, log-normal in DT, and power-law with an exponential cutoff in DLSF networks [15]. Note that the lognormal distribution has an unimodal shape as similar to one in Erdös-Renyi random networks.…”

“…It is better to shorten both the distance and the number of hops, however the constraints are generally conflicted. In the original networks without shortcuts, we note the tendencies [15]: RA networks have a path connected by a few hops but the path length tend to be long including some long-range links, while DT networks have a zig-zag path connected by many hops but each link is short, in addition, DLSF networks have the intermediately balanced properties. Figure 1 shows numerical results in adding shortcuts to the planar networks.…”

“…These results support the usefulness of shortcuts for maintaining both the robust connectivity and the communication efficiency, however the network structures are almost regular and far from the realistic SF. Recently, it has been numerically shown [15] that the robustness is improved by fully random rewirings under a same degree distribution [19] in typical geographical network models: Delaunay triangulation (DT) [20,21], RA, and Delaunay-like scale-free (DLSF) networks [15]. Instead of rewirings, we expect the shortcut effect on the improvement of robustness in such geographical SF networks.…”

“…In addition, the ratio of the shortest path length is bounded by a constant factor to the direct Euclidean distance between any source and terminal [23], while RA network belongs to both SF and planar networks [12,11], however long-range links inevitably appear near the edges of an initial polygon. To reduce the long-range links, Delaunay-like scale-free (DLSF) network has been proposed [15].…”

“…Based on the connection probability according to a penalty of distance r between two nodes and on random triangulation, the generation mechanisms of geographical SF networks have been proposed in lattice-embedded scale-free (LESF) [10] and random Apollonian (RA) [11,12] network models. Unfortunately, the vulnerability of connectivity has been numerically found in both networks [13,14,15]. Moreover, it has been theoretically predicted [13] in a generating function approach to more general networks with any degree distribution that a geographical constraint on linking decreases the tolerance to random failures, since the percolation threshold is increased by the majority of small-order cycles that locally connected with a few hops.…”