Detecting early signs of the 2007-2008 crisis in the world trade

Since 2007, several contributions have tried to identify early-warning signals of the financial crisis. However, the vast majority of analyses has focused on financial systems and little theoretical work has been done on the economic counterpart. In the present paper we fill this gap and employ the theoretical tools of network theory to shed light on the response of world trade to the financial crisis of 2007 and the economic recession of 2008-2009. We have explored the evolution of the bipartite World Trade Web (WTW) across the years 1995-2010, monitoring the behavior of the system both before and after 2007. Our analysis shows early structural changes in the WTW topology: since 2003, the WTW becomes increasingly compatible with the picture of a network where correlations between countries and products are progressively lost. Moreover, the WTW structural modification can be considered as concluded in 2010, after a seemingly stationary phase of three years. We have also refined our analysis by considering specific subsets of countries and products: the most statistically significant early-warning signals are provided by the most volatile macrosectors, especially when measured on developing countries, suggesting the emerging economies as being the most sensitive ones to the global economic cycles.


Additional measures for binary, undirected, bipartite networks
In the following subsection, we provide the explicit definition of two additional topological quantities, in order to corroborate our findings with the analysis of more traditional network quantities. By applying the same kind of analysis presented in the main text, we show that early-warning signals of the crisis are detectable even analysing nestedness and assortativity.
Nestedness. Loosely speaking, the degree of "triangularity" of the observed biadjacency matrix can be quantified according to a number of measures recently proposed under the common name of nestedness. [3][4][5][6] Here we adopt the one proposed in, 3 called NODF, an acronym for "Nestedness metric based on Overlap and Decreasing Fill".
Since the total value of nestedness (NODF t ) is the weighted average of the contribution from rows (NODF r ) and the contribution from columns (NODF c ), we have considered all of them for the present analysis. Naturally, the adopted measure of nestedness does not depend on the rows and columns ordering criterion. 3 If we indicate with N c the number of columns and with N r the number of rows, then               whose row-and column-specific contributions are provided by where T C cc = Assortativity coefficient. The second global quantity we have considered to characterize the WTW evolution is the assortativity measure r ∈ [−1, 1] proposed in, 7 with r = 1 indicating the maximum observable correlation between degrees (thus measuring the strongest tendency of links to connect nodes with similar degrees) and r = −1 indicating the minimum observable correlation between degrees (thus measuring the strongest tendency of links to connect nodes with different degrees). The definition of the assortativity coefficient can be found in. 7 By making it explicit which links contribute to the sums at the numerator and at the denominator, such a coefficient can be rewritten more clearly in terms of the bipartite nodes degrees, as  Let us now carry on the comparison between the aforementioned observed trends and their expected counterparts under the BiCM, by plotting the corresponding z-scores. As shown by fig. S2b, NODF t and NODF c are always consistent with our null model, even if they gradually come closer to the z = −2 border as 2007 approaches. Again, the rate of increase of randomness across and after 2003 is notable. In particular, the total NODF spans one sigma of statistical significance in just four years (2003)(2004)(2005)(2006)(2007), the same range having been spanned across the previous nine years.
As for their observed counterparts, the assortativity z-score is characterized by an opposite trend; in particular, it provides a clear statistical signal in 2003 by crossing the significance bound z = −3. This means that the degree of assortativity of the network becomes less and less significant, to become compatible with the BiCM-induced random value four years before 2007; it then steadily rises from 2003 until 2008 and seems to maintain such a value afterwards.
A more evident signal, confirming the increasingly random character of the network, is provided by the z-score of the row-specific NODF, whose analysis allows us to clearly distinguish two distinct phases, the first one lasting from 1995 to 2003 (characterized by a decreasing trend) and the second one lasting from 2003 to 2010 (characterized, instead, by an almost constant trend). The biennium across 2003 seems to constitute a somehow crucial period, defined by a decrease of statistical significance of two sigmas (from z 3 to z 1); analogously to what already observed for NODF t , the same loss of statistical significance (from z 4 to z 2) was spanned by NODF r in the preceding eight years.
As for motifs, the z-scores trends considered so far agree in pointing out the peculiarity of 2003 as the year in which the network gets through two different regimes, from a "structured" phase, not compatible with our null model, to a increasingly 3/5