21,397 research outputs found
Spreading and shortest paths in systems with sparse long-range connections
Spreading according to simple rules (e.g. of fire or diseases), and
shortest-path distances are studied on d-dimensional systems with a small
density p per site of long-range connections (``Small-World'' lattices). The
volume V(t) covered by the spreading quantity on an infinite system is exactly
calculated in all dimensions. We find that V(t) grows initially as t^d/d for
t>t^*$,
generalizing a previous result in one dimension. Using the properties of V(t),
the average shortest-path distance \ell(r) can be calculated as a function of
Euclidean distance r. It is found that
\ell(r) = r for r<r_c=(2p \Gamma_d (d-1)!)^{-1/d} log(2p \Gamma_d L^d), and
\ell(r) = r_c for r>r_c.
The characteristic length r_c, which governs the behavior of shortest-path
lengths, diverges with system size for all p>0. Therefore the mean separation s
\sim p^{-1/d} between shortcut-ends is not a relevant internal length-scale for
shortest-path lengths. We notice however that the globally averaged
shortest-path length, divided by L, is a function of L/s only.Comment: 4 pages, 1 eps fig. Uses psfi
The structure of borders in a small world
Geographic borders are not only essential for the effective functioning of
government, the distribution of administrative responsibilities and the
allocation of public resources, they also influence the interregional flow of
information, cross-border trade operations, the diffusion of innovation and
technology, and the spatial spread of infectious diseases. However, as growing
interactions and mobility across long distances, cultural, and political
borders continue to amplify the small world effect and effectively decrease the
relative importance of local interactions, it is difficult to assess the
location and structure of effective borders that may play the most significant
role in mobility-driven processes. The paradigm of spatially coherent
communities may no longer be a plausible one, and it is unclear what structures
emerge from the interplay of interactions and activities across spatial scales.
Here we analyse a multi-scale proxy network for human mobility that
incorporates travel across a few to a few thousand kilometres. We determine an
effective system of geographically continuous borders implicitly encoded in
multi-scale mobility patterns. We find that effective large scale boundaries
define spatially coherent subdivisions and only partially coincide with
administrative borders. We find that spatial coherence is partially lost if
only long range traffic is taken into account and show that prevalent models
for multi-scale mobility networks cannot account for the observed patterns.
These results will allow for new types of quantitative, comparative analyses of
multi-scale interaction networks in general and may provide insight into a
multitude of spatiotemporal phenomena generated by human activity.Comment: 9 page
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