24,237 research outputs found
Parametric shortest-path algorithms via tropical geometry
We study parameterized versions of classical algorithms for computing
shortest-path trees. This is most easily expressed in terms of tropical
geometry. Applications include shortest paths in traffic networks with variable
link travel times.Comment: 24 pages and 8 figure
Vulnerability of weighted networks
In real networks complex topological features are often associated with a
diversity of interactions as measured by the weights of the links. Moreover,
spatial constraints may as well play an important role, resulting in a complex
interplay between topology, weight, and geography. In order to study the
vulnerability of such networks to intentional attacks, these attributes must be
therefore considered along with the topological quantities. In order to tackle
this issue, we consider the case of the world-wide airport network, which is a
weighted heterogeneous network whose evolution and structure are influenced by
traffic and geographical constraints. We first characterize relevant
topological and weighted centrality measures and then use these quantities as
selection criteria for the removal of vertices. We consider different attack
strategies and different measures of the damage achieved in the network. The
analysis of weighted properties shows that centrality driven attacks are
capable to shatter the network's communication or transport properties even at
very low level of damage in the connectivity pattern. The inclusion of weight
and traffic therefore provides evidence for the extreme vulnerability of
complex networks to any targeted strategy and need to be considered as key
features in the finding and development of defensive strategies
A graph-based mathematical morphology reader
This survey paper aims at providing a "literary" anthology of mathematical
morphology on graphs. It describes in the English language many ideas stemming
from a large number of different papers, hence providing a unified view of an
active and diverse field of research
Optimal Traffic Networks
Inspired by studies on the airports' network and the physical Internet, we
propose a general model of weighted networks via an optimization principle. The
topology of the optimal network turns out to be a spanning tree that minimizes
a combination of topological and metric quantities. It is characterized by a
strongly heterogeneous traffic, non-trivial correlations between distance and
traffic and a broadly distributed centrality. A clear spatial hierarchical
organization, with local hubs distributing traffic in smaller regions, emerges
as a result of the optimization. Varying the parameters of the cost function,
different classes of trees are recovered, including in particular the minimum
spanning tree and the shortest path tree. These results suggest that a
variational approach represents an alternative and possibly very meaningful
path to the study of the structure of complex weighted networks.Comment: 4 pages, 4 figures, final revised versio
The effects of spatial constraints on the evolution of weighted complex networks
Motivated by the empirical analysis of the air transportation system, we
define a network model that includes geographical attributes along with
topological and weight (traffic) properties. The introduction of geographical
attributes is made by constraining the network in real space. Interestingly,
the inclusion of geometrical features induces non-trivial correlations between
the weights, the connectivity pattern and the actual spatial distances of
vertices. The model also recovers the emergence of anomalous fluctuations in
the betweenness-degree correlation function as first observed by Guimer\`a and
Amaral [Eur. Phys. J. B {\bf 38}, 381 (2004)]. The presented results suggest
that the interplay between weight dynamics and spatial constraints is a key
ingredient in order to understand the formation of real-world weighted
networks
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