Population Extinction Studies By Globalists

Stability of the World Trade Web over Time - An Extinction Analysis

N. Foti, S. Pauls, Daniel N. Rockmore
Comments: 19 page, 6 Figures, 2 Tables
Subjects: General Finance (q-fin.GN); Physics and Society (physics.soc-ph)

Abstract. The World Trade Web (WTW) is a weighted network whose nodes correspond
to countries with edge weights re
ecting the value of imports and/or exports between coun-
tries. In this paper we introduce to this macroeconomic system the notion of extinction
analysis, 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 principled
set of in silico \knockout experiments," akin to those carried out in the investigation of
food webs, but suitably adapted to this macroeconomic network. Broadly, our experiments
show that over time the WTW moves to a \robust yet fragile" con guration where is it
robust under random attacks but fragile under targeted attack. This change in stability
is highly correlated with the connectance of the network. Moreover, there is evidence of
sharp 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 these
results in the context in the post-World War II move towards globalization. Globalization
coincides 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 peak
of 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 can
play 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 economic
networks. 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 explore
the impact of their sudden extinction. Similarly, we cannot remove or disable components of
an economy to discover the downstream e ects. However, given a sensible model for a real
system, we do have the ability to perform simulations, i.e., in silico experiments designed
to shed light on the interdependence of the components and the implications for robustness
to 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 in
particular, the World Trade Web (WTW). The WTW is an economic network summarizing
international trade. Nodes represent countries and country B links to country A with a
weight given by the value (say in US dollars) that country A receives from country B in
exports. With the rise of network analysis the WTW has received a good deal of attention
and analysis (see e.g., [14, 13, 19]).
The recent network analyses of the WTW have provided a wealth of initial information
regarding the interaction of the network structure and the functional architecture of the
Date: April 25, 2011.
1
arXiv:1104.4380v1  [q-fin.GN]  22 Apr 20112 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
WTW. Our goal is to extend this analysis and craft measures robustness stability that
are products of intertwined structure and dynamics. Our core methodology is that of a
knockout experiment, where stability is tested via perturbation of the network by removal
or deprecation of some aspect of the structure. We then observe the system as it returns to
equilibrium and record the e ect.
Examples of studies in which these kinds of knockout experiments have already been done
includes those performed in the context of the World Wide Web (WWW) [2], metabolic
networks [17], protein networks [16], and, of particular relevance to this paper, extinction
analyses conducted on food web models of ecosystems (see e.g., section 4.6 of [11] and the
many 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 and
a directed link is placed to node A from node B if species A eats species B (i.e., if there
is a transfer of resource from B to A). Appropriate dynamics, derived from eld data,
are applied to simulate reasonable population
uctuations. In this setting, stability and
robustness are analyzed via simulated extinction studies in which certain species or groups
of species are removed from the ecosystem. \Evolution" then takes place according to
the simulated extinction dynamics. The system either reaches a new stable con guration
(possibly after one or more consequent extinctions) or collapses entirely. Such studies
address questions of robustness by measures of the existence and strength of extinction
cascades 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 robustness
analysis to the WTW. We provide three types of extinction experiments. First, we consider
an extinction analysis similar to that used in the study of food webs where countries are
sequentially removed from the trade web and their impacts analyzed. Second, we consider
a variant of the rst where rather than removing countries, we instead perturb their im-
port/export pro le. Last, we consider deletion of edges in the network - extinction of trade
relationships - and analyze their impact. We emphasize that in these analyses the results
must be interpreted as consequences of both the network structure of the WTW as well as
the evolution dynamics we place on that structure. In particular, more or less re ned mod-
els of evolution dynamics may yield more or less textured results. We view the dynamics
described below as a parsimonious rst model on which subsequent analyses can build.
For the rst analysis, we de ne maximal extinction analysis (MEA) for the study of
the WTW. The mechanism of MEA is analogous to the kinds of knockout experiments
performed on food webs, but di er in the formulation of the extinction dynamics (i.e., in
the 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 is
a measurement of the extinction power of any given country over any other. Roughly, the
extinction power of country A over country B is the proportion of B's economic activity (as
measured by export data) disrupted by country A's extinction (as well as any extinctions
which are a consequence of A's). We view this as a measure of \economic centrality."
To create an aggregate statistic, we de ne the robustness of a trade network analogously
to that of a food web, namely the proportion of the total income in the network that is
destroyed via node deletion and consequent extinctions to reach the loss of 50% of the
network. This notion of robustness is a measure of stability of the trade network - it
measures the extent to which countries can replace lost imports and lost demand for exportsSTABILITY 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 two
basic results.
(1) There are two sharply di erent periods, around World War I (1914-1919) and World
War II (1939-1948), where the robustness is dramatically higher than at any other
time. In both of these cases, the data sets have substantial holes - many unreported
trade gures - which skew the results. As such, we disregard these two periods from
the qualitative analysis.
(2) Other than the periods around the two world wars, the time period breaks naturally
into two periods, roughly 1870{1975 and 1976{2006. The transition is marked by a
sharp increase in robustness in the 1970s.
(3) Over these periods, robustness follows a generally decreasing trend. The decrease
in the rst period (1870-1938) is statistically signi cant (R2 = 0:26; p = 0:00001 for
a linear t), 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-globalization
policies initiated after the Great Depression [10]. These results support the claim that
increased globalization corresponds to increased robustness. But what are possible expla-
nations for the downward trend after the transition? In [12], we nd a similar extinction
analysis performed on food webs, showing a signi cant positive correlation between robust-
ness of their networks and connectance. Connectance is a measure of the completeness of
the network and is de ned as the proportion of existing connections in the network divided
by the maximum number of possible connections (

n
2

for a network with n nodes). For the
full period, we nd a signi cant negative correlation between robustness and connectance
(R2 = 0:5; p = 1:2 10
21
(
1
Moreover, we nd a number of di erent network statistics .
which also have signi cant negative correlation with robustness in the period after transition
of which we highlight one, maximum trade de cit (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 other
species. We posit that the trade networks have the opposite property due to the nature
of our model dynamics. Extinctions create a ripple e ect in the network, decreasing the
income of countries who have lost exports due to an extinction as well as decreasing the
availability of goods. This ripple e ect can propagate more easily and quickly through a
more highly connected network. The link between high trade de cits and robustness can
also be understood as an aspect of this same reasoning | imbalances between imports
and exports again propagate (and potentially magnify) via our dynamics. The impact of
deleted nodes with negative trade balances (i.e., de cits) propagates primarily in the form
of 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 number
of e ects, some which boost robustness of the network while other that have a negative e ect.
The results point to increased connectance and the existence and rise of trade de cits as
two factors that have negative e ects but that coincide with the policy movement towards
globalization. These factors, in aggregate, have the e ect of dramatically increasing the
robustness at the outset of the policy shift with a slow decline afterwards, as the statistics
associated with the negative e ects grow.
1
This is smaller than the precision tolerance for MATLAB.4 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
For our second analysis, we provide an algorithm for impact analysis of perturbing the
import/export pro le of a given country. Again, the precise details are given below, but
the basic idea is to change the import totals by a xed percentage and the export totals by
another and then to use the similar iterative dynamics as in the maximal extinction analysis
to measure the impact. As an example, we perform this analysis on the 2006 trade web
where we systematically alter each country's export pro le by decreasing exports by 5% and
imports by 30%. This is meant to model a shock to a country's economy which depresses
exports, such as Thailand's economic crash after the currency crises in the late 1990s where
imports and exports dropped by roughly 30% and 5% respectively. The results of the
analysis are measured in the aggregate impact to the world income (due to exports). We
nd two interesting observations from this analysis. First, we recover perhaps unsurprising
major impacts - large players in international trade have, in general, the largest impact, with
the China, Germany, the United States, Japan and France playing central roles. Indeed, this
perturbation of China creates substantial global impact according to this model, causing
at least a 1% drop in income for 94% of countries. Similarly, many of the weakest and/or
most isolated economies are the most vulnerable to such perturbations. For example, many
island nations - Vanuatu, the Soloman Islands, etc. - are among the most vulnerable
as well as countries such as Qatar, Kuwait, the Philippines, Monaco and Liechtenstein.
To examine this longitudinally, we compute the maximum vulnerability percentage - the
maximum over all countries of the percentage of countries whose perturbation results in a
greater than 1% decrease in income of a speci c country. This statistic exhibits a transition
roughly at 1960 and in the two periods (again omitting the periods around the World
Wars), the maximum vulnerability has a signi cant correlation with connectance (1870-
1959, R2 = 0:56; p = 1:9
14
; 1960-2006, R2 = 0:56; p = 1:6
9
). We interpret this connection
as positive relationship between connectance and robustness - as connectance grows, the
maximum vulnerability decreases.
For our third analysis, we consider the removal of edges from the trade network and
investigate their impact. This type of experiment is meant to model situations where a
trade relationship ceases - for example, a war between the two countries. As examples, we
closely analyze the impact of edge deletions for a single years, 2006 and 1965, as well as
more 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 the
world 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 creates
a 4:18% drop in 2006 and a 3:57% drop in 1965. These two snapshots allow us to see some
detailed movement over time. In both cases, links between large economic players are the
most signi cant. However, we see substantial changes between these two years. First, in
2006, a umber of Asian countries such as China, South Korea, Taiwan and Japan gure
prominently in the list of edges with highest impact. In 1965, however, they are largely
absent (expect Taiwan). This is, of course, re
ective of the growth of economic power of
these countries over that period. We also see change in the power of speci c links. For
example, the link between Mexico and the United States creates a 0:52% drop in income in
1965, but in 2006 it is the second most powerful link, creating a 2:87% drop. This re
ects
both the growing economic power of Mexico but also the growing intertwining of the two
economies.
For the longitudinal analysis, we measure the strength of the most powerful edge in each
year in the same terms as well as the same statistic normalized by edge weight. We seeSTABILITY OF THE WORLD TRADE WEB OVER TIME { AN EXTINCTION ANALYSIS 5
a trend that contrasts with the trend shown in the MEA. Speci cally, we see a general
downward trend in the normalized statistic, which we interpret as a growing robustness
of the network in the face of this type of extinction. However, the unweighted statistic
presents more complicated behavior - rst decline with connectance and then growing as
connectance continues to increase. Moreover, we see a transition period after World War II
which 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, and
its dynamics, given by the income model, create networks which have a dual \robust yet
fragile" character. More precisely, the MEA, a targeted attack analysis, shows declining
robustness over time which is correlated with connectance. The perturbation analysis -
again a milder form of targeted attack - shows growing robustness over time with a sharp
jump and then continued rise post World War II. The edge removal analysis shows that
these networks become more and more robust to random attack over time as indicated
by the dropping in
uence over the world income. Again, this measure is correlated with
connectance. However, there are still edges which, if targeted, have consequential e ects.
Together, these shed light on the consequences of globalization and the liberalization
of trade. We see globalization re
ected in increasing connectance - an increasing average
number of trading partners per country. The increased connectance has two opposing
consequences. First, it correlates with decreasing robustness when using the MEA. Second,
it decreases the maximum vulnerability. And third, on average it decreases the power of
individual edges, which increases the overall robustness of the system when considering edge
deletions. These results are all evidence of the claim that these trade networks are \robust
yet fragile" - they are vulnerable to targeted conscious attack but stable under random
attack.
2. Methods
As our basic data source, we use import/export tables available from [6] which detail the
trade relationships between countries from 1870-2006. These tables detail the amount of
goods (in US dollars) imported or exported from one country to another and are presented
as 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 the
dollar amount of exports from country i to country j. While it should be the case that
Mt = N, due to the variation in reporting practices, this is not so. For simplicity, we
analyze only the import matrix M and in this way de ne 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 speci cally, we note that during World War I and II the missing data is substantial and
obvious - many of the world's largest economies at the time make no report of trade between
themselves and their former (and later) large trading partners. This is perhaps unsurprising
as we suspect these countries had higher priorities than import/export reporting in these
periods. But, as the resolution of the data is particularly low during these periods, we
cannot infer much from our analyses during these periods. As such, we exclude them from
the interpretive analysis. However, we can still consider them as abstract networks from
the point of view of attempting to nd relations between network statistics and robustness.6 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
Any sort of extinction analysis in the WTW requires the de nition of the consequences
of node removal. Our working assumptions are as follows:
(1) Node deletion creates an \excess" of dollars (supply) given by the goods that the
deleted countries would have purchased from other countries.
(2) Node deletion creates an unmet demand in the form of goods the other countries
previously purchased from deleted countries.
2.1. Income model. To understand the e ect of the removal of one or more nodes, we
need to instantiate dynamics on the network to model how the trade network rebalances
trade after such a shock. To do so, we introduce an income model on the network. The
basic 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 is
deleted it both decreases demand and supply for its trading partners but also increases
demand for other countries via the unmet demand of its trading partners.
As a rst step in concretely modeling this dynamic, we de ne a country's propensity to
spend based on the in-degree and the out-degree of the original import matrix. Recall their
de nition:
De nition 2.1. Let M be an import matrix. Then, the in-degree for node i is
DM(i) =
X
j
Mji
:
Similarly, the out-degree for node i is
OM(i) =
X
j
Mij :
In the context of the import matrix, the in-degree DM(i) is simply the total dollar value
of goods imported by other countries from country i. The out-degree OM(i) is the total
dollar value of goods imported by country i from other countries. For our purposes, the
in-degree gives a measure of the income of country i while the out-degree gives a measure
of its expenditures.
We de ne the propensity to spend for country i as
(1) i =
)
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 that
the internal economy of the country is producing excess dollars (by some means) that are
used to fund additional import spending. To record this we de ne,
(2) i =
)
OM(i)  DM(i) if DM(i) < OM(i)
0 otherwise
Next, we de ne 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 Markov
chain associated to M. Note that with these de nitions,
OM(i) = iDM(i) + i
M = diag(OM)m
(4)STABILITY OF THE WORLD TRADE WEB OVER TIME { AN EXTINCTION ANALYSIS 7
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 vector .
of incomes It
is given by
(5) It = diag( )MT
t 1 +
where = ( 1; : : : ; n)
T
, = ( 1; : : : ; n)
T
and 1 = (1; : : : ; 1)
T
To iterate we calculate .
the 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 an
import matrix and ; and m are derived from M, then M (and the resulting income
associated with M) is a xed point of this iteration. Moreover, if we x m; ; and , we
can solve the system for xed points, giving the equilibrium income as
I = (diag( )mT
 Id)
1

where Id is the identity matrix
2
(Note that in a limiting case where = (1; : : : ; 1 .
T
= ;
(0; : : : ; 0)
T
the equilibrium income is simply the xed vector of mT
It is in this sense that .
we 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 via
the following algorithm:
(1) Let M0 be the initial import matrix and calculate ; and m from (1),(2), and (3).
Let I0 be the unshocked income vector given by (5) using M0.
(2) Let M1 be a modi ed version of M0 encoding the desired shock. Adjust ; and
m to re
ect the modi cation (this will be detailed in each speci c case below). Let
t = 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 ; and m are xed for the itera-
tion steps of the simulation. We envision the simulation as an approximation of a short time
rebalancing. Thus, we do not allow the fundamental constants | the pro le of countries
that a country trades with and in what proportion, the propensity to spend and the amount
of internal income | to change. Second, we have an explicit assumption of substitutability.
We recognize the shortcomings: imports and exports of wildly di erent items are lumped
into the aggregate statistics. Analogous assumptions are made for extinction models for
food webs - prey are assumed to be interchangeable (and available) for a given predator
species. In our setting, this assumption provides the easiest path of rebalancing the system
after a shock | thus providing the network with the best possible chance of recovery. In
addition, one of our assumptions in the model dynamics helps mitigate the assumption of
substitutability: 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
2
The invertibility of the matrix is equivalent to the fact that diag( )mT
does not have a xed vector.
This is true in most cases because the maximum eigenvalue of mT
is 1 and the 1. The only case where
this isn't true is if = 1. In that case, the mT
 Id is not invertible, but (mT
 Id)I = can still be
solved so long as is not a multiple of the constant vector.8 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
relationships | buying one good from many di erent countries | this is a proxy for the
availability of substitutable goods from existing trading partners. This is particularly true
of the larger economies in each network which, as we see in the simulation, have the most
impact on its stability.
2.2. Shocks via node removal. In our investigations, we consider a shock created by
deleting nodes from the network. Using this framework, to create Mi
1
from M0 = M by
deleting node i, we simply set Mir and Mri equal to zero for all r. Essentially, this preserves
the number of nodes in the network representation of the matrix (and allows for the matrix
multiplication steps in the algorithm above), but changes the connection structure.
We rst perform a maximal extinction analysis, carried out according to the following
algorithm:
(1) Fix the initial import matrix M and its associated income matrix I. For the rst
iteration, let M0 = M, I0 = I.
(2) For each i, delete node i from M0 to form M1. Adjust m by removing the i
th
row
and column and renormalizing. Adjust by setting (i) = 0 and by setting
(i) = 0.
(3) Calculate fM0; Mi
1
; : : : ; Mi
5
g; fI0; I
i
1
; : : : ; I
i
5
g via the income model.
3
(4) Calculate
P ow(i) = 1
P
j
I
i
5
(j)
P
j
I0(j)
:
We call this the total power of node i. It represents the percent of total income left
after node deletion and rebalancing. Note: sometimes this number is larger than
one, 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 max
k
P ow(k)
Add this node index to the list, D, of deleted nodes.
(6) Letting M0 = M
j
5
and I0 = I
j
5
Repeat steps 2 5 this until the total income falls .
below 50% of the total income of the original import matrix, i.e.
P
k
I
j
5
(k)
P
k
I(k)
< 0:5
From this algorithm, we measure robustness by computing the percentage of the initial
income we must remove to reach the 50% threshold:
R =
P
j2D I(j)
P
k
I(k)
From its de nition, 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 removal
of an entire country from the import-export economy. While this may seems drastic and
unrealistic, it provides the most direct way of measuring the impact and power of a particular
country over the entire system. Moreover, as the model is inherently linear, we would nd
similar results if we instead removed only a portion of a given country's import/export
pro le. 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.STABILITY OF THE WORLD TRADE WEB OVER TIME { AN EXTINCTION ANALYSIS 9
example, prey are assumed available in su cient quantity to sustain predator populations
until extinction.
2.3. Shocks via node perturbation. The shocks described in the previous section are
both the most drastic types of shocks as well as the one of easiest to understand. In this
section, we present a more complicated type of shock based on perturbation of nodes rather
than their entire removal. Such perturbation can be seen as models for intrinsic or extrinsic
shocks to economies of individual countries which (temporarily) e ect their import/export
pro le. Good examples of this are some of south-east asian economies during the currency
crises of the 1990s. For example, Thailand enjoy rapid growth in the late 1980s and earlier
1990s - 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 consequence
of this crisis that we see is a drop in the value of exports from and a substantial drop in
imports to Thailand in the late nineties.
To model such types of shocks, we simply create a perturbation of the network created
by a xed manipulation of a speci c node's import and export values. For example, we can
create a two parameter perturbation of imports and exports. If we x a; b 2 [0; 1] and a
node i, we de ne
(M1)jk =
8
><
:<
(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; ; and accordingly. For m, we scale the i
th
row
by a and then i
th
column by b and renormalize. We leave unchanged except to multiply
the i
th
entry by
b
a
We leave unchanged. The choice to leave unchanged is re
ectie of .
our desire to produce a model which re
ects 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 linearity
of the dynamics. When a = b, this model provides a method by which to evaluate test
hypotheses.
2.4. Shocks via edge removal. Similarly to the previous section, we also model shocks
created by the termination of a trade relationship. In this case, we simply remove an
edge from the import/export matrix. This models a situation where a trade relationship
is completely dissolved - for example, due to war, broken treaty, etc. If we denote the two
countries 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 and are unchanged. Then, following the algorithm in Section 2.1, we
compute the impact.
3. Results
Figure 1 shows the results of the robustness computation under the maximal extinction
analysis over time. Outside of the years around World War I and II, we see that there
are two regimes split at roughly 1975. Before the mid seventies, the robustness is low and
decreasing over time. In the seventies, we see a rapid increase in robustness, and then the10 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
1880 1900 1920 1940 1960 1980 2000
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Year
Robustness
R
2
=0.26, p=0.00001,
slope=-0.0015
R
2
=0.07,p=0.17
slope = -0.0019
R
2
=0.33, p=0.0006
slope = -0.0021
Figure 1. Robustness scores for the WTW over time. Solid lines denote
mean robustness of 50 trials of null models, dotted lines are the 95% and 5%
cuto s. Filled circles show years with robustness scores that fall outside the
5  95% interval.
.
resumption of the decreasing trend. In the period around World War II, we see a sharp
increase in robustness. As stated above, this is due to the sparseness of the reported data
rather 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, the
transition is not nearly so sharp.
To better understand the signi cance of these results, we compare the robustness mea-
surements to an appropriate null model. For the null model, we use a randomization of the
import matrix and repeat our maximal extinction analysis on the result. Repeating this
multiple 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{2006
plotted with the mean, minimum and maximum of robustness scores of 50 runs of each null
model. The robustness scores are coded by shape to indicate their signi cance. The scores
shown as lled circles are either larger than 95% or smaller than 5% of the corresponding
scores for the 50 null models. Thus, according to this threshold, these robustness scores
are signi cantly di erent than randomized null models with the same number of nodes and
total degree.
We can interpret this as follows. In the periods from 1870-1913 and 1920-1939, some
aspect of the structure of the networks creates higher robustness than expected at random
in a number of years. Overall, we see slow decline of robustness over time. We then see a
similar picture of decline between 1949 and 1975, with a transition in 1975-1976. Shortly
after the transition, the robustness increases to a level signi cantly above that of comparable
random (Erdos-Renyi) networks but then again begins a decline. In the rst periods, theSTABILITY OF THE WORLD TRADE WEB OVER TIME { AN EXTINCTION ANALYSIS 11
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Connectence
Robustness
1880 1900 1920 1940 1960 1980 2000
0
0.25
0.5
Connectence
R
2
=0.50, p =1.2 x 10
-21
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 signi cant (R2 = 0:26; p = 10
5
) while in the period from
1949 to 1975, it is not signi cant (R2 = 0:07; p = 0:17). In the last period, 1976-2006, the
trend is again signi cant (R2 = 0:33; p = 6 10
4
). To attempt to understand the decline
over time, we consider statistics with potential predictive power. Following the food web
literature, we consider connectance,
C =
# of edges
(# of nodes)
2
:
We also consider the maximum trade de cit, i.e., the maximum of DM  OM for a given
import matrix M.
In Figure 2, we plot the robustness against connectance. The inset graph shows the plot
of the connectance over the entire time period, 1870{2006. We also plot the best linear
t of the data and provide the R2
and p values for the regression. The t is statistically
signi cant with R2 = 0:5 and p = 1:2 10
21
In this computation we include the periods .
around both World Wars. We do this as we are merely attempting to link the summary
statistics of robustness and connectance and are not (yet) interpreting the results in terms
of actual trade relationships. So, as the trade webs in these periods are simply incomplete
trade webs, they are still appropriate for inclusion in this analysis. We again note that the
sharp drop in connectance around the World Wars is likely due to the spareness of the data
and the resulting incomplete trade web. We also note the smaller drop in connectance in the
early 1960s. This corresponds to a sudden increase in the number of countries in the trade
webs which, in turn, is a consequence of the wave of former colonial countries gaining their
independence. Generally, as these countries gained independence, they entered the world
trade network slowly - rst trading with geographic neighbors and their former colonizing
country.12 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
1950 1960 1970 1980 1990 2000 2010
0.3
0.35
0.4
0.45
0.5
0.55
Year
1950 1960 1970 1980 1990 2000 2010
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
x 10
5
Year
Maximum Trade Deficit
1950 1960 1970 1980 1990 2000 2010
0
0.5
1
1.5
2
2.5
3
x 10
5
Year
Maximum Trade Surplus
Figure 3. Plots of connectance, maximum trade de cits and maximum
trade surpluses from 1950{2006.
As mentioned in the introduction, potentially one way to understand this correlation
is 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 longer
has any prey (except, possibly, itself ). In our situation, the income model provides simple
linear dynamics associated with the fundamental principles of supply and demand. These
dynamics are one type of analogue to population dynamics on food webs. They allow for
the shock associated with node removal to propagate through the network, much like a
contagion. As higher connectance generally implies faster propagation speeds through a
network, it is not surprising that we see a signi cant negative correlation.
A similar argument applies as a possible explanation of the correlation between the
maximum trade de cit and robustness. The deletion of a node with a substantial de cit
again propagates through the network but the de cit itself creates unused supply which
is not balanced (overall), by unmet demand. Irrespective of the network topology, this
creates downward pressure incomes. The jump in the 1970s and then decline in robustness
over time tracks the increased connectance and the existence (and increasing size) of trade
imbalances. The regression analysis for robustness in terms of maximum trade de cit yields
a signi cant result (R2 = 0:39; p = 10
4
). It seems plausible that these are linked with
a move towards policies of increasing globalization. As we see in Figure 3, the e ects of
these changes on maximum trade de cits and maximum trade surpluses are relatively mild
at rst before rapidly growing. The connectance decreases for a time before assuming an
upward trend. Thus, one conclusion we may draw is that the positive e ects on robustness
are 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 considered
the 2006 import/export matrix and perturbed each node sequentially with model parametersSTABILITY OF THE WORLD TRADE WEB OVER TIME { AN EXTINCTION ANALYSIS 13
No data
0.0 - 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 data
0.0 - 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 percentage
and the vulnerability percentage for the WTW in 2006. Hatched countries
are ones where no data was available for the computation.
= 0:7; = 0:95. In other words we modeled a 30% drop in imports and a 5% drop in
exports and analyzed the impact according to the model. These percentages roughly match
the drops seen in the late nineties for Thailand as a result of their currency crisis. To create
an aggregate statistic to help visualize the results of the simulation, for each country, we
counted the number of other countries whose income decreased by 1% or more when we
simulated the drop in imports and exports. If we divide this count by the total number
of countries, we call the result the power percentage. We also calculate the vulnerability
percentage, which is the number of countries who perturbation create a 1% or more decrease
in income of a given country divided by the number of countries.
An initial review of the results show us something relatively unsurprising, that the largest
players in the world economy - the United Stated, Germany, China, the United Kingdom, the
Netherlands, France, Japan, Italy, Canada, and Belgium - have the most impact. Somewhat
more surprising is the list of most vulnerable countries, which include a number of southeast
Asian countries, South American Countries, Japan, Australia, and Russia. Figure 4 shows
two world maps. The left map is colored by the power percentage while the map of the
right hand side is colored according to the vulnerability percentage. This map shows that
a 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 groups
of countries - notably those listed above - are especially vulnerable due to their patterns of
trade.
To get a sense of this method applied over time, we repeat the same experiment for each
year in our data set and calculate the maximum power and vulnerability percentages for
each year. Figure 5 shows the results where, in each case, we have omitted the scores from
the periods around the two World Wars due to data sparseness. In computing the maximum
power percentage over this time period, we nd it is relatively uniform and large - always
above 90%. In Figure ?? (a), we see that the maximum vulnerability percentage decreases
over time, which we interpret as growing robustness of the system. We also note the
transition after World War II which echoes the transition found in the MEA. In investigating
the 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 connectance14 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
1860 1880 1900 1920 1940 1960 1980 2000 2020
0
0.1
0.2
0.3
0.4
0.5
Maximum Vulnerability Percentage
Year
(a) Vulnerability Percentage
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
0.1
0.2
0.3
0.4
0.5
0.6
Connectance
Maximum vulnerability percentage
R
2
 = 0.56, p = 1.9× 10
-14
R
2
 = 0.56, p = 1.6 × 10
-9
(b) Vulnerability vs. Connectance
Figure 5. (a) Vulnerability Percentage over time. The solid line is mean
percentage over 100 null model trials while the dotted lines are the 5% and
95% cuto s from the null models. The circles in black are years where the
vulnerability percentage is outside the 5  95% range. (b) Vulnerability
percentage plotted against connectance.
and the maximum vulnerability percentage. However, there are two distinct regimes which
amplify our earlier observation of a transition - the black circles are years after 1960 while the
white circles are before. In this experiment, we interpret this as linking growing connectance
with growing robustness as measured by the maximum vulnerability. This interpretation
dovetails with the analysis of the relationship of connectance and the MEA. For the MEA, we
found that growing connectance decreases robustness when measuring cascading extinctions.
On the other hand, growing connectance increases robustness from the point of view of the
maximum vulnerability. This is an aspect of what are often called \robust yet fragile"
networks - those which are fragile in the face of speci c (targeted) attack but stable in the
face of nonspeci c (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 half
percent 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 other
analyses, a more nuanced picture appears with closer analysis. For example, the links
between the United States and Canada and between the United States and Mexico are the
most powerful by this metric. While this is a consequence of the dominance of the United
States in the world economy, it also re
ects the impact of the tighter integration of North
American economies due to the general liberalization of trade as well as the free trade
agreements put in place in the 1990s. This is exhibited by the strength of these trade ties
which, while not the largest ties in the world economy, have the most aggregate impact. To
see this more clearly, we compare the results from 2006 to those of the same type of edge
removal for the year 1965. Table 2 again shows the edges with negative impact larger than
half a percent. While we see the same basic pattern, ties including the largest economic
players have the largest impact, there are also interesting changes. The most stark is the
change in predominance of the link between Mexico and the United States. In 2006, it is
has 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 hasSTABILITY OF THE WORLD TRADE WEB OVER TIME { AN EXTINCTION ANALYSIS 15
Country 1 Country 2 Percent change
Canada United States of America -4.18
Mexico United States of America -2.87
Germany Netherlands -1.03
Germany France -1.03
China United States of America -0.96
Japan United States of America -0.92
United Kingdom United States of America -0.87
France Belgium -0.82
Belgium Netherlands -0.81
Germany United States of America -0.80
Germany Belgium -0.79
Japan China -0.72
Italy Germany -0.72
Germany United Kingdom -0.72
Italy France -0.70
Spain France -0.67
South Korea China -0.62
France United Kingdom -0.61
France United States of America -0.55
Taiwan China -0.54
Austria Germany -0.53
South Korea United States of America -0.52
Table 1. The results of shock via edge deletion for the 2006 trade web.
two contradictory e ects showing general robustness - impacts generally decline - tempered
by weaknesses shown by the much higher impact of speci c edges.
This type of result echoes the \robust yet fragile" results found in network theory when
subjecting networks to either targeted or random attack. For example, networks with
power 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 of
targeted attack. Our methodology shows the same type of two pronged results - random
edge deletion has little e ect but there are some edges which, if targeted, create substantial
e ects. We emphasize, however, that the e ect here is not directly linked to the degree
distribution 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 trade
webs from 1870-2006 (excluding the periods around World War I and II) and found the
edge that had the maximum negative impact. Figure 6 shows the results. On the top
left hand side, we plot the percentage of the total world income that is removed due to
the deletion of the edge with the most impact. On the bottom left hand side, we plot
the percentage as a multiple of the percentage of the world income encoded in that edge's
weight. On the top left hand side, we see a di erent aspect of the same split we see in the
maximal extinction analysis. In general, there is a downward trend in the impact of removal
of a single edge. But, after World War II, there is a period, roughly 1949-1975, where the
impacts are higher than before World War II but fairly erratic. After a transition in the16 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
Country 1 Country 2 Percent Change
Canada United States of America -3.57
Germany France -2.22
Germany Netherlands -1.99
Taiwan United States of America -1.86
Hungary Germany -1.48
Germany Belgium -1.44
Belgium Netherlands -1.37
United Kingdom United States of America -1.16
Germany United States of America -1.07
France Belgium -1.03
Malaysia United Kingdom -0.97
Germany Switzerland -0.91
United Kingdom Canada -0.90
Germany United Kingdom -0.88
Russia Germany -0.87
Hungary France -0.85
Zambia United Kingdom -0.79
Netherlands United Kingdom -0.75
Russia United Kingdom -0.73
Poland Germany -0.70
France United Kingdom -0.62
Finland United Kingdom -0.61
Ireland United Kingdom -0.59
France Netherlands -0.58
Malaysia Taiwan -0.58
Hungary United States of America -0.56
Finland Germany -0.55
France United States of America -0.53
Mexico 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 the
previous observations. Before World War II, the largest impact of a single edge was generally
larger than simply removing the trade income of that edge from the world income. From
1949-1970, the impact is generally smaller than this removal. Then, upon a transition in the
late 1960s and early 1970s, impacts are again greater than simple removal and growing over
time. The right hand side shows the relationship between the connectance and these two
measures. We see that in the unweighted version, connectance has a quadratic relationship
with the maximum edge impact, implying that growing connectance is rst correlated with
decreasing impact but later with increasing impact. The bottom righthand graph clari es
this - the weighted impact is negatively correlated with connectance, i.e. as connectance
grows the edge impact decreases. Viewing the results together shows us the most plausible
implication - that connectance increase robustness in the sense of edge impact but that its
e ect is mitigated by edge weight.STABILITY OF THE WORLD TRADE WEB OVER TIME { AN EXTINCTION ANALYSIS 17
1860 1880 1900 1920 1940 1960 1980 2000 2020
0
2
4
6
8
10
12
Year
Maximum negative edge impact
(a) Maximum edge impact (percent of total income)
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
0
2
4
6
8
10
12
Connectance
Maximum negative edge impact
y = 110x
2
-84x+20
R
2
 = 0.3, p = 9.1 × 10
-10
(b) Connectance vs. Edge Impact
1880 1900 1920 1940 1960 1980 2000
0
1
2
3
4
5
6
Year
Maximum negativeedge impact as
a multiple of edge weight
(c) Maximum edge impact as multiple of edge
weight
0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
1
2
3
4
5
Connectance
Maximum negative impact as
a multiple of edge weight
R
2
 = 0.2, p = 2.6 × 10
-7
(d) Connectance vs. weighted edge impact
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 the
percentage of the total world income that is removed due to the deletion
of the edge with the most impact (top) and the same impact normalized
by edge weight (bottom). The right hand side shows a comparison of these
statistics with connectance.
In the context of the results of the MEA, we see that the edge removal analysis shows
greater fragility of the network before World War II with a coherent drop and then steady
increase after World War II. Again, this can be plausibly explained in terms of increased
international trade and trends toward globalization. Generally, with increased trade and
increases in the number of trade partners - in other words, increasing connectance - we see
a general drop in the power of any given edge. In contrast, the MEA shows that increasing
connectance is correlated with decreasing robustness - deletion of all of a country's trading
partners can create a substantial e ect rippling throughout the network.18 NICK FOTI
y
, SCOTT PAULS

, AND DANIEL N. ROCKMOREy; ;#
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 economic
networks. As a rst example, these tools, versions of the extinction analyses used in studies
of food webs, are applied to the World Trade Web. The analysis reveals a trend and a
transition. The trend shows a strong correlation between connectance and our robustness
measures - a negative correlation with the MEA robustness, a negative correlation with the
maximum vulnerability percentage (i.e. a positive correlation with a corresponding measure
of 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. In
the case of the MEA, robustness sharply increases and then resumes the downward trend
correlated 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 by
edge weight.
In the context of the move towards globalization which began after World War II, we see
these results as evidence of the multifaceted impact of globalization on the stability of the
WTW. We view the transitions in robustness as an indication of the positive and negative
aspects of globalization | more trade partnerships and multiple partnerships for the same
goods allow the system to recover if speci c avenues of trade are removed. But, as the policy
shift continues, negative implications grow. While higher connectance provides the bene ts
described above, it also provides shorter paths for impacts to travel and propagate through
the network. This balances gives the network it \robust yet fragile" characterization - a
general increasing stability due to the higher connectance engendered by globalization with
speci c fragility to targeted shocks.
This methodology is highly adaptable, providing an interesting laboratory for explore the
impact of policy changes, such as a move towards globalization, on the WTW.
References
[1] D. Aikman, P. Alessandri, B. Eklund, P. Gai, S. Kapadia, E. Martin, N. Mora, G. Sterne, and M. Willi-
son. Funding liquidity risk in a quantitative model of systemic stability. Bank of England Working
Paper, (372), 2009.
[2] R eka Albert, Hawoong Jeong, and Albert-L aszl o Barab asi. Error and attack tolerance of complex net-
works. Nature, 406:378{382, 2000.
[3] P. Alessandri, P. Gai, S. Kapadia, N. Mora, and C. Puhr. Towards a framework for quantifying systemic
stability. International Journal of Central Banking, 5:47{81, 2009.
[4] Stefano Allesina and Mercedes Pascual. Googling food webs: Can an eigenvector measure species'
importance for coextinctions? PLoS Comput Biol, 5(9):e1000494, 09 2009.
[5] A.-L Barabasi and R. Albert. Science, 1999.
[6] Katherine Barbieri, Omar Keshk, and Brian Pollins. Correlates of war project trade data set codebook,
version 2.01. 2008. Online: http://correlatesofwar.org.
[7] M. Boss, M. Summer, and S. Thurner. Contagion
ow through banking networks. Lecture Notes in
Computer Science, 3038:1070{1077, 2004.
[8] Moore C and Newman ME. Epidemics and percolation in small-world networks. Phys Rev E, 61:5678{
5682, 2000.
[9] John C. Doyle, David L. Alderson, Lun Li, Steven Low, Matthew Roughan, Stanislav Shalunov, Reiko
Tanaka, and Walter Willinger. The robust yet fragile nature of the internet. PNAS, 102(41):1449714502,
2005.
[10] G erard Dum enil and Dominique L evy. The Crisis of Neoliberalism. Harvard University Press, Cam-
bridge, MA, 2010.STABILITY OF THE WORLD TRADE WEB OVER TIME { AN EXTINCTION ANALYSIS 19
[11] J. A. Dunne. The network structure of food webs. In M. Pascual and J. A. Dunne, editors, Ecological
Networks: Linking Structure to Dynamics in Food Webs, pages 27{86. Santa Fe Institute Studies on
the Sciences of Complexity Series: Oxford University Press, New York, 2006.
[12] J. A. Dunne, R. J. Williams, and N. D. Martinez ND. Network structure and biodiversity loss in food
webs: robustness increases with connectance. Ecol Lett, 5:558{567, 2002.
[13] G. Fagiolo, J. Reyes, and S. Schiavo. On the topological properties of the world trade web: A weighted
network analysis. Physica A, 387:3868{3873, 2008.
[14] G. Fagiolo, J. Reyes, and S. Schiavo. Multinetwork of international trade: A commodity-speci c analysis.
Phys. Rev. E, 79:036115, 2009.
[15] J. Jacobs. The Nature of Economies. Vintage, 2001.
[16] H. Jeong, S. P. Mason, A-L. Barab asi, and Z. N. Oltvai. Lethality and centrality in protein networks.
Nature, 411:41, 2001.
[17] H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and A-L. Barab asi. The large-scale organization of
metabolic networks. Nature, 407:651{654, 2000.
[18] F. Schweitzer, G. Fagiolo, D. Sornette, F. Vega-Redondo, A. Vespignani, and D. R. White. Economic
networks: The new challenges. Science, 325, 2009.
[19] M. A. Serrano and M. Boguna. Topology of the world trade web. Physical Review E, 68:015101, 2003.
[20] Xiao Fan Wang and Guanrong Chen. Synchronization in scale-free dynamical networks: robustness
and fragility. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on,
49(1):54 {62, jan 2002.
[21] S.-H. Yook, H. Jeong, and A.-L. Barabasi. Proc. Natl. Acad. Sci. USA, 2002.