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Systemic Risk: Robustness and
Fragility in Trade Networks.
Scott Pauls
Department of Mathematics
Dartmouth College
WPI, 12/6/13
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Credit where credit is due
The first part is joint with N. Foti and D.
Rockmore.
Stability of the world trade network over time: an extinction
analysis, JEDC, 37:9 (2013), 1889-1910.
The second part is joint with D. Rockmore
and is work in progress.
WPI, 12/6/13
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is “robust yet fragile”
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
WPI, 12/6/13
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Systemic Risk and “normal
accidents”
Perrow, 1984:
interactive
complexity & normal accidents
tight coupling
WPI, 12/6/13
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Stability and robustness
Robustness
Dynamics
Structure
WPI, 12/6/13
We use network models:
actors are nodes, relationships are edges.
We construct dynamics to model exchanges
between the actors.
We define robustness in terms of responses to
shocks.
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is “robust yet fragile”
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
WPI, 12/6/13
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World Trade Web
nodes are countries.
edges are directed and weighted, giving the dollars that flow from country i to country j for traded goods.
dynamics are given by the Income-Expenditure model.
WPI, 12/6/13
Katherine Barbieri, Omar M. G. Keshk, and Brian Pollins. 2009. “TRADING DATA: Evaluating our Assumptions
and Coding Rules.” Conflict Management and Peace Science. 26(5): 471-491.
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Modeling paradigm: extinction
analysisFood webs:
nodes are species
edges are feeding
relationships
dynamics are quite simple.
If a species no longer has anything to feed on, it goes extinct.
WPI, 12/6/13
Dunne, J.A., R.J. Williams, and N.D. Martinez. 2002.
Food-web structure and network theory: The role of connectance and
size.
PNAS, vol. 99, no. 20, pp. 12917-12922.
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Dynamics
We consider trade relations analogously to plumbing – country Ahas n trading partners and has a preference to trade with them according to fixed proportions:
In a given year, the country has a certain amount of income it gains from selling its goods to others and, according to a fixed ratio, spends a portion of that income on imports, spreading it proportionally according to the table.
If the country spends more than it takes in, we assume that the balance is funded by debt.
WPI, 12/6/13
1 2 3 4 5 6 7 … n
A 5% 25% 1% 10% 7% 13% 2% … 6%
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Dynamics
The proportionality table, extended to all countries forms a Markov chain which governs the basic flow of money and goods through the trading network.
We denote the spending ratio by 𝛼 and call it the propensity to spend.
We denote the debt by 𝛽.
WPI, 12/6/13
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Income-Expenditure model
In/out-strength:
𝐼𝑀 𝑖 =
𝑗
𝑀𝑗𝑖 , 𝑂𝑀 𝑖 =
𝑗
𝑀𝑖𝑗
Relationships:
𝑂𝑀 𝑖 = 𝛼𝑖𝐼𝑀 𝑖 + 𝛽𝑖𝑀 = 𝑑𝑖𝑎𝑔 𝑂𝑀 𝑚
Iterative model:
𝐸𝑡 = 𝑑𝑖𝑎𝑔 𝛼 𝑀𝑡𝑇 ⋅ 1 + 𝛽
𝑀𝑡+1 = 𝑑𝑖𝑎𝑔 𝐸𝑡 𝑚
WPI, 12/6/13
propensity to spend
debt
Markov model
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Attacks on the system
Edge deformation: policy decisions, sharp trade evolution.
Bilateral edge deletion: war, collapse of trade agreement.
Node deformation: internal collapse (e.g. bhatcollapse in the 1980s)
Node deletion: unrealistic but useful as a type of worst case scenario
Maximal Extinction Analysis (MEA): really a worst case scenario!
WPI, 12/6/13
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Power and robustness
Given an attack, a , we measure two things:
𝑃𝑜𝑤𝑒𝑟(𝑎) = 1 − 𝑘𝐸5 𝑘
𝑘𝐸0 𝑘
𝑅𝑡𝑦𝑝𝑒 = 1 − max𝑎∈𝑡𝑦𝑝𝑒
𝑃𝑜𝑤𝑒𝑟(𝑎)
WPI, 12/6/13
Total initial $$
Total $$ after
rebalancing
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is “robust yet fragile”
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
WPI, 12/6/13
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WPI, 12/6/13
WTWs are
“robust yet fragile”
Left hand side:
TARGETED ATTACK
The strength of maximal
attacks of each type. Colored
bars (and circles) indicate
significance.
Right hand side:
RANDOM ATTACK
Circles indicate the proportion
of all possible attacks which
are not significant.
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When is an attack significant?
An attack is deemed significant if
Thus, an attack is significant if it produce
some second (or higher) order damage to
the system.
WPI, 12/6/13
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is “robust yet fragile”
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
WPI, 12/6/13
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The role of connectance
WPI, 12/6/13
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The role of connectance
WPI, 12/6/13
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A closer look
WPI, 12/6/13
U.S./Canada
link
U.S.
deformation
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Discussion
WPI, 12/6/13
We see evidence that increased connectance has two effects related to risk in the system.
1. On one hand, denser connections allow for more paths through which shocks may be mitigated.
2. But, on the other, denser connection patterns provide more paths along which collapse can spread.
These two are in tension.
Further, we see an additional wrinkle related to connectance coupled with the topology of the network.
3. Denser connections allow for propagation of shocks which, while possibly mitigated overall, can have adverse impact on individual countries.
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Emergent and Systemic risk
In our model, the tension is resolved in different ways depending on the size of the shock.
Systemic risk
a. Smaller shocks are easily absorbed into the system (and sometimes result in income increases!).
b. But, there is a tipping point above which the larger shocks spark a substantial contagion effect.
Emergent Risk
c. Even with smaller shocks, we see evidence that mere participation in the WTW brings new risk.
d. Large shocks amplify this risk.
WPI, 12/6/13
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is “robust yet fragile”
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
WPI, 12/6/13
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Income-Expenditure model
(revised)
Relationships:
𝑂𝑀 𝑖 = 𝛼𝑖𝐼𝑀 𝑖𝑀 = 𝑑𝑖𝑎𝑔 𝑂𝑀 𝑚
Iterative model:
𝐸𝑘 = 𝑑𝑖𝑎𝑔 𝛼 𝑀𝑘𝑇 ⋅ 1
𝑀𝑘+1 = 𝑑𝑖𝑎𝑔 𝐸𝑘 𝑚
WPI, 12/6/13
𝐸𝑘 = 𝐴𝑀𝑘𝑇 ⋅ 1
= 𝐴 𝑑𝑖𝑎𝑔 𝐸𝑘−1 𝑚𝑇 ⋅ 1
= 𝐴𝑚𝑇𝐸𝑘−1= 𝐴𝑚𝑇𝐴𝑀𝑘−1
𝑇 ⋅ 1= 𝐴𝑚𝑇 2𝐸𝑘−2⋮= 𝐴𝑚𝑇 𝑘𝐸0
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Asymptotic Power
𝑃𝑜𝑤 = 1 −𝐸𝑘𝑇1
𝐸0𝑇1
= 1 −𝐸0𝑇 𝑚𝐴 𝑘1
𝐸0𝑇1
Under mild assumptions on 𝑚𝐴, as 𝑘 → ∞,
𝑃𝑜𝑤 = 1 − lim𝑘→∞𝜆𝑘𝐸0𝑇 𝑣 𝑣𝑇1
𝐸0𝑇1
where (𝜆, 𝑣) is the lead eigenvalue/eigenvector pair for 𝑚𝐴.
WPI, 12/6/13
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Consequences
𝑃𝑜𝑤 = 1 − lim𝑘→∞𝜆𝑘𝐸0𝑇 𝑣 𝑣𝑇1
𝐸0𝑇1
If 𝜆 < 1, the power is asymptotically equal to one, i.e. we expect the entire trade network to collapse in infinite time.
If 𝜆 > 1, the deformation asymptotically creates an infinite amount of money (!).
When 𝜆 = 1, the power of the deformation is governed by 𝐸0 ⋅ 𝑣.
WPI, 12/6/13
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𝜆 over time
WPI, 12/6/13
1960 1965 1970 1975 1980 1985 1990 1995 20000.998
0.9985
0.999
0.9995
1
1.0005
Year
For our empirical networks, 𝜆 is almost always essentially equal to
one.
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Eigenvector centrality
When 𝜆 = 1, the power of the deformation is governed by 𝐸0 ⋅ 𝑣.
The vector 𝑣 can be interpreted as the eigenvector centrality of the network given by 𝑚𝐴.
As a consequence, we see that shocks are the most powerful when 𝐸0 is close to orthogonal to 𝑣. In other words, if the expenditures are happening in patterns that avoid the most central nodes, the power can be significant.
WPI, 12/6/13
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Eigenvectors for empirical networks
WPI, 12/6/13
1965 1970 1975 1980 1985 1990 1995
10
20
30
40
50
60
70
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
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Eigenvector centrality over time:
Examples
WPI, 12/6/13
1960 1980 20000
0.5
1Canada
1960 1980 20000
0.5
1USA
1960 1980 20000
0.5
1Mexico
1960 1980 20000
0.5
1India
1960 1980 20000
0.5
1China
1960 1980 20000
0.5
1UK
1960 1980 20000
0.5
1USSR/Russia
1960 1980 20000
0.5
1Japan
1960 1980 20000
0.5
1France,Monac
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Edge deformation
Given the importance of the lead eigenvalue to the stability of the system, we study its perturbation under an edge deformation.
Fix 𝑖, 𝑗. Let 𝑀 be a trade matrix and 𝑀(𝜖) be the matrix which is identical to 𝑀 except for the 𝑖𝑗𝑡ℎ entry which is replaces with 𝑀𝑖𝑗(1 + 𝜖). Let 𝑚 and 𝑚 𝜖 be the associated Markov chains and let 𝐴 be the diagonal matrix of 𝛼′𝑠 . We define a deformation of 𝑚𝐴 by 𝑚 𝜖 𝐴.
Theorem: Under this deformation, we have
𝜆′ 0 = 𝑚𝑖𝑗𝑣𝑖
𝑘≠𝑗
𝑚𝑖𝑘(𝑣𝑗𝛼𝑗 − 𝑣𝑘𝛼𝑘) .
WPI, 12/6/13
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Edge deformation
Theorem: Under this deformation, we have
𝜆′ 0 = 𝑚𝑖𝑗𝑣𝑖
𝑘≠𝑗
𝑚𝑖𝑘(𝑣𝑗𝛼𝑗 − 𝑣𝑘𝛼𝑘) .
WPI, 12/6/13
Centrality of i
and the %
sent from i to
j.
The relative
centralities,
amplified by the 𝛼make this either
positive or
negative
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Example: 1997
(𝜆′ > 0)
WPI, 12/6/13
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Asian economic crisis
WPI, 12/6/13
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Example: 1997 (𝜆′ < 0)
WPI, 12/6/13
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is “robust yet fragile”
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
WPI, 12/6/13
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𝑘-amplification
Given the importance of the matrix 𝑚𝐴, we focus on its further interpretation.
We can think of 𝐴 as an amplification matrix.
𝐴𝑖𝑖 = 𝛼𝑖 is the amplification factor for node i. It measures the amplification of changes to the income of node i in the next step.
WPI, 12/6/13
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𝑘-amplification
What happens after one step?
𝛼𝑖
𝑘
𝑚𝑖𝑘𝛼𝑘 = 𝐴𝑚𝐴 ⋅ 1
After k steps:
𝐴 𝑚𝐴 𝑘 ⋅ 1
So, we define this as the 𝑘-amplification vector. The 0-amplification is simply 𝛼.
WPI, 12/6/13
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Local and global effects
We can use the 𝑘-amplifications as potential
explanatory variables for the power of
deformations. The larger the k, the more steps
we include.
Hence, lower k-amplifications measure local
properties of the network near a node while
higher k-amplifications incorporate more global
network information.
WPI, 12/6/13
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Local and global effects
Considering the significant power as our dependent variable and the k-amplifications of the source and target nodes of an edge deformation as independent variables, we perform simple regression.
Findings:
1. The k-amplifications of the source node i have almost no explanatory power.
2. Alone, the 0-amplification of node j has substantial power for years 1962-1997 (𝑅2 ∈ 0.86,0.96 , 𝑝 ≈ 0). But for 1998-2000, the power drops significantly 𝑅2 = 0.76,0.05, 0.14 , 𝑝 ≈ 0 .
3. For the last three years, global properties of the network contribute substantially to the explanatory power.
WPI, 12/6/13
i j
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Local and global effects
WPI, 12/6/13
1960 1965 1970 1975 1980 1985 1990 1995 20000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Comparison of local and global models
Year
R2
Full model
0-amplification only
During the
Asian
economic
crisis, global
structure
plays a large
role.
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Summary of results
1. The WTW is robust yet fragile.
2. Increasing globalization, as witnessed through connectance, mitigates risk initially but then amplifies it.
3. The extent to which deformations harm the global stability of the network depends on the interaction of the eigenvector centrality of the fundamental matrix and the amplification.
4. The k-amplifications demonstrate the interaction between local and global geometric properties and their effect on stability. Empirically, we see that in one time of substantial crisis, the global effects dominate. Thus, we have evidence of emergent risk in such periods.
WPI, 12/6/13