The Multilayer Model for Sea-Locked Countries

Abstract

Our model of trade in sea locked countries is grounded on the concept that the flow of trade is characterised by integrated layers of networks. The Multilayer model is composed of three categories of layers: physical, economic and sociological, which will determine the main outputs of the analyses. These layers are interactive, so to comprehensively study their interrelationships and impacts on international and national trade, we have to construct the model based on two distinct approaches. First, horizontal layers are studied through the use of complex network theory (see inter alia, the works of Barabasi and Albert 1999; Watts and Strogatz 1998; Erdős and Rényi 1959). Thereafter, vertical integration is analysed through the spatial interaction approach of Wilson (1967, 1970). In this way we are able to investigate both horizontal and vertical interdependency among layers and determine the impacts of natural and man-made shocks, such as (for the SPICs case study) the introduction of a specific policy for the reduction of transport cost, the maintenance of infrastructure and the provision of inter-island shipping services.

Appendix: The Dijkstra Algorithm

This Appendix formalises the Dijkstra shortest path algorithm (1959). In describing the algorithm we keep a consistent notation with that used throughout this book (readers are asked to return to Sect. 4.3 for a definition of the variables). The set S_ij , k of links with minimum cost between source node i and target node j is said to be the shortest path if its transport cost is minimum among all i-to-j paths. Dijkstra’s Algorithm is based on the following assumptions

• All link costs c_uv , k are non-negative;

• The number of vertices is finite;

• The source is a single node, but the target may be all other nodes;

• We assume we have no loops or parallel edges.

Dijkstra’s Algorithm marks the vertices as permanent or temporary vertices. The label of a node j is denoted β(j) and we define

$$ \upgamma \left(\mathrm{j}\right)=\left\{\begin{array}{c}1\kern0.5em \mathrm{if}\ \mathrm{the}\ \mathrm{label}\ \mathrm{is}\ \mathrm{permanent}\\ {}0\ \mathrm{if}\ \mathrm{the}\ \mathrm{label}\ \mathrm{is}\ \mathrm{temporary}\kern0.5em \end{array}\right. $$

A permanent label β(j) expresses the weight of the shortest directed i-to-j path. A temporary label β(j) gives an upper limit to this weight (can be ∞). Furthermore, we denote

$$ \uppi \left(\mathrm{j}\right)=\left\{\begin{array}{c}\mathrm{the}\ \mathrm{predecessor}\ \mathrm{of}\ \mathrm{node}\ \mathrm{p}\ \mathrm{on}\ \mathrm{the}\ \mathrm{shortest}\ \mathrm{path}\ \mathrm{i}-\mathrm{j},\\ {}0\ \mathrm{otherwise}\kern20.75em \end{array}\right. $$

In this way we can construct the directed path with the lowest weights. The Dijkstra Algorithm can be formalised in the following steps

We see that the algorithm is correct as follows. We denote (for every step):

$$ \mathrm{V}1=\left\{\mathrm{permanently}\ \mathrm{labelled}\ \mathrm{vertices}\Big\}\right. $$

$$ \mathrm{V}2=\left\{\mathrm{temporarily}\ \mathrm{labelled}\ \mathrm{vertices}\Big\}\right. $$

(V1, V2) is a cut with the completely scanned vertices on one side and other vertices on the other side.

In the next chapter we provide a thorough discussion of the Port Attractiveness Index, which is a critical component in the analysis of the shipping network.