Spatial Embedding and Complexity: The Small-World is Not Enough

The 'order for free' exhibited by some classes of system has been exploited by natural selection in order to build systems capable of exhibiting complex behaviour. Here we explore the impact of one ordering constraint, spatial embedding, on the dynamical complexity of networks. We apply a measure of functional complexity derived from information theory to a set of spatially embedded network models in order to make some preliminary characterisations of the contribution of space to the dynamics (rather than mere structure) of complex systems. Although our measure of dynamical complexity hinges on a balance between functional integration and segregation, which seem related to an understanding of the small-world property, we demonstrate that smallworld structures alone are not enough to induce complexity. However, purely spatial constraints can produce systems of high intrinsic complexity by introducing multiple scales of organisation within a network.