35,119 research outputs found
Geometric Crossing-Minimization - A Scalable Randomized Approach
We consider the minimization of edge-crossings in geometric drawings of graphs G=(V, E), i.e., in drawings where each edge is depicted as a line segment. The respective decision problem is NP-hard [Daniel Bienstock, 1991]. Crossing-minimization, in general, is a popular theoretical research topic; see Vrt\u27o [Imrich Vrt\u27o, 2014]. In contrast to theory and the topological setting, the geometric setting did not receive a lot of attention in practice. Prior work [Marcel Radermacher et al., 2018] is limited to the crossing-minimization in geometric graphs with less than 200 edges. The described heuristics base on the primitive operation of moving a single vertex v to its crossing-minimal position, i.e., the position in R^2 that minimizes the number of crossings on edges incident to v.
In this paper, we introduce a technique to speed-up the computation by a factor of 20. This is necessary but not sufficient to cope with graphs with a few thousand edges. In order to handle larger graphs, we drop the condition that each vertex v has to be moved to its crossing-minimal position and compute a position that is only optimal with respect to a small random subset of the edges. In our theoretical contribution, we consider drawings that contain for each edge uv in E and each position p in R^2 for v o(|E|) crossings. In this case, we prove that with a random subset of the edges of size Theta(k log k) the co-crossing number of a degree-k vertex v, i.e., the number of edge pairs uv in E, e in E that do not cross, can be approximated by an arbitrary but fixed factor delta with high probability. In our experimental evaluation, we show that the randomized approach reduces the number of crossings in graphs with up to 13 000 edges considerably. The evaluation suggests that depending on the degree-distribution different strategies result in the fewest number of crossings
Non-crossing frameworks with non-crossing reciprocals
We study non-crossing frameworks in the plane for which the classical
reciprocal on the dual graph is also non-crossing. We give a complete
description of the self-stresses on non-crossing frameworks whose reciprocals
are non-crossing, in terms of: the types of faces (only pseudo-triangles and
pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a
geometric condition on the stress vectors at some of the vertices.
As in other recent papers where the interplay of non-crossingness and
rigidity of straight-line plane graphs is studied, pseudo-triangulations show
up as objects of special interest. For example, it is known that all planar
Laman circuits can be embedded as a pseudo-triangulation with one non-pointed
vertex. We show that if such an embedding is sufficiently generic, then the
reciprocal is non-crossing and again a pseudo-triangulation embedding of a
planar Laman circuit. For a singular (i.e., non-generic) pseudo-triangulation
embedding of a planar Laman circuit, the reciprocal is still non-crossing and a
pseudo-triangulation, but its underlying graph may not be a Laman circuit.
Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal
arise as the reciprocals of such, possibly singular, stresses on
pseudo-triangulation embeddings of Laman circuits.
All self-stresses on a planar graph correspond to liftings to piece-wise
linear surfaces in 3-space. We prove characteristic geometric properties of the
lifts of such non-crossing reciprocal pairs.Comment: 32 pages, 23 figure
Statistical analysis of 22 public transport networks in Poland
Public transport systems in 22 Polish cities have been analyzed. Sizes of
these networks range from N=152 to N=2881. Depending on the assumed definition
of network topology the degree distribution can follow a power law or can be
described by an exponential function. Distributions of paths in all considered
networks are given by asymmetric, unimodal functions. Clustering, assortativity
and betweenness are studied. All considered networks exhibit small world
behavior and are hierarchically organized. A transition between dissortative
small networks N=500 is observed.Comment: 11 pages, 17 figures, 2 tables, REVTEX4 forma
Short and random: Modelling the effects of (proto-)neural elongations
To understand how neurons and nervous systems first evolved, we need an
account of the origins of neural elongations: Why did neural elongations (axons
and dendrites) first originate, such that they could become the central
component of both neurons and nervous systems? Two contrasting conceptual
accounts provide different answers to this question. Braitenberg's vehicles
provide the iconic illustration of the dominant input-output (IO) view. Here
the basic role of neural elongations is to connect sensors to effectors, both
situated at different positions within the body. For this function, neural
elongations are thought of as comparatively long and specific connections,
which require an articulated body involving substantial developmental processes
to build. Internal coordination (IC) models stress a different function for
early nervous systems. Here the coordination of activity across extended parts
of a multicellular body is held central, in particular for the contractions of
(muscle) tissue. An IC perspective allows the hypothesis that the earliest
proto-neural elongations could have been functional even when they were
initially simple short and random connections, as long as they enhanced the
patterning of contractile activity across a multicellular surface. The present
computational study provides a proof of concept that such short and random
neural elongations can play this role. While an excitable epithelium can
generate basic forms of patterning for small body-configurations, adding
elongations allows such patterning to scale up to larger bodies. This result
supports a new, more gradual evolutionary route towards the origins of the very
first full neurons and nervous systems.Comment: 12 pages, 5 figures, Keywords: early nervous systems, neural
elongations, nervous system evolution, computational modelling, internal
coordinatio
Planar Ising model at criticality: state-of-the-art and perspectives
In this essay, we briefly discuss recent developments, started a decade ago
in the seminal work of Smirnov and continued by a number of authors, centered
around the conformal invariance of the critical planar Ising model on
and, more generally, of the critical Z-invariant Ising model on
isoradial graphs (rhombic lattices). We also introduce a new class of
embeddings of general weighted planar graphs (s-embeddings), which might, in
particular, pave the way to true universality results for the planar Ising
model.Comment: 19 pages (+ references), prepared for the Proceedings of ICM2018.
Second version: two references added, a few misprints fixe
Influences of degree inhomogeneity on average path length and random walks in disassortative scale-free networks
Various real-life networks exhibit degree correlations and heterogeneous
structure, with the latter being characterized by power-law degree distribution
, where the degree exponent describes the extent
of heterogeneity. In this paper, we study analytically the average path length
(APL) of and random walks (RWs) on a family of deterministic networks,
recursive scale-free trees (RSFTs), with negative degree correlations and
various , with an aim to explore the
impacts of structure heterogeneity on APL and RWs. We show that the degree
exponent has no effect on APL of RSFTs: In the full range of
, behaves as a logarithmic scaling with the number of network nodes
(i.e. ), which is in sharp contrast to the well-known double
logarithmic scaling () previously obtained for uncorrelated
scale-free networks with . In addition, we present that some
scaling efficiency exponents of random walks are reliant on degree exponent
.Comment: The definitive verion published in Journal of Mathematical Physic
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