2,979 research outputs found
Micro-Macro Analysis of Complex Networks
Complex systems have attracted considerable interest because of their wide range of applications, and are often studied via a \u201cclassic\u201d approach: study a specific system, find a complex network behind it, and analyze the corresponding properties. This simple methodology has produced a great deal of interesting results, but relies on an often implicit underlying assumption: the level of detail on which the system is observed. However, in many situations, physical or abstract, the level of detail can be one out of many, and might also depend on intrinsic limitations in viewing the data with a different level of abstraction or precision. So, a fundamental question arises: do properties of a network depend on its level of observability, or are they invariant? If there is a dependence, then an apparently correct network modeling could in fact just be a bad approximation of the true behavior of a complex system. In order to answer this question, we propose a novel micro-macro analysis of complex systems that quantitatively describes how the structure of complex networks varies as a function of the detail level. To this extent, we have developed a new telescopic algorithm that abstracts from the local properties of a system and reconstructs the original structure according to a fuzziness level. This way we can study what happens when passing from a fine level of detail (\u201cmicro\u201d) to a different scale level (\u201cmacro\u201d), and analyze the corresponding behavior in this transition, obtaining a deeper spectrum analysis. The obtained results show that many important properties are not universally invariant with respect to the level of detail, but instead strongly depend on the specific level on which a network is observed. Therefore, caution should be taken in every situation where a complex network is considered, if its context allows for different levels of observability
Loop Quantum Mechanics and the Fractal Structure of Quantum Spacetime
We discuss the relation between string quantization based on the Schild path
integral and the Nambu-Goto path integral. The equivalence between the two
approaches at the classical level is extended to the quantum level by a
saddle--point evaluation of the corresponding path integrals. A possible
relationship between M-Theory and the quantum mechanics of string loops is
pointed out. Then, within the framework of ``loop quantum mechanics'', we
confront the difficult question as to what exactly gives rise to the structure
of spacetime. We argue that the large scale properties of the string condensate
are responsible for the effective Riemannian geometry of classical spacetime.
On the other hand, near the Planck scale the condensate ``evaporates'', and
what is left behind is a ``vacuum'' characterized by an effective fractal
geometry.Comment: 19pag. ReVTeX, 1fig. Invited paper to appear in the special issue of
{\it Chaos, Solitons and Fractals} on ``Super strings, M,F,S,...Theory''
(M.S. El Naschie and C.Castro, ed
On Fuzzy Concepts
In this paper we try to combine two approaches. One is the theory of knowledge graphs in which concepts are represented by graphs. The other is the axiomatic theory of fuzzy sets (AFS).
The discussion will focus on the idea of fuzzy concept. It will be argued that the fuzziness of a concept in natural language is mainly due to the difference in interpretation that people give to a certain word. As different interpretations lead to different knowledge graphs, the notion of fuzzy concept should be describable in terms of sets of graphs. This leads to a natural introduction of membership values for elements of graphs. Using these membership values we apply AFS theory as well as an alternative approach to calculate fuzzy decision trees, that can be used to determine the most relevant elements of a concept
Tensor models and hierarchy of n-ary algebras
Tensor models are generalization of matrix models, and are studied as models
of quantum gravity. It is shown that the symmetry of the rank-three tensor
models is generated by a hierarchy of n-ary algebras starting from the usual
commutator, and the 3-ary algebra symmetry reported in the previous paper is
just a single sector of the whole structure. The condition for the Leibnitz
rules of the n-ary algebras is discussed from the perspective of the invariance
of the underlying algebra under the n-ary transformations. It is shown that the
n-ary transformations which keep the underlying algebraic structure invariant
form closed finite n-ary Lie subalgebras. It is also shown that, in physical
settings, the 3-ary transformation practically generates only local
infinitesimal symmetry transformations, and the other more non-local
infinitesimal symmetry transformations of the tensor models are generated by
higher n-ary transformations.Comment: 13 pages, some references updated and correcte
Spectral dimension of a quantum universe
In this paper, we calculate in a transparent way the spectral dimension of a
quantum spacetime, considering a diffusion process propagating on a fluctuating
manifold. To describe the erratic path of the diffusion, we implement a minimal
length by averaging the graininess of the quantum manifold in the flat space
case. As a result we obtain that, for large diffusion times, the quantum
spacetime behaves like a smooth differential manifold of discrete dimension. On
the other hand, for smaller diffusion times, the spacetime looks like a fractal
surface with a reduced effective dimension. For the specific case in which the
diffusion time has the size of the minimal length, the spacetime turns out to
have a spectral dimension equal to 2, suggesting a possible renormalizable
character of gravity in this regime. For smaller diffusion times, the spectral
dimension approaches zero, making any physical interpretation less reliable in
this extreme regime. We extend our result to the presence of a background field
and curvature. We show that in this case the spectral dimension has a more
complicated relation with the diffusion time, and conclusions about the
renormalizable character of gravity become less straightforward with respect to
what we found with the flat space analysis.Comment: 5 pages, 1 figure, references added, typos corrected, title changed,
final version published in Physical Review
Geometry from Matrices via D-branes
In this paper, we give a map from matrices to a commutative geometry from a
bound state of a D2-brane and N D0-branes. For this, tachyons in auxiliary
unstable D-brane system describing the bound state play crucial roles. We found
the map obtained in this way coincides with the recent proposals. We also
consider the map from the geometry to matrices in a large N limit and argue
that the map is a matrix regularization of geometry.Comment: 15 pages, corrected typos, comments added, minor correction
Hausdorff dimension of a quantum string
In the path integral formulation of quantum mechanics, Feynman and Hibbs
noted that the trajectory of a particle is continuous but nowhere
differentiable. We extend this result to the quantum mechanical path of a
relativistic string and find that the ``trajectory'', in this case, is a
fractal surface with Hausdorff dimension three. Depending on the resolution of
the detecting apparatus, the extra dimension is perceived as ``fuzziness'' of
the string world-surface. We give an interpretation of this phenomenon in terms
of a new form of the uncertainty principle for strings, and study the
transition from the smooth to the fractal phase.Comment: 18 pages, non figures, ReVTeX 3.0, in print on Phys.Rev.
Disentangling agglomeration and network externalities : a conceptual typology
Agglomeration and network externalities are fuzzy concepts. When different meanings are (un)intentionally juxtaposed in analyses of the agglomeration/network externalities-menagerie, researchers may reach inaccurate conclusions about how they interlock. Both externality types can be analytically combined, but only when one adopts a coherent approach to their conceptualization and operationalization, to which end we provide a combinatorial typology. We illustrate the typology by applying a state-of-the-art bipartite network projection detailing the presence of globalized producer services firms in cities in 2012. This leads to two one-mode graphs that can be validly interpreted as topological renderings of agglomeration and network externalities
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