International Journal of Bifurcation and Chaos, Vol. 19, No. 2 (2009) 695–702 c  World Scientific Publishing Company TOPOLOGICAL PROPERTIES OF THE CONTACT NETWORK OF GRANULAR MATERIALS ROBERTO AR ´ EVALO, IKER ZURIGUEL and DIEGO MAZA Departamento de F´ ısica, Facultad de Ciencias, Universidad de Navarra, E-31080 Pamplona, Spain Received March 14, 2008; Revised May 14, 2008 The force networks of different granular ensembles are defined and their topological properties studied using the tools of complex networks. In particular, for each set of grains compressed in a square box, a force threshold is introduced that determines which contacts conform the network. Hence, the topological characteristics of the network are analyzed as a function of this parameter. The characterization of the structural features thus obtained, may be useful in the understanding of the macroscopic physical behavior exhibited by this class of media. Keywords : Granular materials; force network; random graph. 1. Introduction Granular materials are being widely studied by the physics community since they exhibit unusual and distinctive properties [Duran, 1999]. These mate- rials are composed of macroscopic particles that interact by a dissipative contact force and can be thought of as displaying gas, liquid and solid phases. A suitable model for the study of granular materials is to consider each grain as a hard sphere, ignoring fragmentation and moving the effect of deformation to the dissipative term. As pointed in [Anikeenko et al., 2007], the wide applicability of these model to the study of liquids, glasses and colloids implies a paramount importance of the geometrical prop- erties of the packing of hard spheres in determin- ing the physics exhibited by the materials analyzed. The geometry of granular packing has been inves- tigated i.e. by [Anikeenko et al., 2007] and [Aste et al., 2005] using Voronoi–Delaunay partitioning to identify structures in the former and volume dis- tributions in the latter. In the present work we propose, in the same line of those and other works [Ostojic et al., 2006], a structural study of granular packing but using tools specifically developed in the frame of complex networks. As will be explained later we define for each packing a network of contacts, see Fig. 1, of which the topological properties are studied afterwards. The contact topology of a granular packing can be studied as a graph where particles are nodes and the interacting force pairs edges. This approach has important advantages. On the one hand, it is a quantitative tool as are not other ideas proposed in the granular community, namely, “force chains” [Peters et al., 2005]; it is an abstract point of view that allows to reach very primitive concepts such as connectivity over which to elaborate more complex definitions; and, finally, the field of complex net- works provides us with a great amount of concepts and algorithms among which we can choose the most suitable for our proposes of characterization. The remainder of this work is structured as fol- lows: in Sec. 2 we explain the numerical method used and the protocol followed to obtain the sam- ples that we study. In Sec. 3, the topological properties analyzed are defined and the results are presented for several different conditions. Finally, in Sec. 4 we summarize our results and draw some conclusions. 695