94,426 research outputs found
Dimension Theory of Graphs and Networks
Starting from the working hypothesis that both physics and the corresponding
mathematics have to be described by means of discrete concepts on the
Planck-scale, one of the many problems one has to face in this enterprise is to
find the discrete protoforms of the building blocks of continuum physics and
mathematics. A core concept is the notion of dimension. In the following we
develop such a notion for irregular structures like (large) graphs and networks
and derive a number of its properties. Among other things we show its stability
under a wide class of perturbations which is important if one has 'dimensional
phase transitions' in mind. Furthermore we systematically construct graphs with
almost arbitrary 'fractal dimension' which may be of some use in the context of
'dimensional renormalization' or statistical mechanics on irregular sets.Comment: 20 pages, 7 figures, LaTex2e, uses amsmath, amsfonts, amssymb,
latexsym, epsfi
Identifying the underlying structure and dynamic interactions in a voting network
We analyse the structure and behaviour of a specific voting network using a
dynamic structure-based methodology which draws on Q-Analysis and social
network theory. Our empirical focus is on the Eurovision Song Contest over a
period of 20 years. For a multicultural contest of this kind, one of the key
questions is how the quality of a song is judged and how voting groups emerge.
We investigate structures that may identify the winner based purely on the
topology of the network. This provides a basic framework to identify what the
characteristics associated with becoming a winner are, and may help to
establish a homogenous criterion for subjective measures such as quality.
Further, we measure the importance of voting cliques, and present a dynamic
model based on a changing multidimensional measure of connectivity in order to
reveal the formation of emerging community structure within the contest.
Finally, we study the dynamic behaviour exhibited by the network in order to
understand the clustering of voting preferences and the relationship between
local and global properties.Comment: 20 pages, 10 figures, 3 tables, submitted to Physica
Pregeometric Concepts on Graphs and Cellular Networks as Possible Models of Space-Time at the Planck-Scale
Starting from the working hypothesis that both physics and the corresponding
mathematics have to be described by means of discrete concepts on the
Planck-scale, one of the many problems one has to face is to find the discrete
protoforms of the building blocks of continuum physics and mathematics. In the
following we embark on developing such concepts for irregular structures like
(large) graphs or networks which are intended to emulate (some of) the generic
properties of the presumed combinatorial substratum from which continuum
physics is assumed to emerge as a coarse grained and secondary model theory. We
briefly indicate how various concepts of discrete (functional) analysis and
geometry can be naturally constructed within this framework, leaving a larger
portion of the paper to the systematic developement of dimensional concepts and
their properties, which may have a possible bearing on various branches of
modern physics beyond quantum gravity.Comment: 16 pages, Invited paper to appear in the special issue of the Journal
of Chaos, Solitons and Fractals on: "Superstrings, M, F, S ... Theory" (M.S.
El Naschie, C. Castro, Editors
Regular quantum graphs
We introduce the concept of regular quantum graphs and construct connected
quantum graphs with discrete symmetries. The method is based on a decomposition
of the quantum propagator in terms of permutation matrices which control the
way incoming and outgoing channels at vertex scattering processes are
connected. Symmetry properties of the quantum graph as well as its spectral
statistics depend on the particular choice of permutation matrices, also called
connectivity matrices, and can now be easily controlled. The method may find
applications in the study of quantum random walks networks and may also prove
to be useful in analysing universality in spectral statistics.Comment: 12 pages, 3 figure
Resolving structural variability in network models and the brain
Large-scale white matter pathways crisscrossing the cortex create a complex
pattern of connectivity that underlies human cognitive function. Generative
mechanisms for this architecture have been difficult to identify in part
because little is known about mechanistic drivers of structured networks. Here
we contrast network properties derived from diffusion spectrum imaging data of
the human brain with 13 synthetic network models chosen to probe the roles of
physical network embedding and temporal network growth. We characterize both
the empirical and synthetic networks using familiar diagnostics presented in
statistical form, as scatter plots and distributions, to reveal the full range
of variability of each measure across scales in the network. We focus on the
degree distribution, degree assortativity, hierarchy, topological Rentian
scaling, and topological fractal scaling---in addition to several summary
statistics, including the mean clustering coefficient, shortest path length,
and network diameter. The models are investigated in a progressive, branching
sequence, aimed at capturing different elements thought to be important in the
brain, and range from simple random and regular networks, to models that
incorporate specific growth rules and constraints. We find that synthetic
models that constrain the network nodes to be embedded in anatomical brain
regions tend to produce distributions that are similar to those extracted from
the brain. We also find that network models hardcoded to display one network
property do not in general also display a second, suggesting that multiple
neurobiological mechanisms might be at play in the development of human brain
network architecture. Together, the network models that we develop and employ
provide a potentially useful starting point for the statistical inference of
brain network structure from neuroimaging data.Comment: 24 pages, 11 figures, 1 table, supplementary material
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