# Methodology for Simulation and Analysis of Complex Adaptive Supply Network Structure and Dynamics Using Information Theory

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## Abstract

**:**

## 1. Introduction

#### 1.1. Complex Adaptive Supply Networks

#### 1.2. Information Dynamics

_{x}, α

_{y}, and α

_{z}. Lizier [15] describes mutual information as a measure of the information contained in X about Y (or vice versa) and the conditional mutual information as the mutual information between X and Y when Z is known.

_{n}

^{(l)}, provides about the next state of a destination (or target) process, X

_{(n+1)}, in the context of the destination’s past, X

_{n}

^{(k)}. In this equation k and l are the history lengths of X and Y, respectively, while n is the current time index. Other useful definitions of transfer entropy describe it as a measure of deviation from independence [17] or as an observed correlation between two processes rather than a direct effect [18].

_{n}

^{(k)}and the past realizations y

_{n}

^{(k)}to quantify the information contained in the source, Y, about the next state of the destination, X, at time step n+1. The local transfer entropy measure gives insight into network structure dynamics and how the correlation and influence or two processes are changing over that time series. This is because local transfer entropy provides a time history of transfer entropy values which quantifies the observed correlation (or influence) between two processes.

#### 1.3. Transfer Entropy Application

## 2. Materials and Methods

- Create conceptual network graph
- Simulate production data on network
- Apply transfer entropy to production data for static analysis
- Apply local information transfer to production data for dynamic analysis

#### 2.1. Conceptual Static Supply Network Simulation

#### Static Network, Structure Validation

#### 2.2. Conceptual Dynamic Supply Network Simulation

#### 2.2.1. Dynamic Network, Static Structure Validation

#### 2.2.2. Dynamic Network, Dynamic Structure Validation

## 3. Results

#### 3.1. Real-World, Dynamic Supply Network

#### Real-World Network, Static Structure

#### 3.2. Real-World Network, Dynamic Structure

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Choi, T.Y.; Dooley, K.J.; Rungtusanatham, M. Supply networks and complex adaptive systems: Control versus emergence. J. Oper. Manag.
**2001**, 19, 351–366. [Google Scholar] [CrossRef] - Surana, A.; Kumara, S.; Greaves, M.; Raghavan, U.N. Supply-chain networks: A complex adaptive systems perspective. Int. J. Prod. Res.
**2005**, 43, 4235–4265. [Google Scholar] [CrossRef] - Pathak, S.D.; Day, J.M.; Nair, A.; Sawaya, W.J.; Kristal, M.M. Complexity and adaptivity in supply networks: Building supply network theory using a complex adaptive systems perspective. Decis. Sci.
**2007**, 38, 547–580. [Google Scholar] [CrossRef] - Mari, S.I.; Lee, Y.H.; Memon, M.S.; Park, Y.S.; Kim, M. Adaptivity of complex network topologies for designing resilient supply chain networks. Int. J. Ind. Eng.
**2015**, 22, 102–116. [Google Scholar] - Li, G.; Yang, H.; Sun, L.; Ji, P.; Feng, L. The evolutionary complexity of complex adaptive supply networks: A simulation and case study. Int. J. Prod. Econ.
**2010**, 124, 310–330. [Google Scholar] [CrossRef] - Giannoccaro, I. Adaptive supply chains in industrial districts: A complexity science approach focused on learning. Int. J. Prod. Econ.
**2015**, 170, 576–589. [Google Scholar] [CrossRef] - Giannoccaro, I. Assessing the influence of the organization in supply chain management using NK simulation. Int. J. Prod. Econ.
**2011**, 131, 263–272. [Google Scholar] - Capaldo, A.; Giannoccaro, I. Interdependence and network-level trust in supply chain networks: A computational study. Ind. Mark. Manag.
**2015**, 44, 180–195. [Google Scholar] [CrossRef] - Capaldo, A.; Giannoccaro, I. How does trust affect performance in the supply chain? The moderating role of interdependence. Int. J. Prod. Econ.
**2015**, 166, 36–49. [Google Scholar] [CrossRef] - Hearnshaw, E.J.; Wilson, M.M. A complex network approach to supply chain network theory. Int. J. Oper. Prod. Manag.
**2013**, 33, 442–469. [Google Scholar] [CrossRef] - Bellamy, M.A.; Basole, R.C. Network analysis of supply chain systems: A systematic review and future research. Syst. Eng.
**2013**, 16, 235–249. [Google Scholar] [CrossRef] - Basole, R.C.; Bellamy, M.A. Supply network structure, visibility, and risk diffusion: A computational approach. Decis. Sci.
**2014**, 45, 753–789. [Google Scholar] [CrossRef] - Bellamy, M.A.; Ghosh, S.; Hora, M. The influence of supply network structure on firm innovation. J. Oper. Manag.
**2014**, 32, 357–373. [Google Scholar] [CrossRef] - Shannon, C.E.; Weaver, W. The Mathematical Theory of Communication; University of Illinois Press: Urbana, IL, USA, 1949; p. 1. [Google Scholar]
- Lizier, J.T. JIDT: An information-theoretic toolkit for studying the dynamics of complex systems. 2014; arXiv: 1408.3270. [Google Scholar]
- Schreiber, T. Measuring information transfer. Phys. Rev. Lett.
**2000**, 85, 461. [Google Scholar] [CrossRef] [PubMed] - Lizier, J.T.; Prokopenko, M.; Zomaya, A.Y. A framework for the local information dynamics of distributed computation in complex systems. In Guided Self-Organization: Inception; Springer: Berlin, Germany, 2014; Volume 9, pp. 115–158. [Google Scholar]
- Lizier, J.T.; Prokopenko, M. Differentiating information transfer and causal effect. Eur. Phys. J. B
**2010**, 73, 605–615. [Google Scholar] [CrossRef] - Gomez, C.; Lizier, J.T.; Schaum, M.; Wollstadt, P.; Grutzner, C.; Uhlhaas, P.; Freitag, C.M. Reduced predictable information in brain signals in autism spectrum disorder. Front. Neuroinform.
**2014**, 8. [Google Scholar] [CrossRef] [PubMed] - Lungarella, M.; Sporns, O. Mapping information flow in sensorimotor networks. PLoS Comput. Boil.
**2006**, 2, e144. [Google Scholar] [CrossRef] [PubMed] - Vicente, R.; Wibral, M.; Lindner, M.; Pipa, G. Transfer entropy—A model-free measure of effective connectivity for the neurosciences. J. Comput. Neurosci.
**2011**, 30, 45–67. [Google Scholar] [CrossRef] [PubMed] - Dimitrov, A.G.; Lazar, A.A.; Victor, J.D. Information theory in neuroscience. J. Comput. Neurosci.
**2011**, 30, 1–5. [Google Scholar] [CrossRef] [PubMed] - Ver Steeg, G.; Galstyan, A. Information transfer in social media. In Proceedings of the 21st International Conference on World Wide Web, New York, NY, USA, 16–20 April 2012; pp. 509–518.
- Ver Steeg, G.; Galstyan, A. Information-theoretic measures of influence based on content dynamics. In Proceedings of the Sixth ACM International Conference on Web Search and Data Mining, New York, NY, USA, 4–8 February 2013; pp. 3–12.
- Rosenblum, M.G.; Cimponeriu, L.; Bezerianos, A.; Patzak, A.; Mrowka, R. Identification of coupling direction: Application to cardiorespiratory interaction. Phys. Rev. E
**2002**, 65, 041909. [Google Scholar] [CrossRef] [PubMed] - Marschinski, R.; Kantz, H. Analysing the information flow between financial time series. Eur. Phys. J. B
**2002**, 30, 275–281. [Google Scholar] [CrossRef] - Knuth, K.H.; Gencaga, D.; Rossow, W.B. Information-theoretic methods for identifying relationships among climate variables. 2014; arXiv: 1412.6219. [Google Scholar]
- Runge, J.; Petoukhov, V.; Kurths, J. Quantifying the strength and delay of climatic interactions: The ambiguities of cross correlation and a novel measure based on graphical models. J. Clim.
**2014**, 27, 720–739. [Google Scholar] [CrossRef] - Wang, F.; Chen, W.; Wu, F.; Zhao, Y.; Hong, H.; Gu, T.; Bao, H. A visual reasoning approach for data-driven transport assessment on urban roads. In Proceedings of the 2014 IEEE Conference on Visual Analytics Science and Technology (VAST), Paris, France, 25–31 October 2014.
- Doyle, L.R.; McCowan, B.; Hanser, S.F.; Chyba, C.; Bucci, T.; Blue, J.E. Applicability of information theory to the quantification of responses to anthropogenic noise by southeast Alaskan humpback whales. Entropy
**2008**, 10, 33–46. [Google Scholar] [CrossRef] - Doyle, L.R. Quantification of information in a one-way plant-to-animal communication system. Entropy
**2009**, 11, 431–442. [Google Scholar] [CrossRef] - Prokopenko, M.; Boschetti, F.; Ryan, A.J. An information-theoretic primer on complexity, self-organization, and emergence. Complexity
**2009**, 15, 11–28. [Google Scholar] [CrossRef] - Feistel, R.; Ebeling, W. Entropy and the self-organization of information and value. Entropy
**2016**, 18, 193. [Google Scholar] [CrossRef] - Miller, J.M.; Wang, X.R.; Lizier, J.T.; Prokopenko, M.; Rossi, L.F. Measuring information dynamics in swarms. In Guided Self-Organization: Inception; Prokopenko, M., Ed.; Springer: Berlin, Germany, 2014; pp. 343–364. [Google Scholar]
- Nicolis, G.; Nicolis, C. Stochastic resonance, self-organization and information dynamics in multistable systems. Entropy
**2016**, 18, 172. [Google Scholar] [CrossRef] - Rosas, F.; Ntranos, V.; Ellison, C.J.; Pollin, S.; Verhelst, M. Understanding interdependency through complex information sharing. Entropy
**2016**, 18, 38. [Google Scholar] [CrossRef] - Lizier, J.T. The Local Information Dynamics of Distributed Computation in Complex Systems. Ph.D. Thesis, The University of Sydney, New South Wales, Australia, 2010. [Google Scholar]
- Rodewald, J.; Colombi, J.; Oyama, K.; Johnson, A. Using information-theoretic principles to analyze and evaluate complex adaptive supply network architectures. Procedia Comput. Sci.
**2015**, 61, 147–152. [Google Scholar] [CrossRef] - Van der Aalst, W.; Weijters, T.; Maruster, L. Workflow mining: Discovering process models from event logs. IEEE Trans. Knowl. Data Eng.
**2004**, 16, 1128–1142. [Google Scholar] [CrossRef] - Van der Aalst, W.M.; van Dongen, B.F. Discovering workflow performance models from timed logs. In Engineering and Deployment of Cooperative Information Systems; Han, Y., Tai, S., Wikarski, D., Eds.; Springer: Berlin, Germany, 2002; pp. 45–63. [Google Scholar]

**Figure 6.**Local transfer entropy moving averages (15 time units) for significant links in dynamic supply network.

**Figure 9.**Local transfer entropy moving averages (four time units) for all potential node E suppliers in real-world supply network.

**Table 1.**Transfer entropy values as edge weights in adjacency matrix for static supply network (significant values in bold).

Node | A | B | C | D | E | F | G | H | I | J |
---|---|---|---|---|---|---|---|---|---|---|

A | 0 | 0.000 | 0.000 | 0.000 | 0.887 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

B | 0.000 | 0 | 0.000 | 0.000 | 0.000 | 0.882 | 0.000 | 0.000 | 0.000 | 0.000 |

C | 0.000 | 0.000 | 0 | 0.000 | 0.000 | 0.000 | 0.371 | 0.000 | 0.000 | 0.000 |

D | 0.000 | 0.000 | 0.000 | 0 | 0.000 | 0.001 | 0.000 | 0.390 | 0.000 | 0.000 |

E | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 | 0.386 | 0.001 | 0.000 | 0.000 |

F | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 | 0.377 | 0.000 | 0.000 |

G | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | 0.000 | 0 | 0.000 | 0.999 | 0.000 |

H | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.000 | 0.304 |

I | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 | 0.302 |

J | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0 |

**Table 2.**p-values for corresponding edges in adjacency matrix for static supply network (significant values in bold).

Node | A | B | C | D | E | F | G | H | I | J |
---|---|---|---|---|---|---|---|---|---|---|

A | 1 | 0.375 | 0.612 | 0.989 | 0.000 | 0.455 | 0.129 | 0.287 | 0.252 | 0.780 |

B | 0.556 | 1 | 0.763 | 0.978 | 0.736 | 0.000 | 0.868 | 0.317 | 0.280 | 0.046 |

C | 0.315 | 0.937 | 1 | 0.074 | 0.016 | 0.204 | 0.000 | 0.643 | 0.686 | 0.005 |

D | 0.688 | 0.860 | 0.814 | 1 | 0.367 | 0.566 | 0.807 | 0.000 | 0.202 | 0.143 |

E | 0.197 | 0.598 | 0.122 | 0.612 | 1 | 0.369 | 0.000 | 0.456 | 0.135 | 0.584 |

F | 0.407 | 0.441 | 0.002 | 0.917 | 0.537 | 1 | 0.145 | 0.000 | 0.822 | 0.944 |

G | 0.645 | 0.635 | 0.386 | 0.612 | 0.815 | 0.203 | 1 | 0.001 | 0.000 | 0.034 |

H | 0.646 | 0.164 | 0.387 | 0.055 | 0.437 | 0.921 | 0.775 | 1 | 0.058 | 0.000 |

I | 0.399 | 0.252 | 0.059 | 0.361 | 0.627 | 0.696 | 0.900 | 0.410 | 1 | 0.000 |

J | 0.609 | 0.199 | 0.870 | 0.113 | 0.670 | 0.046 | 0.298 | 0.273 | 0.816 | 1 |

**Table 3.**Transfer entropy values as edge weights in adjacency matrix for dynamic supply network (significant values in bold).

Node | A | B | C | D | E |
---|---|---|---|---|---|

A | 0 | 0.002 | 0.005 | 0.072 | 0.005 |

B | 0.003 | 0 | 0.003 | 0.051 | 0.001 |

C | 0.005 | 0.004 | 0 | 0.020 | 0.002 |

D | 0.002 | 0.009 | 0.001 | 0 | 0.743 |

E | 0.002 | 0.008 | 0.006 | 0.008 | 0 |

**Table 4.**p-values for corresponding edges in adjacency matrix for dynamic supply network (significant values in bold).

Node | A | B | C | D | E |
---|---|---|---|---|---|

A | 1 | 0.431 | 0.155 | 0.000 | 0.159 |

B | 0.254 | 1 | 0.358 | 0.000 | 0.793 |

C | 0.160 | 0.245 | 1 | 0.000 | 0.400 |

D | 0.467 | 0.026 | 0.639 | 1 | 0.000 |

E | 0.538 | 0.034 | 0.076 | 0.045 | 1 |

**Table 5.**Transfer entropy values as edge weights in adjacency matrix for real-world supply network (significant values in bold).

Node | A | B | C | D | E |
---|---|---|---|---|---|

A | 0 | 0.052 | 0.033 | 0.018 | 0.024 |

B | 0.026 | 0 | 0.008 | 0.025 | 0.006 |

C | 0.017 | 0.034 | 0 | 0.022 | 0.024 |

D | 0.022 | 0.003 | 0.086 | 0 | 0.011 |

E | 0.011 | 0.002 | 0.026 | 0.028 | 0 |

**Table 6.**p-values for corresponding edges in adjacency matrix for real-world supply network (significant values in bold).

Node | A | B | C | D | E |
---|---|---|---|---|---|

A | 1 | 0.014 | 0.067 | 0.220 | 0.141 |

B | 0.118 | 1 | 0.518 | 0.122 | 0.587 |

C | 0.234 | 0.061 | 1 | 0.157 | 0.136 |

D | 0.157 | 0.790 | 0.001 | 1 | 0.409 |

E | 0.412 | 0.867 | 0.114 | 0.101 | 1 |

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**MDPI and ACS Style**

Rodewald, J.; Colombi, J.; Oyama, K.; Johnson, A. Methodology for Simulation and Analysis of Complex Adaptive Supply Network Structure and Dynamics Using Information Theory. *Entropy* **2016**, *18*, 367.
https://doi.org/10.3390/e18100367

**AMA Style**

Rodewald J, Colombi J, Oyama K, Johnson A. Methodology for Simulation and Analysis of Complex Adaptive Supply Network Structure and Dynamics Using Information Theory. *Entropy*. 2016; 18(10):367.
https://doi.org/10.3390/e18100367

**Chicago/Turabian Style**

Rodewald, Joshua, John Colombi, Kyle Oyama, and Alan Johnson. 2016. "Methodology for Simulation and Analysis of Complex Adaptive Supply Network Structure and Dynamics Using Information Theory" *Entropy* 18, no. 10: 367.
https://doi.org/10.3390/e18100367