Analogies Between Complex Systems and Phases of Matter

The behavior of a complex system in a changing environment is strongly affected by the system's architecture. We present an analogy between the major phases of matter (solid, liquid, gas) and three major generic architectures of complex systems: tree structures, layered structures and grid networks. This analogy is realized using a graph-based formalism, with nodes and edges in a given configuration. Solid materials are akin to tree structures, especially when we consider that most solids actually have cracks. Solids with cracks between their components can be modeled by nodes (representing each component) and their interconnection, leading to a tree structured hierarchy. Gases made up of molecules can be modeled by nodes (the molecules) with local interconnections representing nearby molecules in space, thus forming a grid network. Liquids can form layers as in a mixture of oil and water. We represent this by connections that are densely horizontal within layers as well as sparsely vertical between layers. A key issue for complex systems is the ease by which they may be changed, which we call the system’s flexibility. Our definition of flexibility indicates that tree structures, like solids, are relatively inflexible and that grid networks, like gases, are extremely flexible, possibly leading to loss of control and chaotic behavior. Like liquids, layered systems are intermediate in flexibility and controllability. Solids, even with cracks, are relatively difficult to modify, whereas gases change internal form so quickly that they can only be constrained; not controlled. Liquids are intermediate in their ability to change form internally. Just as heating solids can lead to liquids, and heating liquids can result in gases, we shall present transformations in the interconnection structure of systems, analogous to heating, that change tree structures into layered ones and layered structures into networks.