Banking Supervision (BCBS) - BOT

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Systemic Import Analysis (SIA) � Application of Entropic Eigenvector Centrality (EEC) Criterion

for A Priori Ranking of Financial Institutions in terms of Regulatory-Supervisory Concern,

with Demonstrations on Stylised Small Network Topologies and Connectivity Weights

Poomjai Nacaskul, Ph.D.1

Abstract � This paper presents a simple method for quantifying relative import amongst

financial institutions in terms of the systemic risk they bear on any highly interconnected

financial system to which they belong. W.r.t. the macroprudential framework comprising

risk�based supervision and systemic�stability regulation, this paper distinguishes three

levels of analysis based on network models of systemic phenomena, and appeals to the

connectionist principle whereby an entity is deemed �systemically important� if it materially

impacts more of other �systemically important� entities. This recursive definition essentially

amounts to (Bonacich�s) Eigenvector Centrality (BEC) concept, yet BEC is not apt for

macroprudential applications for a number of reasons. This paper thus proposes a weighted-

network extension based on (information) entropy consideration, the Entropic Eigenvector

Centrality (EEC) criterion, for assessing systemic implications of individual system entities,

as per Systemic Import Analysis (SIA), and demonstrates its usage on a number of stylised

small network topologies and connectivity weights, such as may represent channels of risk

propagation amongst economic agents, esp. regulated and supervised financial institutions.

1. Introduction

1.1 Macroprudential Framework & Network Modelling

Prior to the global financial (nee subprime) crisis of 2007, bank regulators/supervisory agencies

worldwide were already well committed to the so-called risk�based supervision modelwhose root

may be traced back to the US Office of the Comptroller of the Currency (OCC) [1] and latterly

championed by the likes of (UK) Financial Services Authority (FSA) [2] and Basel Committee on

1 Team Executive – Quantitative Models & Financial Engineering Team, Financial Institutions Strategy Department, Bank of Thailand [[email protected]] & Faculty – MBA Program, Mahanakorn University of Technology.

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Banking Supervision (BCBS)and central bankers in general welcomed of the instruments of

macroprudential surveillance. In particular, some central banks, especially those concurrently mandated

payment system oversight, notably Oesterreichische Nationalbank (OeNB), the Austrian central bank,

perhaps having correctly identified payment/settlement/clearing circle as a network of liquidity transfer

entities, began to formulate systemic contagion in the language of network models (and implicitly graph

theory) [3]. Meanwhile, yet other central banks, especially with recent experience of asset bubbles,

notably Banco de España (BdE), the Spanish central bank, perhaps having wisely acknowledged the

impact of asset inflation on the conduct of monetary policy, began to devise microprudential measures to

specifically address systemic procyclicality [4].

With the coming (and hopefully passing) of this crisis, the very word �systemic� and all things

network-centric gained the much needed publicity [5][6][7][8], as policy reformers came to a general

consensus that one needs a proper macroprudential frameworkalbeit the very concept preceded this

crisis involving some kind of �systemic regulator�, i.e. an official body in charge of financial stability

(work) in the purely systemic sense. International policy discussions grew around the cross@sectional vs.

time dimensions of systemic risk [9], i.e. contagion vs. procyclicality, as disparate national regulators

scrambled to determine who exactly is in charge of systemic�stability regulation. In so doing,

methodological perspectives pioneered by the likes of OeNB and BdE came to the fore, gaining currency

and credibility within policy and academic circles alike.

1.2 Risk-Based Supervision & Systemic Importance

In many ways, the task is not entirely new. Prudential supervision (retroactively termed

microprudential supervision, to be sure) has always been about minimising the probability and extent of

bank failures. In the days when banking mostly meant retails, when system risk largely concerned

cheque@clearing circles, regulators rightly focused on banks on basis of financial assets and/or customer

base, hence the focus on size, as encapsulated by the �too big to fail� credo. The well@documented moral

hazard effects notwithstanding [10], modern rectification has been to heuristically append �in the systemic

sense� to the clause. Presumably, banks can be less certain about whether they are deemed �big in the

systemic sense�parenthetically, the now widely accepted phrase is �too interconnected to fail�

[11]and therefore somewhat less certain also about the extent to which they benefit from any form of

implicit guarantees. A way out is seemingly found around the usual compromise between nourishing

natural system stability and engendering endemic market discipline.

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Now, with respect to risk@based supervision, a once common mistake amongst the general public,

even banking professionals, was to assume this simply meant placing risk management as objective of

bank examination, whence relegating the assessment of banks� Capital-Asset-Management-Earnings-

Liquidity (CAMEL) [12] to more archaic usage. At best, this is half the truth, and the less useful half at

that, for just as banks have always been in the business of taking and managing risks, so too have

supervisors always been charged with monitoring and ensuring banks� ability to manage risks they took

on. In other words, a methodological shift toward greater reliance on quantitative risk analysis method,

even one capable of integrating risk measurement across banking products and services, is only the half of

the deal. And not only is it obviously implied, easily assumed, and widely understood, but, as a function

of the state of knowledge and technology, this is almost bound to happen sooner or later anyway. The

other, essential half of risk@based supervision fundamentally expounds an adaptive shift of supervisory

resources toward supervised entities of greatest risk implication. To a degree, the extent of systemic risk

implication of any one financial institution can be (if somewhat circularly) defined by how much resource

a supervisory body ought to, plans to, or is seen to devote to it, relative to other supervised entities.

Arguably, risk@based supervision introduced a new analysis problem to the fold, that of

identifying, and preferably relatively ranking, quantitatively if possible, the likely impact each bank�s

failure vis-à-vis dynamic (in)stability of the systemic whole. Ideally, a supervisory body should have an

estimate, perhaps in some form of weight vectors, each element of which signifies the �systemic import�

for each bank under supervision, relative to the others, so that it is possible to strategically procure and

operationally allocate supervisory resourcesbe it in terms of manpower, on@site examination hours,

risk modelling expertise, database/computational endowment, research focus, goodwill, even willingness

to stake the supervisory body�s own institutional credibility, etc.accordingly.

It�s worth pointing out that the intent here is to construct an a priori metric of systemic import

with which to direct supervisory resources, hence much less model precision is required than would be the

case, say, if one wishes to stress test the financial system using failure(s) of one or more member bank(s)

as the stress scenarios [13], to factor in systemic risk exposure in the pricing of firm capitals [14], to

obtain such network statistics as �average path length�, clustering coefficient, and so on [15], to calibrate

system parameters from market data [16], or to evaluate the extent to which homogeneity/heterogeneity

amongst constituent entities effect fragility/robustness of the overall system [17].

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1.3 Three Levels of Network Analyses

Given the rapidly growing body of literatures on network models of systemic phenomena, it is

useful to distinguish three levels of network methodologies and the types of analysis they support.

Static Network Analysis @ at this level all we know of the network is essentially captured by some

form of matrix of connectivities. System entities correspond to the �graph-theoretic� notion of nodes,

while connections correspond to edgeswhich may be directed (with arrows) or undirected (without an

arrow)whence the modelled system in total can be drawn as a graph.2 Here we will be working with

directed graphs, also called digraphs. The resultant �structural� diagram (the collective specification as to

which nodes have edges connecting to/from which nodes), or (network) topology, may resemble any of a

number of stylised configurations. Well-known topologies include �ring�, �star�, �mesh�, �cascade�, and

�fully interconnected� networks.

At this point it is then possible to apply mathematical techniques to a priori extract a number of

salient features inherent to the system without actually allowing for any kind of �behavioural� modelling.

For regulatory-supervisory applications, each node may correspond to an individual bank, hence each

edge an interbank exposure (generally on a gross basis). Alternatively, a node may represent the banking

sector as a whole, with another to represent the real sector, and so on, so that procyclicality mechanism,

for instance, can be modelled as a flow of causation via the edge(s) connecting the two. This paper is

essentially confined to just this level of analysis.

Dynamic Network Analysis @ at this intermediate level each node is endowed with some kind of

state variable(s), each edge is endowed with propagation or update rule, and therefore the entire graph can

be said in a simulation environment to behave dynamically, with the collective state of the system

(comprising individual entity states) changing over time as external shocks are applied and propagated

through the network, i.e. via such linkages as credit lines or indeed any form of interbank exposures

[18][19], causing individual nodes to update along the way. Robustness analysis can also be performed by

simulating failure(s) of individual node(s), i.e. to see which/whether other node(s) will also succumb.

2 Mathematically speaking, a graph is the same as a network, although some authors may choose to

reserve the term �network� for a graph whose edges are associated with quantities, such as to signify

varying connectivity �weights� or �strengths�.

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Agent-Driven Network Analysis @ at this advanced level, each entity, while still represented as a

node in a graph, is best thought of as a semi-autonomous decision making agent [20][21] capable to

choosing the best responses to various input stimuli, even able to modify its connectivities vis-à-vis other

system entities. Here, not only are state variables changeable, but so is the very network topology itself. It

is fair to say that regulatory-supervisory research, comprising theoretical/methodological works, as well as

empirical findings in case countries [22][23][24][25][26][27][28][29][30], has barely (if at all) penetrated/

progressed up to this level of analysis.

2. Methodology

2.1 Bonacich�s Eigenvector Centrality (BEC)

Fortunately, static network analysis is well within our immediate grasp, especially as the issue of

quantifying relative importance has already been much explored [31][32], collectively under the notion of

(Node) Centrality, particularly (Bonacich�s) Eigenvector Centrality (BEC) [33], and applications abound

in contexts other than banking, especially epidemiology [34][35]. In the world of internet, for instance, not

only might a website be judged commercially significant or intellectually influential according to the

number of visits it gets on average each day, but also by the numbers of other websites that have links

dedicated to it. Yet on reflection a better measure is necessarily recursive: a website is considered

important if many important websites provide links to it [36][37]. Another example can be found in sports.

In a league, an ex post measure of a given team�s accomplishment is clearly its point accumulation, which

reflects the number of other teams it has beaten, tied with, or lost to. Yet a better ex ante measure of

strength would take into account about whether those whom it has beaten had also been among the

stronger teams, i.e. teams that are have beaten other strong teams, and so on, recursively.

Parenthetically, other than BEC, there exist at least three other forms of node centrality, namely

degree centrality, closeness centrality, and betweenness centrality. Of these, only degree centrality will

play further role/be related to the rest of the paper, especially as it is unclear how crucial the notion of

shortest path (on which the definition of closeness centrality as well as that of betweenness centrality rest)

can be when it comes to macroprudential concerns, especially as in financial network, finding the shortest

(i.e. least costly) path to liquidity may not be particularly relevant when there is a threat of systemic crisis.

Given a closed system of 2>n entities (nodes) there are 2n possible connections (edges), of

which )1( −nn connect between 2 different entities. Define as the adjacency matrix nnA ×∈ }1,0{ whose

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ijth entry equals 1 if entity i matters to entity j in some way, 0 if otherwise. For instance bank i might be

borrowing O/N from bank j, hence liquidity problem in bank i may well transpire to become liquidity

problem in bank j. We shall work mostly with asymmetric A , i.e. entity i may matter to entity j but not the

other way around (ijth entry equals 1 while ji

th entry equals 0). Often, they matter to each other. For

instance, while bank i is borrowing O/N from bank j, there is also an outstanding swap which currently

marks@to@market in favour of bank i, in which case the disfavoured bank j poses a counterparty risk to

bank i. Netting out bilateral exposures reduces the amount of information unnecessarily. Let�s keep, by

definition, the diagonal entries strictly 0.

Now, we would like to define systemic import vector nℜ∈v as a vector each of whose entries

signifies how systemically important the corresponding system entity is, relative to all other system

entities. This is taken to mean that the �systemic import� of entity i is proportional to the sum of �systemic

imports� all the entries to which entity i matters, hence, using entries of A to denote whether the

�systemic import� of entity j contributes to the �systemic import� of entity i, and so on):

{

0,

,,1,

1

,,1,

,,1,

,1,11,11

,,11,

>

=

⇓

=++++∝

k

v

v

v

AAA

AAA

AAA

k

v

v

v

nivAvAvAv

n

i

A

nninn

niiii

ni

n

i

nnijjiii

M

M

44444 344444 21LL

MOM

LL

MOM

LL

M

M

444444444 3444444444 21KLL

v

(1)

Clearly, any solution to this recursive equation will be an eigenvector corresponding to one of the

eigenvalues, themselves solutions to the characteristic polynomial thus:

( { ) { ( )44 344 21

polynomialsticcharacterivector

eigenvalueeigen

IAIAkA 0det0, =−⇒=−⇒=≠∀−−

λλ vvv0v (2)

2.2 Problems/Limitations with BEC

While this general approach is intuitively appealing, adapting it for macroprudential applications

in general calls for a couple of modifications, some of which are necessary, while others are desired

features that will enhance the method�s usefulness.

1. Guaranteeing the right interpretation requires that 0v ≥ .

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2. Guaranteeing that the solution is unique, i.e. it is always possible to specify which amongst

the solution eigenvectors { }nee ,,1 K to take as v .

3. Guaranteeing meaningful comparison requires that unless the network is completely

homogeneous and every way symmetric, elements of v cannot all have the same value,

hence considered a �degenerate�3 solution.

4. Incorporating pair-wise information with regards to the relative connectivity �strengths� or

�weights� requires working with a matrix of interval-scale elements, which A is not.

Let�s take care of the 1st and 2

nd points first. Here we elect to modify the adjacency matrix A by

replacing the zero entries each with a �small� value 0>ε . In this paper, we let 0001.0=ε , or 1 basis

point. By keeping all entries strictly positive, we can then invoke the celebrated Perron�Frobenius

theorem [38] to guarantee that the largest eigenvalue of A (largest in an absolute value sense, i.e. the

spectral radius of A ), called the Perron�Frobenius eigenvalue or the Perron root, FrobeniusPerron −−λ , will

be unique, and that its corresponding eigenvector, FrobeniusPerron −−e , will also be strictly positive (same

sign). This takes care of the 1st and 2

nd issues. In fact, in our application specifying 0v > makes better

sense than merely 0v ≥ , as no financial institution can be deemed to be of absolutely no systemic

consequence whatsoever.

The 3rd point remains problematic. As a matter of fact, BEC, as it stands, cannot resolve relative

importance when the network is characterised by an adjacency matrix that is mathematically regular. This

happens when all nodes have the same degree (defined as the number of edges to which it is connected).

For then the eigenvector corresponding to the Perron root will be �degenerate� (all elements have the same

value). Note that with a digraph there are in degree, the number of edges connected to and pointing to it,

as well as out degree, the number of edges connected to and pointing from it. In short, eigenvector

centrality is useless when we are faced with a network of financial institutions whose representative

adjacency matrix is regular in terms of the out degree.

Turning to the 4th point, replacing zero entries with ε notwithstanding, the adjacency matrix still

only posits strictly binary relationseither entity i does or does not matter to entity j while with

macroprudential framework, the degree of contagion risk to which bank i incurs upon bank j ought to take

3 From now on, this paper shall use the word �degenerate� to describe a vector whose elements are all the

same, so as to suggest that it vector cannot be used as a criterion to distinguish centrality scores.

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into account the amount of risk exposures (on/off balance-sheet liabilities accounted in monetary units,

varying drawdown probability, even varying extent of reputational linkages). This calls for a non@negative

real number representation, in short a generalisation from �unweighted� to weighted network application,

whence replacing nnA ×∈ }1,{ε by the (weighted) connectivity matrix nnC ×∞∈ ),[ε , i.e. a non@negative

square and generally asymmetric matrix each of whose entries signifies not only whether entity i matters

to entity j in an �all-or-nothing� sense, but by how much, relative to other entities which may or may not

also matter to entity j.4 One way of viewing the key conceptual difference between using adjacency vs.

connectivity matrix is that while the former purely expresses the effect of network topology, the latter can

be thought of as a fully-connected network (with zero weight assigned to every �absent� edge). Of course,

there might still be cases when a supervisory body has rather restricted information and is only prepared to

designate whether bank i systemically matters to bank j in an �all-or-nothing� sense, but as ),[}1,{ ∞∈⊂∈

at least the generality is there when required.

2.3 Entropic Eigenvector Centrality (EEC)

Turning to weighted networks [39] we find that the issue of how to construct analogous measures

is far from resolved. A simple replacement of A with C in the BEC calculation, for instance, enables one

to incorporate connectivity weights in a straight forward manner, and yet, to give a simple contra example,

one can no longer distinguish between a node with 1 outgoing connection of weight 100 (in some unit)

and a node with 100 outgoing connection of weight 1 (same unit), and yet, all else being equal, the latter

node (with out degree 100) should be systemically more critical than the former (with out degree of 1). To

an extent, the situation can be rectified by constructing a measure that mixes in the degree parameter. But

while such a fix is available for other notions of relative importance amongst nodes [40], no such fix is

available specifically for BEC, which is what we (motivated by macroprudential application) are most

interested in. In essence, we would like to use C instead of A but at the same time correct for the above

equivalence, again so that, all else being equal, a node with 100 outgoing connection of weight 1 will be

deemed more systemically important than one with 1 outgoing connection of weight (100 by virtue of

greater out degree by a factor of 100).

4 Note how the mathematical notion of �weighted� does not require that the weights sum to one, as with

�asset allocation� weights. In fact, normalising the rows of the weighted connectivity matrix effectively

makes it regular in terms of the out degree, i.e. the �degenerate� case. In any event, some authors may

choose to avoid the confusion by using the word connectivity �strength� instead of �weight�.

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Noting how (information) entropy neatly distinguishes between the two extreme situations, and

any other cases in between, this paper proposes the Entropic Eigenvector Centrality (EEC) criterion. As

EEC extends from BBC, the �systemic import� ranking from either system should be similar. In other

words, in most cases one would expect elements of �BEC vectors� and those of �EEC vectors� to have

high Spearman�s rank correlation (rho).

The notable exception of course is when the connectivity matrix involved is regular, in which

case EEC can resolve systemic importance in favour of nodes with higher out degree; where as, BEC

would yield a �degenerate� eigenvector/ranking.

In essence, one can think of EEC as relying on BEC as much as possible, but then invokes degree

centrality whenever relying on BEC alone leads to �degenerate� result. This is indeed the intuition that is

motivated from macroprudential application viewpoint.

EEC is formally constructed as follows:

1. Given nnC ×∞∈ ),[ε , create a row-normalised matrix nnW ×−∈ ]1,[ εε by dividing each

element with the corresponding row sum:

∑=

==∀n

jij

ijij

c

cwnji

1

,,,1, K (3)

2. For each row, calculate the row entropy:

( )∑=

−==∀n

jijiji wwni

1

ln,,,1 τK (4)

3. Note that the row entropy is maximised when all n connection weights5 are equal, hence:

( ) ( )n

n

jnnni 1

1

11max lnln,,,1 −=−==∀ ∑=

τK (5)

4. For each row, use this to scale the row entropy iτ and create an entropy adjustment

multiplier:

5 Strictly speaking, we work with no self-connection (an edge that extends to and from the same node),

hence there are 1−n possible out connections, but working with n is more general (and simple).

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( )

( )n

n

jijij

ii

ww

mni1

1

max ln

ln

11,,,1∑=+=+==∀

ττ

K (6)

5. Multiply this to all row elements of C to obtain a new matrix nnP ×∞∈ ),[ε :

( )

( ) ij

n

n

jijij

iji

ijiij c

ww

ccmpnji

+=

+===∀

∑=

1

1

max ln

ln

11,,,1,ττ

K (7)

6. Solve the characteristic polynomial defined w.r.t. this matrix P , and define the EEC

�systemic import� vector EECv as the eigenvector corresponding to the Perron root of P :

( )

444444444444 8444444444444 76

44 844 76

FrobeniusPerronFrobeniusPerron

FrobeniusPerron

FrobeniusPerronEEC

polynomialsticcharacteri

toingcorrespondreigenvecto

IP

−−−−

−−

−−

≡

=

⇓

=−

λ

λ

e

e

ev ,

0det

(8)

7. The EEC criterion is the ranking of �systemic import� based on elements of EECv :

{ }

{ }

⇔

⇔

EECi

i

EECi

i

vlowest

vhighest

importlysystemical

EECCentralityrEigenvectoEntropic

minarg

maxarg

''

:)(

MMM (9)

Elements of EEC vector can thus be called systemic import scores, although here one has to be

extra careful when applying interpretation beyond that of ranking. For example, it would be rather

precarious to claim that entity i is twice as systemically important as entity j simply because

numerically EECj

EECi vv 2= . In other words, a supervisory body would be well advised to use EEC in

guiding the planning of supervisory resources, but rather ill advised to use the EEC vector as a budget

allocation device!

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3. Numerical Examples

An ensemble of 9>n system entities presents a useful template for demonstrating the systemic

network effects, not so small that the task became trivial, not so big that it becomes difficult to glean

insights from the solution.

Nodes are represented as follows: �, �, �, �, �, �, �, �, . Edges are represented

as arrows, either (pointing from lower-number node to higher-number node) or (pointing from

higher-number node to higher-number node). For example �,�� indicates that the 5th

node has 3 outgoing connections, one edge pointing to the 1st node, one pointing to the 2

nd node, and one

to the 9th node. Connectivity weights are written as numbers next to the arrows. For example,

�,�8,1�2 signifies that the 5th node has 3 outgoing connections, one edge of strength 8

pointing to the 1st node, one of strength 1 pointing to the 2

nd node, and one of strength 2 to the 9

th node.

All eigenvalue & eigenvector solutions are found courtesy of the excellent �bluebit� online matrix

calculator [http://www.bluebit.gr/matrix-calculator/default.aspx]. Figures plot with Mathematica.

3.1 Pure Topology, Adjacency Matrix

We begin a dozen base cases constructed with a modified adjacency matrix 99}1,0001.0{ ×∈A .

Results for �Simple Ring� are given in Figure 1 (for unidirectional connections, i.e. with 1 edge

for each pair: from node i to node j) and in Figure 2 (for bidirectional connections, i.e. with 2 edges for

each pair: one from node i to node j and another from node j back to node i). In each cases the network is

perfectly homogeneous and symmetric, signifying that, by the very design of it, no system entity is more

systemically important than any other, hence the BEC and EEC vectors in all such cases are �degenerate�.

Results for �Simple Star� are given in Figure 3 (for unidirectional connections, outward from the

nucleus of the star) and Figure 4 (for bidirectional connections, outward and inward), where both BEC and

EEC vectors indicate systemic dominance by the central nodes, especially in cases where connections go

strictly one way outward.

Results for �2-D Mesh� and �Dense 2-D Mesh) topology are given in Figure 5 (unidirectional

connections), Figure 6 (bidirectional connections), Figure 7 (dense, unidirectional connections), and

Figure 8 (dense, bidirectional connections). Mesh topologies are useful when system entities are

organised, loosely speaking, in a �geographical� manner. As such, it is also of interest to macroprudential

application especially when retail entities are concerned. Note how there are radial symmetry for mesh

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networks with bidirectional connections, with the centre-most node naturally being the most systemically

important.

For completeness, �Fully Interconnected� topology is depicted in Figure 9. Once again, with

perfect homogeneity and symmetry, no individual entity can be deemed systemically more important than

the rest. Similar to this is the �Checker Board� topology, Figure 10, where every other connection is

skipped. Here, all nodes are similarly systemically important, the only difference being that with 9 nodes,

roughly half are connected to 4 nodes each, with each of the rest connected to 5 nodes, hence two levels of

systemic importance in the BEC and EEC vectors.

Finally, the �Randomly Connected (Sparse)� and �Randomly Connected (Dense)� networks are

depicted, respectively, in Figure 11 and Figure 12. Note that with sparse network, there is an opportunity

for systemic importance scores to vary greatly; where as, with the dense network, more closely resembling

�Fully Interconnected� topology, individual systemic importance scores tend to average out.

3.2 Mixed Topology, Adjacency Matrix

Here are a dozen more cases still based on modified adjacency matrix, but with additional

complexities in how the topologies are defined.

Figure 13 �2-Deep Ring� presents a bidirectional ring as well, except that all connections are 2-

deep, i.e. each node is connected to either side to their neighbouring nodes, as well as to their neighbours�

neighbours further out. In contrast, Figure 14 �3-Depth Ring� alternates 1-deep, 2-deep, and 3-deep nodes,

resulting in a 3-fold rotational symmetry, as reflected in both BEC and EEC vectors with elements

following a triple sequence of high, medium, and low centrality scores.

Figure 15 �Fan-Blade Star� also presents a star topology as well, except there is a 4-fold

symmetry, with unidirectional connections from the nucleus node out to two other nodes in succession

before looping back. This resembles the situation where a bank has net borrowers (loan clients) and net

lenders (deposit clients), and the former holds trade receivables against the latter. Note, rather

unsurprisingly, how the nodes receiving connections from the nucleus node are less systemically

important than the nodes connecting to the nucleus node. In contrast, Figure 16 �3-Nuclei Star� creates a

3-fold symmetry by utilising 3 nodes as equally important nuclei, with the remaining 6 as equally less

important nodes, as reflected in the corresponding BEC and EEC vectors.

Figure 17 and Figure 18 present results from superimposing ring and star topologies together,

hence the dominance of nucleus node in each instance is reduced (when compared to �Simple Star�). Of

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more interest to macroprudential application are the �Ring of Stars� topology, illustrated in Figure 19 (for

unidirectional connections) and in Figure 20 (for bidirectional connections). One can imagine how bigger

�tier-1� banks are related to one another, each of which serves as a nucleus entity relative to smaller �tier-

2� banks.

Figure 21 �Full Cascade� presents us with one of the most uneven, yet neatly ordered, systemic

importance scores, the highest awarded to the 1st node, which is connected to all remaining 8 nodes, then

less for the 2nd node, which is connected to the remaining 7, and so on. Figure 22 �3-Block Cascade�, with

3 blocks of 3 nodes (arranged in a similar but smaller cascade pattern), results in a 3-fold rotational

symmetry. One can expect cascade arrangements to arise naturally in real applications, although not a full

one, unless it happens by design. Figure 23 �3-Tier Cascade� is similar but puts the 3 nodes with 3

outgoing connections each together, followed by the 3 nodes with 2 outgoing connections each, and

finally the 3 nodes with 1 outgoing connection each. Here the 3-fold rotational symmetry is destroyed.

Figure 24 �Inner & Outer Circles� also presents an interesting case from macroprudential

application viewpoint. Here the inner circles of 5 nodes are fully interconnected, while the 4 outer nodes

form a ring, each connected 2 both ways to one of the inner-circle node.

3.3 Stylised Weighted Connectivity Matrix

This is where the superiority of EEC over BEC criterion is demonstrated.

Let�s take some of the network topologies examined earlier, only this time let�s endow them with

uneven connectivity weights. For simplicity, we use integer weights, so that a weight of 2 (or 3, and so on)

can be represented graphically by 2 (or 3, and so on) edges pointing from the same node to the same node.

First recall Figure 4 �Simple Star (Bidirectional)�. Let�s see what happens when each of the

nucleus node�s 8 outgoing edges has weight 1, while each of the 8 incoming edges has weight 8, so that all

9 nodes have equal out degree of 9 each. This is presented in Figure 25 �Simple Star (Bidirectional,

Norm.). Clearly, BEC criterion is not able to resolve the difference between the nucleus node and the rest,

as the BEC vector becomes �degenerate�.

Now recall Figure 14 �3-Depth Ring�, Figure 22 �3-Block Cascade�, and Figure 23 �3-Tier

Cascade� and do the same. In each case assign connectivity weights so that the entries from each row of

the respective weighted connectivity matrix sums to the same total. The results are presented in Figure 26

�3-Depth Ring (Norm.)�, Figure 27 �3-Block Cascade (Norm.), and Figure 28 �3-Tier Cascade (Norm.)�.

In each case, BEC vector is �degenerate�; whereas, the EEC vector retains (near) the systemic import

14

ranking from the respective original topologies. Figure 23 �3-Tier Cascade� and Figure 28 �3-Tier Cascade

(Norm.)� are particularly worth noting, as there is no 3-fold symmetry, and yet in they two corresponding

EEC vectors have near perfect rank correlation. For the former, systemic import ranking goes from 1st, the

most systemically important node, to 2nd to 9

th to 3

rd to 8

th to 4

th to 6

th to 5

th and finally to 7

th, the least

systemically important node. For the latter, systemic import ranking goes from 1st, the most systemically

important node, to 2nd to 3

rd to 9

th to 4

th to 8

th to 6

th to 5

th and finally to 7

th, the least systemically important

node.

Finally, for added interest, Figure 29 �Fan-Blade Star (Rand. Wt.) #1� and Figure 30 �Fan-Blade

Star (Rand. Wt.) #2� are two connectivity-weight randomisation instances based on the �Fan-Blade Star�

topology depicted in Figure 15. Similarly, Figure 31 �Randomly Connected (Sparse) #1� and Figure 32

�Randomly Connected (Sparse) #2� are two connectivity-weight randomisation instances based on the

�Randomly Connected (Sparse)� topology depicted in Figure 11.

In all cases it is quite assuring to note how BEC vector and EEC vector have high rank

correlation, and where there are differences in terms of out degree, EEC-based rankings would generally

favour high out degree more than BEC-based rankings.

4. Concluding Remarks

While this is strictly a methodology paper, the problem is motivated by application, and

demonstration cases are chosen to represent stylised patterns that may arise in real application. It is

precisely due to application motivation that we have chosen to pursue one form of centrality over the

others.

The main contribution to Systemic Import Analysis (SIA) is the tool, Entropic Eigenvector

Centrality (EEC), an extension of the well-known and seminal Bonacich�s Eigenvector Centrality (BEC)

to network with weighted connectivity matrix that gives similar results to BEC but has the added benefit

of using out degree information to resolve systemic importance when connectivity weights alone lead to

�degenerate� result.

It remains to be seen whether numerically, the EEC �systemic import� vector, which is based on

Static Network Analysis, correlates highly with other measures of relative systemic importance especially

a posteriori measures based on Dynamic Network Analysis, or even Agent-Driven Network Analysis.

This is the direction of research we intend to pursue.

15

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17

Connectivity (outbound) BEC Vector

|λ| = 1.0008

EEC Vector

|λ| = 1.0045

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

� 11.11 11.11

1

2

3

4

5

6 7

8

9

SimpleRingHUnidirectionalL

� 11.11 11.11

Figure 1: Connectivity & Centrality for a �Simple Ring (Unidirectional)� Network


|λ| = 2.0007

EEC Vector

|λ| = 2.6351

��, 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

�� 11.11 11.11

1

2

3

4

5

6 7

8

9

SimpleRingHBidirectionalL

�,� 11.11 11.11

Figure 2: Connectivity & Centrality for a �Simple Ring (Bidirectional)� Network

18


|λ| = 0.0287

EEC Vector

|λ| = 0.0567

��,�,�,�,�,�,�, 97.20 97.20

� 0.35 0.35

� 0.35 0.35

� 0.35 0.35

� 0.35 0.35

� 0.35 0.35

� 0.35 0.35

� 0.35 0.35

1

2

3

4

5 6

7

8

9

SimpleStarHUnidirectionalL

0.35 0.35

Figure 3: Connectivity & Centrality for a �Simple Star (Unidirectional)� Network


|λ| = 2.8289

EEC Vector

|λ| = 3.9539

��,�,�,�,�,�,�, 26.12 32.99

�� 9.23 8.38

�� 9.23 8.38

�� 9.23 8.38

�� 9.23 8.38

�� 9.23 8.38

�� 9.23 8.38

�� 9.23 8.38

1

2

3

4

5 6

7

8

9

SimpleStarHBidirectionalL

� 9.23 8.38

Figure 4: Connectivity & Centrality for a �Simple Star (Bidirectional)� Network

19


|λ| = 0.2519

EEC Vector

|λ| = 0.3308

��,� 76.95 77.06

��,� 9.69 9.68

�� 0.82 0.68

��,� 9.69 9.68

��,� 1.60 1.74

� 0.20 0.21

�� 0.82 0.68

� 0.20 0.21

12 3

4 5 6

7 89

2-D MeshHUnidirectionalL

0.04 0.06

Figure 5: Connectivity & Centrality for a �2-D Mesh (Unidirectional)� Network


|λ| = 2.829

EEC Vector

|λ| = 4.2082

��,� 8.58 7.77

��,� 12.13 12.41

�� 8.58 7.77

��,� 12.13 12.41

�,��,� 17.16 19.25

�,�� 12.13 12.41

�� 8.58 7.77

�,�� 12.13 12.41

12 3

4 5 6

7 89

2-D MeshHBidirectionalL

�,� 8.58 7.77

Figure 6: Connectivity & Centrality for a �2-D Mesh (Bidirectional)� Network

20


|λ| = 0.4227

EEC Vector

|λ| = 0.612

��,�,� 67.09 68.33

��,�,�,� 22.92 22.86

��,� 3.98 3.33

��,�,� 4.03 3.70

��,�,�, 1.40 1.29

��, 0.27 0.25

�� 0.21 0.13

� 0.08 0.07

1

2

3

4

5

6

7

8

9

Dense2-D MeshHUnidirectionalL

0.02 0.04

Figure 7: Connectivity & Centrality for a �Dense 2-D Mesh (Unidirectional)� Network


|λ| = 4.8288

EEC Vector

|λ| = 8.3251

��,�,� 8.58 7.88

��,�,�,� 12.13 12.38

��,� 8.58 7.88

�,��,�,� 12.13 12.38

�,�,�,��,�,�, 17.16 18.95

�,�,��, 12.13 12.38

�,�� 8.58 7.88

�,�,�,�� 12.13 12.38

1

2

3

4

5

6

7

8

9

Dense2-D MeshHBidirectionalL

�,�,� 8.58 7.88

Figure 8: Connectivity & Centrality for a �Dense 2-D Mesh (Bidirectional)� Network

21


|λ| = 8.0001

EEC Vector

|λ| = 15.5718

��,�,�,�,�,�,�, 11.11 11.11

��,�,�,�,�,�, 11.11 11.11

�,��,�,�,�,�, 11.11 11.11

�,�,��,�,�,�, 11.11 11.11

�,�,�,��,�,�, 11.11 11.11

�,�,�,�,��,�, 11.11 11.11

�,�,�,�,�,��, 11.11 11.11

�,�,�,�,�,�,�� 11.11 11.11

1

2 3

4

5

6

7

8

9

Fully Interconnected

�,�,�,�,�,�,�,� 11.11 11.11

Figure 9: Connectivity & Centrality for a �Fully Interconnected� Network


|λ| = 4.4726

EEC Vector

|λ| = 7.5203

��,�,�,� 10.56 10.41

��,�,�, 11.80 11.99

��,�,� 10.56 10.41

�,��,�, 11.80 11.99

�,��,� 10.56 10.41

�,�,��, 11.80 11.99

�,�,�� 10.56 10.41

�,�,�,�� 11.80 11.99

1

2

3

45

6 78

9

CheckerBoard

�,�,�,� 10.56 10.41

Figure 10: Connectivity & Centrality for a �Checker Board� Network

22


|λ| = 0.1421

EEC Vector

|λ| = 0.1898

� 0.07 0.11

��,� 4.55 5.03

�� 0.56 0.61

� 0.07 0.11

� 0.07 0.11

�,�� 61.02 66.61

�� 29.05 23.54

�� 0.56 0.61

1

2

3

45

6

7

8 9

RandomlyConnectedHSparseL

� 4.05 3.28

Figure 11: Connectivity & Centrality for a �Randomly Connected (Sparse)� Network


|λ| = 7.2322

EEC Vector

|λ| = 13.7598

��,�,�,�,�,�, 11.01 10.98

��,�,�,�,�,�, 12.15 12.39

�,��,�,�, 9.33 8.93

�,�,��,�,�,�, 12.15 12.39

�,�,��,�,�, 11.01 10.98

�,�,�,�,��,�, 12.15 12.39

�,�,�,�,��, 10.67 10.56

�,�,�,�,�,� 9.38 8.98

1

2

3

4

5

6

7 8

9

Randomly ConnectedHDenseL

�,�,�,�,�,�,�,� 12.15 12.39

Figure 12: Connectivity & Centrality for a �Randomly Connected (Dense)� Network

23


|λ| = 4.0005

EEC Vector

|λ| = 6.5269)

��,�,�, 11.11 11.11

��,�, 11.11 11.11

�,��,� 11.11 11.11

�,��,� 11.11 11.11

�,��,� 11.11 11.11

�,��,� 11.11 11.11

�,��, 11.11 11.11

�,�,�� 11.11 11.11

1

23

4

5

6

7

8

9

2-DeepRing

�,�,�,� 11.11 11.11

Figure 13: Connectivity & Centrality for a �2-Deep Ring� Network


|λ| = 4.1568

EEC Vector

|λ| = 6.9612

��,�,�,�,�, 16.04 17.39

��,�, 10.83 10.64

�� 6.46 5.31

�,�,��,�,� 16.04 17.39

�,��,� 10.83 10.64

�� 6.46 5.31

�,�,�,��, 16.04 17.39

�,�,�� 10.83 10.64

1

2

3

4

56

7

8

9

3-Depth Ring

�,� 6.46 5.31

Figure 14: Connectivity & Centrality for a �3-Depth Ring� Network

24


|λ| = 1.5881

EEC Vector

|λ| = 1.8742

��,�,�,� 19.58 23.30

�� 7.77 6.69

�� 12.33 12.48

�� 7.77 6.69

�� 12.33 12.48

�� 7.77 6.69

�� 12.33 12.48

� 7.77 6.69

1

2

3 4

5

6

7

8

9

Fan-BladeStar

� 12.33 12.48

Figure 15: Connectivity & Centrality for a �Fan-Blade Star� Network


|λ| = 4.2431

EEC Vector

|λ| = 7.0046

��,�,�,�,�, 13.81 14.58

��,�,�,�,�, 13.81 14.58

��,�,�,�,�, 13.81 14.58

�,�,�� 9.76 9.37

�,�,�� 9.76 9.37

�,�,�� 9.76 9.37

�,�,�� 9.76 9.37

�,�,�� 9.76 9.37

1

2

3

4

5

6

7

8

9

3-Nuclei Star

�,�,� 9.76 9.37

Figure 16: Connectivity & Centrality for a �3-Nuclei Star (Bidirectional)� Network

25


|λ| = 1.0015

EEC Vector

|λ| = 1.006

��,�,�,�,�,�,�, 49.97 65.94

�� 6.25 4.26

�� 6.25 4.26

�� 6.25 4.26

�� 6.25 4.26

�� 6.25 4.26

�� 6.25 4.26

� 6.25 4.26

1

2 3

4

5

67

8

9

SimpleStar& RingHUnidirectionalL

� 6.25 4.26

Figure 17: Connectivity & Centrality for a �Simple Ring & Star (Unidirectional)� Network


|λ| = 4.0004

EEC Vector

|λ| = 6.5637

��,�,�,�,�,�,�, 20.00 22.87

��, 10.00 9.64

�,�� 10.00 9.64

�,�� 10.00 9.64

�,�� 10.00 9.64

�,�� 10.00 9.64

�,�� 10.00 9.64

�,�� 10.00 9.64

1

2 3

4

5

67

8

9

SimpleRing& StarHBidirectionalL

�,�,� 10.00 9.64

Figure 18: Connectivity & Centrality for a �Simple Ring & Star (Bidirectional)� Network

26


|λ| = 1.0008

EC Vector

|λ| = 1.5024

��,�,� 33.31 33.31

� 0.01 0.01

� 0.01 0.01

��,�,� 33.31 33.31

� 0.01 0.01

� 0.01 0.01

��, 33.31 33.31

� 0.01 0.01

1

2

3

4

56

7 8

9

Ringof StarsHUnidirectionalL

0.01 0.01

Figure 19: Connectivity & Centrality for a �Ring of Star (Unidirectional)� Network


|λ| = 2.7325

EEC Vector

|λ| = 4.0686

��,�,�,� 19.24 22.32

�� 7.05 5.51

�� 7.05 5.51

��,�,� 19.24 22.32

�� 7.05 5.51

�� 7.05 5.51

�,��, 19.24 22.32

�� 7.05 5.51

1

2

3

4

56

7 8

9

Ringof StarsHBidirectionalL

� 7.05 5.51

Figure 20: Connectivity & Centrality for a �Ring of Star (Bidirectional)� Network

27


|λ| = 0.5609

EEC Vector

|λ| = 0.9133

��,�,�,�,�,�,�, 64.07 68.08

��,�,�,�,�,�, 23.03 21.52

��,�,�,�,�, 8.27 6.93

��,�,�,�, 2.97 2.28

��,�,�, 1.07 0.78

��,�, 0.38 0.27

��, 0.14 0.09

� 0.05 0.04

1

2 3

4

5

6

7

8

9

Full Cascade

0.02 0.02

Figure 21: Connectivity & Centrality for a �Full Cascade� Network


|λ| = 2.1485

EEC Vector

|λ| = 2.925

��,�,� 15.52 17.11

��,� 10.59 10.35

�� 7.23 5.87

��,�,� 15.52 17.11

��,� 10.59 10.35

�� 7.23 5.87

��, 15.52 17.11

�� 10.59 10.35

1

2

3

4

5

6

7

8 9

3-Block Cascade

� 7.23 5.87

Figure 22: Connectivity & Centrality for a �3-Block Cascade� Network

28


|λ| = 1.8013

EEC Vector

|λ| = 2.2695

��,�,�,� 24.37 30.49

��,�,�,� 17.99 19.83

��,�,�,� 13.06 12.64

��,� 6.91 5.49

��,� 5.93 4.45

��,� 6.50 5.00

�� 4.18 2.64

� 7.52 5.97

1

23

4

5

6

7

8

9

3-Tier Cascade

� 13.53 13.49

Figure 23: Connectivity & Centrality for a �3-Tier Cascade� Network


|λ| = 4.3468

EEC Vector

|λ| = 7.3397

��,�,�,� 13.89 14.17

��,�,�,� 15.09 15.94

�,��,�,� 15.09 15.94

�,�,��,� 15.09 15.94

�,�,�,�� 15.09 15.94

��, 6.43 5.52

�,�� 6.43 5.52

�,�� 6.43 5.52

1

2

3

4

5

6

7

8

9

Inner& OuterCircles

�,�,� 6.43 5.52

Figure 24: Connectivity & Centrality for a �Inner & Outer Circles� Network

29


|λ| = 8.0005

EEC Vector

|λ| = 11.1648

��,�,�,�,�,�,�, 11.11 14.85

�8� 11.11 10.64

�8� 11.11 10.64

�8� 11.11 10.64

�8� 11.11 10.64

�8� 11.11 10.64

�8� 11.11 10.64

�8� 11.11 10.64

1

2

3

4

5 6

7

8

9

SimpleStarHBidirectional, Norm.L

�8 11.11 10.64

Figure 25: Connectivity & Centrality for a �Simple Star (Bidirectional, Norm.)� Network


|λ| = 12.0005

EEC Vector

|λ| = 19.3743

�2�,�,�,�,�, 11.11 12.49

�3�3�,�, 11.11 11.19

�6�6� 11.11 9.65

�,�,�2�2�,�,� 11.11 12.49

�,�3�3�,� 11.11 11.19

�6�6� 11.11 9.65

�,�,�,�2�2�, 11.11 12.49

�,�,�3�3 11.11 11.19

1

2

3

4

56

7

8

9

3-DepthRingHNorm.L

�,�6 11.11 9.65

Figure 26: Connectivity & Centrality for a �3-Depth Ring (Norm.)� Network

30


|λ| = 6.0007

EEC Vector

|λ| = 7.9131

�2�,�,� 11.11 12.64

�3�,� 11.11 11.10

�6� 11.11 9.59

�2�,�,� 11.11 12.64

�3�,� 11.11 11.10

�6� 11.11 9.59

�2�2�, 11.11 12.64

�3�3 11.11 11.10

1

2

3

4

5

6

7

8 9

3-Block CascadeHNorm.L

�6 11.11 9.59

Figure 27: Connectivity & Centrality for a �3-Block Cascade (Norm.)� Network


|λ| = 4.0007

EEC Vector

|λ| = 4.9349

��,�,�,�,�, 11.11 15.00

��,�, 11.11 13.65

�� 11.11 12.25

�,�,��,�,� 11.11 10.10

�,��,� 11.11 9.38

�� 11.11 9.55

�,�,�,��, 11.11 8.02

�,�,�� 11.11 9.88

1

23

4

5

6

7

8

9

3-Tier CascadeHNorm.L

�,� 11.11 12.17

Figure 28: Connectivity & Centrality for a �3-Tier Cascade (Norm.)� Network

31


|λ| = 8.2038

EEC Vector

|λ| = 9.6188

�6,9,4,9�,�,�,� 23.89 27.92

�1� 0.36 0.30

�1� 2.91 2.92

�4� 7.10 6.05

�5� 14.56 14.53

�4� 8.52 7.26

�6� 17.47 17.43

�6 10.65 9.07

1

2

3 4

5

6

7

8

9

Fan-BladeStarHRand. Wt.L Ò1

�5 14.56 14.53

Figure 29: Connectivity & Centrality for a �Fan-Blade Star (Rand. Wt. 1)� Network


|λ| = 6.5038

EEC Vector

|λ| = 7.5412

�7,9,7,1�,�,�,� 20.59 24.01

�2� 6.82 5.93

�7� 22.17 22.31

�4� 5.85 5.08

�3� 9.50 9.57

�2� 2.92 2.54

�3� 9.50 9.57

�9 13.15 11.42

1

2

3 4

5

6

7

8

9

Fan-BladeStarHRand. Wt.L Ò2

�3 9.50 9.57

Figure 30: Connectivity & Centrality for a �Fan-Blade Star (Rand. Wt. 2)� Network

32


|λ| = 0.5141

EEC Vector

|λ| = 0.6798

� 0.02 0.03

�1,6�,� 4.26 4.31

�1� 0.05 0.06

� 0.02 0.03

� 0.02 0.03

�,�3,7�9 76.03 80.20

�8�9 18.22 14.15

�9� 0.36 0.41

1

2

3

45

6

7

8 9

RandomlyConnectedHSparse, Rand. Wt.L Ò1

�9 1.03 0.78

Figure 31: Connectivity & Centrality for a �Randomly Connected (Sparse, Rand. Wt. 1)� Network


|λ| = 0.5028

EEC Vector

|λ| = 0.6279

� 0.02 0.03

�7,3�,� 0.65 0.88

�5� 0.22 0.27

� 0.02 0.03

� 0.02 0.03

�,�1,2�9 49.49 53.27

�1�9 46.90 42.81

�1� 0.06 0.07

1

2

3

45

6

7

8 9

RandomlyConnectedHSparse, Rand. Wt.L Ò2

�6 2.62 2.60

Figure 32: Connectivity & Centrality for a �Randomly Connected (Sparse, Wt. 2)� Network

Banking Supervision (BCBS) - BOT

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