| Complex systems have adaptive, self organizing components and emergent behavior. Furthermore, the elements of such systems empirically show local, heterogeneous structures rather than global, homogeneous ones. This makes them much more difficult to study, predict or engineer compared to other systems, for which traditional tools of modeling and engineering were developed. To be able to engineer complex systems, we need to develop methods to describe the structures underlying such systems with coarse grained models that we can understand and use to make predictons. In other words, we need to build maps of the heterogeneous structures in complex systems. Here we present a methodology for achieving this goal. For the subset of emergent systems that can be represented by complex networks, we use network positionality to group statistically equivalent units together. This allows us to forge block models that describe the heterogeneity. If the system does not directly have a network representation, it is possible to create mappings between other representations and complex networks. We present a stochastic mapping between time series and networks to demonstrate this possibility. Finally, for complex systems that do not allow themselves to be portrayed as networks, we show that mathematical models can capture the heterogeneous biases in the interactions among components of the system. Armed with such maps of the underlying heterogenous structures, we can get one step closer to engineering complex systems and controlling their behavior. |
