Assessing Globalization and Regionalization Through Network Indices

Abstract

The authors use network analysis (NA) in order to study the impact of globalization and regionalization on the entire structure of trade flows. NA focuses on trade flows as a network, therefore emphasizing the relationship between countries – the nodes – and the network structure itself. According to the authors, this approach is particularly fit to offer a unified view of the system’s properties, help develop trade policies, and analyze changes in the world trading system. This is why the authors present the changes in the trade networks by studying indices that describe the network’s characteristics: density, closeness, betweenness and degree distribution. The authors conclude that network completeness is indeed achieved in some sub-regional components; however the level of heterogeneity between countries has increased.

Annexes

1.1 Annex A.1: Definition of a Network

A network consists of a graph plus some additional information on the vertices or the lines of the graph. In its general form, a network:

$$ N=\left( V, L, W, P\right) $$

(15.1)

consists of a graph G = (V,L), where V = (1, 2, …. n) is a set of vertices and L is a set of lines between pairs of vertices. In simple graphs, L is a binary variable, and L _ij ϵ (0,1) denotes the link between two vertices i and j, taking a value of 1 if there exists a link between i and j and 0 otherwise. Another convenient way (Vega-Redondo 2007) of representing simple graphs is through its adjacency matrix, a V × V-dimensional matrix denoted by a _ij such that:

$$ {a}_{ij}=\kern0.5em \left\{\begin{array}{cc}\hfill 1\hfill & \hfill if\ \left( i, j\right)\in L\hfill \\ {}\hfill 0\hfill & \hfill otherwise\hfill \end{array}\right. $$

Therefore, two vertices are said to be adjacent (or at just one-step distance) if they are connected by a line. The concept of geodesic distance in networks refers to the number of steps needed to connect two vertices V _i and V _j , named δ_ij. The shortest the distance between two vertices the closest is the connection between them.

Weighted networks add to simple graph some additional information on the lines of the graph. The additional information is contained in the line value function W, where line values are positive weights associated to each line, usually indicating the strength of the relation. In the ij case, w _ij is the link’s weight. The additional information on the vertices is contained in the vertex value function P, assembling different properties or characteristics of the vertices.

The size of a network is expressed by the number of vertices n and the number of lines m. The set of vertices that are connected to any given V _i defines its neighborhood where d ≥ 0 denotes the number of neighbors of V _i . Since, in simple directed graphs, a vertex can be both a sender and a receiver, the indegree of a vertex is the number of arcs it receives, and the outdegree is the number of arcs it sends. The notion of neighborhood is associated to the one of clustering. The clustering coefficient of a vertex V _i is the proportion of a vertex’s neighbors which are neighbors of each other. The clustering coefficient for the network as a whole can be derived taking a weighted or an unweighted average across vertices in the network.

1.2 Annex A.2: Structural Properties and Centrality of a Network

The density of a network is the number of lines in a simple network, expressed as a proportion of the maximum possible number of lines. It is defined by:

$$ \gamma =\frac{m}{m_{m ax}} $$

(15.2)

where in a simple directed graph, m _max  = n(n − 1).

Accordingly, a complete network is a network with maximum density.

The position of every vertex in a network is measured in terms of centrality.¹¹ The simplest measure of centrality of V _i is the number of its neighbors, i.e. its degree centrality, ${\ C}_i^d= d.$ ¹²

The degree centralization of a network is defined in relative terms (like most other measures of centralization) looking at the unevenness of the distribution of links among nodes in the network. The minimum degree for any component of the network is 0 and the maximum possible degree is n−1. If ${C}_i^{d^{*}}$ is the centrality of the vertex that attains the maximum centrality score, the variation in the degree of vertices is the summed absolute differences between the centrality scores of the vertices and the maximum centrality score among them. So, as the maximum sum of degree centrality absolute differences is (n−2)(n−1), the degree centralization of a network is defined as:

$$ {C}^d=\frac{\sum_{i=1}^n\left|{C}_i^d-{C}_i^{d^{*}}\right|}{\left( n-1\right)\left( n-2\right)} $$

(15.3)

and the higher the variation in the degree of vertices, the higher the centralization of a network. The degree centralization of any regular network is 0, while a pure star has a degree centralization of 1.

The notion of geodesic distance is the base of a second definition of centrality: closeness centrality. The closeness centrality of a vertex V _i is the number of other vertices divided by the sum of all distances between V _i and all others vertices:

$$ {C}_i^c=\frac{n-1}{\sum_{i\ne j}^{n-1}{\delta}_{i j}} $$

(15.4)

At the network level, if ${C}_i^{c^{*}}$ is the centrality of the vertex that attains the maximum closeness centrality score, the closeness centralization of a network is (Freeman 1979; Goyal 2007):

$$ {C}^c=\frac{\sum_{i=1}^n\left|{C}_i^c-{C}_i^{c^{*}}\right|}{\left( n-1\right)\left( n-2\right)/\left(2 n-3\right)} $$

(15.5)

A third notion of centrality often used in the literature is based on the intuition that a vertex V _i is central if it is essential in the indirect link between V _j and V _k . A vertex that is located on the geodesic distance between many pairs of vertices plays a central role in the network, because it is necessary for all periphery vertices in order to be mutually reachable. This concept of centrality is called betweenness centrality. The betweenness centrality of vertex V _i is the proportion of all geodesic distances between pairs of other vertices that include this vertex (Vega-Redondo 2007):

$$ {C}_i^b=\sum_{j\ne k}\frac{\delta_{j k}^i}{\delta_{j k}} $$

(15.6)

where δ _jk is the total number of shortest paths joining any two vertices V _j and V _k , and ${\delta}_{jk}^i$ is the number of those paths that connect V _j and V _k through V _i . The core of a star network has maximum betweenness centrality, because all geodesic distances between pairs of other vertices include the core. In contrast, all other vertices have minimum betweenness centrality, because they are not located between other vertices.

The betweenness centralization is the variation in the betweenness centrality of vertices divided by the maximum variation in betweenness centrality scores possible in a network of the same size:

$$ {C}^b=\frac{\sum_{i=1}^n\left|{C}_i^b-{C}_i^{b^{*}}\right|}{n^2-\left( n-1\right)/\left(2 n-1\right)} $$

(15.7)

1.3 Annex A.3: Network Analysis of Regional Trade Flows

1.3.1 Binary Network Analysis

The simplest indicator that can be used to analyze the structure of a regional trade network is the intra-regional node degree (IND _i ),¹³ that is the number of regional partner countries of each country i, which can be expressed in absolute terms, or as an intra-regional density index (IDI _i ), that is as a ratio of the total number of possible regional partner countries (n − 1):

$$ {IDI}_i={IND}_i/\left( n{\textstyle\ \hbox{--}\ }1\right) $$

(15.8)

The intra-regional density index can be computed also for the entire region r, where it measures to what extent the actual number of trade linkages corresponds to its maximum potential level:

$$ {IDI}_r={\varSigma}_i{IND}_i/\left[ n\left( n{\textstyle\ \hbox{--}\ }1\right)\right] $$

(15.9)

The density of a regional trade network can be compared with a pre-defined external benchmark area o that can be a set of other regions or the rest of the world made of m countries. Denoting with END _i the extra-regional node degree, that is the number of country i’s trading partners located in the external benchmark, a relative intra-regional density index (RIDI _i ) can be defined as:

$$ {RIDI}_i=\left({IDI}_i{\textstyle\ \hbox{--}\ }{EDI}_i\right)/\left({IDI}_i+{EDI}_i\right) $$

(15.10)

where: EDI _i = END _i /m

RIDI _i ranges between −1 and 1and is equal to zero if IDI _i = EDI _i (geographic neutrality). At the regional level:

$$ {RIDI}_r=\left({IDI}_r{\textstyle\ \hbox{--}\ }{EDI}_r\right)/\left({IDI}_r+{EDI}_r\right) $$

(15.11)

where: EDI _r = Σ _i END _i /(n∙m)

Another indicator frequently used in the BNA of the world trade network is the average nearest neighbor degree (ANND _i ), which is simply the average node degree of country i’s partners. In our context, to reduce the complexity of notation, we will replace the phrase nearest neighbor with partner, and define an intra-regional average partner degree (IAPD _i ) as follows:

$$ {\mathbf{IAPD}}_{\mathbf{i}}\kern0.28em =\kern0.28em \left({\boldsymbol{A}}_{\left(\mathbf{i}\right)}\cdot \boldsymbol{A}\cdot \mathbf{1}\right)/{\mathbf{IND}}_{\mathbf{i}} $$

(15.12)

where A _(i) is the ith row of the adjacency matrix A describing the network and 1 is a unitary vector. The maximum level of IAPD _i is reached when all country i’s regional partners’ IDI _j are equal to one, that is when all the possible n(n − 1) trade linkages exist. This allows us to define an average intra-regional partner density index (IPDI _i ) as follows:

$$ {IPDI}_i={IAPD}_i/\left( n{\textstyle\ \hbox{--}\ }1\right) $$

(15.13)

At the regional level IAPD _r and IPDI _r can simply be computed as the averages of the corresponding country indicators.

An extra-regional average partner degree (EAPD _i ) and an extra-regional partner density index (EPDI _i ) can be defined as follows:

$$ {EAPD}_i=\left({\boldsymbol{EA}}_{(i)}\cdot \boldsymbol{OA}\cdot 1\right)/{END}_i $$

(15.14)

$$ {EPDI}_i={EAPD}_i/\left( m{\textstyle\ \hbox{--}\ }1\right) $$

(15.15)

where EA is the n×m adjacency matrix of linkages between the region’s members and the benchmark area’s countries, and OA is the m × m adjacency matrix of linkages among the benchmark area’s countries.

Finally, a relative intra-regional partner density index (RIPDI _i ), ranging from −1 to 1 with a neutrality threshold of zero, can be computed as:

$$ {RIPDI}_i=\left({IPDI}_i{\textstyle\ \hbox{--}\ }{EPDI}_i\right)/\left({IPDI}_i+{EPDI}_i\right) $$

(15.16)

The fact that a country has a certain average partner degree does not necessarily imply that all its partners are connected between each other. In order to capture this feature of the network, a third indicator has been developed, named binary clustering coefficient (BCC _i ), aimed at measuring to what extent a country’s partners tend to cluster into triangles, that is to trade between each other. BCC has also been used to detect a possible hierarchic structure of the network.

The intra-regional binary clustering coefficient (IBCC _i ) can be defined as:

$$ {IBCC}_i={\left({\boldsymbol{A}}^3\right)}_{i i}/\left[{IND}_i\left({IND}_i-1\right)\right] $$

(15.17)

where (A ³) _ii is the i-th entry on the main diagonal of A A A. Given IND _i , IBCC _i measures the actual number of bilateral linkages between country i′s regional partners, relative to its potential.¹⁴

Another useful concept is the degree of centrality, which is used to assess to what extent trade linkages tend to concentrate towards one or more hub countries. The maximum degree of centralization is reached in a star network, where only one country is connected with all the others, whereas each of the others is connected only with the center of the network.

Several indicators have been proposed to measure the centrality of a node and the centralization of a network. At the country level, intra-regional node centrality (INC _i ) can simply be measured as:

$$ {INC}_i=\left(1{\textstyle\ \hbox{--}\ }{IBCC}_i\right) $$

(15.18)

INC _i measures to what extent a country is connected to regional partners that are not connected between each other.

At the network level, an intra-regional centralization index (ICI _r ) can be defined as:

$$ {ICI}_r= \max [{INC}_i]={\varSigma}_{\mathsf{i}}( \max [{IND}_i]{\textstyle \text{\ \hbox{--}\ }}{IND}_i)/[(n{\textstyle \text{\ \hbox{--}\ }}1)(n{\textstyle \text{\ \hbox{--}\ }}2)] $$

(15.19)

This indicator measures the network’s actual centralization as a proportion of its theoretical maximum, defined by the number of missing linkages in the corresponding star network, which is equal to (n − 1)(n − 2).¹⁵

1.3.2 Weighted Network Analysis

The weighted network analysis (WNA) of international trade represents the intensity of linkages among the network nodes through the actual matrix of their bilateral trade flows (W) expressed in absolute or relative terms.¹⁶

Apart from the difference between the respective matrices, indicators used in WNA are similar to those used in BNA. In our context, node degree is replaced by intra-regional node value (INV _i ), which is the value of a country’s total trade with its region. However, since there is no given maximum value for trade, a density index similar to that used in BNA cannot be easily defined, and there are several options to build a normalized INV _i .

If we refer to the geographic neutrality criterion mentioned in the text (Sect. 15.3.1), we can introduce intensity and revealed trade preference indices into the context of WNA. Since INV _i refers to intra-regional trade, we can define extra-regional node value (ENV _i ) as the total value of country i’s trade with the benchmark area, and the density index of BNA can be replaced by a homogeneous (country-size-independent) trade intensity index HI _ir . More precisely:

$$ {HI}_{ir}={S}_{ir}/{V}_{or} $$

(15.20)

where:

$$ \begin{array}{l}{S}_{i r}={INV}_i/\left({INV}_i+{ENV}_i\right)\\ {}{V}_{or}={\varSigma}_k{ENV}_k/{\varSigma}_k\left({ENV}_k+{INV}_k\right)\end{array} $$

and k = 1, …, m refers to countries of the benchmark area o.

HI _ir is higher (lower) than one if country i’s intra-regional trade share is higher (lower) that the share of region r in the benchmark area’s trade.

In a similar way, a homogeneous extra-regional trade intensity index (HE _ir ) can be defined as:

$$ {HE}_{ir}=\left(1{\textstyle\ \hbox{--}\ }{S}_{ir}\right)/\left(1{\textstyle\ \hbox{--}\ }{V}_{or}\right) $$

(15.21)

and finally the relative intra-regional revealed trade preference index (RIRTP _ir ) can be computed as:

$$ {RIRTP}_{ir}=\left({HI}_{ir}{\textstyle\ \hbox{--}\ }{HE}_{ir}\right)\left({HI}_{ir}+{HE}_{ir}\right) $$

(15.22)

This index measures unambiguously if intra-regional trade is more or less intense than what implied by the geographic neutrality criterion.

Other WNA indicators can be used to better illustrate the topology of regional trade networks in terms of connectivity and centralization, taking into account not only direct bilateral linkages between a country and its partners, but also linkages among the latter.

Reminding that the importance of a node in a network depends not only on its own degree, but also on the degree of its partners, we can adapt the binary indicator of IAPD _i to WNA in several ways.

The first possibility is to compute an intra-regional weighted average partner degree (IWAPD _i ) through the following formula:

$$ {IWAPD}_i=\left({\boldsymbol{W}}_{(i)}\boldsymbol{A}1\right)/{INV}_i $$

(15.23)

where W _(i) is the i-th row of the weight matrix W.

A similar indicator could be built for extra-regional partners and the two indicators could be compared as for the previous ones. However, IWAPD _i , although weighting each partner with its trade value, is still to be considered as a binary indicator, since its unit of measurement remains the number of partners.

A more appropriate WNA equivalent of the binary IAPD _i is the intra-regional average partner value (IAPV _i ), which is the average value of a country’s regional partners’ intra-regional trade:

$$ {IAPV}_i=\left({\boldsymbol{A}}_{(i)}\boldsymbol{W}1\right)/{IND}_i $$

(15.24)

The maximum level of IAPV _i can be defined as follows:

$$ Max{(IAPV)}_i={\varSigma}_k{INV}_k/{IND}_i $$

(15.25)

where k = 1, … IND _i are the possible regional partners of country i ranked according to their total trade value. This implies that Σ _k INV _k necessarily grows less than proportionally than IND _i . It is also important to note that, for any given IND, the list of possible partners changes across countries, because it cannot include country i. As a consequence, Max(IAPV) _i is negatively related to INV _i and will be reached only if the actual regional trade partners of country i happen to be those with the highest total trade value.

This definition allows us to build an intra-regional normalized average partner value (INAPV _i ) as the ratio between IAPV _i and its maximum.

At the regional level IAPV _r and INAPV _r can simply be computed as the weighted averages of the corresponding country indicators.

The binary concept of clustering into triangles can easily be adapted to WNA. The intra-regional weighted clustering coefficient (IWCC _i ) is defined as follows:

$$ {IWCC}_i={{\left({\boldsymbol{W}}^{\left[1/3\right]}\right)}^3}_{i i}/\left[{IND}_i\left({IND}_i-1\right)\right] $$

(15.26)

where W ^[1/3] is the matrix obtained by raising each element of the W matrix to 1/3 and (W ^[1/3])³ _ii is the i-th entry on the main diagonal of W ^[1/3] · W ^[1/3] · W ^[1/3]. IWCC _i measures the intensity of trade among country i′s regional partners relative to the total number of their potential connections. So, it is positively related to the actual density of these connections (IBCC) and to their intensity.

For any given IND _i , the maximum level of IWCC _i is not scale-independent. This problem can be solved by dividing each element in the W matrix by their maximum, which results into an intra-regional normalized weighted clustering coefficient (INWCC _i ), ranging from 0 to 1:

$$ {INWCC}_i={{\left({\boldsymbol{NW}}^{\left[1/3\right]}\right)}^3}_{i i}/\left[{IND}_i,\left({IND}_i-1\right)\right] $$

(15.27)

where NW is the matrix of trade flows within region r normalized with respect to their maximum.

1.3.3 Assortative Mixing (Homophily) and Trade Regionalization

This application of network analysis is aimed at detecting the degree of regionalization of the world trade network (Iapadre and Plummer 2011), and is based on the weighted equivalent of the binary assortativity coefficient proposed by Newman (2003a, b).

The starting point is a matrix of international trade flows classified by regions, reporting intra-regional trade values on the main diagonal and inter-regional flows in the remaining cells. The resulting intra-regional assortativity coefficient (IAC) is:

$$ \mathbf{I}\mathbf{A}\mathbf{C}=\left(\mathbf{Tr}\left(\boldsymbol{R}\right)\ {\textstyle \mathbf{\hbox{--}}}\ ||{\mathbf{R}}^{\mathbf{2}}||\right)/\left(\mathbf{1}{\textstyle\ \mathbf{\hbox{--}}}\ \left|\right|{\boldsymbol{R}}^{\mathbf{2}}\left|\right|\right) $$

where R is the matrix of intra- and inter-regional trade flows, divided by their total, Tr is the trace operator, and ‖R ²‖ is the sum of all the elements of matrix R ².

IAC is equal to zero in the case of geographic neutrality, which is when regions trade among each other in proportion to their total trade values, and reaches a maximum value of one in the limiting case of no inter-regional trade. On the other hand, in the limiting case of no intra-regional trade, the minimum (negative) value of IAC is equal to – ‖R ²‖/(1 − ‖R ²‖).¹⁷