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**Initial distribution of the shortest path length**

Figure 3.5a plots the distributions of L showing that the FC archetype had signicantly shorter paths than the other two archetypes. This directly resulted from the fact that all node-pairs were directly connected, not requiring rerouting through a hub node. A Kolmogorov-Smirnov test (KS-test) comparing the distributions of L for the SH and DH archetypes rejected the null hypothesis that the distributions were similar with p = 0:0047.

The lower mean and wider spread in both tails of the DH archetype was explained by the structure of GD1 . Intra-hub paths (paths between nodes that shared a common hub node) in the DH archetype were generally shorter than intra-hub paths of the SH archetype.

This was because in the DH archetype the placement of the two hubs on the grid eectively split the grid and therefore the intra-hub nodes were closer to their respective hubs. On the other hand, the inter-hub movements were longer as they had to be rerouted through two hubs. Intra-hub paths accounted for 45% of the network, inter-hub for 38% and the remainder of the paths were links between one hub and a node assigned to the other hub and vice versa. Paths in this last group were not distinctly longer or shorter between the two archetypes. As the majority of the links in the DH archetype had shorter lengths, L was lower.

The diameter of a network is the length of its longest shortest path denoted by max (L). Figure 3.5b plots the distributions of max (L), where once again the FC archetype showed far shorter paths than the others. This time, the KS-test failed to reject the null hypothesis that the distributions of max (L) for DH and SH archetypes were similar with p = 0:29.

The FC archetype was thus most ecient by a large margin, whereas the clear winner between the two hub archetypes was instance-specic, with a slight prejudice in favour of the DH archetype.

**Initial distributions of the shortest path set sizes**

The shortest path set size (Pij) for two directly connected nodes was the combinatorial product of the number of columns and rows of G2 traversed when moving from one node to the other (refer to Figure 3.4). Furthermore, Pij for any indirectly connected node-pair was the combinatorial product of the shortest path set sizes of all the directly connected node-pairs that constituted the logical path in G1K.

The sum of the shortest path set sizes of an instance were denoted by P i;j2Sij ;i6=jPij and P i;j2Sij ;i6=j Pij , when considering only the direct connections (SDij) and full network (Sij), respectively. Figure 3.6a shows the box plots of the distributions of the sums over the directly connected node-pairs (SDij) in each instance, according to the archetypes. The distributions had very long tails as conrmed by the high kurtosis values. A kurtosis value of 3 indicates that a distribution is mesokurtic, being no more likely to produce outliers than a normal distribution. A value greater than three indicates a leptokurtic distribution that has a greater degree of \tailedness » than the normal distribution and kurtosis less than 3 indicates a platykurtic distribution with a smaller likelihood than the February 1, 2018 34

Quantifying supply chain vulnerability using a multilayered complex network perspective normal distribution to produce outliers. For the FC archetype all nodes were directly connected thus the relevant distributions in Figures 3.6a and 3.6b are the same. For the hub archetypes the distributions in Figure 3.6b were disproportionately wider due to the fact that 83.3% of the node-pairs were indirectly linked in these archetypes and each of these sets SIij was the product of the

set sizes of its component sets SDij . The DH archetype had the smallest set sizes for directly connected node-pairs, owing to the fact that its directly connected nodes were closer together on the grid. Interestingly, the kurtosis of SDij for the DH archetype also showed that it was far more likely to produce outliers than the other two archetypes.

1 Introduction

1.1 Turbulence is the new normal .

1.1.1 Supply chain vulnerability

1.1.2 Building resilient supply chains

1.1.3 Quantitative tools for a new management mindset

1.1.4 Problem statement

1.2 A complex network theory perspective

1.2.1 Multilayered complex network theory

1.2.2 Supply chain applications

1.2.3 Road network applications

1.3 Road vulnerability studies .

1.4 Topological vulnerability studies using Complex Network Theory

1.5 Research design

1.6 Research methodology

1.6.1 Multilayered network formulation and theoretical datasets

1.6.2 Link-based targeted attack simulation

1.6.3 Link-based random error simulation and statistical tests .

1.6.4 Case study validation

1.7 Thesis overview

2 Literature review

2.1 Modelling the complex adaptive nature of supply chains

2.2 Relevant applications of Complex Network Theory to road networks

2.3 Common design parameters for topological vulnerability studies

2.3.1 Metrics to assess network damage

2.3.2 Targeted attack strategies

2.4 Conclusion

3 Multilayered network instances

3.1 Conceptual structure of the multilayer network

3.1.1 Urban road network (physical ) layer

3.1.2 Supply chain (logical ) layer .

3.2 Multilayered network formulation

3.2.1 Generic multilayered formulation

3.2.2 Customised multilayered formulation

3.3 Sample generation of multilayered network instances

3.4 Formulation of the collection of shortest path sets Quantifying supply chain vulnerability using a multilayered complex network perspective

3.5 Shortest path statistics of the initial datasets

3.5.1 Initial distribution of the shortest path length

3.5.2 Initial distributions of the shortest path set sizes

4 Discovering vulnerability characteristics

4.1 Link-based targeted attack simulations

4.1.1 Dening network damage

4.1.2 Prioritising links for removal .

4.1.3 Overall link betweenness (Overall-B) .

4.1.4 Elemental link betweenness (Elemental-B) .

4.1.5 Overall link salience (Overall-S)

4.2 Results of link-based targeted attack simulations

4.2.1 Eectiveness of simulation strategies

4.2.2 Evolution of the prioritisation metrics

4.2.3 Characteristics that made M vulnerable

5 Development and analysis of vulnerability metrics

6 Statistical validation of vulnerability metrics

7 Case study: Real-life networks from South Africa

8 Case study: Link-based random error simulation

9 Conclusion and future work

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Quantifying supply chain vulnerability using a multilayered complex network perspective