440 research outputs found
On the non-ergodicity of the Swendsen-Wang-Kotecky algorithm on the kagome lattice
We study the properties of the Wang-Swendsen-Kotecky cluster Monte Carlo
algorithm for simulating the 3-state kagome-lattice Potts antiferromagnet at
zero temperature. We prove that this algorithm is not ergodic for symmetric
subsets of the kagome lattice with fully periodic boundary conditions: given an
initial configuration, not all configurations are accessible via Monte Carlo
steps. The same conclusion holds for single-site dynamics.Comment: Latex2e. 22 pages. Contains 11 figures using pstricks package. Uses
iopart.sty. Final version accepted in journa
Blocking nonorientability of a surface
AbstractLet S be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in S is a blockage if every one-sided simple closed curve in S crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in S with sufficiently large face-width is “roughly” equal to one-half of the minimum number of intersections of a blockage with the graph. The conjecture was disproved by Mohar (Discrete Math. 182 (1998) 245) and replaced by a similar one. In this paper, it is proved that the conjectures in Mohar (1998) and Robertson and Thomas (J. Graph Theory 15 (1991) 407) hold up to a constant error term: For any graph G embedded in S, the orientable genus of G differs from the conjectured value at most by O(g2), where g is the genus of S
Enhancing the spectral gap of networks by node removal
Dynamics on networks are often characterized by the second smallest
eigenvalue of the Laplacian matrix of the network, which is called the spectral
gap. Examples include the threshold coupling strength for synchronization and
the relaxation time of a random walk. A large spectral gap is usually
associated with high network performance, such as facilitated synchronization
and rapid convergence. In this study, we seek to enhance the spectral gap of
undirected and unweighted networks by removing nodes because, practically, the
removal of nodes often costs less than the addition of nodes, addition of
links, and rewiring of links. In particular, we develop a perturbative method
to achieve this goal. The proposed method realizes better performance than
other heuristic methods on various model and real networks. The spectral gap
increases as we remove up to half the nodes in most of these networks.Comment: 5 figure
Local chromatic number of quadrangulations of surfaces
The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four.
Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces
replacing the condition of the graph being not bipartite by a more technical condition of
an odd quadrangulation. This paper investigates when these general results are true for the
local chromatic number instead of the chromatic number. Surprisingly, we find out that
(unlike in the case of the chromatic number) this depends on the genus of the surface. For
the non-orientable surfaces of genus at most four, the local chromatic number of any odd
quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5
or higher.
We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of
arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for
the usual chromatic number
Finding a subdivision of a prescribed digraph of order 4
The problem of when a given digraph contains a subdivision of a fixed digraph F is considered.Bang-Jensen et al. [2] laid out foundations for approaching this problem from the algorithmic pointof view. In this paper we give further support to several open conjectures and speculations about algorithmiccomplexity of finding F-subdivisions. In particular, up to 5 exceptions, we completely classify forwhich 4-vertex digraphs F, the F-subdivision problem is polynomial-time solvable and for which it is NPcomplete.While all NP-hardness proofs are made by reduction from some version of the 2-linkage problemin digraphs, some of the polynomial-time solvable cases involve relatively complicated algorithms
Analysis of Nonlinear Synchronization Dynamics of Oscillator Networks by Laplacian Spectral Methods
We analyze the synchronization dynamics of phase oscillators far from the
synchronization manifold, including the onset of synchronization on scale-free
networks with low and high clustering coefficients. We use normal coordinates
and corresponding time-averaged velocities derived from the Laplacian matrix,
which reflects the network's topology. In terms of these coordinates,
synchronization manifests itself as a contraction of the dynamics onto
progressively lower-dimensional submanifolds of phase space spanned by
Laplacian eigenvectors with lower eigenvalues. Differences between high and low
clustering networks can be correlated with features of the Laplacian spectrum.
For example, the inhibition of full synchoronization at high clustering is
associated with a group of low-lying modes that fail to lock even at strong
coupling, while the advanced partial synchronizationat low coupling noted
elsewhere is associated with high-eigenvalue modes.Comment: Revised version: References added, introduction rewritten, additional
minor changes for clarit
Diffusion dynamics on multiplex networks
We study the time scales associated to diffusion processes that take place on
multiplex networks, i.e. on a set of networks linked through interconnected
layers. To this end, we propose the construction of a supra-Laplacian matrix,
which consists of a dimensional lifting of the Laplacian matrix of each layer
of the multiplex network. We use perturbative analysis to reveal analytically
the structure of eigenvectors and eigenvalues of the complete network in terms
of the spectral properties of the individual layers. The spectrum of the
supra-Laplacian allows us to understand the physics of diffusion-like processes
on top of multiplex networks.Comment: 6 Pages including supplemental material. To appear in Physical Review
Letter
Box representations of embedded graphs
A -box is the cartesian product of intervals of and a
-box representation of a graph is a representation of as the
intersection graph of a set of -boxes in . It was proved by
Thomassen in 1986 that every planar graph has a 3-box representation. In this
paper we prove that every graph embedded in a fixed orientable surface, without
short non-contractible cycles, has a 5-box representation. This directly
implies that there is a function , such that in every graph of genus , a
set of at most vertices can be removed so that the resulting graph has a
5-box representation. We show that such a function can be made linear in
. Finally, we prove that for any proper minor-closed class ,
there is a constant such that every graph of
without cycles of length less than has a 3-box representation,
which is best possible.Comment: 16 pages, 6 figures - revised versio
Synchronization in small-world systems
We quantify the dynamical implications of the small-world phenomenon. We
consider the generic synchronization of oscillator networks of arbitrary
topology, and link the linear stability of the synchronous state to an
algebraic condition of the Laplacian of the graph. We show numerically that the
addition of random shortcuts produces improved network synchronizability.
Further, we use a perturbation analysis to place the synchronization threshold
in relation to the boundaries of the small-world region. Our results also show
that small-worlds synchronize as efficiently as random graphs and hypercubes,
and more so than standard constructive graphs
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