Distributed Graphs and Graph Transformation

Abstract

The new approach of distributed graphs and graph transformation developed in this article allows us to use structured graph transformation on two abstraction levels: the network and the local level. The network level contains the description of topological structures of a system. The local level covers the description of states and their transitions in local systems. Local state transitions may depend on others using suitable synchronization mechanisms. The main distribution concepts of categorical graph grammars presented by Schneider are combined with the algebraic approach to distributed graph transformation introduced by Ehrig et al. Modeling of distributed systems by this new kind of distributed graph transformation offers a clear and elegant description of dynamic networks, distributed actions as well as communication and synchronization using a graphical notation. Moreover, distributed graph transformation offers the possibility to describe splitting and joining of local graphs as well as parallel transformations in local systems. The formalization of distributed graph transformation is performed by means of category theory. A distributed graph is formalized by a diagram in the category GRAPH of graphs and total graph morphisms. A distributed transformation step is characterized by a double-pushout in the category DISTR(GRAPH) of distributed graphs and distributed graph morphisms. A pushout over distributed graph morphisms cannot always be constructed componentwise in each local part. But especially the componentwise construction of a distributed transformation reflects distributed computations best. Thus, we present the necessary conditions for componentwise construction of pushouts in DISTR(GRAPH) and show that they are also sufficient. These conditions are summarized in the distributed gluing condition. Moreover, these conditions are needed for componentwise construction of pushout complements which are used to characterize the first step of a distributed graph transformation.