Population Destruction X

STABILITY OF THE WORLD TRADE WEB OVER TIME – AN

EXTINCTION ANALYSIS

NICK FOTI†, SCOTT PAULS∗, AND DANIEL N. ROCKMORE†,∗,#

Abstract. The World Trade Web (WTW) is a weighted network whose nodes correspondto countries with edge weights reflecting the value of imports and/or exports between coun-tries. In this paper we introduce to this macroeconomic system the notion of extinctionanalysis, a technique often used in the analysis of ecosystems, for the purposes of investi-gating the robustness of this network. In particular, we subject the WTW to a principledset of in silico “knockout experiments,” akin to those carried out in the investigation offood webs, but suitably adapted to this macroeconomic network. Broadly, our experimentsshow that over time the WTW moves to a “robust yet fragile” configuration where is itrobust under random attacks but fragile under targeted attack. This change in stabilityis highly correlated with the connectance of the network. Moreover, there is evidence ofsharp change in the structure of the network in the 1960s and 1970s, where the most mea-sures of robustness rapidly increase before resuming a declining trend. We interpret theseresults in the context in the post-World War II move towards globalization. Globalizationcoincides with the sharp increase in robustness but also with a rise in those measures (e.g.,connectance and trade imbalances) which correlate with decreases in robustness. The peakof robustness is reached after the onset of globalization policy but before the negative im-pacts are substantial. In this way we anticipate that knockout experiments like these canplay an important role in the evaluation of the stability of economic systems.

1. Introduction

In this paper we introduce new methods to articulate measures of robustness in economicnetworks. In these and other living complex systems researchers interested in studying ro-bustness generally do not have the luxury of performing in vivo experiments to test hypothe-ses. Consider the example of an ecosystem: we cannot remove species in order to explorethe impact of their sudden extinction. Similarly, we cannot remove or disable components ofan economy to discover the downstream effects. However, given a sensible model for a realsystem, we do have the ability to perform simulations, i.e., in silico experiments designedto shed light on the interdependence of the components and the implications for robustnessto sudden component failure. The integration of network methodology to economic situa-tions is of great current interest ([18]), particularly with regard to stability and contagion([1, 3, 7]). This is the sort of methodology that we aim to bring to economic systems, and inparticular, the World Trade Web (WTW). The WTW is an economic network summarizinginternational trade. Nodes represent countries and country B links to country A with aweight given by the value (say in US dollars) that country A receives from country B inexports. With the rise of network analysis the WTW has received a good deal of attentionand analysis (see e.g., [14, 13, 19]).

The recent network analyses of the WTW have provided a wealth of initial informationregarding the interaction of the network structure and the functional architecture of the

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WTW. Our goal is to extend this analysis and craft measures robustness stability thatare products of intertwined structure and dynamics. Our core methodology is that of aknockout experiment, where stability is tested via perturbation of the network by removalor deprecation of some aspect of the structure. We then observe the system as it returns toequilibrium and record the effect.

Examples of studies in which these kinds of knockout experiments have already been doneincludes those performed in the context of the World Wide Web (WWW) [2], metabolicnetworks [17], protein networks [16], and, of particular relevance to this paper, extinctionanalyses conducted on food web models of ecosystems (see e.g., section 4.6 of [11] and themany references therein as well as [4]).Food webs are network models of ecological systems(see [11] and references therein) where the nodes represent species in the ecosystem anda directed link is placed to node A from node B if species A eats species B (i.e., if thereis a transfer of resource from B to A). Appropriate dynamics, derived from field data,are applied to simulate reasonable population fluctuations. In this setting, stability androbustness are analyzed via simulated extinction studies in which certain species or groupsof species are removed from the ecosystem. “Evolution” then takes place according tothe simulated extinction dynamics. The system either reaches a new stable configuration(possibly after one or more consequent extinctions) or collapses entirely. Such studiesaddress questions of robustness by measures of the existence and strength of extinctioncascades induced by the knockout of particular species.

The analogies that might be drawn between food webs and economies are of great inter-est [15]. In this paper we bring the idea of extinction analyses and consequent robustnessanalysis to the WTW. We provide three types of extinction experiments. First, we consideran extinction analysis similar to that used in the study of food webs where countries aresequentially removed from the trade web and their impacts analyzed. Second, we considera variant of the first where rather than removing countries, we instead perturb their im-port/export profile. Last, we consider deletion of edges in the network - extinction of traderelationships - and analyze their impact. We emphasize that in these analyses the resultsmust be interpreted as consequences of both the network structure of the WTW as well asthe evolution dynamics we place on that structure. In particular, more or less refined mod-els of evolution dynamics may yield more or less textured results. We view the dynamicsdescribed below as a parsimonious first model on which subsequent analyses can build.

For the first analysis, we define maximal extinction analysis (MEA) for the study ofthe WTW. The mechanism of MEA is analogous to the kinds of knockout experimentsperformed on food webs, but differ in the formulation of the extinction dynamics (i.e., inthe hypotheses that dictate the consequences of node removal for the reduced network).Our methodology is described in detail in the next section. The output of the simulation isa measurement of the extinction power of any given country over any other. Roughly, theextinction power of country A over country B is the proportion of B’s economic activity (asmeasured by export data) disrupted by country A’s extinction (as well as any extinctionswhich are a consequence of A’s). We view this as a measure of “economic centrality.”

To create an aggregate statistic, we define the robustness of a trade network analogouslyto that of a food web, namely the proportion of the total income in the network that isdestroyed via node deletion and consequent extinctions to reach the loss of 50% of thenetwork. This notion of robustness is a measure of stability of the trade network - itmeasures the extent to which countries can replace lost imports and lost demand for exports

STABILITY OF THE WORLD TRADE WEB OVER TIME – AN EXTINCTION ANALYSIS 3

given a shock in the system. In tracking robustness over this period 1870–2006, we see twobasic results.

(1) There are two sharply different periods, around World War I (1914-1919) and WorldWar II (1939-1948), where the robustness is dramatically higher than at any othertime. In both of these cases, the data sets have substantial holes - many unreportedtrade figures - which skew the results. As such, we disregard these two periods fromthe qualitative analysis.

(2) Other than the periods around the two world wars, the time period breaks naturallyinto two periods, roughly 1870–1975 and 1976–2006. The transition is marked by asharp increase in robustness in the 1970s.

(3) Over these periods, robustness follows a generally decreasing trend. The decreasein the first period (1870-1938) is statistically significant (R2 = 0.26, p = 0.00001 fora linear fit), the decrease in the second (1947-1974) is not (R2 = 0.07, p = 0.17),while the decrease in the third period (1975-2006) is (R2 = 0.33, p = 0.0006).

The transition coincides with a commonly understood move away from the anti-globalizationpolicies initiated after the Great Depression [10]. These results support the claim thatincreased globalization corresponds to increased robustness. But what are possible expla-nations for the downward trend after the transition? In [12], we find a similar extinctionanalysis performed on food webs, showing a significant positive correlation between robust-ness of their networks and connectance. Connectance is a measure of the completeness ofthe network and is defined as the proportion of existing connections in the network dividedby the maximum number of possible connections (

(n2

)for a network with n nodes). For the

full period, we find a significant negative correlation between robustness and connectance(R2 = 0.5, p = 1.2 × 10−21)1. Moreover, we find a number of different network statisticswhich also have significant negative correlation with robustness in the period after transitionof which we highlight one, maximum trade deficit (R2 = 0.39, p = 0.0001).

In food webs, the hypothesis is that increased connectance aids robustness by provid-ing, on average, more feeding opportunities for species in the face of the removal of otherspecies. We posit that the trade networks have the opposite property due to the natureof our model dynamics. Extinctions create a ripple effect in the network, decreasing theincome of countries who have lost exports due to an extinction as well as decreasing theavailability of goods. This ripple effect can propagate more easily and quickly through amore highly connected network. The link between high trade deficits and robustness canalso be understood as an aspect of this same reasoning — imbalances between importsand exports again propagate (and potentially magnify) via our dynamics. The impact ofdeleted nodes with negative trade balances (i.e., deficits) propagates primarily in the formof decreased demand which, in turn, lowers the aggregate income.

We interpret these results in terms of the trend of increased globalization since the 1970s.Our results provide evidence for the hypothesis that globalization brings with it an numberof effects, some which boost robustness of the network while other that have a negative effect.The results point to increased connectance and the existence and rise of trade deficits astwo factors that have negative effects but that coincide with the policy movement towardsglobalization. These factors, in aggregate, have the effect of dramatically increasing therobustness at the outset of the policy shift with a slow decline afterwards, as the statisticsassociated with the negative effects grow.

1This is smaller than the precision tolerance for MATLAB.


For our second analysis, we provide an algorithm for impact analysis of perturbing theimport/export profile of a given country. Again, the precise details are given below, butthe basic idea is to change the import totals by a fixed percentage and the export totals byanother and then to use the similar iterative dynamics as in the maximal extinction analysisto measure the impact. As an example, we perform this analysis on the 2006 trade webwhere we systematically alter each country’s export profile by decreasing exports by 5% andimports by 30%. This is meant to model a shock to a country’s economy which depressesexports, such as Thailand’s economic crash after the currency crises in the late 1990s whereimports and exports dropped by roughly 30% and 5% respectively. The results of theanalysis are measured in the aggregate impact to the world income (due to exports). Wefind two interesting observations from this analysis. First, we recover perhaps unsurprisingmajor impacts - large players in international trade have, in general, the largest impact, withthe China, Germany, the United States, Japan and France playing central roles. Indeed, thisperturbation of China creates substantial global impact according to this model, causingat least a 1% drop in income for 94% of countries. Similarly, many of the weakest and/ormost isolated economies are the most vulnerable to such perturbations. For example, manyisland nations - Vanuatu, the Soloman Islands, etc. - are among the most vulnerableas well as countries such as Qatar, Kuwait, the Philippines, Monaco and Liechtenstein.To examine this longitudinally, we compute the maximum vulnerability percentage - themaximum over all countries of the percentage of countries whose perturbation results in agreater than 1% decrease in income of a specific country. This statistic exhibits a transitionroughly at 1960 and in the two periods (again omitting the periods around the WorldWars), the maximum vulnerability has a significant correlation with connectance (1870-1959, R2 = 0.56, p = 1.9−14; 1960-2006, R2 = 0.56, p = 1.6−9). We interpret this connectionas positive relationship between connectance and robustness - as connectance grows, themaximum vulnerability decreases.

For our third analysis, we consider the removal of edges from the trade network andinvestigate their impact. This type of experiment is meant to model situations where atrade relationship ceases - for example, a war between the two countries. As examples, weclosely analyze the impact of edge deletions for a single years, 2006 and 1965, as well asmore coarsely analyze the impact of edge deletion for our entire range of data, 1870-2006.For the single year deletions, we measure the strength of the edge as the percentage of theworld income which is removed consequent to the deletion. We see that, in both cases,the strongest trade link is between the United States and Canada, whose deletion createsa 4.18% drop in 2006 and a 3.57% drop in 1965. These two snapshots allow us to see somedetailed movement over time. In both cases, links between large economic players are themost significant. However, we see substantial changes between these two years. First, in2006, a umber of Asian countries such as China, South Korea, Taiwan and Japan figureprominently in the list of edges with highest impact. In 1965, however, they are largelyabsent (expect Taiwan). This is, of course, reflective of the growth of economic power ofthese countries over that period. We also see change in the power of specific links. Forexample, the link between Mexico and the United States creates a 0.52% drop in income in1965, but in 2006 it is the second most powerful link, creating a 2.87% drop. This reflectsboth the growing economic power of Mexico but also the growing intertwining of the twoeconomies.

For the longitudinal analysis, we measure the strength of the most powerful edge in eachyear in the same terms as well as the same statistic normalized by edge weight. We see


a trend that contrasts with the trend shown in the MEA. Specifically, we see a generaldownward trend in the normalized statistic, which we interpret as a growing robustnessof the network in the face of this type of extinction. However, the unweighted statisticpresents more complicated behavior - first decline with connectance and then growing asconnectance continues to increase. Moreover, we see a transition period after World War IIwhich echoes the similar transition found by the MEA in the mid-seventies.

Taken together, these analyses provide us with a more complete picture of the robust-ness of trade webs as a consequence of their network structure and evolution dynamics.This combination of the topology of the network, as measured by the connectance, andits dynamics, given by the income model, create networks which have a dual “robust yetfragile” character. More precisely, the MEA, a targeted attack analysis, shows decliningrobustness over time which is correlated with connectance. The perturbation analysis -again a milder form of targeted attack - shows growing robustness over time with a sharpjump and then continued rise post World War II. The edge removal analysis shows thatthese networks become more and more robust to random attack over time as indicatedby the dropping influence over the world income. Again, this measure is correlated withconnectance. However, there are still edges which, if targeted, have consequential effects.

Together, these shed light on the consequences of globalization and the liberalizationof trade. We see globalization reflected in increasing connectance - an increasing averagenumber of trading partners per country. The increased connectance has two opposingconsequences. First, it correlates with decreasing robustness when using the MEA. Second,it decreases the maximum vulnerability. And third, on average it decreases the power ofindividual edges, which increases the overall robustness of the system when considering edgedeletions. These results are all evidence of the claim that these trade networks are “robustyet fragile” - they are vulnerable to targeted conscious attack but stable under randomattack.

2. Methods

As our basic data source, we use import/export tables available from [6] which detail thetrade relationships between countries from 1870-2006. These tables detail the amount ofgoods (in US dollars) imported or exported from one country to another and are presentedas matrices which we denote as M for the import matrix and N for the export matrix.Thus, Mij is the dollar amount of imports to country i from country j while Nij is thedollar amount of exports from country i to country j. While it should be the case thatM t = N , due to the variation in reporting practices, this is not so. For simplicity, weanalyze only the import matrix M and in this way define the weighted directed WTW.

This data is collected and aggregated from a number of sources and has, in some years,substantial missing data. In general, completeness of the data reported increases with time.More specifically, we note that during World War I and II the missing data is substantial andobvious - many of the world’s largest economies at the time make no report of trade betweenthemselves and their former (and later) large trading partners. This is perhaps unsurprisingas we suspect these countries had higher priorities than import/export reporting in theseperiods. But, as the resolution of the data is particularly low during these periods, wecannot infer much from our analyses during these periods. As such, we exclude them fromthe interpretive analysis. However, we can still consider them as abstract networks fromthe point of view of attempting to find relations between network statistics and robustness.


Any sort of extinction analysis in the WTW requires the definition of the consequencesof node removal. Our working assumptions are as follows:

(1) Node deletion creates an “excess” of dollars (supply) given by the goods that thedeleted countries would have purchased from other countries.

(2) Node deletion creates an unmet demand in the form of goods the other countriespreviously purchased from deleted countries.

2.1. Income model. To understand the effect of the removal of one or more nodes, weneed to instantiate dynamics on the network to model how the trade network rebalancestrade after such a shock. To do so, we introduce an income model on the network. Thebasic economic assumption is that as a country’s income from selling its exports increases,its demand for products (in the form of imports) increases as well. Thus, if a country isdeleted it both decreases demand and supply for its trading partners but also increasesdemand for other countries via the unmet demand of its trading partners.

As a first step in concretely modeling this dynamic, we define a country’s propensity tospend based on the in-degree and the out-degree of the original import matrix. Recall theirdefinition:

Definition 2.1. Let M be an import matrix. Then, the in-degree for node i is

DM (i) =∑j

Mji.

Similarly, the out-degree for node i is

OM (i) =∑j

Mij .

In the context of the import matrix, the in-degree DM (i) is simply the total dollar valueof goods imported by other countries from country i. The out-degree OM (i) is the totaldollar value of goods imported by country i from other countries. For our purposes, thein-degree gives a measure of the income of country i while the out-degree gives a measureof its expenditures.

We define the propensity to spend for country i as

(1) αi =

{OM (i)DM (i) if DM (i) ≥ OM (i)

1 otherwise

In the case when a country spends more than it earns (DM (i) < OM (i)) we assume thatthe internal economy of the country is producing excess dollars (by some means) that areused to fund additional import spending. To record this we define,

(2) βi =

{OM (i)−DM (i) if DM (i) < OM (i)

0 otherwise

Next, we define a stochastic matrix m associated to the import matrix M via

(3) mij = Mij/OM (i).

We call m the propensity to import matrix. In mathematical terms, it is simply the Markovchain associated to M . Note that with these definitions,

OM (i) = αiDM (i) + βi

M = diag(OM )m(4)


where diag(v) for a vector v is the diagonal matrix with the entries of v along the diagonal.We now model the iterative buying and selling among trade partners as a two-step up-

dating rule. Let the import matrix at time t be denoted as the matrix Mt. Then, the vectorof incomes It is given by

(5) It = diag(α)MTt 1 + β

where α = (α1, . . . , αn)T , β = (β1, . . . , βn)T and 1 = (1, . . . , 1)T . To iterate we calculatethe next import matrix from the income:

(6) Mt+1 = diag(It)m.

We can then repeat, forming It+1,Mt+2, . . . . Note that from (4), we see that if M is animport matrix and α, β and m are derived from M , then M (and the resulting incomeassociated with M) is a fixed point of this iteration. Moreover, if we fix m,α, and β, wecan solve the system for fixed points, giving the equilibrium income as

I = −(diag(α)mT − Id)−1β

where Id is the identity matrix2. Note that in a limiting case where α = (1, . . . , 1)T , β =(0, . . . , 0)T the equilibrium income is simply the fixed vector of mT . It is in this sense thatwe see the income model as a perturbation of a Markov process.

From this we can describe how the income model responds to shocks on the system viathe following algorithm:

(1) Let M0 be the initial import matrix and calculate α, β and m from (1),(2), and (3).Let I0 be the unshocked income vector given by (5) using M0.

(2) Let M1 be a modified version of M0 encoding the desired shock. Adjust α, β andm to reflect the modification (this will be detailed in each specific case below). Lett = 1.

(3) Calculate It via (5).(4) Calculate Mt+1 via (6).(5) Increment t and repeat the last two steps.

Some comments are in order. First, we emphasize that α, β and m are fixed for the itera-tion steps of the simulation. We envision the simulation as an approximation of a short timerebalancing. Thus, we do not allow the fundamental constants — the profile of countriesthat a country trades with and in what proportion, the propensity to spend and the amountof internal income — to change. Second, we have an explicit assumption of substitutability.We recognize the shortcomings: imports and exports of wildly different items are lumpedinto the aggregate statistics. Analogous assumptions are made for extinction models forfood webs - prey are assumed to be interchangeable (and available) for a given predatorspecies. In our setting, this assumption provides the easiest path of rebalancing the systemafter a shock — thus providing the network with the best possible chance of recovery. Inaddition, one of our assumptions in the model dynamics helps mitigate the assumption ofsubstitutability: as we do not allow countries to form new trading partners during the simu-lation, countries cannot secure available goods from other countries but only more (or less)goods from their original partners. As the majority of countries have multifaceted trading

2The invertibility of the matrix is equivalent to the fact that diag(α)mT does not have a fixed vector.This is true in most cases because the maximum eigenvalue of mT is 1 and the α ≤ 1. The only case wherethis isn’t true is if α = 1. In that case, the mT − Id is not invertible, but (mT − Id)I = −β can still besolved so long as β is not a multiple of the constant vector.


relationships — buying one good from many different countries — this is a proxy for theavailability of substitutable goods from existing trading partners. This is particularly trueof the larger economies in each network which, as we see in the simulation, have the mostimpact on its stability.

2.2. Shocks via node removal. In our investigations, we consider a shock created bydeleting nodes from the network. Using this framework, to create M i

1 from M0 = M bydeleting node i, we simply set Mir and Mri equal to zero for all r. Essentially, this preservesthe number of nodes in the network representation of the matrix (and allows for the matrixmultiplication steps in the algorithm above), but changes the connection structure.

We first perform a maximal extinction analysis, carried out according to the followingalgorithm:

(1) Fix the initial import matrix M and its associated income matrix I. For the firstiteration, let M0 = M , I0 = I.

(2) For each i, delete node i from M0 to form M1. Adjust m by removing the ith rowand column and renormalizing. Adjust α by setting α(i) = 0 and β by settingβ(i) = 0.

(3) Calculate {M0,Mi1, . . . ,M

i5}, {I0, Ii1, . . . , Ii5} via the income model.3

(4) Calculate

Pow(i) = 1−∑

j Ii5(j)∑

j I0(j).

We call this the total power of node i. It represents the percent of total income leftafter node deletion and rebalancing. Note: sometimes this number is larger thanone, signifying that the node deletion results in a net increase in income.

(5) Find the node j so that the total power is maximized:

j = arg maxk

Pow(k)

Add this node index to the list, D, of deleted nodes.

(6) Letting M0 = M j5 and I0 = Ij5 . Repeat steps 2−−5 this until the total income falls

below 50% of the total income of the original import matrix, i.e.∑k I

j5(k)∑

k I(k)< 0.5

From this algorithm, we measure robustness by computing the percentage of the initialincome we must remove to reach the 50% threshold:

R =

∑j∈D I(j)∑k I(k)

From its definition, 0 < R ≤ 0.5. Higher (resp. lower) values of R correspond to higher(resp. lower) robustness of the network to the maximal extinction process.

The removal of entire nodes is meant to create a ”worst-case scenario” - the removalof an entire country from the import-export economy. While this may seems drastic andunrealistic, it provides the most direct way of measuring the impact and power of a particularcountry over the entire system. Moreover, as the model is inherently linear, we would findsimilar results if we instead removed only a portion of a given country’s import/exportprofile. Similarly coarse assumptions are used in robustness analysis for food webs - for

3We repeated this procedure with 10 and 50 iterations as well. The results were qualitatively the same.


example, prey are assumed available in sufficient quantity to sustain predator populationsuntil extinction.

2.3. Shocks via node perturbation. The shocks described in the previous section areboth the most drastic types of shocks as well as the one of easiest to understand. In thissection, we present a more complicated type of shock based on perturbation of nodes ratherthan their entire removal. Such perturbation can be seen as models for intrinsic or extrinsicshocks to economies of individual countries which (temporarily) effect their import/exportprofile. Good examples of this are some of south-east asian economies during the currencycrises of the 1990s. For example, Thailand enjoy rapid growth in the late 1980s and earlier1990s - a so-called “tiger” economy - but rapidly declined due to a crises in the bhat in 1997-1998. The crises brought massive unemployment and economic hardships. One consequenceof this crisis that we see is a drop in the value of exports from and a substantial drop inimports to Thailand in the late nineties.

To model such types of shocks, we simply create a perturbation of the network createdby a fixed manipulation of a specific node’s import and export values. For example, we cancreate a two parameter perturbation of imports and exports. If we fix a, b ∈ [0, 1] and anode i, we define

(M1)jk =

(M0)jk if j, k 6= i

a(M0)ik if j = i

b(M0)ji if k = i

Using this M1, we must then adjust m,α, and β accordingly. For m, we scale the ith rowby a and then ith column by b and renormalize. We leave α unchanged except to multiplythe ith entry by b

a . We leave β unchanged. The choice to leave β unchanged is reflectie ofour desire to produce a model which reflects short term response to shocks.

Next, we follow the algorithm in Section 2.1 to compute the impact. As we noted above,if a = b, the results will be proportional to the full extinction of a node due to the linearityof the dynamics. When a = b, this model provides a method by which to evaluate testhypotheses.

2.4. Shocks via edge removal. Similarly to the previous section, we also model shockscreated by the termination of a trade relationship. In this case, we simply remove anedge from the import/export matrix. This models a situation where a trade relationshipis completely dissolved - for example, due to war, broken treaty, etc. If we denote the twocountries involved by i and j, then

(M1)kl =

{(M0)kl if (k, l) 6= (i, j), (j, i)

0 if (k, l) 6= (i, j), (j, i)

With this M1, we then adjust m by deleting the same entries and renormalizing the matrix.The values of α and β are unchanged. Then, following the algorithm in Section 2.1, wecompute the impact.

3. Results

Figure 1 shows the results of the robustness computation under the maximal extinctionanalysis over time. Outside of the years around World War I and II, we see that thereare two regimes split at roughly 1975. Before the mid seventies, the robustness is low anddecreasing over time. In the seventies, we see a rapid increase in robustness, and then the


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R2=0.07,p=0.17slope = −0.0019

R2=0.33, p=0.0006slope = −0.0021

Figure 1. Robustness scores for the WTW over time. Solid lines denotemean robustness of 50 trials of null models, dotted lines are the 95% and 5%cutoffs. Filled circles show years with robustness scores that fall outside the5− 95% interval.

.

resumption of the decreasing trend. In the period around World War II, we see a sharpincrease in robustness. As stated above, this is due to the sparseness of the reported datarather than related to an actual economic change. We note that around World War I,we have similar sparseness. But, in that case, while we see an uptick in robustness, thetransition is not nearly so sharp.

To better understand the significance of these results, we compare the robustness mea-surements to an appropriate null model. For the null model, we use a randomization of theimport matrix and repeat our maximal extinction analysis on the result. Repeating thismultiple times provides a family of null models for a given import matrix and a correspond-ing distribution of robustness scores. Figure 1 shows the robustness scores from 1870–2006plotted with the mean, minimum and maximum of robustness scores of 50 runs of each nullmodel. The robustness scores are coded by shape to indicate their significance. The scoresshown as filled circles are either larger than 95% or smaller than 5% of the correspondingscores for the 50 null models. Thus, according to this threshold, these robustness scoresare significantly different than randomized null models with the same number of nodes andtotal degree.

We can interpret this as follows. In the periods from 1870-1913 and 1920-1939, someaspect of the structure of the networks creates higher robustness than expected at randomin a number of years. Overall, we see slow decline of robustness over time. We then see asimilar picture of decline between 1949 and 1975, with a transition in 1975-1976. Shortlyafter the transition, the robustness increases to a level significantly above that of comparablerandom (Erdos-Renyi) networks but then again begins a decline. In the first periods, the


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Figure 2. Plots of robustness vs. connectance over the period 1870–2006.The inset graph show the time series for connectance for 1870–2006.

downward trend is statistically significant (R2 = 0.26, p = 10−5) while in the period from1949 to 1975, it is not significant (R2 = 0.07, p = 0.17). In the last period, 1976-2006, thetrend is again significant (R2 = 0.33, p = 6× 10−4). To attempt to understand the declineover time, we consider statistics with potential predictive power. Following the food webliterature, we consider connectance,

C =# of edges

(# of nodes)2.

We also consider the maximum trade deficit, i.e., the maximum of DM − OM for a givenimport matrix M .

In Figure 2, we plot the robustness against connectance. The inset graph shows the plotof the connectance over the entire time period, 1870–2006. We also plot the best linearfit of the data and provide the R2 and p values for the regression. The fit is statisticallysignificant with R2 = 0.5 and p = 1.2× 10−21. In this computation we include the periodsaround both World Wars. We do this as we are merely attempting to link the summarystatistics of robustness and connectance and are not (yet) interpreting the results in termsof actual trade relationships. So, as the trade webs in these periods are simply incompletetrade webs, they are still appropriate for inclusion in this analysis. We again note that thesharp drop in connectance around the World Wars is likely due to the spareness of the dataand the resulting incomplete trade web. We also note the smaller drop in connectance in theearly 1960s. This corresponds to a sudden increase in the number of countries in the tradewebs which, in turn, is a consequence of the wave of former colonial countries gaining theirindependence. Generally, as these countries gained independence, they entered the worldtrade network slowly - first trading with geographic neighbors and their former colonizingcountry.


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Figure 3. Plots of connectance, maximum trade deficits and maximumtrade surpluses from 1950–2006.

As mentioned in the introduction, potentially one way to understand this correlationis by considering our model dynamics, particularly in contrast to the food web analogue.Maximal extinction analysis for food webs includes no population dynamics. After a tar-geted extinction, consequent extinctions in the network only occur if a species no longerhas any prey (except, possibly, itself). In our situation, the income model provides simplelinear dynamics associated with the fundamental principles of supply and demand. Thesedynamics are one type of analogue to population dynamics on food webs. They allow forthe shock associated with node removal to propagate through the network, much like acontagion. As higher connectance generally implies faster propagation speeds through anetwork, it is not surprising that we see a significant negative correlation.

A similar argument applies as a possible explanation of the correlation between themaximum trade deficit and robustness. The deletion of a node with a substantial deficitagain propagates through the network but the deficit itself creates unused supply whichis not balanced (overall), by unmet demand. Irrespective of the network topology, thiscreates downward pressure incomes. The jump in the 1970s and then decline in robustnessover time tracks the increased connectance and the existence (and increasing size) of tradeimbalances. The regression analysis for robustness in terms of maximum trade deficit yieldsa significant result (R2 = 0.39, p = 10−4). It seems plausible that these are linked witha move towards policies of increasing globalization. As we see in Figure 3, the effects ofthese changes on maximum trade deficits and maximum trade surpluses are relatively mildat first before rapidly growing. The connectance decreases for a time before assuming anupward trend. Thus, one conclusion we may draw is that the positive effects on robustnessare eventually mitigated by the negative consequences of the other changes to the network.This is one plausible explanation for the peak of robustness, before a steady decline.

As an example of the second methodology, shocks via node perturbation, we consideredthe 2006 import/export matrix and perturbed each node sequentially with model parameters


No data0.0 - 1.0%1.0 - 4.2%4.2 - 8.3%8.3 - 11.4%

11.4 - 18.8%18.8 - 26.6%26.6 - 44.3%44.3 - 63.5%63.5 - 82.8%

(a) Power Percentage

No data0.0 - 0.5%0.5 - 1.5%1.5 - 2.5%2.5 - 3.5%3.5 - 4.5%4.5 - 5.7%5.7 - 6.7%6.7 - 7.8%7.8 - 8.8%

(b) Vulnerability Percentage

Figure 4. Two world maps with coloring indicating the power percentageand the vulnerability percentage for the WTW in 2006. Hatched countriesare ones where no data was available for the computation.

α = 0.7, β = 0.95. In other words we modeled a 30% drop in imports and a 5% drop inexports and analyzed the impact according to the model. These percentages roughly matchthe drops seen in the late nineties for Thailand as a result of their currency crisis. To createan aggregate statistic to help visualize the results of the simulation, for each country, wecounted the number of other countries whose income decreased by 1% or more when wesimulated the drop in imports and exports. If we divide this count by the total numberof countries, we call the result the power percentage. We also calculate the vulnerabilitypercentage, which is the number of countries who perturbation create a 1% or more decreasein income of a given country divided by the number of countries.

An initial review of the results show us something relatively unsurprising, that the largestplayers in the world economy - the United Stated, Germany, China, the United Kingdom, theNetherlands, France, Japan, Italy, Canada, and Belgium - have the most impact. Somewhatmore surprising is the list of most vulnerable countries, which include a number of southeastAsian countries, South American Countries, Japan, Australia, and Russia. Figure 4 showstwo world maps. The left map is colored by the power percentage while the map of theright hand side is colored according to the vulnerability percentage. This map shows thata small number of economies hold substantial power over the world trade web. Moreover,most countries are vulnerable to perturbations of their trading partners and some groupsof countries - notably those listed above - are especially vulnerable due to their patterns oftrade.

To get a sense of this method applied over time, we repeat the same experiment for eachyear in our data set and calculate the maximum power and vulnerability percentages foreach year. Figure 5 shows the results where, in each case, we have omitted the scores fromthe periods around the two World Wars due to data sparseness. In computing the maximumpower percentage over this time period, we find it is relatively uniform and large - alwaysabove 90%. In Figure ?? (a), we see that the maximum vulnerability percentage decreasesover time, which we interpret as growing robustness of the system. We also note thetransition after World War II which echoes the transition found in the MEA. In investigatingthe results of the MEA, we looked at the relationship between connectance and robustness.We do the same here (Figure 5 (b)), showing a strong correlation between connectance


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(b) Vulnerability vs. Connectance

Figure 5. (a) Vulnerability Percentage over time. The solid line is meanpercentage over 100 null model trials while the dotted lines are the 5% and95% cutoffs from the null models. The circles in black are years where thevulnerability percentage is outside the 5 − 95% range. (b) Vulnerabilitypercentage plotted against connectance.

and the maximum vulnerability percentage. However, there are two distinct regimes whichamplify our earlier observation of a transition - the black circles are years after 1960 while thewhite circles are before. In this experiment, we interpret this as linking growing connectancewith growing robustness as measured by the maximum vulnerability. This interpretationdovetails with the analysis of the relationship of connectance and the MEA. For the MEA, wefound that growing connectance decreases robustness when measuring cascading extinctions.On the other hand, growing connectance increases robustness from the point of view of themaximum vulnerability. This is an aspect of what are often called “robust yet fragile”networks - those which are fragile in the face of specific (targeted) attack but stable in theface of nonspecific (random) attack.

As an example of shocks via edge removal, we again use the 2006 trade web and system-atically deleted single edges. Table 1 shows the edge removals that had greater than a halfpercent negative impact on aggregate world income.

We again see the impacts dominated by the largest economic players - the United States,European countries, China, South Korea, Taiwan and Japan. But, as with the our otheranalyses, a more nuanced picture appears with closer analysis. For example, the linksbetween the United States and Canada and between the United States and Mexico are themost powerful by this metric. While this is a consequence of the dominance of the UnitedStates in the world economy, it also reflects the impact of the tighter integration of NorthAmerican economies due to the general liberalization of trade as well as the free tradeagreements put in place in the 1990s. This is exhibited by the strength of these trade tieswhich, while not the largest ties in the world economy, have the most aggregate impact. Tosee this more clearly, we compare the results from 2006 to those of the same type of edgeremoval for the year 1965. Table 2 again shows the edges with negative impact larger thanhalf a percent. While we see the same basic pattern, ties including the largest economicplayers have the largest impact, there are also interesting changes. The most stark is thechange in predominance of the link between Mexico and the United States. In 2006, it ishas the second largest impact of −2.87% while in 1965 is the last on this list with −0.52%impact. One conclusion we can draw is that the move towards liberalized trade policies has


Country 1 Country 2 Percent changeCanada United States of America -4.18Mexico United States of America -2.87

Germany Netherlands -1.03Germany France -1.03

China United States of America -0.96Japan United States of America -0.92

United Kingdom United States of America -0.87France Belgium -0.82

Belgium Netherlands -0.81Germany United States of America -0.80Germany Belgium -0.79

Japan China -0.72Italy Germany -0.72

Germany United Kingdom -0.72Italy France -0.70Spain France -0.67

South Korea China -0.62France United Kingdom -0.61France United States of America -0.55Taiwan China -0.54Austria Germany -0.53

South Korea United States of America -0.52Table 1. The results of shock via edge deletion for the 2006 trade web.

two contradictory effects showing general robustness - impacts generally decline - temperedby weaknesses shown by the much higher impact of specific edges.

This type of result echoes the “robust yet fragile” results found in network theory whensubjecting networks to either targeted or random attack. For example, networks withpower laws degree distributions ([9, 5, 21, ?]) and/or small world characteristics ([20, 8])have the property that they are very robust to random attack, but fragile in the face oftargeted attack. Our methodology shows the same type of two pronged results - randomedge deletion has little effect but there are some edges which, if targeted, create substantialeffects. We emphasize, however, that the effect here is not directly linked to the degreedistribution but is a product of the interaction of the network structure and the dynamics.

To see how this measure changes over time, we ran the same experiment on all tradewebs from 1870-2006 (excluding the periods around World War I and II) and found theedge that had the maximum negative impact. Figure 6 shows the results. On the topleft hand side, we plot the percentage of the total world income that is removed due tothe deletion of the edge with the most impact. On the bottom left hand side, we plotthe percentage as a multiple of the percentage of the world income encoded in that edge’sweight. On the top left hand side, we see a different aspect of the same split we see in themaximal extinction analysis. In general, there is a downward trend in the impact of removalof a single edge. But, after World War II, there is a period, roughly 1949-1975, where theimpacts are higher than before World War II but fairly erratic. After a transition in the


Country 1 Country 2 Percent ChangeCanada United States of America -3.57

Germany France -2.22Germany Netherlands -1.99Taiwan United States of America -1.86

Hungary Germany -1.48Germany Belgium -1.44Belgium Netherlands -1.37

United Kingdom United States of America -1.16Germany United States of America -1.07France Belgium -1.03

Malaysia United Kingdom -0.97Germany Switzerland -0.91

United Kingdom Canada -0.90Germany United Kingdom -0.88

Russia Germany -0.87Hungary France -0.85Zambia United Kingdom -0.79

Netherlands United Kingdom -0.75Russia United Kingdom -0.73Poland Germany -0.70France United Kingdom -0.62Finland United Kingdom -0.61Ireland United Kingdom -0.59France Netherlands -0.58

Malaysia Taiwan -0.58Hungary United States of America -0.56Finland Germany -0.55France United States of America -0.53Mexico United States of America -0.52

Table 2. Results of shock via edge deletion for the 1965 trade web.

1970s, a resumption of the downward trend. The right hand side provides context for theprevious observations. Before World War II, the largest impact of a single edge was generallylarger than simply removing the trade income of that edge from the world income. From1949-1970, the impact is generally smaller than this removal. Then, upon a transition in thelate 1960s and early 1970s, impacts are again greater than simple removal and growing overtime. The right hand side shows the relationship between the connectance and these twomeasures. We see that in the unweighted version, connectance has a quadratic relationshipwith the maximum edge impact, implying that growing connectance is first correlated withdecreasing impact but later with increasing impact. The bottom righthand graph clarifiesthis - the weighted impact is negatively correlated with connectance, i.e. as connectancegrows the edge impact decreases. Viewing the results together shows us the most plausibleimplication - that connectance increase robustness in the sense of edge impact but that itseffect is mitigated by edge weight.


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Figure 6. Plots of the maximal impact of a single edge removal, 1870–2006(omitting the period around the world wars). The left hand side shows thepercentage of the total world income that is removed due to the deletionof the edge with the most impact (top) and the same impact normalizedby edge weight (bottom). The right hand side shows a comparison of thesestatistics with connectance.

In the context of the results of the MEA, we see that the edge removal analysis showsgreater fragility of the network before World War II with a coherent drop and then steadyincrease after World War II. Again, this can be plausibly explained in terms of increasedinternational trade and trends toward globalization. Generally, with increased trade andincreases in the number of trade partners - in other words, increasing connectance - we seea general drop in the power of any given edge. In contrast, the MEA shows that increasingconnectance is correlated with decreasing robustness - deletion of all of a country’s tradingpartners can create a substantial effect rippling throughout the network.


4. Conclusion

We introduce the concept of extinction power and the related techniques of maximal ex-tinction, perturbation, and edge extinction analysis to examine the robustness in economicnetworks. As a first example, these tools, versions of the extinction analyses used in studiesof food webs, are applied to the World Trade Web. The analysis reveals a trend and atransition. The trend shows a strong correlation between connectance and our robustnessmeasures - a negative correlation with the MEA robustness, a negative correlation with themaximum vulnerability percentage (i.e. a positive correlation with a corresponding measureof stability), and a negative correlation with edge extinction robustness. The transition,exhibited in each case in the 1960s and 1970s, shows a rapid transition in these metrics. Inthe case of the MEA, robustness sharply increases and then resumes the downward trendcorrelated with increasing connectance. In the perturbation analysis, the maximum vulner-ability sharply drops and then continues a decline associated with increasing connectance.In the edge extinction, we see a sharp increase in maximum negative impact normalized byedge weight.

In the context of the move towards globalization which began after World War II, we seethese results as evidence of the multifaceted impact of globalization on the stability of theWTW. We view the transitions in robustness as an indication of the positive and negativeaspects of globalization — more trade partnerships and multiple partnerships for the samegoods allow the system to recover if specific avenues of trade are removed. But, as the policyshift continues, negative implications grow. While higher connectance provides the benefitsdescribed above, it also provides shorter paths for impacts to travel and propagate throughthe network. This balances gives the network it “robust yet fragile” characterization - ageneral increasing stability due to the higher connectance engendered by globalization withspecific fragility to targeted shocks.

This methodology is highly adaptable, providing an interesting laboratory for explore theimpact of policy changes, such as a move towards globalization, on the WTW.

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