Resilient Design of Complex Engineered Systems

This paper presents a complex network and graph spectral approach to calculate the resiliency of complex engineered systems. Resiliency is a key driver in how systems are developed to operate in an unexpected operating environment, and how systems change and respond to the environments in which they operate. This paper deduces resiliency properties of complex engineered systems based on graph spectra calculated from their adjacency matrix representations, which describes the physical connections between components in a complex engineered systems. In conjunction with the adjacency matrix, the degree and Laplacian matrices also have eigenvalue and eigenspectrum properties that can be used to calculate the resiliency of the complex engineered system. One such property of the Laplacian matrix is the algebraic connectivity. The algebraic connectivity is defined as the second smallest eigenvalue of the Laplacian matrix and is proven to be directly related to the resiliency of a complex network. Our motivation in the present work is to calculate the algebraic connectivity and other graph spectra properties to predict the resiliency of the system under design.

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Resilient Design of Complex Engineered Systems Against Cascading Failure

Volume 12: Systems and Design ◽

10.1115/imece2013-63308 ◽

2013 ◽

Author(s):

Hoda Mehrpouyan ◽

Brandon Haley ◽

Andy Dong ◽

Irem Y. Tumer ◽

Chris Hoyle

Keyword(s):

Complex Networks ◽

System Design ◽

Electrical Power ◽

Electrical Power System ◽

Complex Engineered Systems ◽

Failure Propagation ◽

Resilient Design ◽

Engineered Systems ◽

Design Characteristics ◽

Energy Signal

This paper describes an approach commonly used with complex networks to study the failure propagation in an engineered system design. The goal of the research is to synthesize and illustrate system design characteristics that results from possible impact of the underlying design methodology based on cascading failures. Further, identifying the most vulnerable component in the design or system design architectures that are resilient to such dissemination of failures provide additional property improvement for resilient design. The paper presents a case study based on the ADAPT (Electrical Power System) EPS testbed at NASA Ames as a subsystem for the Ramp System of an Infantry Fighting Vehicle (IFV). A popular methodology based on the adjacency matrix, which is commonly used to represent edge connections between nodes in complex networks, has inspired interest in the use of similar methods to represent complex engineered systems. This is made possible, by defining the connections between components as a flow of energy, signal, and material and constraining physical connection between compatible components within complex engineered systems. Non-linear dynamical system (NLDS) and epidemic spreading models are used to compare the failure propagation mean time transformation. The results show that coupling, modularity, and module complexity all play an important part in the design of robust large complex engineered systems.

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Spectral Characterization of Hierarchical Modularity in Product Architectures1

Journal of Mechanical Design ◽

10.1115/1.4025490 ◽

2013 ◽

Vol 136 (1) ◽

Cited By ~ 25

Author(s):

Somwrita Sarkar ◽

Andy Dong ◽

James A. Henderson ◽

P. A. Robinson

Keyword(s):

Adjacency Matrix ◽

Product Architecture ◽

Modular Organization ◽

Design Management ◽

Spectral Approach ◽

Graph Theoretic ◽

Complex Engineered Systems ◽

System Resilience ◽

Engineered Systems

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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The normalized distance Laplacian

Special Matrices ◽

10.1515/spma-2020-0114 ◽

2021 ◽

Vol 9 (1) ◽

pp. 1-18

Author(s):

Carolyn Reinhart

Keyword(s):

Spectral Radius ◽

Characteristic Polynomial ◽

Adjacency Matrix ◽

Connected Graph ◽

Distance Matrix ◽

Laplacian Matrix ◽

Diagonal Matrix ◽

The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

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On minimum algebraic connectivity of graphs whose complements are bicyclic

Open Mathematics ◽

10.1515/math-2019-0119 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1490-1502 ◽

Cited By ~ 1

Author(s):

Jia-Bao Liu ◽

Muhammad Javaid ◽

Mohsin Raza ◽

Naeem Saleem

Keyword(s):

Connected Graph ◽

Diffusion Processes ◽

Laplacian Matrix ◽

Group Differences ◽

Algebraic Connectivity ◽

Connected Graphs ◽

Bicyclic Graph ◽

Smallest Eigenvalue ◽

Unique Graph ◽

The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

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On Consensus Index of Triplex Star-Like Networks: A Graph Spectra Approach

Symmetry ◽

10.3390/sym13071248 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1248

Author(s):

Da Huang ◽

Jian Zhu ◽

Zhiyong Yu ◽

Haijun Jiang

Keyword(s):

Algebraic Connectivity ◽

Multi Agent Systems ◽

Graph Spectra ◽

Asymptotic Behaviors ◽

Agent Systems ◽

First Order ◽

Laplacian Spectra ◽

Multi Agent ◽

Consensus Index

In this article, the consensus-related performances of the triplex multi-agent systems with star-related structures, which can be measured by the algebraic connectivity and network coherence, have been studied by the characterization of Laplacian spectra. Some notions of graph operations are introduced to construct several triplex networks with star substructures. The methods of graph spectra are applied to derive the network coherence, and some asymptotic behaviors of the indices have been derived. It is found that the operations of adhering star topologies will make the first-order coherence increase a constant value under the triplex structures as parameters tend to infinity, and the second-order coherence have some equality relations as the node related parameters tend to infinity. Finally, the consensus related indices of the triplex systems with the same number of nodes but non-isomorphic graph structures have been compared and simulated to verify the results.

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Forensic Engineering Use Of Fault Tree Analysis

Journal of the National Academy of Forensic Engineers ◽

10.51501/jotnafe.v23i2.662 ◽

2006 ◽

Vol 23 (2) ◽

Author(s):

Frank H. Johnson ◽

DeWitt William E.

Keyword(s):

Fault Tree ◽

Fault Tree Analysis ◽

Tree Analysis ◽

Complex Engineered Systems ◽

Track Record ◽

Forensic Engineering ◽

Fire And Explosion ◽

Engineered Systems ◽

Analytical Tools

Analytical Tools, Like Fault Tree Analysis, Have A Proven Track Record In The Aviation And Nuclear Industries. A Positive Tree Is Used To Insure That A Complex Engineered System Operates Correctly. A Negative Tree (Or Fault Tree) Is Used To Investigate Failures Of Complex Engineered Systems. Boeings Use Of Fault Tree Analysis To Investigate The Apollo Launch Pad Fire In 1967 Brought National Attention To The Technique. The 2002 Edition Of Nfpa 921, Guide For Fire And Explosion Investigations, Contains A New Chapter Entitled Failure Analysis And Analytical Tools. That Chapter Addresses Fault Tree Analysis With Respect To Fire And Explosion Investigation. This Paper Will Review The Fundamentals Of Fault Tree Analysis, List Recent Peer Reviewed Papers About The Forensic Engineering Use Of Fault Tree Analysis, Present A Relevant Forensic Engineering Case Study, And Conclude With The Results Of A Recent University Study On The Subject.

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Spectral property of certain class of graphs associated with generalized Bethe trees and transitive graphs

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm0802260f ◽

2008 ◽

Vol 2 (2) ◽

pp. 260-275 ◽

Cited By ~ 2

Author(s):

Yi-Zheng Fan ◽

Li Shuang-Dong ◽

Dong Liang

Keyword(s):

Spectral Property ◽

Adjacency Matrix ◽

Rooted Tree ◽

Laplacian Matrix ◽

Transitive Graph ◽

Tridiagonal Matrices ◽

Equal Degree ◽

Transitive Graphs ◽

Structure Properties

A generalized Bethe tree is a rooted tree for which the vertices in each level having equal degree. Let Bk be a generalized Bethe tree of k level, and let T r be a connected transitive graph on r vertices. Then we obtain a graph Bk?T r from r copies of Bk and T r by appending r roots to the vertices of T r respectively. In this paper, we give a simple way to characterize the eigenvalues of the adjacency matrix A(Bk ? T r) and the Laplacian matrix L(Bk?T r) of Bk?T r by means of symmetric tridiagonal matrices of order k. We also present some structure properties of the Perron vectors of A(Bk?T r) and the Fiedler vectors of L(Bk ? T r). In addition, we obtain some results on transitive graphs.

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Bounds of Laplacian Energy of a Hypercube Graph

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.10.21287 ◽

2018 ◽

Vol 7 (4.10) ◽

pp. 582

Author(s):

K. Ameenal Bibi ◽

B. Vijayalakshmi ◽

R. Jothilakshmi

Keyword(s):

Lower Bounds ◽

Adjacency Matrix ◽

Laplacian Matrix ◽

Positive Semidefinite ◽

Laplacian Energy ◽

Eigen Values ◽

Degree Matrix

Let Qn denote the n – dimensional hypercube with order 2n and size n2n-1. The Laplacian L is defined by L = D where D is the degree matrix and A is the adjacency matrix with zero diagonal entries. The Laplacian is a symmetric positive semidefinite. Let µ1 ≥ µ2 ≥ ....µn-1 ≥ µn = 0 be the eigen values of the Laplacian matrix. The Laplacian energy is defined as LE(G) = . In this paper, we defined Laplacian energy of a Hypercube graph and also attained the lower bounds.

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