Multidimensional Multilevel Networks in the Science of the Design of Complex Systems

Assembling parts under relations to form wholes is the Fundamental Construct of Multilevel Systems. Intermediate assemblies themselves become assemblies, defining multilevel structure. If the set of parts exists at one level, the set structured by an assembly relation exists at a higher level. The structured set with its assembly relation defines a multidimensional simplex at this higher level. A simplex is a multidimensional generalisation of a link in a network: a relation between three things being a triangle (two-dimensional simplex), a relation between four things being a tetrahedron (three-dimensional simplex), and so on. Sets of simplices have a multidimensional connectivity that can be analysed by Q-analysis, and a more refined method called star-hub analysis. Star-hub systems also have a Galois lattice structure. Connectivity between simplices acts as a kind of relatively static backcloth for more dynamic patterns of numbers called the system traffic. Relationships between numerical mappings constitute the Order-I dynamics of the system, while changes in the backcloth constitute Order-II dynamics in the system. Multidimensional connectivity constrains the horizontal intra-level Order-I dynamics and the Order-I inter-level dynamics. Order-II dynamics concern the building of structure and the annihilation of structure, and are discrete and non-linear. A theory of design is presented using this multidimensional multilevel network theory. Designers build structures in bottom-up and top-down fashion. Top-down involves hypothesising sets of parts and relationship to aggregate into higher level abstract constructs. Bottom-up involves assembling real things into realisable structures under explicit relations. As top-down meets bottom-up, abstractions are instantiated with tangibilities, and eventually the whole design becomes grounded in tangible things. This is the blueprint stage at which the design can be fabricated. To achieve the blueprint it is necessary to follow a dynamic creative process, the design process, which is sensitive to initial conditions, computationally irreducible, path-dependent and characterised by emergence and coevolution between the designed system and the requirements and specification of that system. This is a science of the artificial: if the designer creates a system that did not exist before, they are the first person to accumulate and synthesise knowledge about that system. Thus the designer acts as a scientist, by building the representation of the system, making hypotheses about the system within the language being constructed, performing experiments on the system, and synthesising this into a theory of the system and its dynamics. Many scientists interested in complex artificial systems are motivated by the possibility of using that scientific knowledge to manipulate the system, either by designing new systems, or modifying and managing the behaviour of existing systems. Thus not only are the designers of artificial systems scientists, but the scientists of artificial systems are designers. During the meetings of the Embracing Complexity in Design cluster it has become clear that designers across the disciplines share a culture based on the creation of new systems and the management of existing systems. In particular the design process is common to all design domains, from graphic design through architecture through software to engineering design. This culture informs the particular design process, supporting creativity and divergence, and leading to convergence and delivery of results. It is suggested that scientists of the artificial would benefit from accepting that they are acting as designers, and that complexity science has much to learn from the design community.