Thomas Nowotny and Manfred Requardt 1998 J. Phys. A: Math. Gen. 31 2447 doi:10.1088/0305-4470/31/10/018
Dimension theory of graphs and networks
Thomas Nowotny and Manfred Requardt
Show affiliationsStarting 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 such as (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.
- PACS
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02.10.Ox Combinatorics; graph theory
64.60.A- Specific approaches applied to studies of phase transitions
07.05.Mh Neural networks, fuzzy logic, artificial intelligence
- MSC
- Subjects
- Dates
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Issue 10 (13 March 1998)
Received 14 August 1997, in final form 13 November 1997
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Dimension theory of graphs and networks
Thomas Nowotny and Manfred Requardt 1998 J. Phys. A: Math. Gen. 31 2447
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