Propagation algorithms on graphs for physical applications

The purpose of this paper is to introduce newpropagation algorithms on graphs in order to predict some macroscopic properties of heterogeneous media from information on their microstructure modeled by graphs. Among the physical properties of interest, the following were studied in our approach: fracture toughness, sound propagation, and diffusion in composites. The idea is to replace a continuous medium by a discrete one modeled by a graph. After giving the edges of this graph an appropriate value (according to the studied physical phenomenon), we calculate for each vertex its distance to a source of propagation. Depending on the cases, the efficient computation of suchdistance functions is realized by three different types of algorithms: the first is based on a simple breadth-first scanning of the vertices starting from the source, this being implemented via a queue of pixels. The second algorithm actually simulates the propagation of a wave inside the medium under study, this wave being modeled by a chain. Lastly, asequential algorithm is introduced, which allows the computation of distance transformation on very general graphs, provided their structure fulfills some regularity conditions. Various types of informations are deduced from the distance functions: length and location of the shortest paths, distribution of distances, etc. They are related to the physical properties of the materials considered. The interest of these propagation algorithms is illustrated by the determination of the coefficient of diffusion of polymeric thermoplastic alloys. s.