Country centrality in the international multiplex network

Our main objective in building the international multiplex has been to identify two cross sections with a sufficient number of nodes and layers to compare the structure of the multiplex before and after the Great Recession.

From the larger dataset showed in Fig. 12 in the Appendix we have first selected a set of layers with non overlapping characteristics, avoiding duplicates and partial sources.

The selection of the two years in the dataset to be compared is based on two criteria: first network measures often evolve slowly, thus a sufficiently long period is needed to capture some changes; second depending on the choice of the years some layers might not be present in the final selection, due to lack of data. To partially mitigate this second issue we have relaxed the constraint on some significant layers including them in the final cross sections even when they are missing in a given year, but data are available for an interval around it.

By choosing years 2003 and 2010 as reference points we obtain the results shown in the last two columns of Table 1, t₀ and t₁, where we can see that the two cross sections are scattered: migration stock, FDI and aid layers are not available for the 2003 reference year⁵. In the rest of the paper we will occasionally refer to the cross section in 2003 as first cross section and to the crosssection in 2010 as second cross section.

Our main goal in the construction of the international multiplex was to include as many countries as possible. By looking at the column nodes of Table 1 one can see that the number of reported nodes for some layers differs across time. To obtain a stable and large set of countries we proceeded in two steps. First, we have identified non missing observations: if the node is the target or the source of at least one non-zero edge in any of the years available for a given layer it belongs to the dataset. Second we have calculated the common subset of nodes for all layers. Some layers have been discarded from the analysis since they do not have enough observations. A more detailed discussion of our two steps strategy is provided in the Appendix, section Data selection.

As a result we selected a multiplex of 19 layers with the same set of 112 nodes in each of them, sampled at two time periods around 2003 and 2010. Table 1 shows the features of our final selection and in “Summary statistics” section we describe the content of each layer. By looking at the distribution of weights and at the density of each layer we can see that, even though balancing our two cross sections on the same set of nodes and layers ensures that we can make meaningful comparisons before and after the Crisis, some further steps are required to normalize the content of the dataset, across multiple and heterogeneous dimensions. In “Filtering” section we explain how we have filtered the edges to make layers comparable. For additional details on how we processed the data before filtering we refer to section “Data preprocessing” in the Appendix.

Table 1 reports the list of selected networks in our multiplex, with the short name of each layer used in our dataset in the first column and some summary statistics for each layer of the multiplex: number of nodes and edges in each of the cross sections analyzed, symmetry and weight checks on the edges and actual years used to construct the database.

Layers can be classified in six categories according to the type of international relationship they represent:

Table 1 shows that trade layers are constituted mostly by monetary flows, hence they are represented by directed weighted graphs where edges are flows of money from source country to target country. The network of trade agreements (fta_wto) is one exception: it is a symmetric and unweighted layer with edges equal to 1 when two country have signed a free trade agreement.

Financial layers too have edges representing monetary flows, such as FDI, TPI and M&A. It is worth noticing that FDI flows are reported twice, since we integrate OECD data with a separate source (FDI_Greenfield) collected in Kirkegaard (2013) focusing on specific type of firms which are difficult to measure in the official FDI statistic. Given the extent of this second source and the relevance of its content (which differs significantly from OECD) we added it as a separate source. As for BIS flows, which represent data collected from the Bank of International Settlements on loans among international banks, we selected only the claims layer, which is also automatically adjusted for changes in exchange rates. Finally, flows of international aid constitute another exception. These specific flows of capital are named Official Development Assistance (ODA) Disbursments and are measured by the OECD. While in the other layers edges represented monetary payments from source countries to target ones, here the relation represents voluntary donation from source countries to foster the development of their counterparts. Hence ODA flows represent a reversed measure of the development of a countries: central nodes (in terms of incoming edges) are those which are more underdeveloped. For this reason in our final multiplex the aid layer has been reversed (more details on this in the section “Data preprocessing” in the Appendix).

The third category of layers concerns the movement of people. Layers of this group can be divided in temporary migration (tourism and residencies lasting less than one year) and permanent stock of migrants. Both measures are collected by the United Nations statistical division. A third measure, flows of migrants among countries, has been produced for every five years from 1990 to 2010 by Abel and Sander (2014) through estimates based on the UN stock data⁶.

Knowledge flows are represented by citations of patents and papers by countries. Patent citations come from the NBER database and refer to the country of patent inventors. Paper citations instead have been collected by Pan et al. (2012) and represent an average over a five year period. These layers have a common unit of measurement: each edge between two nodes represents the number of papers/patents of the destination country cited by the source one.

The last two layers are cable route capacity and diplomatic alliances. The first is a measure of the common telecommunication infrastructure linking two countries: each edge represents how many terabyte of data can travel between two countries via cable connection. Hence by construction this layer is weighted and symmetric. The alliances layer is symmetric too but it is unweighted: each edge represents an international diplomatic alliance between two countries. This is one of the measures of diplomatic interaction from the Correlates of War project⁷.

The layers of the multiplex are very different and there is a need to standardize them before proceeding in any further analysis. In fact, the distribution of weights of layers in Table 2 (more details in Fig. 13 and 14 in the Appendix) shows that they have very different mean value, range and skewness. Moreover, in many of our sources there is a bias in the reporting countries: due to their economic (and sometimes territorial) size, developed countries take part in exchanges with other countries more frequently.

To solve these issues we apply an hypergeometric filter on the data, as in Riccaboni et al. (2013). The reasoning goes like this: since edge weights are affected by the size of nodes, a filter must be applied to the edges in order to distinguish the significant ones. Following Riccaboni et al. (2013) and Armenter and Koren (2014) we take as as starting point a null model where we assume that edges are randomly assigned to all nodes with probability proportional to their connectivity. The resulting probability distribution is an hypergeometric one. Next we fix a significance threshold and we test if the weights of the actual edges in our layers passes the threshold. If not so, we discard them. The outcome of this procedure is a filtered layer where insignificant edges are removed. This approach is used to standardize the layers using the probabilities created with the filter, solving at the same time both the size bias and the heterogeneity of layers. In a nutshell, we replace heterogeneous links with the probability of them being more intense than expected under a random null model.

The final effect of the hypergeometric filter is to reduce the density variance by smoothing out denser layers and leaving almost untouched sparse layers. This follows our assumption that more connected nodes (in denser layers) have a greater share of irrelevant edges, while less connected ones have few relevant links which pass the filter. Figures 15 and 16 in the Appendix demonstrate this by showing the amount of observations eliminated by the filter in every layer and how this quantity is positively correlated with the layer density before filtering.

Our choice of the filter was made to reduce the variation in density across layers to make them comparable. However, we have not controlled for variation in other possible features, such as the clustering or the weak-tie structure, which might have been altered as a consequence of our procedure.

Other filters could have been chosen to preserve different characteristics of the networks. Some examples are the disparity filter of (Serrano et al. 2009), the GloSS filter of Radicchi et al. (2011) or the Maximum Entropy approach proposed in Gemmetto et al. (2017). While some of these methods are certainly better to avoid unwanted alterations in the structure of the filtered networks, as demonstrated in Gemmetto et al. (2017), we see two possible difficulties in our case. First, the disparity filter relies on a different theoretical assumption about the null model used to validate the network, a uniform distribution of links, whereas in our case we have assumed an hypergeometric one. Second, in the other two cases, the full information on the network structure is used to construct the null model. Even though this is is principle a desired property of the filter, we cannot rule out some reporting error in our data which could make the above mentioned methods less reliable. In our approach, by using only information on a given country, we avoid spreading the reporting error to the full specification of the null model.

In the next subsections we show some general network statistics calculated on the filtered multiplex using a very conservative filtering threshold.⁸

Given that our interest is in the aggregated multiplex and not in the single layers composing it, we have simplified the presentation of the single layer analysis by computing network statistics on the unweighted version of our graphs, without taking into consideration those assortativity and clustering measures which rely on the directionality of edges. We also do not perform a full fledged analysis of similarity of layers, as in Bródka et al. (2018).

Table 2 reports the network statistics for the cross section in 2003 of our multiplex. A similar table is provided in the Appendix for the cross section of year 2010 (Table A2). To help visualizing some properties of layers, we show in Fig. 1 the rank of each layer with respect to each statistic (left hand side plot) and the correlation matrix measured on these rankings (right hand side plot). A lighter (darker) color in the left panel means that the selected layer (y-axis) ranks lower (higher) with respect to the selected statistics (x-axis). A lighter (darker) color in the right panel means that the two selected layers have lower (higher) correlation with respect to their position in the statistics ranking (i.e. if correlation is high the two layers rank similar in the same network statistic).

While we see some patterns of similarity, no clear cluster structure emerges: only few layers have a distinct profile (the symmetric layers) while all the others cannot be grouped in a clear manner.

This is evident in plot (a) of Fig. 1: layers rank similarly only regarding some particular statistics, but look very different in all the others. For this reason we do not perform a proper cluster analysis, which will only increase the complexity of the exposition, and we inspect only the macro structure of the layers.

We identify two macro groups of layers with somewhat similar ranking in the statistics. The first one comprehends trade and migration data with the paper citations layer and the three symmetric layers. These last three also form another more compact ensemble in the picture. The other big group is mainly composed by financial layers together with mobility, papers collaboration and patent citation data. While the division in two groups is not clear-cut, it approximately corresponds to the division of layers by typology we introduced at the beginning with layer of different types behaving differently. This suggests that integrating these different data sources may be beneficial.

Moving to the network statistics graph (left hand side of Fig. 1) the picture is more detailed and we can observe differences in the previously identified groups. The first one is the more defined: migration and trade data, together with the citation layer, are usually denser, both in the simple and in the bilateral sense, they have bigger average size and diameter of their bigger strongly connected components, longer average short path length and larger size of their weakly connected components. On the other hand, they have fewer components, their degree (including outdegree and indegree) is usually more skewed to the right and they have lower values of network centralization and weighted asymmetry. These characteristics are also shared by the next three layers (cow_alliances, fta_wto, totIC) which are also less dense.

Other layers exhibit an opposite behavior. There is a set of layers (FDI, arms, value) which is highly anti-correlated with the previous ones: they all rank higher in the left hand side statistics, i.e they have greater number of weakly and strongly connected components and degrees skewed to the left. Moreover they show lower rankings in the statistics where previous layers showed higher rankings: they are less dense, have lower average clustering coefficient and their biggest weakly connected component is smaller and has lower average short path length. Moreover their largest strongly connected components have smaller diameter and size.

Finally, the remaining layers have less sparkly defined characteristics: while, on average, they have opposite behavior with respect to the initial ones the differences are not so clear. In particular, like the first layers, they have higher density and higher average size of their biggest weakly connected component. However, for the remaining statistics, they differ from the first set of layers.

The (single layer) PageRank of a node is a measure of node centrality that accounts for the importance of the node by looking at its centrality and the centrality of all the in-neighbours pointing to it.

Formally, given an unweighted network of N nodes, the PageRank of node i can be defined as (cfr. Halu et al. 2013):

Where d is the so called damping factor, A_ij is an element of the unweighted adjacency matrix of our network A and g_j is the out-degree of node j (i.e the number of its out-neighbors) when its greater than 0, 1 otherwise. Hence a random walker will choose among the out-neighbours of node j with probability d and with probability 1−d will switch to one of all the other nodes in the network. Starting with a uniform distribution and running iteratively the algorithm we should obtain a stable distribution of the PageRank for all nodes.

When we move from single to multiple layer networks, nodes will have multiple dimensions over which nodes share links, hence the PageRank centrality of nodes is affected both by the single-layer centrality and by the centrality of a layer in the multiplex. Therefore, a rank of the layers of the multiplex is needed to compute the multiplex version of the PageRank.

Following Rahmede et al. (2018) we summarize the multiplex from two perspectives: first as a colored network with links of different colors having different influences, second as a bipartite network of nodes and layers. Then we use both dimensions to obtain a generalized multiplex PageRank. This multilayer version of the PageRank algorithm can be defined for directed and undirected networks as well as weighted and binary networks, hence the adjacency matrix A^α will refer without distinction to any of these types of layers.

From the colored network perspective we obtain matrix G as the sum of adjacency matrices across M layers weighted by their respective influence z^α. So, given layers α=1,2,…M, the elements of G are given by:

From the bipartite view of a multiplex we obtain the M×N incidence matrices Bⁱⁿ and B^out representing the normalized in-strength and out-strength (respectively in-degree and out-degree for binary networks) of each node i in every layer α:

Here \(W^{\alpha }=\sum _{i=1}^{N} \sum _{j=1}^{N} A^{\alpha }_{ij}\) represents the sum of all edge weights in layer α for directed weighted networks or twice this number if the layer is undirected. For unweighted networks it becomes the total number of link (directed case) or its double (undirected case). Similarly, for undirected multiplex networks \(B^{in}_{\alpha i}\) and \(B^{out}_{\alpha i}\) are identical, since the A matrix is symmetric.

Our specification of the multiplex PageRank works on the assumption that, given a certain node in the multiplex, its centrality will be affected both by the centrality of nodes pointing to it and by the influence of the layers to which these in-neighbors of the node belong. A random walker moves from node j to a neighbour of i along all layers with a probability \(\tilde {d}\) and proportionally to G_ij. Otherwise with probability \(1-\tilde {d}\) it jumps randomly on another node of G. The stable distribution we get at the end of this process is the multiplex PageRank centrality of the nodes.

Similarly to the single layer PageRank equation, we get the multilayer PageRank equation:

Where \(\tilde {d}\) is the damping factor and:

Here θ(·) is the Heaviside step function. Equation 4 is a function of the set of layers’ influences. To avoid calculating all these values, we couple the above equation with another one aimed at determining the influence of layers. This time we define layersŠ influence (relevance) as a function of the centrality of their nodes. Hence:

Here \(\mathcal {N}\) represents a normalization constant. A more flexible specification of this equation can be obtained by introducing some tuning parameters:

where a regulates the effect of total weight of layers W^α on the influence: with a=1 layers with higher W^α become more influential, while with a=0 the layer influence is rescaled with respect to W^α.

The s parameters instead indicates if more influential layers are those with fewer (s=−1) or more central nodes (s=1).

Finally once s is settled the parameter γ allows us to suppress or enhance the contribution of low centrality nodes: with s=1 values of γ>1 (γ<1) suppress (enhance) their contribution. Conversely, when s=−1 values of γ>1 (γ<1) enhance (suppress) the contribution of less central nodes.

Solving simultaneously the two coupled Eqs. 4 and 9, given a set of parameters a,s,γ, allows us to assign a centrality X_i to each node and an influence z^α to each layer α of the multiplex.

The (single layer) hubs and authority scores of a set of nodes are two measures of centrality which depend one on the other recursively. For a node to have an high authority score it is necessary to have many high hub nodes pointing to it. Similarly, for a node to have an high hub score it must have a lot of high authority score nodes to point to.

Given the adjacency matrix A of an (unweighted) graph, the hub and authority scores are defined as:

Usually, the algorithm is calculated by setting all scores to one and iterating for a sufficient number of times, which requires after each step to normalize to one the sum of all the scores.

The generalization of the algorithm for multilayer networks provided by Arrigo and Tudisco (2018) includes layers and time stamps as dimensions to be used to compute the centrality. This results in five scores: two for nodes (hub and centrality scores), two for layers (broadcasting and receiving scores) and one for the time dimension. In our case the last score would not be used since we have only two cross sections.

Similarly to the MultiRank, a node receives an high hub score if it belongs to an high broadcasting layer and has many links towards high authority nodes in layers with high receiving capabilities. Conversely, high authority would be awarded to nodes in high receiving layers with high hub nodes from high broadcasting layers pointing to them.

Finally the definition of the broadcasting and receiving scores of the layers follow when the focus is toward layers instead of nodes.

For each layer α the multilayer generalization of the HITS algorithm reads as:

where b and r are the broadcasting and receiving scores of each layer. It is straightforward to notice that the overall hub centrality of a node in the multiplex is \(h_{i}= \sum _{\alpha } b^{\alpha } a_{i}^{\alpha }\) while its overall authority score is \(a_{i}= \sum _{\alpha } r^{\alpha } h_{i}^{\alpha }\). Since we are dealing with a multiplex, our formulation of the layer scores does not allow for some layer to not be connected to others: in other words the inter-layer network in our case is fully connected.

As we can see, there are no parameters to choose in the MD-HITS, hence we will always have one and only one ranking of nodes and layers (according to the score we are interested in). To obtain the centralities of nodes and layers we need to solve the recursive equations defined above, which requires us to solve the eigenvector problem on the whole tensor which represents the multiplex. This can be done by finding an adequate multi-homogeneous map on the adjacency tensor and its unique Perron eigenvector (cfr. Definition 1 in Arrigo and Tudisco, 2018). In Arrigo and Tudisco (2018) the authors then demonstrate the existence, uniqueness and maximality of the MD-HITS measure and provide a converging and fast parallel algorithm to compute it. MD-HITS is an ideal candidate algorithm to compute centrality in a multiplex, since it exists and is unique regardless of connectivity of the graph, while other centrality measures based on eigenvectors require the graph to be strongly connected, a property which is rarely satisfied.

In the next section we analyze the centrality of countries in the international multiplex.

Reporting agencies collect data on a fixed set of countries in different years. Hence an empty report would not imply the absence of the country from the dataset but for a given year only. To translate in network theory terms: there may be some nodes in a network whose edge weights with respect to all other nodes are 0 at a certain time t_i, but are still present in the network with positive edge weights at other t_j with j≠i. These nodes belong to the network and we call them isolated nodes at time t_i. On the other hand, missing nodes don’t have link to other nodes at any t.

To tell if the absence of a certain node from a specific year means its absence in the overall layer we needed to define when a node is part of our dataset. To do that we have made the following assumption: if a node is present as origin or destination at least once in the whole reported set of years for a certain layer, that node has to be considered as existing in the dataset.¹³ To calculate the set of existing nodes in each layer we have used the whole span of its observations in our dataset, hence not only years 2003 and 2010.

Finally, to get a constant set of nodes across all the multiplex, the existing nodes of each layer must match across all the others. The result of this process is a set of 112 nodes common to all our layers. By referring to the nodes column in Table 1 we can get a good approximation of this process: we define the set of nodes for each layer as the union of the existing nodes in the years 2003 and 2010. Hence their size corresponds to the maximum number between t0 and t1 in Table 1 and ranges from 211 in the free trade agreement layer to 124 in the alliances layer. This leaves us with 19 different sets of nodes, one for each layer, partially overlapping. To obtain a common set of countries for the whole multiplex we calculate the subset of all the individual layers’ node sets. Hence our final set is a group of 119 countries and the difference between 124 and 119 is given by five countries belonging only to the alliances layer and not to the rest of the multiplex, which we have removed.

It must be noted that this process leads automatically to select the smallest set of common countries across all layers. To avoid reducing this number too much we had to select a minimum threshold of countries to include in the multiplex with the consequence of eliminating some layers which would have had observations in the required years but with a reduced number of nodes (Fig. 12 in the Appendix show some of the layers we have eliminated with this criterion).

Our final result is a multiplex of 19 layers with 112 nodes in each of them, observed at two “rugged” time snapshots, i.e. with not exactly the same starting and ending year for all layers. Balancing our two cross sections on the same selection of nodes and layers ensures that we can make meaningful comparisons over time, but some further steps are required before moving on to the actual analysis.

First we have preprocessed our data to remove inconsistent observations. While in principle network weights do not have strict requirements to satisfy, in our case we needed to impose some restrictions: since our network layers are mostly directed, negative weights have no meaningful interpretation. They should instead be rewritten as positive edges in the reverse direction: if country A receives a negative flow from country B, it actually means that country B is the source of a positive flow towards A. Hence we have reversed negative flows when possible, i.e when the whole layer showed a consistent pattern of reversed flows and there were no duplicates after reversion. If instead most negative weights seemed randomly distributed or reverse flows were already recorded in the dataset we have dismissed the negative values as computational errors.¹⁴

Another type of requirements we imposed is to have edge weights not smaller than one, in order to avoid problems with eventual filtering operations. As one can see from the minimum weight column in Table A1 several layers have fractional values. Most of the time this was due to the unit of scale employed, hence it has been sufficient to rescale them with the right multiplier. This is the case for FDI, FDI_Greenfield, BIS_flow_claims, aid, value, serv_exp and totIC layers.

The other cases of fractional weights are the results of citations of multi-authored works which are attributed to certain countries only with the fraction related to a specific author. To deal with this we have added by default a single citation to all the fractional citations¹⁵.

The last restriction we have imposed on our data regards the way we intend to interpret our results. Since the basic stylized fact we want to recover from the MultiRank and MD-HITS algorithms is the division of countries in two groups, developed and underdeveloped countries (a north-south view of the word), we need our layers to be oriented in the right way: the concentration of flows toward some countries must reflect their centrality as developed nodes in the network. To achieve this we have inverted the international aid layer, as we said before, and the migration stock one which, by construction, tells us how many persons from the source country migrated to the destination one. On the contrary in a more north-centric view of the world we will need to know how many migrants the source country hosted from the destination one, since migration flows usually points from south countries to north ones.

Results of the MultiRank algorithm with respect to changes in each of the three parameters a,s,γ (for nodes only the top 18 elements are shown) are reported in Fig. 18a and b. We can see that choosing a different combination of parameters alters the final ranking of nodes and layers and that some combinations experience more abrupt changes than others. While for the MD-HITS we don’t need to provide any parameters in order to obtain the final rankings, for the MultiRank we need to choose how to specify them. In what follows we will show how we choose our two configurations.

In Fig. 5 one can see the two rankings of layers which are the results of the previous assumptions. Now the second step in the evaluation of the two configurations is to move to the node rankings deriving from them and compare their stability with respect to the parameter γ which from Fig. 18 seems to create disruptions in the rankings even when keeping fixed the other two parameters. We would like to have two configurations of the MultiRank which are not heavily affected by the choice of γ in at least the majority of the range of its values.

Finally the third criterion to evaluate our choice of parameters is by comparing the similarity of the node rankings with respect to the one stemming from pcGDP. We would like our choice of parameters to have a good correlation (in rank) with pcGDP which will mean that our centrality measures are capturing a good signal from the underlying data. However we do not aim to completely reconstruct the ranking of nodes by pcGDP, otherwise our measures will not have an informative content.

Our test for these last two criteria is shown in Fig. 19. Here we have plotted the Spearman rank correlation coefficient (ρ) between the nodes ranking resulting from pcGDP and the one resulting from different centrality algorithms while we let the parameter γ take different values (x-axis). The coloured lines are 4 different configuration of the MultiRank while the dashed lines are the MD-HITS hubs and authority scores (here called MultiHub and MultiAuth) which by definition are not affected by γ. For robustness we have added the single layer version of the three previous measures (PageRank, Hubs and Authority) calculated on the aggregate multiplex obtained by summing over all the layers the corresponding entries of their adjacency matrices, which are constant too.

We observe in fact that in the 2003 cross section the combinations with a=1 are rewarded, while for a=0 the Spearman correlation with pcGDP ranking is lower. On the contrary in the second cross section combination with s=−1 are rewarded irrespective of the magnitude of a.

Even though in these last cases we obtain a better fit with respect to pcGDP, still choosing s=−1 which corresponds to assume that low centrality nodes are more important is difficult to justify without further evidence and does not account for a sufficient improvement in fitting in cross section 2010. Moreover the combination a=1, s=−1 which would be ideal in both cross section is the more affected by the choice of γ in cross section 2010, hence making it too volatile. Finally for γ=1 has the same fitting to pcGDP ranking as our second choice a=0, s=1.

Hence we have two configurations of the MultiRank which are worth analyzing since they stem from sensible assumptions on the data and are sufficiently stable with respect to γ. Once we fix the parameter γ to 1, one of them, (a=1,s=1), has the best fitting with respect to pcGDP in cross section 2003, while the other (a=0,s=1) represents a good second best option in cross section 2010. Similar trade-offs could have arises by choosing one of the other combinations, which however have stronger implication that needs further empirical justifications.