Worldwide spreading of economic crisis

We model the spreading of a crisis by constructing a global economic network and applying the Susceptible-Infected-Recovered (SIR) epidemic model with a variable probability of infection. The probability of infection depends on the strength of economic relations between the pair of countries, and the strength of the target country. It is expected that a crisis which originates in a large country, such as the USA, has the potential to spread globally, like the recent crisis. Surprisingly we show that also countries with much lower GDP, such as Belgium, are able to initiate a global crisis. Using the {\it k}-shell decomposition method to quantify the spreading power (of a node), we obtain a measure of ``centrality'' as a spreader of each country in the economic network. We thus rank the different countries according to the shell they belong to, and find the 12 most central countries. These countries are the most likely to spread a crisis globally. Of these 12 only six are large economies, while the other six are medium/small ones, a result that could not have been otherwise anticipated. Furthermore, we use our model to predict the crisis spreading potential of countries belonging to different shells according to the crisis magnitude.


I. INTRODUCTION
A global economic crisis, such as the recent 2008-2009 crisis, is certainly due to a large number of factors. In today's global economy, with strong economic relations between countries, it is important to investigate how a crisis propagates from the country of origin to other countries in the world. Indeed, several significant crises in the past few decades have been originated in a single country. However, it is still not clear how and to what extent domestic economies of other countries may be affected by this spreading, due to the interdependence of economies [1]. Here, we use a statistical physics approach to deal with the modern economy, as it has been done successfully in the recent years for the case of financial markets and currencies [2][3][4][5][6][7][8][9][10]. More precisely, we view the global economy by means of a complex network [11][12][13], where the nodes of the network correspond to the countries and the links to their economic relations. For both networks we are able to locate a nucleus of countries that are the most likely to start a global crisis, and to sort the remaining countries crisis spreading potential according to their "centrality". Initially, a crisis is triggered in a country and propagates from this country to others. The propagation probability depends on the strength of the economic ties between the countries involved and on the strength of the economy of the target country.
Our results show that, besides the large economies, even smaller countries have the potential to start a significant crisis outbreak.
The CON is a network that connects 206 countries around the globe, using as links the ownership relations within large companies. If companies listed in country A have subsidiary corporations in country B, there is a link connecting these two countries directed from country A to country B. The weight of the link, w AB , equals the number of the subsidiary corporations in country B controlled by companies of country A. Next, if companies from country B have subsidiary corporations in country C, then again there is a weighted link, w BC , connecting these two countries directed from B to C, and so on. This way we obtain a network with total 2886 links among 206 nodes (countries). Of these links 685 are bidirectional, meaning that if there is a link from node i to j, as well as a link from node j to i, and the rest 1516 are one directional only.
We assume that the total link weight between a pair of nodes (countries) ij is the sum of all links independently of their direction, w (ij) tot = w ij + w ji . The total link weight represents the strength of the economic ties between two countries in the network. We quantify the total economic strength of a country i by its total node weight,w i tot = j w ij + j w ji , i.e., summing the weights of all links of node i. The probability density distributions of the total node weights and of the total link weights is skewed and heavy tailed, as shown in Fig. S1 in the Supplementary Information. We find an almost linear relation betweenw i tot and the GDP of country i, (as shown in supplementary Fig. S2) which indicates that the total weight of a country in our network is strongly correlated to a traditional economic measure.
The ITN is calculated from the second database after we aggregate the trade actions between all pairs of countries. Using the trading relations between each pair of countries e.g., A and B, we can create a bi-directional network where E AB represents the export of A to B, and E BA represents the export of B to A. Of course E AB is equal to I BA , which stands for the imports of B from A. In accordance to the above notations, the total link weight is given by w (ij) tot = E ij + E ji , but the total node weightw i tot which quantifies the economic strength of a node equals to its GDP value.
To identify the uneven roles of different countries in the global economic network, we use the k-shell decomposition and assign a shell index, k s , to each node. The k-shell is a method identifying how central is a node in the network, the higher its k s the more central role the node is considered to have as a spreader [14][15][16]. The nodes in the highest shell, called the nucleus of the network, represent the most central countries. To determine the k-shell structure we start by removing all nodes having degree k = 1, and we repeat this procedure until we are left only with nodes having k ≥ 2. These nodes constitute shell k s = 1. In a similar way, we remove all nodes with k = 2 until we are left only with nodes having degree k ≥ 3. These nodes constitute k s = 2. We apply this procedure until all nodes of the network have been assigned to one of the k-shells. With this approach we view the network as a layered structure. The outer layers (small k s ) include the loosely connected nodes at the boundary of the network, while in the deeper layers we are able to locate nodes that are more central. An illustration of this structure is shown in Fig. 1.
To identify the nucleus of CON we consider in the k-shell the network having only links with weights above a cut-off threshold w c . By using different threshold values we locate different nuclei of different sizes, as shown in Fig. 2 When performing the k-shell decomposition method on the ITN we locate a nucleus at k s = 10 composed of 11 countries, that is stable and always the same for w c ≥ 5100$M as shown in Fig. 2(B). Actually we may view it as a 12 country core, because the CHELEM database regards Belgium and Luxembourg as one trade zone, and therefore the core consists . This nucleus is almost the same as the nucleus found for CON. There are only two differences, which can be clearly understood due to the complimentary nature of the networks. The first difference is that Sweden (SE) and Switzerland (CH) are now located at shell 9 in ITN, which is one shell below of the core. The second difference is that China (CN) and Russia (RU) are now part of the nucleus, while in the CON they were located one shell and five shells away from the core, respectively. The presence of these countries in the nucleus of ITN, and their absence in the nucleus of CON can be Compared to the other shells, the countries in the nucleus are not only strongly interconnected among themselves, but also have many links to other nodes of the network. This is clearly demonstrated in Fig. 2(C) for CON and in Fig. 2(D) for the ITN. More specifically, for CON the 12 countries of the nucleus have significantly large average degree k = 139 ± 7 compared to the countries in the shells below, like k = 69 ± 9 for shell k s = 10. For ITN the average degree of the nucleus is k = 28 ± 2, while the average degree of shell k s = 9 is k = 12 ± 1. Surprisingly, not all 12 countries have the largest total weights or the largest GDP. Nevertheless, our results suggest that they do play an important role in the global economic network. For CON i.e. we find that six of the G8 members are part of the nucleus (except Russia and Canada that belong to lower shells), while the other six are smaller countries in absolute size (relative to the large countries). This is explained by the fact that these smaller countries do not support only their local economy, but they are a haven for foreign investments, as they attract funds from large countries for taxation purposes, safekeeping, etc. and a problem in such investments can easily lead to a chain reaction in other countries. Countries such as LU and CH, which are headquarters for some of the world's largest companies and subsidiaries, interact very strongly with a very large number of countries. For example, about 95% of all pharmaceutical products of the Swiss industry are not intended for local consumption, but for exporting.
Although, both CON and ITN are based on different databases, it is interesting that most of the countries are located in very close k-shells, supporting the robustness of our approach. This is shown in the scatter plot of Fig. 2(E) which maps the change in the k-shell ranking for both networks. The ranking is done in units of distance from the nucleus. We find that most of the countries are located in almost the same distance from the nucleus for both networks, since 82% of the countries are inside the two lines ( Fig. 2(E)), representing distance ≤ 1. The few countries that are in very different k-shells are usually large countries (in terms of population), which are very active in terms of trade but do not own many large corporations with global status, i.e, Turkey (TR) and India (IN).
It should be noted that when we apply the clique percolation method (CPM) [17], both CON for w c ≥ 100 and ITN for w c ≥ 5100$M, we obtain that the strongest connected communities are the same as the nuclei found using the k-shell method, as shown in supple- Next we study how an economic crisis can propagate on these networks. An economic crisis is a very complex phenomenon that cannot be reduced to an "all or nothing" situation.
In order to get some insight on the mechanisms of crisis spreading and on the role of the network topology in this spreading, we applied a Susceptible-Infected-Recovered (SIR)-type model. The SIR model has been used successfully to model spreading of epidemics in various networks [18,19]. The basic characteristic of SIR is that it usually assumes a fixed probability for neighbour-to-neighbour infection. In our case we assume a probability which depends on the economic weights of the links and the strength of the targeted country (see Eq. 1).
Initially all nodes are in the susceptible (S) state. We chose one node (country) and set it to be in a crisis (infected) state (I). During the first time step it will infect all its neighbouring nodes with probability calculated from Eq. 1, and the status of the infected nodes switches from S to I. This process is repeated with all infected nodes trying to infect during each time step their susceptible (S) neighbours. After each time step, the status of the original infected nodes changes to recovered (R) and can no longer infect or become infected. In our economic crisis epidemics the recovery means that a set of successful measures has been applied and the country overcomes the crisis. The simulation stops when there are no more infected nodes, or when all the nodes have been infected.
We assume for both, CON and ITN, that the probability p ij that node i infects its neighbouring node j depends on the total weight of the link, the total weight (strength) of the targeted node, and on the magnitude m of the crisis, as follows The ratio w (ij) tot /w j tot represents the relative economic dependence of country j on country i, which we consider as a factor in the probability that country j will be infected by country i. The factor m represents the strength of the crisis and can obtain any positive value.
In economic relations directionality plays an important role. For CON i.e., a crisis in a subsidiary could create some problems in the mother company, but if there is a crisis in the mother company its effect on the subsidiaries is always significantly more severe. To account for directionality, we study the case when we keep only one link, directed or undirected, for each pair of connected nodes. To this end we calculate for all pairs of nodes the quantity If T is smaller than a certain threshold value, which is a parameter, then the link will be regarded as undirected. Otherwise, the link is directed pointing from i to j if w ij > w ji , and from j to i if w ij < w ji . In all cases the weight of the link is w In the current study we set the threshold value of T to be zero. By increasing this value, we increase the percentage of undirected links in the resulting network, and if we set T = 1 then the entire network is undirected. Our results are not sensitive to the value of T. We find that for all values of T < 0.5 the crisis epidemics is nearly the same as shown in supplementary Fig. S4.
In Fig. 3  In Fig. 4(A) we show the world countries colored according to the shell they belong in CON, and in Fig. 4(B) we show the countries affected by the economic crisis that started in 2008 in USA, which has spread globally. In Fig. 4(C) we show a prediction of the model for CON. We simulate an infection of a crisis starting in the US, using m = 4.5, which leads to roughly the same percentage of infected countries as the actual crisis of 2008-2009 (∼ 90% of the world countries reported economic slowdown due to the crisis). The percentage of correctly predicted countries infected by the crisis using the model is above the average percentage of the prediction using random selection. A quantitative agreement between the prediction power of our model and the actual infection strength is shown in Fig. 4(D).
Considering the example of BE (ranked 29th according to its total GDP), we find that a crisis originating in this country with magnitude m = 4.5 is able to affect for CON almost 60% of the world countries (average result of 50 realizations), while the worst case

II. DISCUSSION
We model the way an economic crisis could spread globally, depending to the country of origin. We first create two global networks, which describe strong interaction patterns of the world economy. The first network (CON) is based on the world's largest companies and all their subsidiaries and links together 206 countries, and the second network (ITN) is created using aggregated trade data linking together 82 countries. A crisis is triggered with a controlled magnitude and propagates from one country to another with a probability that depends on the strength of the economic ties between the countries involved and on the strength of the economy of the target country. Furthermore, using the k-shell decomposition method we are able to identify the role of the different countries in a world crisis, according to the shell they belong, and we identify the 12 most effective countries for crisis spreading.
We find that although a recent global economic crisis originated in the USA, this might not be always the case and even smaller countries have the potential to start similar crisis outbreaks. In retrospect, we know that this has happened several times in the past (e.g. the crisis in Indonesia), and is happening these days as well, with a crisis in Greece 2 threatening to cause an avalanche effect to other European economies. In part III of the supplementary information we examine in more detail the case study of Greece, where we show that the fears of contagion of other major European economies by the Greek crisis are justified. We report almost 40% probability of infection for the major EU countries only through the contagion channels modeled using CON and ITN i.e. not taking into acount the loans Greece received from other countries. This is in contrast to the common belief that smaller countries partake only in crises that are locally contained, but do not spread out to affect the larger countries.
Thus, our findings show the predicting power of the network approach using the k-shell ranking methodology. This analysis is important when establishing international commerce rules, policies and legislations, trade treaties and alliances, as they can have a serious impact on monetary policies and international affairs.

PART I
Results about Corporate Ownership Network (CON).
As described in the article, we construct CON by using the ORBIS database.

PART II
Results about the International Trade Network (ITN). To be consistent with the notation used about CON, we define that ij ij w E ≡ .
Therefore, the total link weight between a pair of nodes (countries) ij is calculated by ( ) ij tot ij ji w E E = + . This is the actual measure of the total trade flow between node I and node j. Furthermore, we calculate the trade balance (a quantity describing the balance of payment between each pair of countries) . Note that the trade balance is the numerator of Eq. (2) and is the factor that creates the directionality of the network.
In order to study the propagation of crises in the ITN, we applied the SIR model that is described in the manuscript. Similar to the implementation we used for CON, we assume that the probability ij p that node i (a country in crisis) infects node j (a country not yet affected) is given by As it is presented here, CON is more sensitive in comparison to the ITN, and it predicts an easier propagation of a crisis to countries in the neighbour of Greece (South-Eastern Europe). These countries traditionally have strong economic ties with Greece.