Systemic Risk: Robustness and

Fragility in Trade Networks.

Scott Pauls

Department of Mathematics

Dartmouth College

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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.

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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

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Systemic Risk and “normal

accidents”

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 “robust yet fragile”

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. “TRADING DATA: Evaluating our Assumptions

and Coding Rules.” 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 – 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 … n

A 5% 25% 1% 10% 7% 13% 2% … 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 𝛼 and call it the propensity to spend.

We denote the debt by 𝛽.

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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

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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:

𝑃𝑜𝑤𝑒𝑟(𝑎) = 1 − 𝑘𝐸5 𝑘

𝑘𝐸0 𝑘

𝑅𝑡𝑦𝑝𝑒 = 1 − max𝑎∈𝑡𝑦𝑝𝑒

𝑃𝑜𝑤𝑒𝑟(𝑎)

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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 “robust yet fragile”

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

“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.

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 “robust yet fragile”

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 “robust yet fragile”

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:

𝑂𝑀 𝑖 = 𝛼𝑖𝐼𝑀 𝑖𝑀 = 𝑑𝑖𝑎𝑔 𝑂𝑀 𝑚

Iterative model:

𝐸𝑘 = 𝑑𝑖𝑎𝑔 𝛼 𝑀𝑘𝑇 ⋅ 1

𝑀𝑘+1 = 𝑑𝑖𝑎𝑔 𝐸𝑘 𝑚

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𝐸𝑘 = 𝐴𝑀𝑘𝑇 ⋅ 1

= 𝐴 𝑑𝑖𝑎𝑔 𝐸𝑘−1 𝑚𝑇 ⋅ 1

= 𝐴𝑚𝑇𝐸𝑘−1= 𝐴𝑚𝑇𝐴𝑀𝑘−1

𝑇 ⋅ 1= 𝐴𝑚𝑇 2𝐸𝑘−2⋮= 𝐴𝑚𝑇 𝑘𝐸0

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.

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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.

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𝜆 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, 𝜆 is almost always essentially equal to

one.

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.

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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 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) .

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Edge deformation

Theorem: Under this deformation, we have

𝜆′ 0 = 𝑚𝑖𝑗𝑣𝑖

𝑘≠𝑗

𝑚𝑖𝑘(𝑣𝑗𝛼𝑗 − 𝑣𝑘𝛼𝑘) .

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Centrality of i

and the %

sent from i to

j.

The relative

centralities,

amplified by the 𝛼make this either

positive or

negative

Example: 1997

(𝜆′ > 0)

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Asian economic crisis

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Example: 1997 (𝜆′ < 0)

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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

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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.

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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.

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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.

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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.

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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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