5,028 research outputs found
Random Geometric Graphs
We analyse graphs in which each vertex is assigned random coordinates in a
geometric space of arbitrary dimensionality and only edges between adjacent
points are present. The critical connectivity is found numerically by examining
the size of the largest cluster. We derive an analytical expression for the
cluster coefficient which shows that the graphs are distinctly different from
standard random graphs, even for infinite dimensionality. Insights relevant for
graph bi-partitioning are included.Comment: 16 pages, 10 figures. Minor changes. Added reference
What exactly are the properties of scale-free and other networks?
The concept of scale-free networks has been widely applied across natural and
physical sciences. Many claims are made about the properties of these networks,
even though the concept of scale-free is often vaguely defined. We present
tools and procedures to analyse the statistical properties of networks defined
by arbitrary degree distributions and other constraints. Doing so reveals the
highly likely properties, and some unrecognised richness, of scale-free
networks, and casts doubt on some previously claimed properties being due to a
scale-free characteristic.Comment: Preprint - submitted, 6 pages, 3 figure
Dynamic programming for graphs on surfaces
We provide a framework for the design and analysis of dynamic
programming algorithms for surface-embedded graphs on n vertices
and branchwidth at most k. Our technique applies to general families
of problems where standard dynamic programming runs in 2O(k·log k).
Our approach combines tools from topological graph theory and
analytic combinatorics.Postprint (updated version
Anomalous behavior in an effective model of graphene with Coulomb interactions
We analyze by exact Renormalization Group (RG) methods the infrared
properties of an effective model of graphene, in which two-dimensional massless
Dirac fermions propagating with a velocity smaller than the speed of light
interact with a three-dimensional quantum electromagnetic field. The fermionic
correlation functions are written as series in the running coupling constants,
with finite coefficients that admit explicit bounds at all orders. The
implementation of Ward Identities in the RG scheme implies that the effective
charges tend to a line of fixed points. At small momenta, the quasi-particle
weight tends to zero and the effective Fermi velocity tends to a finite value.
These limits are approached with a power law behavior characterized by
non-universal critical exponents.Comment: 42 pages, 7 figures; minor corrections, one appendix added (Appendix
A). To appear in Ann. Henri Poincar
Selection, tinkering and emergence in complex networks: crossing the land of tinkering
Complex biological networks have very different origins than technologic ones. The latter involve extensive design and, as engineered structures, include a high level of optimization. The former involve (in principle) contingency and structural constraints, with new structures being incorporated through tinkering with previously evolved modules or units. However, the observation of the topological features of different biological nets suggests that nature can have a limited repertoire of ”attractors” that essentially optimize communication under some basic constraints of cost and architecture or that allow the biological nets to reach a high degree of homeostasis. Conversely, the topological features exhibited by some technology graphs indicate that tinkering and internal constraints play a key role, in spite of the ”designed” nature of these structures. Previous scenarios suggested to explain the overall trends of evolution are re-analyzed in light of topological patterns.Peer ReviewedPostprint (author's final draft
Combinatorial and topological phase structure of non-perturbative n-dimensional quantum gravity
We provide a non-perturbative geometrical characterization of the partition
function of -dimensional quantum gravity based on a coarse classification of
riemannian geometries. We show that, under natural geometrical constraints, the
theory admits a continuum limit with a non-trivial phase structure parametrized
by the homotopy types of the class of manifolds considered. The results
obtained qualitatively coincide, when specialized to dimension two, with those
of two-dimensional quantum gravity models based on random triangulations of
surfaces.Comment: 13 page
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