Closure operators associated with networks

Year 2017, Volume: 46 Issue: 1, 91 - 101, 01.02.2017

Josef \v{s}lapal , John L. Pfaltz

Abstract

We study network (i.e., undirected simple graph) structures by investigating associated closure operators and the corresponding closed sets.
To describe the dynamic behavior of networks, we employ continuous transformations and neighborhood homomorphisms between them. These transformations and homomorphisms are then studied. In particular, the problem of preserving generators by continuous transformations and that of preserving minimal dominating sets by neighborhood homomorphisms are dealt with.

Keywords

Network, Closure operator, Transformation

References

• Agnarsson, G. and Greenlaw, R. Graph Theory: Modeling, Applications and Algorithms (Prentice Hall, Upper Saddle River, NJ, 2007).
• Bourqui R., Gilbert, F., Simonetto, P., Zaidi, F., Sharan, U. and Jourdan, F. Detecting structural changes and command hierarchies in dynamic social networks, in: 2009 Advances in Social Network Analysis and Mining, 8388 (Athens, Greece, 2009).
• Godsil, Ch. and Royle, G. Algebraic Graph Theory (Springer, New York, 2001).
• Harary, F. Graph Theory (Addison-Wesley, 1969).
• Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. (editors) Domination in Graphs, Advanced Topics (Marcel Dekker, New York, 1998).
• Haynes, T.W., Hedetniemi, S.T. and Slater, P.J. Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
• Jankovic, D. and Hamlett, T.R. New topologies from old via ideals, Amer. Math. Monthly 97 (4), 295310, 1990.
• Koshevoy, G.A. Choice functions and abstract convex geometries, Mathematical Social Sciences, 38 (1), 3544, 1999.
• McKee, T.A. and McMorris, F.R. Topics in Intersection Graph Theory, SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics (Philadelphia, PA, 1999).
• Monjardet, B. Closure operators and choice operators: a survey, in: Fifth International Conference on Concept Lattices and their Applications (Montpellier, France, 2007).
• Monjardet, M and Raderinirina, V. The duality between the antiexchange closure operators and the path independent choice operators on a finite set, Mathemetical Social Sciences 41(2), 131150, 2001.
• Newman, M.E.J. The structure of function of complex networks, SIAM Review 45, 167256, 2003.
• Oystein Ore. Mappings of closure relations, Annals of Math. 47 (1), 5672, 1946.
• Pfaltz, J.L. and ’lapal, J. Transformations of discrete closure systems, Acta Math. Hungar. 138 (4), 386405, 2013.
• Pfaltz, J.L. Closure lattices, Discrete Mathematics 154, 217236, 1996.
• Pfaltz, J.L. Logical implication and causal dependency, in: Schärfe, H., Hitzler, P. and Ohrstrom, P. (editors) Conceptual Structures: Inspiration and Application, Lecture Notes in Articial Intelligence 4068, 145157, 2006.
• Pfaltz, J.L. Establishing logical rules from empirical data, Intern. Journal on Articial Intelligence Tools 17 (5), 9851001, 2008.
• Pfaltz, J.L. Mathematical continuity in dynamic social networks, in: Datta, A., Shulman, S., Zheng, B., Lin, S., Sun, A. and Lim, E.-P (editors) Third International Conference on Social Informatics 2011, Lecture Notes Comput. Sci. 6984, 3650, 2011.
• Slapal, J. Direct arithmetics of relational systems, Publ. Math. Debrecen 38, 3948, 1991.
• Slapal, J. Convenient closure operators on $\mathbb{Z}^2$, in: Wiederhold, P. and Barneva, R.P. (editors) Combinatorial Image Analysis, Lecture Notes Comput. Sci. 5852, 425436, 2009.
• Slapal, J. A Galois correspondence for digital topology, in: Denecke, K., Erné, M. and Wismath, S.L. (editors) Galois Connections and Applications, 413424 (Kluwer Academic Publishers, Dordrecht, 2004).