Dr. Kimmo SoramäkiFounder and CEOFNA, www.fna.fi

Center for Financial Studies at the Goethe UniversityPhD Mini-course Frankfurt, 25 January 2013

Financial Networks

III. Centrality and Systemic Importance

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Agenda for today

• Centrality and Network Core• Developing SinkRank• Analyzing and visualizing cross-border banking

exposures

Centrality in Networks

Degree: number of links

Closeness: distance from/to other nodes via shortest paths

Betweenness: number of shortest paths going through the node

Eigenvector: nodes that are linked byother important nodes are more central, probability of a random process

Common centrality metrics

Centrality aims to summarize some notion of importance. Operationalizing the concept is more challenging.

Centrality depends on network process• Trajectory

– Geodesic paths (shortest paths)– Any path (visit a given node once)– Trails (visit a given link once)– Walks (free movement)

• Transmission – Parallel duplication– Serial duplication – Transfer

Borgatti (2005). Centrality and network flow . Social Networks 27, pp. 55–71.

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Path based centrality measures

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Closeness

• The Farness of a node is defined as the sum of its distances to all other nodes

• The Closeness of a node is defined as the inverse of the farness

• Needs a connected graph (or component)

• Directed/undirected

• Weighed/un-weighted

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Betweenness Centrality

• Measures the number of shortest paths going through a vertex or an arc

• Algorithm– For each pair of vertices (s,t),

compute the shortest paths between them

– For each pair of vertices (s,t), determine the fraction of shortest paths that pass through the vertex in question

– Sum this fraction over all pairs of vertices (s,t).

• Directed/undirected; Weighed/unweighted

Freeman, Linton (1977). "A set of measures of centrality based upon betweenness". Sociometry 40: 35–4

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Calculating BWC and Closeness

# Load sample networkloada -file pathnetwork-a.txt[delimiter=tab] -preserve falseloadvp -file pathnetwork-v.txt[delimiter=tab]

# Calculate unweighted and undirected Betweenness centralitybwc -direction undirected

# Calculate unweighted and undirected Closeness centralitycloseness -direction undirected

# Set values for arcscalcap -e [?random:uniform:1,5:123?] -saveas weight

# Calculate weighed and undirected Betweenness centralitybwc -direction undirected -p weight -saveas bwc-weighted

# Calculate unweighted and undirected Closeness centralitycloseness -direction undirected -p weight -saveas closeness-weighted

# Visualizeviz -arrows false -vsize bwc

Cut Edge/Arc

Cut edge or bridge is an edge whose deletion increases the number of connected components

Tarjan ('74) provides a linear time algorithm

Cut Points/Vertices

# Add network ‘CutPoint' to database. addn -n CutPoint -preserve false

# Add vertices and arcs to network.adda -a v1-v2adda -a v1-v5adda -a v2-v3adda -a v3-v4adda -a v4-v5adda -a v3-v6adda -a v6-v8adda -a v8-v7adda -a v6-v7

# Identify cut arc and vertexcutarccutvertex

# Visualizeviz -vcolor cutvertex -vsizedefault 10 -vlabel vertex_id -awidthdefault 2 -arrows false -fontsize 25 -

saveas CutPointViz

Cut points are the end vertices of a cut arc (if their degree is not 1)

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Walk based centrality measures

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Sample network

A B C

A 0 1 2

B 1 0 0

C 0 1 0

Adjacency matrix

Transition matrix :Right stochastic Left stochastic

A B C

A 0 1/3 2/3

B 1 0 0

C 0 1 0

adda -a A-C -preserve falseadda -a C-Badda -a A-Badda -a B-Asetap -p value -value 1setap -a A-C -p value -value 2

A B C

A 0 1 0

B 1/3 0 0

C 2/3 1 0

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Degree

• Local measure• Can be calculated for all types of networks

• Undirected, outgoing and incoming direction• Weighted degree = Strength

A B C

degree 3 3 2

out-degree 2 1 1

in-degree 1 2 1

strength 4 3 3

out-strength 3 1 1

in-strength 1 2 2

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Eigenvector Centrality (EVC)

• Connections are not equal, a connection to a more important node is more important

• We make centrality (xi) proportional to the average of the centrality (e.g. degree) of i’s network neighbors:

where λ is a constant and A the adjacency matrix (Aij =1 if link i-j exists, and 0 otherwise)

• Defining a vector of centralities x=(x1, x2, ..., xn), we can rewrite

lx=Ax• We see that x is an eigenvector of the adjacency matrix with

eigenvalue λ

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• All entries of x are positive for eigenvector associated with the largest eigenvalue (Perron–Frobenius theorem). The entry xi gives EVC for node i.

• Adjacency matrix A can also contain weights instead of 0-1 links -> weighed EVC

• The graph can be directed (asymmetric A) -> directed EVC

• Can contain loops (self-links, Ai=j)

• The graph must be strongly connected!Can be calculated only for GSCC.

EVC - Properties

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Markov chains

• Markov chains are memoryless randomprocesses that undergo transitions fromone state to another

• We describe a Markov chain as follows: We have a set of states, S = {s1, s2,.. sn}

• A process starts in one of these states and moves successively from one state to another. Each move is called a step.

• If the chain is currently in state si, it moves to state sj at the next step with a probability denoted by pij

• The probabilities pij are called transition probabilities and a matrix T specifying pij's a transition matrix

(100%)

(100%)

(33.3%)

(66.6%)

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State probability vector

• Let q(t)=(q1(t),, q2

(t) , ... , qn(t)) be the state probability vector

whose ith component is the probability that the chain is in state i at time t.

• Markov chain is fully defined by q(0) and T

q(t)=q(t−1)T=q(0)Tt

• q(t) is also called the distribution of the chain at time t

• Question: at which probabilities do we find a random process at states si when t is large?

• An important node would have the process visit it often

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Stationary probability vector

• A stationary probability vector π is defined as a vector that does not change under application of the transition matrix

π= πT

• For any – irreducible (~ strongly connected component)– aperiodic (~ process does not visit nodes at determined

intervals)– positive-recurrent (~ process re-enters each node

eventually)

Markov Chain there exists a unique stationary probability vector (Fundamental Theorem of Markov Chains)

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Simple way of Calculating

• The distribution vector after 1 step is the matrix product, q(0)T

• The distribution one step later, obtained by again multiplying by T, is given by (q(0)T)T = q(0)T2.

• Similarly, the distribution after t steps can be obtained by multiplying q(0) on the right by T t times, or multiplying q(0) by Tt.

• Distribution After t Steps: q(t)=q(0)Tt

• EVC = elements of q(t) for a large t

• Power iteration -method

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Combining iterative and Markov chain interpretations

• The Perron–Frobenius theorem says that in a stochatic matrix, the largest absolute eigenvalue is always 1

• Transition matrix can be right (T) or left (T') stochastic

• As a result we have:

lπ=Aπ (Eigenvector)π=πT (Markov Chain)

1π=T'π (Largest Eigenvector, i.e. 1, of left stochastic transition matrix)

Most networks are not strongly connected

• EVC can be calculated only for “Giant Strongly Connected Component” (GSCC)

• Due to need for irreducible, aperiodic, positive-recurrent Markov Chain

• Solution: PageRank and the 'Random Surfer" -model

PageRank

• Solves the problem with a “Damping factor” a which is used to modify the transition matrix (S)– Gi,j= aTi,j + (1- )/a N

• Effectively allowing the random process out of dead-ends (dangling nodes), but at the cost of introducing error

• Effect of a– =0 -> a Centrality of each node is 1/N

– =1 -> a Eigenvector Centrality

– Commonly =0.85 a is used =0 (0.333, 0.333, 0.333)a=0.85 (0.368, 0.374, 0.258)a=1 (0.375, 0.375, 0.250)a

A B

C

(100%)

(100%)

(33.3%)

(66.6%)

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Calculating EVC and PageRank

A B C

EVC 0.375 0.375 0.250

PageRank 0.368 0.374 0.258

PageRank-0 0.375 0.375 0.250

PageRank-1 0.333 0.333 0.333

CheiRank 0.397 0.388 0.215

CheiRank-0 0.400 0.400 0.200

CheiRank-1 0.333 0.333 0.333

# Create sample networkadda -a A-C -preserve falseadda -a C-Badda -a A-Badda -a B-Asetap -p value -value 1setap -a A-C -p value -value 2

# Calculate weighted and directed EVCevc -p value -saveas EVC

# Calculate PageRank (default alpha=0.15)# Note: This relates to 0.85 in slidespagerank -p value -saveas PageRank

# Calculate PageRank (alpha=0)pagerank -p value -alpha 0 -saveas PageRank-0

# Calculate PageRank (alpha=1)pagerank -p value -alpha 1 -saveas PageRank-1

#Calculate same for CheiRankcheirank -p value -saveas CheiRankcheirank -p value -alpha 0 -saveas CheiRank-0cheirank -p value -alpha 1 -saveas CheiRank-1

# save results in a csv filesavev -file walkcentrality.csv

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Final notes on PageRank/EVC

• Undirected vs. Directed– PageRank generally in-

direction– out-direction = CheiRank

• Unweighted vs. Weighted– 0/1 or real values in A/T

Important Important and Fragile

Unimportant Fragile

PageRank

Chei

Rank

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Identifying the core

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Maximum Clique

• A graph may contain many complete subgraphs ("cliques"), i.e. sets of nodes where each pair of nodes is connected

• The largest of these is called 'Maximum Clique'• One way of finding the 'core'

# Create random networkrandom -nv 30 -na 120 -preserve false -seed 123

# Identify maximum undirected clique# 0 - no clique, 1 - maximum clique, 2... smaller cliquesmaxclique -direction any

# Set color property of nodes in clique as redsetvp -p color -value red -e maxclique=1

# Visualizeviz -vcolor color -vsizedefault 8 -arrows false

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Newman Modularity

• Method for detecting modules (also called groups, clusters or communities)

• Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules.

Newman, M. E. J. (2006). "Modularity and community structure in networks". PROCEEDINGS- NATIONAL ACADEMY OF SCIENCES USA 103 (23): 8577–8696.

# Create random treetree -nv 30 -preserve false -seed 123

# Identify communities with Newman's modularity algorithmnewman

# Visualizeviz -vcolor newman -vsizedefault 8 -arrows false

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Craig - von Peter Core

• Interbank markets are tiered in a core-periphery structure

• Determines the optimal set of core banks that achieves the best structural match between observed structure and perfectly tiered structure# Create network with core-periphery structurecomplete -nv 3 -preserve false -directed falseadda -a 00001-00004adda -a 00002-00005adda -a 00003-00006

# Calculate corecvpcore

# Set color property of nodes in clique as redsetvp -p color -value red -e cvpcore=true

# Visualizeviz -vcolor color -vsizedefault 8 -arrows false

Ben Craig and Goetz von Peter (2010). Interbank tiering and money center banks, BIS Working Papers No 322.

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Developing a centrality metric for Payment Systems

SinkRank

Discussion Paper, No. 2012-43 | September 3, 2012 http://www.economics-ejournal.org/economics/discussionpapers/2012-43

Interbank Payment Systems

• Provide the backbone of all economic transactions

• Banks settle claims arising from customers transfers, own securities/FX trades and liquidity management

• Target 2 settled 839 trillion in 2010

Systemic Risk in Payment Systems

• Credit risk has been virtually eliminated by system design (real-time gross settlement)

• Liquidity risk remains– “Congestion” – “Liquidity Dislocation”

• Trigger may be– Operational/IT event– Liquidity event– Solvency event

• Time scale is intraday, spillovers possible

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Process in payment systems

Transfer along walks

Distance to Sink

From B

From C

1

2

1

To A

From A

From CTo B

From A

From BTo C

• Markov chains are well-suited to model transfers along walks

• Absorbing Markov Chains give distances:

(100%)

(100%)

(33.3%)

(66.6%)

SinkRank

• SinkRank is the average distance to a node via (weighted) walks from other nodes

• We need an assumption on the distribution of liquidity in the network at time of failure

– Assume uniform -> unweighted average

– Estimate distribution -> PageRank -weighted average

– Use real distribution ->Real distribution are used as weights

SinkRanks on unweighted networks

SinkRank – effect of weights

Uniform(A,B,C: 33.3% )

“Real”(A: 5% B: 90% C:5%)

Note: Node sizes scale with 1/SinkRank

PageRank(A: 37.5% B: 37.5% C:25%)

How good is it?

Experiments

• Design issues

– Real vs. artificial networks?– Real vs. simulated failures?– How to measure disruption?

• Approach taken

1. Create artificial data with close resemblance to the US Fedwire system (BA-type, Soramäki et al 2007)

2. Simulate failure of a bank: the bank can only receive but not send any payments for the whole day

3. Measure “Liquidity Dislocation” and “Congestion” by non-failing banks

4. Correlate 3. (the “Disruption”) with SinkRank of the failing bank

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Barabási–Albert (BA) model

• Based on Barabási–Albert (BA) model • The BA algorithm generates random scale-free networks

and is based on two forces: growth an preferential attachment:– The network begins with an initial network of m0 (>2) nodes. – New nodes are added to the network one at a time. – Each new node is connected to existing nodes with a probability that

is proportional to the number of links that the existing nodes already have.

• Instead of links, we generate payments (multiple links between pairs of nodes)

• We use lower preferential attachment accumulation than the BA model

Generated data

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Measures

• Congestion: duration of delays in the system aggregated over all banks

• Liquidity Dislocation: the average reduction in available funds of the other banks due to the failing bank

• Disruption: duration-weighted sum of Congestion and Liquidity Dislocation

• -> Carry out counterfactual simulations with generated data - failing banks and measuring impact

Distance from Sink vs. Disruption

Highest disruption to banks whose liquidity is absorbed first (low Distance to Sink)

Relationship between Failure Distance and Disruption when the most central bank fails

Distance to Sink

SinkRank vs. Disruption

Relationship between SinkRank and Disruption

Highest disruption by banks who absorb liquidity quickly from the system (low SinkRank)

Implementing SinkRank

Implementation example

available at www.fna.fi

Blog, Library and Demos at www.fna.fi

Dr. Kimmo Soramäki kimmo@soramaki.netTwitter: soramaki