66,410 research outputs found
Spectral comparison of large urban graphs
The spectrum of an axial graph is proposed as a means for comparison between spaces,
particularly for measuring between very large and complex graphs. A number of methods have
been used in recent years for comparative analysis within large sets of urban areas, both to
investigate properties of specific known types of street network or to propose a taxonomy of urban
morphology based on an analytical technique. In many cases, a single or small range of predefined,
scalar measures such as metric distance, integration, control or clustering coefficient have
been used to compare the graphs. While these measures are well understood theoretically, their
low dimensionality determines the range of observations that can ultimately be drawn from the data.
Spectral analysis consists of a high dimensional vector representing each space, between which
metric distance may be measured to indicate the overall difference between two spaces, or
subspaces may be extracted to correspond to certain features. It is used for comparison of entire
urban graphs, to determine similarities (and differences) in their overall structure.
Results are shown of a comparison of 152 cities distributed around the world. The clustering of
cities of similar properties in a high dimensional space is discussed. Principal and nonlinear
components of the data set indicate significant correlations in the graph similarities between cities
and their proximity to one another, suggesting that cultural features based on location are evident in
the city form and that these can be quantified by the proposed method. Results of classification
tests show that a city’s location can be estimated based purely on its form.
The high dimensionality of the spectra is beneficial for its utility in data-mining applications that can
draw correlations with other data sets such as land use information. It is shown how further
processing by supervised learning allows the extraction of relevant features. A methodological
comparison is also drawn with statistical studies that use a strong correlation between human
genetic markers and geographical location of populations to derive detailed reconstructions of
prehistoric migration. Thus, it is suggested that the method may be utilised for mapping the transfer
of cultural memes by measuring comparison between cities
Self-Assembly of Geometric Space from Random Graphs
We present a Euclidean quantum gravity model in which random graphs
dynamically self-assemble into discrete manifold structures. Concretely, we
consider a statistical model driven by a discretisation of the Euclidean
Einstein-Hilbert action; contrary to previous approaches based on simplicial
complexes and Regge calculus our discretisation is based on the Ollivier
curvature, a coarse analogue of the manifold Ricci curvature defined for
generic graphs. The Ollivier curvature is generally difficult to evaluate due
to its definition in terms of optimal transport theory, but we present a new
exact expression for the Ollivier curvature in a wide class of relevant graphs
purely in terms of the numbers of short cycles at an edge. This result should
be of independent intrinsic interest to network theorists. Action minimising
configurations prove to be cubic complexes up to defects; there are indications
that such defects are dynamically suppressed in the macroscopic limit. Closer
examination of a defect free model shows that certain classical configurations
have a geometric interpretation and discretely approximate vacuum solutions to
the Euclidean Einstein-Hilbert action. Working in a configuration space where
the geometric configurations are stable vacua of the theory, we obtain direct
numerical evidence for the existence of a continuous phase transition; this
makes the model a UV completion of Euclidean Einstein gravity. Notably, this
phase transition implies an area-law for the entropy of emerging geometric
space. Certain vacua of the theory can be interpreted as baby universes; we
find that these configurations appear as stable vacua in a mean field
approximation of our model, but are excluded dynamically whenever the action is
exact indicating the dynamical stability of geometric space. The model is
intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice
Strong-coupling scales and the graph structure of multi-gravity theories
In this paper we consider how the strong-coupling scale, or perturbative
cutoff, in a multi-gravity theory depends upon the presence and structure of
interactions between the different fields. This can elegantly be rephrased in
terms of the size and structure of the `theory graph' which depicts the
interactions in a given theory. We show that the question can be answered in
terms of the properties of various graph-theoretical matrices, affording an
efficient way to estimate and place bounds on the strong-coupling scale of a
given theory. In light of this we also consider the problem of relating a given
theory graph to a discretised higher dimensional theory, a la dimensional
deconstruction.Comment: 23 pages, 7 figures; v2: additional references included, and minor
typos corrected; version published in JHE
Extremal black holes in the Ho\v{r}ava-Lifshitz gravity
We study the near-horizon geometry of extremal black holes in the
Ho\v{r}ava-Lifshitz gravity with a flow parameter . For ,
near-horizon geometry of extremal black holes are AdS with
different radii, depending on the (modified) Ho\v{r}ava-Lifshitz gravity. For
, the radius of is negative, which means
that the near-horizon geometry is ill-defined and the corresponding
Bekenstein-Hawking entropy is zero. We show explicitly that the entropy
function approach does not work for obtaining the Bekenstein-Hawking entropy of
extremal black holes.Comment: 18 pages, v2:some points on Lifshitz black holes claified, v3:
version to appear in EJP
Loop-Generated Bounds on Changes to the Graviton Dispersion Relation
We identify the effective theory appropriate to the propagation of massless
bulk fields in brane-world scenarios, to show that the dominant low-energy
effect of asymmetric warping in the bulk is to modify the dispersion relation
of the effective 4-dimensional modes. We show how such changes to the graviton
dispersion relation may be bounded through the effects they imply, through
loops, for the propagation of standard model particles. We compute these bounds
and show that they provide, in some cases, the strongest constraints on
nonstandard gravitational dispersions. The bounds obtained in this way are the
strongest for the fewest extra dimensions and when the extra-dimensional Planck
mass is the smallest. Although the best bounds come for warped 5-D scenarios,
for which the 5D Planck Mass is O(TeV), even in 4 dimensions the graviton loop
can lead to a bound on the graviton speed which is comparable with other
constraints.Comment: 18 pages, LaTeX, 4 figures, uses revte
Non-linear sigma models with anti-de Sitter target spaces
We present evidence that there is a non-trivial fixed point for the AdS_{D+1}
non-linear sigma model in two dimensions, without any matter fields or
additional couplings beyond the standard quadratic action subject to a
quadratic constraint. A zero of the beta function, both in the bosonic and
supersymmetric cases, appears to arise from competition between one-loop and
higher loop effects. A string vacuum based on such a fixed point would have
string scale curvature. The evidence presented is based on fixed-order
calculations carried to four loops (corresponding to O(\alpha'^3) in the
spacetime effective action) and on large D calculations carried to O(D^{-2})
(but to all orders in \alpha'). We discuss ways in which the evidence might be
misleading, and we discuss some features of the putative fixed point, including
the central charge and an operator of negative dimension. We speculate that an
approximately AdS_5 version of this construction may provide a holographic dual
for pure Yang-Mills theory, and that quotients of an AdS_3 version might stand
in for Calabi-Yau manifolds in compactifications to four dimensions.Comment: 44 pages, 4 figures. v2: references adde
- …