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
Bibliography
[1] The Office of the Comptroller of the Currency � OCC (2004), Annual Report, [http://www.occ.treas.gov/annrpt/2004AnnualReport.pdf].
[2] Financial Services Authority � FSA (1998), NRisk Based Approach to Supervision of BanksO, [http://www.fsa.gov.uk/pubs/policy/P10.pdf].
[3] Boss, et al. (2003), NThe Network Topology of the Interbank MarketO, [http://arxiv.org/PS_cache/cond@mat/pdf/0309/0309582v1.pdf].
[4] de Lis, Pages & Saurina (2000), NCredit Growth, Problem Loans and Credit Risk Provisioning in Spain N, Banco de España Working Paper, no.
0018, [www.bde.es/informes/be/docs/dt0018e.pdf].
[5] Roubini, Nouriel (2008), NThe Rising Risk of a Systemic Financial Meltdown: The Twelve Steps to Financial DisasterO, RGE Monitor,
[http://media.rgemonitor.com/papers/0/12_steps_NR].
[6] Freixas, Parigi & Rochet (2008), NSystemic Risk, Interbank Relations and Liquidity Provision by the Central BankO,
[http://www.econ.upf.edu/docs/papers/downloads/440.pdf].
[7] Acharya,Viral V. (2009), NA Theory of Systemic Risk and Design of Prudential Bank RegulationO, Journal of Financial Stability, no.5, pp. 224@
225, [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1334457].
[8] Haldane, Rudolf (2009), NRethinking the Financial NetworkO, [http://www.bankofengland.co.uk/publications/speeches/2009/speech386.pdf].
[9] Borio, Claudio (2003), NTowards a Macroprudential Framework for Financial Supervision and Regulation?O, BIS Working Paper, no. 128,
[http://www.bis.org/publ/work128.pdf].
[10] Freixas, X. & Rochet, J.-C. (1997), Microeconomics of Banking, Cambridge: The MIT Press, [ISBN: 978@0@262@06193@3].
[11] Markose et al. (2010), NToo Interconnected To Fail: Financial Contagion and Systemic Risk in Network Model of CDS and Other Credit
Enhancement Obligations of US BanksO, University of Essex Discussion Paper Series, no. 683, [http://www.essex.ac.uk/economics/discussion@
papers/papers@text/dp683.pdf].
[12] National Credit Union Administration � NCUA (1994), Letter to Credit Unions, no. 161, [http://www.ncua.gov/letters/prior1996/e-
let161.html].
[13] Canedo, J.M.D. & Jaramillo, S.M. (2009), NA Network Model of Systemic Risk: Stress Testing the Banking SystemO, Intelligent Systems in
Accounting, Finance and Management, no.16, pp. 87@110,
[http://www3.interscience.wiley.com/journal/122456972/abstract?CRETRY=1&SRETRY=0].
[14] Cossin, H. & Schellhorn, D. (2007), NCredit Risk in a Network EconomyO, Management Science, vol. 53, no. 10, pp. 1604@1617, [http://www@
scf.usc.edu/~uscsiam/paperHS.pdf].
[15] Hattori,M. & Suda Y. (2007), NDevelopments in a Cross@Border Bank Exposure �Network�O, Bank of Japan Working Paper Series, no.7/2007,
[http://www.boj.or.jp/en/type/ronbun/ron/wps/data/wp07e21.pdf].
[16] Elsinger, Lehar & Summer (2006), NUsing Market Information for Banking System Risk AssessmentO, International Journal of Central
Banking, March, [http://www.ijcb.org/journal/ijcb06q1a4.pdf].
[17] Iori, G. & Jafarey S. (2008), NCriticality in a Model of Banking CrisesO, [http://arxiv.org/PS_cache/cond@mat/pdf/0104/0104080v1.pdf].
[18] Muller, Jeannette (2006), NInterbank Credit Lines as a Channel of ContagionO, Journal of Financial Services Research, vol. 29, no. 1 pp. 37@60,
[http://www.springerlink.com/content/y01583670x1x1g52].
[19] Nier, et al. (2008), NNetwork Models and Financial StabilityO, Bank of England Working Paper Series, no.346,
[http://www.bankofengland.co.uk/publications/workingpapers/wp346.pdf].
[20] Thurner, Hanel &Pichler (2003), NRisk Trading, Network Topology, and Banking RegulationO,
[http://arxiv.org/PS_cache/cond@mat/pdf/0309/0309581v1.pdf].
[21] Fioretti, Guido (2004), NFinancial Fragility in a Basic Agent@Based ModelO, [http://cogprints.org/4341/1/frag.pdf].
[22] Bech, M.L. & Atalay, E. (2008), NThe Topology of the Federal Funds MarketO, ECB Working Paper Series, no. 986/2008,
[http://www.ecb.int/pub/pdf/scpwps/ecbwp986.pdf].
16
[23] Cajueiro, D.O. & Tabak B.M. (2007), NThe Role of Banks in the Brazilian Interbank Market: Does Bank Type Matter?O, Banco Central Do
Brail Working Paper Series, no.130/2007, pp. 1@36, [http://www.bcb.gov.br/pec/wps/ingl/wps130.pdf].
[24] Castrén, O. & Kavonius L.K. (2009), NBalance Sheet Interlinkages and Macro@Financial Risk Analysis in the Euro AreaO, ECB Working Paper
Series, no. 1124/2009, [http://www.ecb.int/pub/pdf/scpwps/ecbwp1124.pdf].
[25] Estraday, D. & Morales, P. (2008), NThe Structure of the Colombian Interbank Market and Contagion RiskO,
[https://editorialexpress.com/cgi@bin/conference/download.cgi?db_name=simposio2008&paper_id=292].
[26] Krznar, Marko (2009), NContagion Risk in the Croatian Banking SystemO, Croatian National Bank Working Paper Series, no. 20,
[http://www.hnb.hr/publikac/istrazivanja/[email protected]].
[27] Lublóy, Ágnes (2005), NDomino Effect in the Hungarian Interbank MarketO, [http://www.efmaefm.org/efma2005/papers/3@agnes_paper.pdf].
[28] Masi, Iori, Caldarelli (2006), NFitness Model for the Italian Interbank Money Market, Physical Review E, vol. 74,
[http://pil.phys.uniroma1.it/~gcalda/docs/pre74_066112.pdf].
[29] Mistrulli, Paolo E. (2007), NAssessing Financial Contagion in the Interbank Market: Maximum Entropy Versus Observed Interbank Lending
Patterns N, [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1001681].
[30] Upper, C. & Worms A. (2002), NEstimating Bilateral Exposures in the German Interbank Market: Is there a Danger of Contagion?O, Economic
Research Centre of the Deutsche Bundesbank Discussion Paper, no.9/2002,
[http://www.bundesbank.de/download/volkswirtschaft/dkp/2002/200209dkp.pdf].
[31] Wei, T.H. (1952), The Algebraic Foundations of Ranking Theory, London: Cambridge University Press.
[32] Kendall, M.G (1955), NFurther Contributions to the Theory of Paired ComparisonsO, Biometrics, vol. 11, no. 1 (March), pp. 43@62,
[http://www.jstor.org/stable/3001479].
[33] Bonacich, P. (1987), NPower and Centrality: A Family of MeasuresO, The American Journal of Sociology, vol. 92, no. 5, pp. 1170@1182,
[http://www.comhealth.ch/pdf/Bonacich_1987.pdf].
[34] Eames, K.T.D. & Keeling, M.J. (2002), NModeling Dynamic and Network Heterogeneities in the Spread of Sexually Transmitted DiseasesO,
Proceedings of the National Academy of Sciences of the United States of America (PNAS), vol. 99, no. 20, pp. 13330@13335,
[http://www.pnas.org/content/99/20/13330.full].
[35] Newman, M.E.J. (2002), NSpread of Epidemic Disease on NetworksO, Physical Review E, vol. 66, no. 016128, pp. 1@11,
[http://www.stat.cmu.edu/~fienberg/Stat36-835/Newman-PhysRev-2002.pdf].
[36] Brin, S. & Page, L. (1998), NThe Anatomy of a Large@Scale Hypertextual Web Search EngineO, Computer Networks and ISDN Systems, vol. 30,
pp. 107@117.
[37] Wilf, Herbert S. (2001), NSearching the Web with EigenvectorsO, The UMAP Journal, vol. 23, no. 1, pp. 101@103,
[http://www.math.upenn.edu/%7Ewilf/website/KendallWei.pdf].
[38] Keener, James P. (1993), NThe Perron@Frobenius Theorem and the Ranking of Football TeamsO, The Society for Industrial and Applied
Mathematics (SIAM) Journal, vol. 35, no. 1 (March), pp. 80@93,
[http://[email protected]/~steele/Courses/956/Ranking/RankingFootballSIAM93.pdf].
[39] Barrat, etal. (2004), NThe Architecture of Complex Weighted NetworksO, Proceedings of the National Academy of Sciences of the United States
of America (PNAS), vol. 101, no. 11, pp. 3747@3752, [http://www.pnas.org/content/101/11/3747.full].
[40] Opsahl, Agneessens & Skvoretz (2010), NNode Centrality in Weighted Networks: Generalizing Degree and Shortest Paths N,
[http://thetore.files.wordpress.com/2010/04/node_centrality_in_weighted_networks1.pdf].
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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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
Connectivity (outbound) BEC Vector
|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