Chapter 7 Conclusion
7.1 Summary
Although much work has been done on modularity, most of it is qualitative and exploratory in nature. The few quantitative studies typically are function-based or design structure matrix-based, and most quantitative measures of modularity are linkage counting methods. This thesis provides a first attempt to quantify modularity from information-theoretic views and provide a more general quantitative measure of modularity.
To quantify modularity, several key characteristics of “modularity” are identified, and several questions about modularity are clarified. Specifically, they are system vs. model, specific meanings of “inter-module” and “intra-module,” relativity of couplings, aggregation of modularity at different levels in a hierarchical structure, application domain of definition, and how to quantify couplings.
The last question is the key to developing quantitative modularity measure.
Discussions in Chapter 2 show that modularity is actually discussed over a model of a system and, therefore, the representation of the model. According to different purposes, different aspects of design processes and products can be modeled with different representations including function structures, design structure matrices (DSM), graph representations, and systems of random vari- ables. The graph representations can be separated into four subclasses according to whether edges of a graph are associated with attributes and whether the attributes are ordered or not. Those different representations are related. Design structure matrices can be mapped to graphs only with ordered attributes, and function structures can be mapped to graphs only with unordered attributes. By viewing adjacency matrices as covariance matrices of gaussian random vectors, a graph only with ordered attributes can be equivalent to a system of gaussian random variables.
The thesis provides two information-theoretic methods, the mutual information-based method and the Minimal Description Length (MDL)-based method to quantify modularity of these different representations. The mutual information-based method views couplings as information flow instead of real physical interactions between systems. The intuition behind this view is that the stronger
the coupling between two systems, the more information about one system can be inferred from the known information of the other system, i.e., the more information flows between them. Information flow can be quantified by mutual information, which is based on randomness and uncertainty. Since most engineering products can be modeled as stochastic systems and therefore have randomness, mutual information-based methods are applied in very general cases, and it is shown that the general existing linkage counting modularity measure is a special case of the mutual information-based modularity measure.
The mutual information-based methods are applied to products which have physical behaviors, but usually this is not the case at the early phase of engineering design, where only function diagrams exist. To exploit the benefits of modularity as early as possible in engineering design processes, an MDL-based modularity measure is proposed for structure modularity of graph structures, which can represent function diagrams. The method views modularity as a kind of regularity, which can be measured by the MDL principle.
The diagram shown in Figure 7.1 summarizes the relations of different modularity quantification methods. Systems of random variables can be quantified by the mutual information-based method;
graphs without attributes, and therefore 0,1-design structure matrices (binary design structure ma- trices), can be quantified by both methods and the demonstration in Chapter 5 shows that the they are equivalent in this case; graphs only with ordered attributes, and therefore function struc- tures, can be quantified by the MDL-based method; and graphs only with ordered attributes, and therefore design structure matrices, can be quantified by the mutual information-based method. Un- fortunately, both methods do not work well for graphs with both unordered and ordered attributes.
To get design structure matrices, it is required to quantify the interactions between different com- ponents or design parameters. Generally the quantifying is done by designers’ subjective knowledge and experience. The mutual information-based method can provide a formal and quantitative way to get design structure matrices.
Both methods have been verified to hierarchically decompose abstract graph structures and a real function structure of an HP printer. Due to the complexity of graph decomposition, genetic algorithms are used, and new genetic operators for tree representation are developed to find the optimal hierarchical decomposition.
The main motivation to quantify modularity is to produce modular engineering designs, especially by evolutionary design. There are many factors affecting the evolution of modular structures, such as genome representation, fitness function, learning, and task structure. The preliminary work in this thesis looks into how the modularity of tasks affect the modularity of products produced by evolutionary computation. Using neural networks as examples, the study shows that the relation between modular tasks and modular topology of feed-forward neural networks is task-dependent and depends on the amount of resources available for the tasks.
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