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

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

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

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