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Research Papers

A Scheme for Numerical Representation of Graph Structures in Engineering Design

[+] Author and Article Information
David F. Wyatt

e-mail: dw274@cam.ac.uk

David C. Wynn

e-mail: dcw24@cam.ac.uk

P. John Clarkson

e-mail: pjc10@cam.ac.uk
Engineering Design Centre,
University of Cambridge
Department of Engineering,
Trumpington Street,
Cambridge CB2 1PZ, UK

1Corresponding author.

Contributed by the Design Theory and Methodology Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 21, 2012; final manuscript received October 13, 2013; published online November 26, 2013. Assoc. Editor: Irem Y. Tumer.

J. Mech. Des 136(1), 011010 (Nov 26, 2013) (13 pages) Paper No: MD-12-1619; doi: 10.1115/1.4025961 History: Received December 21, 2012; Revised October 13, 2013

Graph structures are fundamental in many aspects of design. This paper discusses a way to improve access to design spaces of graph structures, by converting graph structures into numerical values and vice versa. Mathematical properties of such conversions are described, and those that are desirable are identified. A candidate conversion algorithm, Indexed Stacked Blocks, is proposed. Its use and benefits are illustrated through an example graph-structure design problem. The example demonstrates that such conversions allow design spaces of graph structures to be visualized, sampled, and evaluated. In principle, they also allow other powerful numerical techniques to be applied to the design of graph-structure-based systems.

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Figures

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Fig. 2

An outline of the Indexed Stacked Blocks graph numerical representation scheme. The number line is divided into regions corresponding to the number of nodes in the model. Each region is divided into blocks corresponding to different node label sets. Finally, the edge pattern within a model determines its index number within the relevant block

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Fig. 3

The graph structure used to illustrate the model number calculations

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Fig. 4

The geometric analogy for the sequence number for an alphabet with three symbols (a) In the three-dimensional space whose axes are the ai, all the node label sets containing n nodes lie on a two-dimensional plane whose distance from the origin increases with increasing n (b) A face-on view of the n = 4 plane showing the order in which the sequence number progresses through the points in the plane

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Fig. 5

An example of the calculation of the index number for the example graph shown in Figure 3, by reading out the adjacency matrix as a binary string after ordering the nodes according to their labels

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Fig. 6

A diagrammatic representation of the two options for dealing with the node ordering issue (a) accept all orderings: a single graph structure is mapped to three numerical values (dark grey cells), but all numerical values map to a graph structure (all cells are fully shaded light or dark) (b) choose a specific ordering: a single graph structure is only mapped to one numerical value (dark grey cell), but some numerical values do not map to any graph structures (cells containing crosses)

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Fig. 7

Graph structures used in the illustrative examples (a) The graph structure used for the first illustrative example, a repeat of that shown in Figure 3 (b) The graph structure used for the second illustrative example

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Fig. 8

The graph structure generated from the numerical value 999999 in the graph language used in Fig. 7

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Fig. 9

The search tree explored by the synthesis algorithm, with a modified ISB calculation method that ignores the number of leaves when calculating the sequence number. Search paths leading to valid architectures (those that satisfy all constraints) are shown in dark red, while paths explored that lead to invalid architectures are shown in light grey

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Fig. 10

A plot of average graph distance between nodes against estimated average geometric distance from leaves to hubs, for an exponentially-spaced sample of 500 networks with up to 20 hubs and a 5000-network targeted sample around the Pareto-optimal 10-hub network

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