Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-03T14:42:30.064Z Has data issue: false hasContentIssue false
  • English
  • Français

The specific connectivity number of random networks

Part of: Inference from stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Joseph Mecke  and
Dietrich Stoyan
Joseph Mecke*
Affiliation:
University of Jena
Dietrich Stoyan*
Affiliation:
Freiberg University of Mining and Technology
*
∗ Postal address: Faculty of Mathematics and Informatics, University of Jena, 07740 Jena, Germany.
∗∗ Postal address: Institute of Stochastics, Freiberg University of Mining and Technology, 09596 Freiberg, Germany. Email address: stoyan@orion.hrz.tu-freiberg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

A network is a system of segments or edges in ℝd which intersect only in the segment endpoints, which are called vertices. An example is the system of edges of a tessellation. It is possible to give formulas for the specific connectivity number of a random network; in the stationary case, the intensity of the 0-curvature measure is equal to the difference of the intensities of the point processes of vertices and edge centres.

Keywords

Euler characteristicgradientrandom networkspecific connectivity numbertessellation

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 62M30: Spatial processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 576 - 583
Copyright
Copyright © Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Flory, P. J. (1976). Statistical thermodynamics of random networks. Proc. R. Soc. London A 351, 351–380.Google Scholar
Hahn, U. and Stoyan, D. (1998). Unbiased stereological estimation of the surface area of gradient surface processes. Adv. Appl. Prob. 30, 904–920.Google Scholar
Heinrich, G., Straube, E. and Helmis, G. (1988). Rubber elasticity of polymer networks: theories. Adv. Polymer Sci. 85, 33–87.Google Scholar
Klain, D. and Rota, G. C. (1997). Introduction to Geometric Probability. Cambridge University Press.Google Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Mecke, J. (1980). Palm methods for stationary random mosaics. In Combinatorial Principles in Stochastic Geometry, ed. Ambartzumian, R. V.. Armenian Academy of Sciences, Erevan, pp. 124–132.Google Scholar
Mecke, J. (1984). Parametric representation of mean values for stationary random mosaics. Math. Operationsforsch. Statist., Ser. Statist. 15, 437–442.Google Scholar
Mecke, K. (1996). Morphological characteristics of patterns in reaction-diffusion systems. Phys. Rev. E 53, 4794–4800.CrossRefGoogle Scholar
Miles, R. (1976). Estimating aggregate and overall characteristics for embedded aggregates of not-necessarily-convex particles. J. Microscopy 138, 115–125.Google Scholar
Möller, J., (1989). Random tessellations in Rd . Adv. Appl. Prob. 21, 37–73.Google Scholar
Nagel, W., Ohser, J. and Pischang, K. (2000). An integral geometric approach for the Euler–Poincaré characteristic of spatial images. J. Microscopy 198, 54–62.CrossRefGoogle ScholarPubMed
Ohser, J. and Nagel, W. (1996). The estimation of the Euler–Poincaré characteristic from observations on planar sections. J. Microscopy 184, 117–126.CrossRefGoogle Scholar
Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations. Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.CrossRefGoogle Scholar
Ore, O. (1990). Graphs and their Uses, 2nd edn. The Mathematical Association of America, Washington.CrossRefGoogle Scholar
Pothuaud, L. et al. (2000). A new method for three-dimensional skeleton graph analysis of porous media: application of trabecular bone microarchitecture. J. Microscopy 199, 149–161.Google Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.Google Scholar
Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.Google Scholar
Sok, R. M. et al., (2001). Direct and stochastic generation of network models from tomographic images; effect of topology on two phase flow properties. To appear in Transport in Porous Media.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
Tscheschel, A., Stoyan, D. and Hilfer, R. (2000). Erosion-dilation analysis for experimental and synthetic microstructures of sedimentary rock. Physica A 284, 46–58.CrossRefGoogle Scholar
Vogel, H. J. (2000). A numerical experiment on pore size, pore connectivity, water retention, permeability, and absolute transport using network models. Eur. J. Soil Sci. 51, 99–105.CrossRefGoogle Scholar
Weil, W. and Wieacker, J. A. (1993). Stochastic geometry. In Handbook of Convex Geometry, Vol. B, eds. Gruber, P. M. and Wills, J. M.. Elsevier, Amsterdam, pp. 1391–1438.Google Scholar
Zähle, M., (1988). Random cell complexes and generalized sets. Ann. Prob. 16, 1742–1766.Google Scholar