52,382 research outputs found
Optimal paths on the road network as directed polymers
We analyze the statistics of the shortest and fastest paths on the road
network between randomly sampled end points. To a good approximation, these
optimal paths are found to be directed in that their lengths (at large scales)
are linearly proportional to the absolute distance between them. This motivates
comparisons to universal features of directed polymers in random media. There
are similarities in scalings of fluctuations in length/time and transverse
wanderings, but also important distinctions in the scaling exponents, likely
due to long-range correlations in geographic and man-made features. At short
scales the optimal paths are not directed due to circuitous excursions governed
by a fat-tailed (power-law) probability distribution.Comment: 5 pages, 7 figure
Towards Universally Optimal Shortest Paths Algorithms in the Hybrid Model
A drawback of the classic approach for complexity analysis of distributed
graph problems is that it mostly informs about the complexity of notorious
classes of ``worst case'' graphs. Algorithms that are used to prove a tight
(existential) bound are essentially optimized to perform well on such worst
case graphs. However, such graphs are often either unlikely or actively avoided
in practice, where benign graph instances usually admit much faster solutions.
To circumnavigate these drawbacks, the concept of universal complexity
analysis in the distributed setting was suggested by [Kutten and Peleg,
PODC'95] and actively pursued by [Haeupler et al., STOC'21]. Here, the aim is
to gauge the complexity of a distributed graph problem depending on the given
graph instance. The challenge is to identify and understand the graph property
that allows to accurately quantify the complexity of a distributed problem on a
given graph.
In the present work, we consider distributed shortest paths problems in the
HYBRID model of distributed computing, where nodes have simultaneous access to
two different modes of communication: one is restricted by locality and the
other is restricted by congestion. We identify the graph parameter of
neighborhood quality and show that it accurately describes a universal bound
for the complexity of certain class of shortest paths problems in the HYBRID
model
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Universal properties of shortest paths in isotropically correlated random potentials
We consider the optimal paths in a -dimensional lattice, where the bonds
have isotropically correlated random weights. These paths can be interpreted as
the ground state configuration of a simplified polymer model in a random
potential. We study how the universal scaling exponents, the roughness and the
energy fluctuation exponent, depend on the strength of the disorder
correlations. Our numerical results using Dijkstra's algorithm to determine the
optimal path in directed as well as undirected lattices indicate that the
correlations become relevant if they decay with distance slower than 1/r in d=2
and 3. We show that the exponent relation 2nu-omega=1 holds at least in d=2
even in case of correlations. Both in two and three dimensions, overhangs turn
out to be irrelevant even in the presence of strong disorder correlations.Comment: 8 pages LaTeX, eps figures included, typos added, references added,
content change
Betweenness Centrality in Large Complex Networks
We analyze the betweenness centrality (BC) of nodes in large complex
networks. In general, the BC is increasing with connectivity as a power law
with an exponent . We find that for trees or networks with a small loop
density while a larger density of loops leads to . For
scale-free networks characterized by an exponent which describes the
connectivity distribution decay, the BC is also distributed according to a
power law with a non universal exponent . We show that this exponent
must satisfy the exact bound . If the scale
free network is a tree, then we have the equality .Comment: 6 pages, 5 figures, revised versio
On Compact Routing for the Internet
While there exist compact routing schemes designed for grids, trees, and
Internet-like topologies that offer routing tables of sizes that scale
logarithmically with the network size, we demonstrate in this paper that in
view of recent results in compact routing research, such logarithmic scaling on
Internet-like topologies is fundamentally impossible in the presence of
topology dynamics or topology-independent (flat) addressing. We use analytic
arguments to show that the number of routing control messages per topology
change cannot scale better than linearly on Internet-like topologies. We also
employ simulations to confirm that logarithmic routing table size scaling gets
broken by topology-independent addressing, a cornerstone of popular
locator-identifier split proposals aiming at improving routing scaling in the
presence of network topology dynamics or host mobility. These pessimistic
findings lead us to the conclusion that a fundamental re-examination of
assumptions behind routing models and abstractions is needed in order to find a
routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802
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