systemic risk
TRANSCRIPT
Systemic Risk: Robustness and
Fragility in Trade Networks.
Scott Pauls
Department of Mathematics
Dartmouth College
WPI, 12/6/13
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.
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is โrobust yet fragileโ
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
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Systemic Risk and โnormal
accidentsโ
Perrow, 1984:
interactive
complexity & normal accidents
tight coupling
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Stability and robustness
Robustness
Dynamics
Structure
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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.
Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is โrobust yet fragileโ
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
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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.
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Katherine Barbieri, Omar M. G. Keshk, and Brian Pollins. 2009. โTRADING DATA: Evaluating our Assumptions
and Coding Rules.โ Conflict Management and Peace Science. 26(5): 471-491.
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.
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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.
Dynamics
We consider trade relations analogously to plumbing โ 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.
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1 2 3 4 5 6 7 โฆ n
A 5% 25% 1% 10% 7% 13% 2% โฆ 6%
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 ๐ผ and call it the propensity to spend.
We denote the debt by ๐ฝ.
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Income-Expenditure model
In/out-strength:
๐ผ๐ ๐ =
๐
๐๐๐ , ๐๐ ๐ =
๐
๐๐๐
Relationships:
๐๐ ๐ = ๐ผ๐๐ผ๐ ๐ + ๐ฝ๐๐ = ๐๐๐๐ ๐๐ ๐
Iterative model:
๐ธ๐ก = ๐๐๐๐ ๐ผ ๐๐ก๐ โ 1 + ๐ฝ
๐๐ก+1 = ๐๐๐๐ ๐ธ๐ก ๐
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propensity to spend
debt
Markov model
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!
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Power and robustness
Given an attack, a , we measure two things:
๐๐๐ค๐๐(๐) = 1 โ ๐๐ธ5 ๐
๐๐ธ0 ๐
๐ ๐ก๐ฆ๐๐ = 1 โ max๐โ๐ก๐ฆ๐๐
๐๐๐ค๐๐(๐)
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Total initial $$
Total $$ after
rebalancing
Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is โrobust yet fragileโ
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
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WTWs are
โrobust yet fragileโ
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.
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.
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is โrobust yet fragileโ
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
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The role of connectance
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The role of connectance
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A closer look
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U.S./Canada
link
U.S.
deformation
Discussion
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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.
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.
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is โrobust yet fragileโ
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
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Income-Expenditure model
(revised)
Relationships:
๐๐ ๐ = ๐ผ๐๐ผ๐ ๐๐ = ๐๐๐๐ ๐๐ ๐
Iterative model:
๐ธ๐ = ๐๐๐๐ ๐ผ ๐๐๐ โ 1
๐๐+1 = ๐๐๐๐ ๐ธ๐ ๐
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๐ธ๐ = ๐ด๐๐๐ โ 1
= ๐ด ๐๐๐๐ ๐ธ๐โ1 ๐๐ โ 1
= ๐ด๐๐๐ธ๐โ1= ๐ด๐๐๐ด๐๐โ1
๐ โ 1= ๐ด๐๐ 2๐ธ๐โ2โฎ= ๐ด๐๐ ๐๐ธ0
Asymptotic Power
๐๐๐ค = 1 โ๐ธ๐๐1
๐ธ0๐1
= 1 โ๐ธ0๐ ๐๐ด ๐1
๐ธ0๐1
Under mild assumptions on ๐๐ด, as ๐ โ โ,
๐๐๐ค = 1 โ lim๐โโ๐๐๐ธ0๐ ๐ฃ ๐ฃ๐1
๐ธ0๐1
where (๐, ๐ฃ) is the lead eigenvalue/eigenvector pair for ๐๐ด.
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Consequences
๐๐๐ค = 1 โ lim๐โโ๐๐๐ธ0๐ ๐ฃ ๐ฃ๐1
๐ธ0๐1
If ๐ < 1, the power is asymptotically equal to one, i.e. we expect the entire trade network to collapse in infinite time.
If ๐ > 1, the deformation asymptotically creates an infinite amount of money (!).
When ๐ = 1, the power of the deformation is governed by ๐ธ0 โ ๐ฃ.
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๐ over time
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1960 1965 1970 1975 1980 1985 1990 1995 20000.998
0.9985
0.999
0.9995
1
1.0005
Year
For our empirical networks, ๐ is almost always essentially equal to
one.
Eigenvector centrality
When ๐ = 1, the power of the deformation is governed by ๐ธ0 โ ๐ฃ.
The vector ๐ฃ can be interpreted as the eigenvector centrality of the network given by ๐๐ด.
As a consequence, we see that shocks are the most powerful when ๐ธ0 is close to orthogonal to ๐ฃ. In other words, if the expenditures are happening in patterns that avoid the most central nodes, the power can be significant.
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Eigenvectors for empirical networks
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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
Eigenvector centrality over time:
Examples
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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
Edge deformation
Given the importance of the lead eigenvalue to the stability of the system, we study its perturbation under an edge deformation.
Fix ๐, ๐. Let ๐ be a trade matrix and ๐(๐) be the matrix which is identical to ๐ except for the ๐๐๐กโ entry which is replaces with ๐๐๐(1 + ๐). Let ๐ and ๐ ๐ be the associated Markov chains and let ๐ด be the diagonal matrix of ๐ผโฒ๐ . We define a deformation of ๐๐ด by ๐ ๐ ๐ด.
Theorem: Under this deformation, we have
๐โฒ 0 = ๐๐๐๐ฃ๐
๐โ ๐
๐๐๐(๐ฃ๐๐ผ๐ โ ๐ฃ๐๐ผ๐) .
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Edge deformation
Theorem: Under this deformation, we have
๐โฒ 0 = ๐๐๐๐ฃ๐
๐โ ๐
๐๐๐(๐ฃ๐๐ผ๐ โ ๐ฃ๐๐ผ๐) .
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Centrality of i
and the %
sent from i to
j.
The relative
centralities,
amplified by the ๐ผmake this either
positive or
negative
Example: 1997
(๐โฒ > 0)
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Asian economic crisis
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Example: 1997 (๐โฒ < 0)
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Outline
1. Introduction to the problem and our questions.
2. Vocabulary and Methodology
3. The world trade web is โrobust yet fragileโ
4. Connectance and the impact of globalization
5. The fundamental structural matrix and its properties
6. Amplification and systemic risk
WPI, 12/6/13
๐-amplification
Given the importance of the matrix ๐๐ด, we focus on its further interpretation.
We can think of ๐ด as an amplification matrix.
๐ด๐๐ = ๐ผ๐ is the amplification factor for node i. It measures the amplification of changes to the income of node i in the next step.
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๐-amplification
What happens after one step?
๐ผ๐
๐
๐๐๐๐ผ๐ = ๐ด๐๐ด โ 1
After k steps:
๐ด ๐๐ด ๐ โ 1
So, we define this as the ๐-amplification vector. The 0-amplification is simply ๐ผ.
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Local and global effects
We can use the ๐-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.
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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 (๐ 2 โ 0.86,0.96 , ๐ โ 0). But for 1998-2000, the power drops significantly ๐ 2 = 0.76,0.05, 0.14 , ๐ โ 0 .
3. For the last three years, global properties of the network contribute substantially to the explanatory power.
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i j
Local and global effects
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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.
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.
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