31,365 research outputs found
Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2
Deciding whether a given graph has a square root is a classical problem that
has been studied extensively both from graph theoretic and from algorithmic
perspectives. The problem is NP-complete in general, and consequently
substantial effort has been dedicated to deciding whether a given graph has a
square root that belongs to a particular graph class. There are both
polynomial-time solvable and NP-complete cases, depending on the graph class.
We contribute with new results in this direction. Given an arbitrary input
graph G, we give polynomial-time algorithms to decide whether G has an
outerplanar square root, and whether G has a square root that is of pathwidth
at most 2
Square Gravity
We simulate the Ising model on dynamical quadrangulations using a
generalization of the flip move for triangulations with two aims: firstly, as a
confirmation of the universality of the KPZ/DDK exponents of the Ising phase
transition, worthwhile in view of some recent surprises with other sorts of
dynamical lattices; secondly, to investigate the transition of the Ising
antiferromagnet on a dynamical loosely packed (bipartite) lattice. In the
latter case we show that it is still possible to define a staggered
magnetization and observe the antiferromagnetic analogue of the transition.Comment: LaTeX file and 7 postscript figures bundled together with uufile
Random Sierpinski network with scale-free small-world and modular structure
In this paper, we define a stochastic Sierpinski gasket, on the basis of
which we construct a network called random Sierpinski network (RSN). We
investigate analytically or numerically the statistical characteristics of RSN.
The obtained results reveal that the properties of RSN is particularly rich, it
is simultaneously scale-free, small-world, uncorrelated, modular, and maximal
planar. All obtained analytical predictions are successfully contrasted with
extensive numerical simulations. Our network representation method could be
applied to study the complexity of some real systems in biological and
information fields.Comment: 7 pages, 9 figures; final version accepted for publication in EPJ
Structural patterns in complex networks through spectral analysis
The study of some structural properties of networks is introduced from a graph spectral perspective. First, subgraph centrality of nodes is defined and used to classify essential proteins in a proteomic map. This index is then used to produce a method that allows the identification of superhomogeneous networks. At the same time this method classify non-homogeneous network into three universal classes of structure. We give examples of these classes from networks in different real-world scenarios. Finally, a communicability function is studied and showed as an alternative for defining communities in complex networks. Using this approach a community is unambiguously defined and an algorithm for its identification is proposed and exemplified in a real-world network
Combinatorial approach to Modularity
Communities are clusters of nodes with a higher than average density of
internal connections. Their detection is of great relevance to better
understand the structure and hierarchies present in a network. Modularity has
become a standard tool in the area of community detection, providing at the
same time a way to evaluate partitions and, by maximizing it, a method to find
communities. In this work, we study the modularity from a combinatorial point
of view. Our analysis (as the modularity definition) relies on the use of the
configurational model, a technique that given a graph produces a series of
randomized copies keeping the degree sequence invariant. We develop an approach
that enumerates the null model partitions and can be used to calculate the
probability distribution function of the modularity. Our theory allows for a
deep inquiry of several interesting features characterizing modularity such as
its resolution limit and the statistics of the partitions that maximize it.
Additionally, the study of the probability of extremes of the modularity in the
random graph partitions opens the way for a definition of the statistical
significance of network partitions.Comment: 8 pages, 4 figure
Opt: A Domain Specific Language for Non-linear Least Squares Optimization in Graphics and Imaging
Many graphics and vision problems can be expressed as non-linear least
squares optimizations of objective functions over visual data, such as images
and meshes. The mathematical descriptions of these functions are extremely
concise, but their implementation in real code is tedious, especially when
optimized for real-time performance on modern GPUs in interactive applications.
In this work, we propose a new language, Opt (available under
http://optlang.org), for writing these objective functions over image- or
graph-structured unknowns concisely and at a high level. Our compiler
automatically transforms these specifications into state-of-the-art GPU solvers
based on Gauss-Newton or Levenberg-Marquardt methods. Opt can generate
different variations of the solver, so users can easily explore tradeoffs in
numerical precision, matrix-free methods, and solver approaches. In our
results, we implement a variety of real-world graphics and vision applications.
Their energy functions are expressible in tens of lines of code, and produce
highly-optimized GPU solver implementations. These solver have performance
competitive with the best published hand-tuned, application-specific GPU
solvers, and orders of magnitude beyond a general-purpose auto-generated
solver
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