Abstract
Stochastic rule-based models of networks and biological systems are hard to construct and analyse. Refinements help to produce systems at the right level of abstraction, enable analysis techniques and mappings to other formalisms. Rigidity is a property of graphs introduced in Kappa to support stochastic refinement, allowing to preserve the number of matches for rules in the refined system. In this paper: 1) we propose a notion of rigidity in an axiomatic setting based on adhesive categories; 2) we show how the rewriting of rigid structures can be defined systematically by requiring matches to be open maps reflecting structural features which ensure that rigidity is preserved; and 3) we obtain in our setting a notion of refinement which generalises that in Kappa, and allows a rule to be partitioned into a set of rules which are collectively equivalent to the original. We illustrate our approach with an example of a social network with dynamic topology.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Carboni, A., Lack, S., Walters, R.: Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra 84, 145–158 (1993)
Danos, V., Harmer, R., Winskel, G.: Constraining rule-based dynamics with types. Mathematical Structures in Computer Science 23(2), 272–289 (2013)
Danos, V., Feret, J., Fontana, W., Harmer, R., Hayman, J., Krivine, J., Thompson-Walsh, C.D., Winskel, G.: Graphs, rewriting and pathway reconstruction for rule-based models. In: FSTTCS. LIPIcs, vol. 18, pp. 276–288 (2012)
Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Rule-based modelling of cellular signalling. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 17–41. Springer, Heidelberg (2007)
Danos, V., Feret, J., Fontana, W., Krivine, J.: Scalable simulation of cellular signaling networks. In: Shao, Z. (ed.) APLAS 2007. LNCS, vol. 4807, pp. 139–157. Springer, Heidelberg (2007)
Danos, V., Harmer, R., Honorato-Zimmer, R.: Thermodynamic Graph Rewriting. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013 – Concurrency Theory. LNCS, vol. 8052, pp. 380–394. Springer, Heidelberg (2013)
Danos, V., Honorato-Zimmer, R., Jaramillo-Riveri, S., Stucki, S.: Deriving rate equations for site graph rewriting systems. In: Workshop on Static Analysis and Systems Biology, SASB, Seattle (2013)
Danos, V., Laneve, C.: Formal molecular biology. Theor. Comput. Sci. 325(1), 69–110 (2004)
Deeds, E., Krivine, J., Feret, J., Danos, V., Fontana, W.: Combinatorial complexity and compositional drift in protein interaction networks. PloS One e32032 (2012)
Dörr, H.: Efficient Graph Rewriting and Its Implementation. LNCS, vol. 922. Springer, Heidelberg (1995)
Durrett, R., Gleeson, J., Lloyd, A., Mucha, P., Shi, F., Sivakoff, D., Socoloar, J., Varghese, C.: Graph fission in an evolving voter model. Proceedings of the National Academy of Science 109, 3682–3687 (2012)
Ehrig, H., Ehrig, K., Prange, U., Taentzer, G.: Fundamentals of Algebraic Graph Transformation. EATCS Monographs Theor. Comput. Sci. Springer (2006)
Ehrig, H., Pfender, M., Schneider, H.: Graph grammars: An algebraic approach. In: IEEE Symposium on Switching and Automata Theory, pp. 167–180. IEEE (1973)
Feret, J., Danos, V., Harmer, R., Krivine, J., Fontana, W.: Internal coarse-graining of molecular systems. PNAS 106(16), 6453–6458 (2009)
Hayman, J., Heindel, T.: Pattern graphs and rule-based models: The semantics of Kappa. In: Pfenning, F. (ed.) FOSSACS 2013 (ETAPS 2013). LNCS, vol. 7794, pp. 1–16. Springer, Heidelberg (2013)
Heckel, R.: DPO Transformation with Open Maps. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) ICGT 2012. LNCS, vol. 7562, pp. 203–217. Springer, Heidelberg (2012)
Joyal, A., Nielsen, M., Winskel, G.: Bisimulation from open maps. Inf. Comput. 127(2), 164–185 (1996)
Lack, S., Sobociński, P.: Adhesive Categories. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 273–288. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Danos, V., Heckel, R., Sobocinski, P. (2014). Transformation and Refinement of Rigid Structures. In: Giese, H., König, B. (eds) Graph Transformation. ICGT 2014. Lecture Notes in Computer Science, vol 8571. Springer, Cham. https://doi.org/10.1007/978-3-319-09108-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-09108-2_10
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09107-5
Online ISBN: 978-3-319-09108-2
eBook Packages: Computer ScienceComputer Science (R0)