Research Papers

Spectral Characterization of Hierarchical Modularity in Product Architectures1

[+] Author and Article Information
Somwrita Sarkar

Design Lab, Faculty of Architecture,
Design, and Planning,
University of Sydney,
Sydney NSW 2006, Australia
e-mail: somwrita.sarkar@sydney.edu.au

Andy Dong

Faculty of Engineering
and Information Technologies,
University of Sydney,
Sydney NSW 2006, Australia
e-mail: andy.dong@sydney.edu.au

James A. Henderson

Complex Systems Group, School of Physics,
University of Sydney,
Sydney NSW 2006, Australia
e-mail: henderso@physics.usyd.edu.au

P. A. Robinson

Complex Systems Group, School of Physics,
University of Sydney,
Sydney NSW 2006, Australia
Brain Dynamics Centre, Sydney Medical School,
University of Sydney,
Westmead NSW 2145, Australia
e-mail: robinson@physics.usyd.edu.au

A version of this paper appeared in the Proceedings of the 2011 International Design Engineering Technical Conference & Computers in Engineering Conference, 23rd International Conference on Design Theory and Methodology.

2Corresponding author.

Contributed by the Design Theory and Methodology Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 6, 2013; final manuscript received August 15, 2013; published online November 7, 2013. Assoc. Editor: Jonathan Cagan.

J. Mech. Des 136(1), 011006 (Nov 07, 2013) (12 pages) Paper No: MD-13-1111; doi: 10.1115/1.4025490 History: Received March 06, 2013; Revised August 15, 2013

Despite the importance of the architectural modularity of products and systems, existing modularity metrics or algorithms do not account for overlapping and hierarchically embedded modules. This paper presents a graph theoretic spectral approach to characterize the degree of modular hierarchical-overlapping organization in the architecture of products and complex engineered systems. It is shown that the eigenvalues of the adjacency matrix of a product architecture graph can reveal layers of hidden modular or hierarchical modular organization that are not immediately visible in the predefined architectural description. We use the approach to analyze and discuss several design, management, and system resilience implications for complex engineered systems.

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Grahic Jump Location
Fig. 1

Network models: (a) 64 node random graph with p = 0.29 and average degree = 18; (b) 64 node regular graph with degree = 18; (c) typical 128 node Newman-Girvan modular network [24]; (d) finding the 4 modules in the network in (c); (e) typical hierarchical modular graph from Ref, [26] with 3 hierarchical levels

Grahic Jump Location
Fig. 2

Spectra of random and regular models: eigenvalues arranged in descending order for (a) the random graph in Figs 1(a) and 1(b) the regular graph in Fig. 1(b)

Grahic Jump Location
Fig. 3

Spectra of modular and hierarchical modular networks: (a) A 64 node modular network with 4 modules (black) compared to 64 node random network, p = 0.1 (grey). (b) A 1024 node network with nested modules at 5 hierarchical levels with only the first 100 eigenvalues shown. Lines are for visual presentation only.

Grahic Jump Location
Fig. 4

Spectra of networks with unequal modules: (a) Inset: Perfect modular network with 5 modules of sizes 16, 32, 16, 32, 64; Spectrum shows 1 eigenvalue of 64, 2 eigenvalues of 32, and 2 eigenvalues of 16. (b) Left inset: Unperturbed random block modular network, 5 modules with connection probability p = 0.6, and sizes 16, 32, 16, 32, 64; Spectrum (crosses) shows the same eigenvalue pattern with 1 large eigevalue, followed by 2 large eigenvalues, followed by 2 large eigenvalues, but the eigenvalues scale as pN; Right inset: Perturbed random block modular network, with intermodule connectivity pq = 0.1 and same module sizes as left inset; Spectrum (circles) continues to echo the same pattern, though as the intermodule connectivity increases, the eigenvalues move away from the means, showing larger randomness in structure; (c) Inset: Hierarchical modular network of 3 levels, with unequal module sizes: at the coarsest level is a 256 node network, at the next level three 64 node subnetworks further divide into 32 node finest level networks, and a 64 node subnetwork divides into four 16 node finest level subnetworks; There are 3 hierarchical levels, and the finest level shows 10 modules; Spectrum fingerprints this hierarchical modular organization: there are 10 large eigenvalues, and 3 gaps signifying 3 hierarchical levels.

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Fig. 5

Performing module identification for hierarchical networks using the spectral algorithm. Reordered cosine matrices shown for hierarchical network of Fig. 4(c), with (a) k = 4, (b) k = 7, and (c) k = 10. k values correspond to the largest gaps in the spectrum, and at each k value the hierarchical organization of nodes at the three hierarchical levels is shown.

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Fig. 6

Pratt Whitney Aircraft Engine: (a) Spectrum of eigenvalues of aeroengine model compared with spectrum of a 3-level hierarchical network. Inset shows the original adjacency matrix. (b) Original definition of clusters, with 8 predefined subsystems; note that distinct clusters share high density of links and this fails to bring out the latent natural clustering. (c) clusters plotted using 2D reduced vector representation showing 2 main clusters identified by algorithm at highest hierarchical level, using Eq. (9) and k = 2; data plotted without EC and MC systems that are known to be integrative. (d) 3D reduced vector representation showing 3 main clusters at next hierarchical level, using Eq. (9), and k = 3; data plotted with the EC and MC systems that are densely linked to both the Fan-LPC-HPC cluster and the CC-LPT-HPT cluster.

Grahic Jump Location
Fig. 7

Eigenvector and degree centrality of aeroengine components




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