Abstract

Networks with dynamically changing topology can modelled using rule-based graph transformations. If we are interested in the relative timing of different kinds of change, stochastic models are required, adding rate constants to rules controlling their delay once a match has become available. A particularly case are adaptive networks where structural network changes and updates of local states are interdependent, especially if states carry data over numerical or other infinite domains.

Stochastic graph transformation models for complex networks 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 talk: 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 certain structures; 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 on an example of a social network with dynamic topology and discuss applications to the energy-driven approach to rewriting, progressing from the most basic case where data states are Boolean to numerical data states, both emerging as instances of the categorical framework.