EVALUATING THE IMPORTANCE OF BRAZILIAN PORTS USING GRAPH CENTRALITY MEASURES

. This study analyses the importance of Brazilian ports, based on the ﬂow of non-containerized cargo in 2014, considering both national and foreign trades. For that, we study a non-traditional centrality, called layer centrality, which evaluates the importance of ports, based on how well they are connected to inﬂuential ports. This measurement was preliminarily proposed in 2011 and applied to a simple non-weighted network, though herein we extend it to weighted graphs. For comparison purposes, we also apply three traditional measures, namely degree, eigenvector, and ﬂow betweenness centralities. Our ﬁndings show that the most impactful ports are private terminals Ponta da Madeira and Tubar˜ao , although public ports, particularly Santos , are usually impactful for national trades. Moreover, we analysed the map for public ports and suggest a suitable location for a new public port.


INTRODUCTION
Brazilian ports handle over 900 million tons of merchandise annually, which corresponds to more than 90% of the country's exports, according to the Special Secretary of Ports (Secretaria Especial de   ments (e.g. health surveillance, customs requirements, etc.), and entry barriers (e.g. exorbitant start-up costs, scale economy, etc.).
Other laws have also targeted the system's improvement, competitiveness and legal security, such as law 12.815/2013 (ANTAQ, 2016). As a result, the private sector participation has been increasing, and equipment and procedures have been modernized. However, maritime transport is still considerably underused for local transport.
Today, there are 37 organized public ports, controlled by port authority and 169 private terminals, according to the National Waterway Transportation Agency (ANTAQ -Agencia Nacional de Transportes Aquaviários). Private terminals are private enterprises that explore port activities, authorized by the federal government. They include Private Use Terminals (TUPs -Terminais de Uso Privado) and Transhipment Cargo Point (ETC -Estação de Transbordo de Carga), which are port installation outside port areas, used for cargo transhipment. Hereinafter, we refer to public ports, TUPs, and ETCs as ports. Using different methodologies, Padilha and Ng (2012) investigated the spatial evolution of dry ports in the State of São Paulo, and found that they have not been able to develop with the country's economic growth, due to institutional and infrastructural obstacles. Ng et al. (2013) also investigated development of dry ports in four Brazilian States and how institutional framework affects their bureaucratic and logistical roles.

Port Evaluation
According to Lagoudis et al. (2017), most port studies focus on port selection, efficiency, performance and competitiveness. However, this paper adopts a different perspective, as we analyse the importance of each port to the transport network. Among the first papers to adopt this perspective in maritime transport are Ducruet et al. More recently, other papers also have analysed the evolution of port systems, using graph tools. In contrast to such studies, our analysis is static, focusing on the importance of each port to the system. Moreover, we do not use other graph tools, besides centrality measures. On the other hand, we use four different centralities, measuring different types of importance, which also differs from the aforementioned studies.

CENTRALITY MEASURES
In the present study, we apply three traditional centrality measures, namely, degree, eigenvector, and flow betweenness centralities, briefly described in 3.1. Layer centrality is explained separately, in 3.2.
First, however, let us introduce some necessary definitions and concepts. In this work, G(V ; E), or simply G, denotes a simple undirected weighted graph on n vertices where V is the vertex set, V = {v 1 , . . ., v n } and E is the set of weighted edges [v i , v j ], formed by pairs of vertices from V , i.e., v i , v j ∈V . Each edge of G has an associated value, called a weight.
Graphs can be represented by adjacency list or adjacency matrix. In our case, we opted for the representation through the adjacency matrix, which will be more convenient for the purposes of this work (three of the four centrality measures used here can be extracted directly from the matrix). For a weighted graph, as in our case study, the adjacency matrix of G, A (G) = [a ij ], is the square matrix of order n, such that a ij = ω ij , where ω ij is the value associated to the edge if v i and v j are adjacent and a ij = 0, otherwise.
Since edges in our graphs have no orientation, (undirected graphs), a ij = a ji and the matrix is symmetric.

Traditional Centrality Measures
Degree centrality has been proposed by Freeman (1979). It corresponds to the number of direct connections that each element establishes with the others, which is called the vertex's degree. In transport networks, it represents the degree of usage for each element.
The degree centrality of a vertex v i , denoted by C D (v i ), is calculated by summing the values in the corresponding row in the adjacency matrix, as in (1). where In this paper, we also apply the eigenvector centrality, which corresponds to the linear combination of the centralities of connected vertices (hereinafter called neighbours) (Bonacich and Lloyd, 2001). Hence, an element u is regarded as more central than another element v if u's neighbours are more central than v's neighbours, even if u and v have the same degree.
Using the adjacency matrix, the eigenvector centrality of a vertex v i is defined as x i , where x i is the i-th coordinate of the unit positive eigenvector associated to the spectral radius (greatest eigenvalue) of A (G) . In other words, C ev (v i ) = x i , where x = (x 1 , . . ., x n ) satisfies equation (2), or Ax = λ x, in matrix notation. In (2), λ corresponds to the greatest eigenvalue of the adjacency matrix, whereas x correspond to the unit positive eigenvector associated with λ .
Another traditional measure is the betweenness centrality, based on the concept of geodesic path, i.e., the smallest number of edges connecting two vertices in a graph. Basically, this centrality measures the proportion of geodesic paths that pass through each vertex. However, there is no guarantee that two elements choose the geodesic path between them (Everton, 2012). Moreover, this centrality does not consider the edges' values. Let m i, j be the maximum flow from vertex i to vertex j, and m i, j (v) the maximum flow from vertex i to vertex j, passing through v. The flow betweenness centrality is shown in (3).
There are other centrality measures that could be calculated, such as Katz-centrality, originally proposed by Katz (1953), improved by Grindrod and Higham (2013), among others, and based on the premise that vertices are considered important if they are linked to other important vertices or if they are highly linked. Another possibility is the clustering coefficient, calculated for each vertex, as the number of triangles that pass through that vertex, relative to the maximum number of triangles that could pass through that vertex. In our case, it would be necessary to choose from several definitions of weighted clustering coefficient proposed in the literature, e.g., Barrat et al. However, our target is to analyse the Brazilian port system using a recently proposed centrality measure, with desirable properties for our case study, as explained herein. Thus, we compare this centrality only with popular centrality measures that present similar characteristics, to simplify the comparative analysis.

Layer Centrality
Layer centrality, preliminarily introduced in Bergiante et al. (2011), takes the importance of neighbours into consideration, using, however, a simpler and more intuitive method than the eigenvector centrality.
This centrality, unlike the others described in the previous paragraph, does not assign a value to each vertex, but assigns an ordering to the set of vertices, allowing to identify which vertices are more central in the network.
The first step of the layer methodology is to identify the vertex with the smallest degree, calculated in (1), and ranking it in last place. We then exclude, from the adjacency matrix, the row and column associated with this vertex, and we recalculate the remaining vertices' degrees. By doing so, we remove the vertex from the matrix, thus we do not rank it again, but we also eliminate its connections to the other vertices, in which case, the vertices that are connected to this element also loose these connections. Thence, we repeat this process, ranking vertices from bottom to top. If more than one vertex has the same degree, we rank them in the same position, and exclude their rows and columns from the matrix, together. As previously stated, this centrality provides only a ranking. However, this may not be a major limitation, as plenty of studies are based solely on rankings.
The ordering of the vertices is obtained recursively and this procedure could be described with the Algorithm 1.

Algorithm 1
Input: Graph G Output: Final Order Centrality Procedure We should highlight that, when applied to trees, i.e., connected acyclic graphs (Bondy and Murty, 2008), layer centrality resembles the algorithm based on the proof of Jordan's (1869) theorem, for finding the centre of trees (Hedetniemi et al., 1981). In other words, a tree's best-ranked vertex (or two best-ranked vertices) in the layer centrality is proven to be its centre.  In the original set, the vertex with the smallest degree d 0 (i), i.e., vertex a, with degree d 0 (a) = 30, is removed at the first iteration. Then, vertex e presents the smallest degree d 1 (e) = 30, thus being removed at the second iteration. Then, vertex f presents the smallest degree d 2 ( f ) = 60, thus being removed at the third iteration. At this point, vertices b and g present the smallest degree d 3 (b, g) = 45, and are therefore removed at the final iteration. Table 1 shows the final order for this example. Layer centrality may be, in certain cases, intermediary between degree and eigenvector centralities, because it takes into account the vertices' degrees, but it considers, to some extent, neighbours' centralities (Brandão et al., 2015). On the other hand, layer centrality disregards all elements that have already been ranked, thus analysing the network created only between the most impactful elements. This characteristic may be desirable in some situations, for instance, if only ports that operate with significant amount of cargo should be considered for the transport of certain merchandise types, and if the methodology itself should defines this significant amount of cargo, instead of experts.

SCOPES AND METHODOLOGY FOR THE CASE STUDY
In this paper, we study Brazilian ports that traded non-containerized cargo with other national ports, as well as with foreign ports, because they represent a significant amount. In fact, maritime transport plays a much more important role to international trade than to national transport, according to the Special Secretary of Ports in Brazil. Therefore, the analysis would be incomplete without such trades.
We do not consider containerized cargo because it is measured in Twenty-Foot Equivalent Unit (TEU), whereas non-containerized cargoes, i.e., liquid bulk, granular bulk and loose cargo, are measured in tons. Taking both into account would require methods to combine them, and results would vary accordingly. Alternatively, two separate analyses would be necessary, as in Freire-Seoane et al. To avoid such problems, we limit the scope of our study to non-containerized cargo.
We obtained the necessary data to evaluate the importance of each port in Brazil, from the Public Access Information System (SIG -Sistema de Informações Gerenciais de Acesso Público), by ANTAQ. We found 173 Brazilian ports (32 public ports and 141 private terminals) that transported non-containerized cargo with other domestic or foreign ports, from January to December 2014.
In terms of methodology, the first step is to build an adjacency matrix for each approach. For that, we transform the table extracted from ANTAQ, which presents the cargo transported from each origin to each destination, into the adjacency matrix, which indicates the cargo transported, between the port in the row and the port in the column, regardless of which is the origin and which is the destination. Thus, ports correspond to nodes and the amount of cargo transported between each pair of ports corresponds to the weight of the edge. With regard to foreign ports, we group them all into a single node, because our focus is on the Brazilian system. Thus, we do not analyse trades between foreign ports, and neither the importance of each port to the international system.
To calculate eigenvector and flow betweenness centralities, we used the UCINET program (Borgatti et al., 2002), whereas the other measures described in section 3, did not require any particular software.

RESULTS AND DISCUSSIONS
In Table 2, we present the rankings for some Brazilian ports, considering the degree, eigenvector, flow betweeness, and layer centralities, ordered by the first one. Later in this section, we show a map representing all public ports ( Figure 2). With regard to the other three centralities, Ponta da Madeira and Tubarão occupied the first and second place, respectively. In fact, these private terminals traded the greatest amounts of cargo with foreign ports and present a very poor performance nationally, which resulted in low positions in the flow betweenness ranking.
Next, we have Itaguaí, which is also poorly ranked by the flow betweenness centrality, though very well ranked according to the other centralities. This public port presents a poor performance when we only analyse the national flow, although it is, by far, the most relevant public port for international trade. In fact, according to the layer and eigenvector centralities, it is more central than Santos, because of its connections with international ports, despite having traded less cargo, as shown in the degree centrality ranking.
Similarly, Paranaguá, another public port, is better ranked in the layer and eigenvector centralities than in the degree centrality, because of its connections with international ports. On the other hand, the centrality rankings show that Madre de Deus has a greater importance in terms of national trade, than in terms of international trade. In other words, by comparing all four centrality rankings, we obtain relevant information, regarding the type of trade and its comparative amount.
In Table 3, we present the correlation indexes between the degree, eigenvector, layer and flow betweenness centrality rankings. We may observe that the flow betweenness centrality ranking is the least related to the other three rankings, particularly to the eigenvector ranking. The flow betweenness ranking prizes ports that present considerable flow with other national ports. On the other hand, the eigenvector ranking values connections with foreign ports, because they form the most central group -in fact, the ports' eigenvector centralities and the total amount of cargo they exchanged with foreign ports has a correlation index of 99.97%. This is why the eigenvector and flow betweenness rankings present such low correlation.
The degree centrality represents, by definition, the amount of trade, whether it is with national or international ports. This ranking and the layer centrality ranking are highly correlated, as shown in Table 3. However, Table 2 shows that they are more similar towards the end of the rankings. In the beginning, the layer centrality has many similarities with the eigenvector ranking. In fact, the first seven positions of both rankings are exactly the same, and different from the degree ranking.
Thus, in this sense, the layer centrality behaved as an intermediary between the degree ranking, representing the amount of trade, and the eigenvector ranking, representing connection to international ports, in this case study. More specifically, the layer centrality provides an intermediate measure between both traditional centralities for certain ports, such as Bacia Sedimentar de Campos, Almirante Barroso, Trombetas, Madre de Deus, São Francisco do Sul, Praia Mole, Terminal Manaus, and others.
On the other hand, there are cases in which such logic does not apply (hereinafter referred to as divergent results), such as Vitória, Bacia Sedimentar de Santos and Portocel. Vitória is ranked in 20 th place in the degree ranking, in 24 th place in the eigenvector ranking, though in 16 th place in the layer ranking. Analysing the adjacency matrix, we observe that Vitória is very connected to Almirante Barroso (39% of Vitória's total connections), which is very well ranked. A similar situation happens with Bacia Sedimentar de Santos, which is tightly connected (37%) to Madre de Deus, well ranked in the layer ranking. Contrarily, Portocel is ranked 22 nd in the degree ranking, 17 th in the eigenvector ranking, and 32 nd in the layer ranking. This port has many connections with Fibria (29%) and Belmonte (17%), which are poorly ranked.
Therefore, ports connected to many low or medium-ranked ports, tend to occupy lower positions in the layer ranking, when compared to the eigenvector ranking. On the other hand, ports connected to few highly ranked ports tend to occupy higher positions in the layer ranking, when compared to the eigenvector ranking. However, such divergent results are rare in this case study.  Figure 2 shows a map representing the layer ranking for all public ports in the analysis, obtained with the ArcMap software. Since it wouldn't be possible to visualize all 173 ports, Figure 2 represents only public ports. We also show the name for certain states that will be referred to herein. We may observe that the greatest public ports are in the south-eastern region, although the northeastern region presents many average-sized ports. We may also verify that most small ports are very close to other greater ports, so that the vast majority of the coastal states have at least one average-sized public port.
In fact, only three coastal states do not have a public port: Amapá, Piauí, and Sergipe, shown in the map. In the cases of Amapá and Sergipe, the private sector filled the public gap: there are three private terminals in Amapá, namely, Santana, Estação de Santana and Texaco Amapá, ranked, respectively, in 49 th , 105 th and 128 th place in the layer ranking; and two private terminals in Sergipe, namely, Carmópolis and Inácio Barbosa, ranked, respectively, in 39 th and in 84 th place in the layer ranking.
However, there are no ports -public or private -in the state of Piauí. Although this state is historically unproductive, and sustained by the public sector (Galeno, 2011), its economy has been growing considerably, particularly increasing its exports (Cury, 2014). In other words, Piauí seems to be an interesting location for a new port.

CONCLUSION
This study analysed the influence of each port in the Brazilian system nowadays, based on the flow of non-containerized cargo, between January and December 2014. We also considered trades with foreign ports, which were grouped into a single unit. We studied and applied layer centrality, preliminarily introduced in Bergiante et al. (2011), as well as three traditional measures, namely, degree, eigenvector and flow betweenness centralities, for comparison purposes.
As the eigenvector centrality, the layer centrality takes the centrality of neighbours into account, using, however, a much less complex method, and dispensing with computational aid. Such simplicity is particularly interesting for evaluations of public services, since regulatory agencies, non-governmental organizations, and the general public could easily confirm such evaluations.
In our case study, we compared all four centralities and observed that, in most cases, layer centrality behaves as intermediary between the degree and eigenvector centralities, although, in certain cases, it produces divergent results. Such results tend to occur with elements connected to many medium-ranked units, or to few highly ranked units.
We verified that the most impactful ports, according to the degree, eigenvector and layer rankings, were private terminals, namely Ponta da Madeira and Tubarão. These ports traded the greatest amount of non-containerized cargo with foreign ports, but had a very poor performance with other national ports. Thus, despite having the greatest performance according to such centralities, they presented low flow betweenness centrality.
The public ports of Santos and Itaguaí were ranked in third and fourth place, according to the degree, eigenvector and layer rankings. Almost 100% of Itaguaí's trades were with foreign ports, thus, this port also presented low flow betweenness centrality. On the other hand, Santos had a very important role in the national trade and was ranked in first place in the flow betweenness ranking. Many other public ports also presented a very important role nationally.
We also analysed the map for public ports and verified that the greatest public ports are in the southeastern region, although the vast majority of the coastal states have at least one averagesized public port. Furthermore, the private sector filled the public gap for two of the three coastal states without a public port. Only the state of Piauí has no ports, public or private. Although historically unproductive, Piauí has been growing economically, and particularly increasing exports. Thus, it seems to be an appropriate location for a new port.
Furthermore, impactful ports identified herein should receive special attention from government and port managers in maintenance planning, as well as in other infrastructure projects, because they cause the greatest impacts to the Brazilian port system. Future works may include containerized cargo in the analysis, using, for instance, a bi-criteria composition. Moreover, layer centrality studies should be further developed.