Cascades in multiplex financial networks with debts
of different seniority
Charles D. Brummitt
Teruyoshi Kobayashi
January 2015
Discussion Paper No.1502

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANCascades in multiplex financial networks with debts of different seniority
Charles D. Brummitt
Center for the Management of Systemic Risk, Columbia University, New York, NY 10027, USA

Teruyoshi Kobayashi∗
Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe 657-8501, Japan(Dated: January 22, 2015)
The seniority of debt, which determines the order in which a bankrupt institution repays its debts,
is an important and sometimes contentious feature of financial crises, yet its impact on system-widestability is not well understood. We capture seniority of debt in a multiplex network, a graph ofnodes connected by multiple types of edges. Here, an edge between banks denotes a debt contractof a certain level of seniority. Next we study cascading default. There exist multiple kinds ofbankruptcy, indexed by the highest level of seniority at which a bank cannot repay all its debts.
Self-interested banks would prefer that all their loans be made at the most senior level. However,mixing debts of different seniority levels makes the system more stable, in that it shrinks the set of
network densities for which bankruptcies spread widely. We compute the optimal ratio of senior tojunior debts, which we call the optimal seniority ratio, for two uncorrelated Erdős-Rényi networks.
If institutions erode their buffer against insolvency, then this optimal seniority ratio rises; in otherwords, if default thresholds fall, then more loans should be senior. We generalize the analyticalresults to arbitrarily many levels of seniority.
PACS numbers: 89.65.Gh, 89.75.Hc

I.

INTRODUCTION

The global financial crisis of 2007–2009 spurred a flurryof research on financial contagion using a variety of network models [1–7]. In most of these models, financial
institutions interact in just one way, such as making one
type of loan to one another. In practice, however, banks
interact in many ways, all of which can contribute toa crisis: banks issue loans of different durations and ofdifferent levels of riskiness; they trade assets with eachother; and they hold assets in common. Models have only
recently begun to capture more than one of these kinds ofinteractions [8–10]. Relatively unexplored is the seniority
of debts, which determines the order in which bankrupt
institutions repay their debts. Individual creditors pay
close attention to their assets’ seniority levels, but little is known about how the system-wide composition ofseniority levels affects the risk of large crises.
A concurrent and mostly independent thread of literature in physics has studied multiplex networks, orgraphs with multiple sets of edges (or “network layers”) [11, 12]. In threshold models of contagion in multiplex networks [13–17], when a node changes state, it
influences its neighbors in all layers to change state. A
common conclusion from these models is that the system’s vulnerability to large cascades often cannot be understood by examining the vulnerability of any one layer
to large cascades [13–17]. In particular, the interactions
among network layers can make large cascades easier (ifnodes combine influence from multiple types of neighbors
in a disjunctive [13] or weighted [14] way) or more difficult∗ Corresponding author. kobayashi@econ.kobe-u.ac.jp(if nodes combine influence in a conjunctive way [15]).
In this paper, we combine these two strands of research
to study a financial phenomena of interest: how vulnerability to cascading default depends on the seniority ofdebts. Financial institutions (hereafter called “banks”
for simplicity) hold debt in one another; more generally,
they have exposure to one another’s default risk throughinterbank lending and/or security holding. Each debt
contract has a certain level of seniority. When a bankgoes into default (i.e., when a bank’s liabilities exceed
its assets), the bank sells its remaining assets to pay off
whatever liabilities it can in decreasing order of seniority,from most senior liability to most junior liability.
In our model, there exist multiple types of bankruptcy,
indexed by the highest level of seniority at which a bankcannot repay all its debts. Using a technique called thecascade condition [13, 15, 18, 19], we study whether asmall number of initial bankruptcies (of certain types)
triggers a cascade of many bankruptcies.
The model has interesting tradeoffs. Self-interested
banks would prefer to lend at the most senior level inorder to maximize the assets they would recover in theevent of its debtors defaulting. However, if all banks
lend at the most senior level, then the system is especially vulnerable to large cascades of default, in the sensethat cascades are likely large for a large interval of network densities. (This extreme case recovers the models
in [1, 2].)
By contrast, the presence of debts of different seniority
makes the system less vulnerable to bankruptcies spreading widely. For the case of two seniority levels (called“junior” and “senior”), our main result is to derive the“optimal” ratio of senior loans to junior loans. Here,“optimal” means that it minimizes the size of the interval of network densities for which bankruptcies likely

2spread widely. The optimal multiplex networks have on
the order of 50% to 100% more senior loans than junior ones. As banks’ leverage (i.e., assets divided by net
worth) increases, this “optimal seniority ratio” increases,
meaning that more loans should be senior rather thanjunior in order to minimize the set of network densities
such that cascades are likely large. These results indicate
what types of “seniority–multiplex” interbank networks
are most insusceptible to large cascades of default.
The most recent scheme for financial regulation, called
Basel III, emphasizes the liability structure of individual
banks, but it does not take into account how seniority
of debts affect systemic risk. Our results suggest that
seniority of debt could be an important focus of financial regulators and of macro-prudential policies. Furthermore, this work contributes to the rapidly growing areaof research on contagion in multiplex networks by introducing a model in which nodes can “activate” in morethan one way, which we call “multilevel contagion”.

A.

Related literature

Theoretical studies of contagion in financial networks
generally fall into two classes [20]. Some studies consider a particular instance of a financial network, usually
a complicated and in some cases empirically measured
object. Other studies consider probability distributions
over financial networks.
Our study belongs to the latter class, but two papers closely related to our analysis belong to the former [21, 22]. Elsinger [21] and Gourieroux et al. [22]also assume a senior–junior structure of interbank assets. Whereas they obtain a clearing payment vector fora given network (as in Eisenberg and Noe [23]), we study
the cascade condition for ensembles of random networks
in the same spirit as Watts [18] and followup work in
physics [13–15, 19, 24–29]. The former approach can be
useful to regulators faced with a real system on the vergeof a crisis, whereas the latter approach offers elegant theory and indicative suggestions [20]. Although financial
networks change over time, there is evidence that theirstatistical properties (such as degree distributions) canbe somewhat stable [30].

B.
1.

Model

Multiplex network of loans of different seniority

We consider a multiplex network of N nodes and M
layers. Each node represents a financial institution (ora “bank” for simplicity). Each layer is a set of directededges (i.e., ordered pairs of nodes) that denote loans and,more generally, credit exposures between banks. The
layer to which an edge belongs is the loan’s seniority,and the vector of layers is sorted in increasing order of

FIG. 1. (Color online) An example of a bank’s balance sheet.
Assets (left-hand column) consist of external assets, lS = 4
many senior interbank loans, and lJ = 3 many junior interbank loans. Liabilities (right-hand column) consist of external liabilities, bS = 2 many senior interbank liabilities, andbJ = 3 many junior interbank liabilities. Net worth (a.k.a.
equity or, for banks, capital) is the difference between assets
and liabilities.

seniority. When a bank goes bankrupt, it repays its liabilities in decreasing order of seniority; that is, it repays
its most senior liabilities first, its second-most senior liabilities second, etc. For the moment, we consider a twolayer (“duplex”) network, and the M = 2 many levels ofseniority are called “junior” and “senior” (labeled J and
S), respectively.
The number of banks to which a bank lends funds ofseniority α ∈ {J, S} (i.e., the bank’s out-degree in layerα) is denoted by lα . Likewise, the number of banks fromwhich a bank borrows funds of seniority α (i.e., the bank’s
in-degree in layer α) is denoted by bα . For simplicity,
we assume that the sizes of loans (i.e., the notional values) are the same, so we can consider the networks as
unweighted. Thus, the out-degree and the in-degree correspond to the bank’s total volume of lending and borrowing, respectively.
Banks may also hold assets and issue liabilities outside
the banking system. (For instance, external creditors
may lend to banks in the form of demand deposits.) The
balance sheet of a typical bank is illustrated in Fig. 1.

2.

Multiple types of default

In a model with two levels of seniority of debt, there are
two types of defaults (bankruptcies): junior- and seniordefaults. A bank is in junior-default if and only if its
capital (a.k.a. equity or net worth; see Fig. 1) is negative,
meaning that it cannot pay all of its junior debts. A bankis in senior-default if and only if its capital is so far below
zero that it cannot pay its all of its senior debts nor anyof its junior debts. Note that banks in senior-default arenecessarily in junior-default as well.
We assume that when a bank goes into junior-default,
it cannot repay any of its junior-level liabilities, and whena bank goes into senior-default, it cannot pay any of its
junior liabilities nor any of its senior liabilities. Said differently, we assume that loss given default of seniority

3
α is 100% for interbank liabilities of seniority of α andlower. This assumption departs from previous cascade
models of financial contagion, such as that of Gai and
Kapadia [1], in which loss given default is 100% for all
interbank liabilities, no matter their seniority. Table I
makes this assumption concrete: the value of a bank’s
debt is equal to some positive value (say, 1 unit of currency) if the bank is solvent (i.e., if its equity is nonnegative); the value of the bank’s junior debt falls to zero
when the bank becomes insolvent, and the value of its
senior debt falls to zero when the bank’s equity passes a
nonpositive threshold value (derived later to be −bJ ).
TABLE I. Relationship between the equity of a borrower andthe value of its junior debts and senior debts held by its creditors. Here, we assume that junior and senior debts are equallyvaluable when the borrower is solvent, so values of debts areeither zero or $1. If the borrower’s equity is negative but sufficiently large (≥ −bJ ), then it can pay its senior debts but
none of its junior debts; the borrower is in “junior-default”.
For sufficiently negative equity of the borrower, the borrower
cannot pay neither its junior debts nor its senior debts; theborrower is in “senior-default”, and all its debts have value
zero.
statusjunior debt value senior debt value
equity≥0solvent$1$1< 0 junior-default
0$1
00< −bJ senior-default

3.

Thresholds for junior- and senior-default

Initially, all banks are solvent, and a small fractionof banks are chosen to be in junior-default or in seniordefault. These initial defaults could be caused by theloss of external assets. These bank failures cause losses
for their creditors, who may default as a result, which
may cause one or more of their creditors to default, andso on, resulting in a cascade of default.
Consider some point in time during a cascade. For acertain bank, let mJ denote the number of its juniorborrowers that have junior-defaulted, and let mS denote the number of its senior-borrowers that have seniordefaulted. This bank is in junior-default if and only if
its losses mJ + mS exceeds the bank’s equity w, and thisbank is in senior-default if and only if its losses mJ + mS
exceed the sum of its equity w and its junior liabilitiesbJ .
To exploit the solution technique for threshold cascades in networks [13–15, 19, 24], we express these inequalities in terms of losses relative to total out-degreelJ + lS : the bank is in default at layer α ∈ {J, S} if andonly if(w
if α = J
mJ + mS
J +lS> Rα ≡ lw+b
(1)J
lJ + lS
if α = SlJ +lS[In Appendix A, we generalize the thresholds Rα in
Eq. (1) to the general case of M ≥ 1 layers.] Figure 2
illustrates a bank in junior-default and a bank in seniordefault in two ways: as losses on a balance sheet (toprow) and as contagion in a multiplex network (bottomrow).
The dynamics of multilevel default contagion canbe described as a multiplex version of the thresholdmodel [18, 25, 27, 31]. (Other multiplex generalizations
were studied in [13–15].) The “response functions” foreach layer are(S
1 if mlJJ +m+lS > RJ
0 otherwise
(
S −bJ
> RJ
1 if mJ l+m
J +lS,
FS (~l, ~b, m)~ ≡0 otherwise
FJ (~l, ~b, m)
~ ≡

(2a)

(2b)

where ~l ≡ (lJ , lS ), ~b ≡ (bJ , bS ) and m~ ≡ (mJ , mS ). If
the outputs of FJ and FS are both 0, then the bank issolvent; if FJ gives 1, then the bank is in junior-default;
if FS gives 1, then the bank is in senior-default (and injunior-default).
We note in passing that a special case of this model
coincides with a special case of the “multistage complex
contagion” model of Melnik et al. [29]; for details, see
Appendix B.

II. MULTILEVEL CASCADES ON 2-LAYERMULTIPLEX NETWORKS OF JUNIOR AND
SENIOR LOANS

This multiplex model adds a new dimension of complexity to previous threshold models [1–3, 18, 19, 27] byconsidering multiple levels of seniority of debt, so to understand the model we start by simplifying in two otherdimensions, the distribution of networks and the distribution of thresholds.
First, each layer is modeled as an independent, locally treelike, random directed network generated by theconfiguration model. The sparsity and independenceof the layers implies that the number of “overlappingedges” (i.e., junior and senior loans between the same
pair of banks) is negligibly small for sufficiently large networks [32].
Second, following previous studies [1–3], we assume
that every bank has the same junior-default threshold
RJ = w/(lJ + lS ). This assumption implies that allbanks have the same ratio of capital to total interbank
assets. In the real world, banks’ capital-to-asset ratios are not identical, of course, but they are somewhat
similar [33]. It is straightforward to consider heterogeneous capital-to-asset ratios by considering heterogeneous thresholds [3, 19].

4

FIG. 2. (Color online) An example of the balance sheet (and the corresponding multiplex network visualizations) of a bankthat is (a) in junior-default: mJ + mS > RJ (lJ + lS ) ≡ w and (b) in senior-default: mJ + mS > RS (lJ + lS ) ≡ w + bJ . Circularand spiky-shaped nodes are solvent and insolvent (a.k.a. bankrupt, in default), respectively, in that layer. Edges point fromlender to borrower. Senior loans are dashed green arrows; junior loans are solid red arrows. In panel (a), the bank labeled A
suffers losses of mJ + mS = 2 on its assets side (assets are listed on the left-hand side of the bank’s balance sheet) because
mJ = 1 of its (lJ = 3 many) junior debtors are in junior-default and because mS = 1 of its (lS = 4 many) senior debtors are insenior-default. These losses exceed bank A’s equity (w), so bank A cannot pay off all its (bJ many) junior liabilities. Because of
the assumption that loss given default at layer α is 100% for all creditors in that layer, all of A’s junior creditors do not receive
repayments on their loan to A, and we say that A goes into junior-default. However, bank A does not go into senior-default
because it can still pay its senior liabilities (i.e., because mJ + mS ≤ w + bJ ). In panel (b), bank A suffers so many losses(mJ + mS = 5) that it cannot repay its junior creditors nor its senior creditors, so bank A goes into senior-default.

A.

Analytical approximation

Let φJ0 and φS0 denote the probabilities that a node
is initially in junior-default or in senior-default, where0 < φJ0  1 and φS0 ∈ [0, φJ0 ]. Next we adapt tothis model a technique for calculating whether cascades
likely spread widely, the cascade condition, introducedin [19] and generalized to other multiplex network models in [13–15].

For a sufficiently large graph, the fractions of nodes
in junior- and senior-default at the end of the cascade
are well approximated by a fixed point (φJ∞ , φS∞ ) of therecursion equations(α) Jφα(φt , φSt )t+1 = gα
≡ φα
0 + (1 − φ0 )Recursion equations

Consider a bank chosen uniformly at random, and let
the network hang down like a tree from this root node
by doing a breadth-first search along the directions of theedges. We define φαt+1 as the probability that a node t+1
hops from the bottom of the tree is in α-default because
sufficiently many of its debtors (i.e., its out-neighbors)
are in default. The nodes at the bottom of the tree are
initially in α-default with probability φα
0 . To computeα
φα
t+1 from φt , we condition on whether a node located t+
1 hops above the leaves of the tree is initially in α-default,
which occurs independently with probability φα
0 . If thisnode is not initially in α-default, then we condition on
the number of debtors of this node that have defaulted.

X J,loan S,loan J,borrow
plJ
plS
pbJ

lJ +lS ≥1 bJ

×
1.

X

lJ
X

lS
X

lJ
lS
Bm
(φJt )Bm(φSt )Fα (~l, ~b, m),
~
J
S

(3)

mJ =0 mS =0

for α ∈ {J, S}, where pα,borrowand pα,loandenote thebαlα
in- and out-degree distributions, respectively, on layerα ∈ {J, S}. In Eq. (3), we approximated the numbersof neighbors in junior and senior default as independent
Bernoulli random variables because the network layers
are independent and hence the number of overlappingedges is negligibly small [32].
In short, we approximate the multiplex graph as a treeaccording to outgoing edges, and φαt+1 is the probability
that a bank (located t + 1 hops from the leaves of thetree) goes into α-default because of the defaults of its
debtors. Note from Eq. (2) that FS (~l, ~b, m)~ = 1 implies
that FJ (~l, ~b, m)~ = 1, so φSt ≤ φJt for all t ≥ 0.

5
For convenience, we rewrite the recursions (3) as J   (J) J S 
φt+1
g (φt , φt )
=
.
(4)
φSt+1
g (S) (φJt , φSt )Iterating Eq. (4) to a fixed point (φJ∞ , φS∞ ) gives the probabilities that the root node is in α-default (for α ∈ {J, S})
due to the defaults of its debtors. The quantities φJ∞ andφS∞ are the expected fractions of banks in junior-default
and senior-default, respectively, at the end of the cascade.

2.

Multiplex cascade condition

Now we obtain a simple expression that approximately
captures, in the limit of network size approaching infinity,
for which parameters a vanishingly small seed triggers acascade that results in a finite fraction of banks in junioror senior-default. The first-order cascade condition is thelinear instability of Eq. (4) at the origin (0, 0), which
provides a sufficient condition for the numbers of juniorand senior-defaulted banks to grow. The Jacobian matrixof the right-hand side of Eq. (4) is denoted by
JJJ JJSJ ≡.
(5)
JSJ JSSThe first-order cascade condition is that the largest eigenvalue of the Jacobian matrix, λmax (J ), exceeds 1:p
JJJ + JSS + (JJJ − JSS )2 + 4JJS JSJ
λmax (J ) ≡
2
> 1.
(6)
In Appendix C, we simplify the Jacobian matrix andshow that, for this model, its largest eigenvalue equals
its trace.

B.

Cascade regions

Because there are two types of default, there are two
types of “global cascades”: the fraction φJ∞ of nodes injunior-default is large, but the fraction φS∞ of nodes insenior-default could be large or small.

1.

The set of parameters satisfying this junior-only cascade condition (7) is shown in orange in Fig. 3(a). If
the senior-layer is the empty graph (i.e., if hlS i = 0),then we recover standard cascade models on a singlelayer network [1–3, 18, 19, 27]. Creating more seniorloans (i.e., increasing hlS i) shrinks the “junior-only” cascade region. This shrinking occurs because a bank’s equity w = RJ (lJ + lS ) increases with its number of seniorloans lS [recall Eq. (1)]. Thus, as the expected number
hlS i of senior loans per bank increases, a typical bank’s
buffer against junior-default (w) increases, which makesa junior-default less likely to cause other junior-defaults.

Cascades of junior-default (ignoring senior-default)

To start to understand these types of global cascades,
consider contagion only on junior loans. The “junioronly” cascade condition is that a junior-default causes on
average at least one more junior-default, or equivalently
that the top-left entry of the Jacobian matrix, JJJ , exceeds 1. In Appendix C, we show that this junior-onlycascade condition is the inequality
JJJ = E[lJ (11>RJ (lJ +lS ) )] > 1.

(7)

2.

Cascades of senior-default (ignoring junior-default)

In a similar way, consider contagion along senior loans
only. The “senior-only” cascade condition is that asenior-default causes on average at least one more seniordefault, or equivalently that the bottom-right entry of
the Jacobian matrix, JSS , exceeds 1. In Appendix C, weshow that this senior-only cascade condition is
JSS = E[lS (11−bJ >RJ (lJ +lS ) )]= pJ,borrow
E[lS 11>RJ (lJ +lS ) ] > 1.
0

(8)

The first-order cascade condition considers defaults triggered by the default of just one out-neighbor (i.e., defaults due to mJ + mS = 1); thus, it captures seniordefaults of banks that lack junior debts (bJ = 0), which
explains the factor pJ,borrowin Eq. (8). We found that0
capturing defaults caused by two bankrupt creditors (using the second-order cascade condition) was not necessary to approximate simulations well, as shown in Appendix D.
The parameters satisfying this senior-only cascade condition (8) are shown in green in Fig. 3(a). This region
shrinks with the density of junior loans hlJ i for the same
reason that the junior-only cascade region shrinks with
hlS i.
3.

Multiplex cascades

The blue region in Fig. 3(a) is the multiplex cascade region, the set of average out-degrees (hlJ i, hlS i) such thatthe first-order cascade condition [inequality (6)] is satisfied. Note that, as pointed out in [13, page 3], we haveλmax (J ) ≥ max{JJJ , JSS }. Thus, a sufficient condition
for the multiplex cascade condition to hold is that at least
one kind of default α ∈ {J, S} is supercritical (meaningthat Jαα > 1) even in the absence of the other kind ofdefault. Thus, the blue region in Fig. 3(a) contains boththe orange and green regions.
Importantly, however, the blue region contains pairs ofintermediate edge densities (hlJ i, hlS i) outside the junioronly and senior-only cascade regions. In this part of parameter space, neither junior-defaults nor senior-defaults

6
alone suffice to cause global cascades (i.e., JJJ < 1 and
JSS < 1), but a large cascade of defaults can nevertheless occur. These cascades, akin to the “multiplexityfacilitated cascades” in [13, 15], result from the interaction of junior- and senior-defaults.

FIG. 3. (Color online) (a) The multiplex cascade region[the edge-density parameters (hlJ i, hlS i) satisfying Eq. (6),drawn as a blue region with a thick, solid boundary] contains the junior-only and senior-only cascade conditions given
by Eqs. (7) and (8) and drawn as an orange region (with adashed boundary) and green region (with a dotted boundary),
respectively. Moreover, the multiplex cascade region is asymmetric in the densities of junior and senior loans, (hlJ i, hlS i).
The optimal seniority ratio, defined in Eq. (9), is drawn as a
black line. This ratio hlS i/hlJ i ≈ 1.79 minimizes the cascade
window, the the set of mean degrees satisfying the first-ordercascade condition, Eq. (6). The case of equally many juniorand senior loans (on average) is shown with a dashed line,
which has a slightly larger cascade window. (b) The cascade
window as a function of the ratio of senior loans to juniorloans, hlS i/hlJ i, achieves a minimum size at ≈ 1.79.

C.

Optimal seniority ratio

Note that the multiplex cascade region, show in bluein Fig. 3(a), is not symmetric with respect to the averageout-degrees in the two layers, hlJ i and hlS i. This asymmetry of the cascade region results from the asymmetricrole that junior and senior loans play in the response
function [Eq. (1)]. Given this asymmetry, what ratio of
junior and senior loans is optimal in some sense?
Suppose that the average total degree hlJ i + hlS i is
kept constant and that we tune the fraction of loansthat are senior. For some loan densities hlJ i + hlS i,
the multiplex cascade condition (6) holds no matter thefraction of senior loans. For example, the line segment{(hlJ i, hlS i) : 4 = hlJ i + hlS i} is contained within themultiplex cascade region in Fig. 3(a), so defaults spreadwidely no matter the composition of seniority levels. In
other cases, such as the line segment of mean degrees
satisfying 7 = hlJ i + hlS i in Fig. 3(a), the cascade condition (6) holds if and only if the fraction of senior loans
is very large or moderately small. In cases like this one,having an intermediate fraction of senior debts prevents
defaults from spreading widely.
Now suppose that we can set the ratio hlS i/hlJ i of
the average numbers of senior and junior loans per bank,
but the total density of loans hlS i + hlS i can vary. For
instance, the number of loans may increase during an
economic boom and decrease during a recession, yet thefraction of loans that are senior does not change. Thus,
we are considering rays through the origin in Fig. 3(a).
For “seniority ratios” σ ≥ 0, define the cascade window to be the set of average degrees (hlS i, hlJ i) such thathlS i/hlJ i = σ and such that the first-order cascade condition (6) holds. Figure 3(b) shows this cascade window asa function of the seniority ratio hlS i/hlJ i. Note that thesize of the cascade window is minimized at a unique valueof hlS i/hlJ i, which is approximately 1.79 in Fig. 3. We
define the optimal seniority ratio to be this minimizer,
or more precisely
hlS i
optimal :=hlJ i
arg min #{x ≥ 0 : λmax (J )|hlJ i=x,hlS i=σx ≥ 1}.

(9)σ

If this optimal seniority ratio is greater than one (as inFig. 3), then senior debts should be issued more thanjunior debts (on average) for the sake of minimizing thesize of the set of network densities such that bankruptcies
spread widely. That is, if a financial network keeps approximately the same ratio of senior to junior loans butchanges the total number of loans (say, over the course ofa business cycle), then there should be more senior loans
than junior loans in order to minimize the risk of a large
cascade of defaults.
Intuitively, senior loans are stabilizing because they areless significant early in a cascade: a node in junior-default
causes losses (of assets) for its in-neighbors in the junior

7
layer only. Thus, changing some loans from junior to senior makes large cascades less likely, and hence it shrinks
the cascade window. However, not all loans should besenior, because that case is identical to all loans being
junior, which has a dangerously large cascade window[look at the horizontal and vertical axes of Fig. 3(a)]. In
between those extremes lies an optimal ratio of senior tojunior loans. That optimal ratio exceeds one [the dashedblack 45-degree line in Fig. 3(a)] because senior loans areless contagious, in the sense that nodes in junior-default
can still repay their senior debts.
In Fig. 6 of Appendix D, we show that the three typesof cascade regions shown in Fig. 3(a) (junior-only, senioronly, and multiplex) closely agree with the fixed point of
the recursion equations (4) and with numerical simulations of the model. These comparisons also show thatwithin the multiplex cascade region many nodes end up
in junior-default, but many nodes are in senior-default
only when senior loans greatly outnumber junior loans.
In practice, the number of defaults of any type is of primary concern, so our definition of the optimal seniority
ratio in Eq. (9) ignores types of defaults.

D.

Effect of varying the thresholds (ratio of equity
to interbank assets)

How does the cascade region (and its optimal seniority ratio) change as the banks’ threshold RJ changes?
Here we keep the assumption that banks all have thesame junior-default threshold RJ , but we vary RJ . Ifbanks have a smaller capital ratio RJ , then default becomes more likely, so the multiplex cascade region grows,
as shown in Fig. 4(a). Conversely, as RJ increases, thecascade region shrinks, and it can even splinter into two
regions [see the purple regions for RJ = 0.25 in Fig. 4(a)].
In cases like this one with two disjoint cascade regions,the optimal ratio defined in Eq. (9) does not exist because
the set {x ≥ 0 : λmax (J )|hlJ i=x,hlS i=σx ≥ 1} is empty forsome σ. [A sensible definition for the optimal seniority
ratio could be the slope that maximizes the distance from
(1) the ray through the origin with that slope to (2) thetwo cascade regions.]
The asymmetry between junior and senior loans alsochanges with the junior-default threshold RJ . In particular, the optimal seniority ratio increases as RJ decreases,
as shown in Fig. 4(b). In other words, to reduce systemic
risk, banks should increase the fraction of debts that aresenior as their leverage becomes higher.

III.

FIG. 4. (Color online) If banks hold less capital (equity) w
relative to their interbank assets lJ + lS (i.e., if the thresholdfor junior-default, RJ , decreases), then (a) the cascade region (the set of parameters satisfying the first-order cascadecondition (6)] enlarges, and (b) the optimal ratio of senior tojunior loans increases. The first-order cascade condition (6)considers defaults that occur due to one defaulted neighbor(for details, see Appendix C). Thus, when RJ falls below the
inverse of an integer, 1/k, nodes with total degree k and with
one defaulted neighbor will junior-default, so the cascade region changes at RJ = 1/k, and hence the optimal ratio jumps
up at RJ = 1/k.

els in increasing order of seniority, and let R1 denotethe threshold for default at the most junior layer. The
response function for default at level i, derived in Appendix A, is

MORE THAN TWO SENIORITY LEVELSFi (~l, ~b, m)~ =Now we show that mixing among several seniority
levels can further reduce systemic risk (in the sense ofshrinking the cascade region).
We generalize the model to M ≥ 1 seniority levels.
Let (1, 2, . . . , M ) denote the indices of the seniority lev-

(
PM
PM
Pi−1
1 if
k=1 bk > R1
s=1 ls
s=1 ms −.
0 otherwise(10)

By convention, sums of the form

P0

k=1 are zero.

8
A.

layer i ∈ {1, 2, . . . , M } is, from Eq. (11),First-order cascade condition

The cascade condition, a straightforward generalization of the one for M = 2 layers in Sec. II A 2, is that theJacobian matrix J M ∈ RM ×M has largest eigenvalue exceeding one. The (i, j) entry of J M is
JijM = EM
=

i−1
Y

h

lj 11−Pi−1 bk >R1 (PMiα=1 lα

k=1

(11a)

)

h
E
pk,borrow
M lj 11>R1 (PM
0

i
)

(11b)

ps,loan
lj .
ls

(11c)

α=1 lα

k=1

=

i−1
Y
k=1

M
Y

X

pk,borrow
0

" j−1
#
PM
X
hl
iΓ(d1/R
e
−
1,
hl
i)
i
1
i
i=1
JijM -ER =exp −hlk i ,
Γ(d1/R1 e − 1)
k=1

(13)
where
incomplete gamma function Γ (x, y) ≡R ∞ x−1the
−u
u
e
du
and the gamma function Γ(x) ≡ Γ(x, 0).
yFigure 5 shows the resulting cascade region (i.e., theparameters satisfying trJ M -ER > 1) for M ∈ {1, 2, 3, 4}
many Erdős-Rényi layers that each have mean out-degree
hli i = z/M . Here, the parameters are the threshold R1
for default at the most junior level (horizontal axis) andthe mean total out-degree z (vertical axis).

~l:PM lα <1/R1 s=1α=130

Here, products of the form k=1 are defined to be 1,
and the expectations EM are over p1,loan, . . . , plM,loanandl1M1,borrow
M −1,borrow, . . . , pbM −1. The expression in (11b)over pb1
equals that in (11a) because the threshold R1 > 0, so
the indicator in (11a) is 1 only if the bank has no liabilities of seniority k (i.e., bk = 0) for all k = 1, 2, . . . , i −1; furthermore, because a bank’s in- and out-degrees
are independent, the indicator in Eq. (11a) factors
to 1bk =0 ∀ k∈{1,2,...,i−1} 11>R1 (PM. Equation (11c)α=1 lα )writes out the expectation in Eq. (11b) by summing over~l ∈ {0, 1, 2, . . . }M such that PM lα < 1/R1 .
α=1Observe from Eq. (11b) that J M consists of rows thatare linearly dependent. Thus, J M has rank one and only
one non-zero eigenvalue. Because the trace of a matrix
equals the sum of its eigenvalues, the largest eigenvalueof J M is its trace, which, from Eq. (11c), is given by
trJ M =M i−1
XY
i=1 k=1

pk,borrow
0

X

M
Y

ps,loanli . (12)ls~l:PM lα <1/R1 s=1
α=1

Therefore, we can analytically express λmax (J M ) by using the degree distributions and the threshold value R1
on the most junior layer. This expression immediately
gives the first-order cascade condition, trJ M > 1.

B.

Distinguishing more seniority levels reduces riskof large cascades of default

If we realize that not all loans have the same seniority level, how does vulnerability to large cascades
change? For simplicity, we consider “splitting” an ErdősRényi random graph into M independent, identically distributed layers (as in [13]). That is, we compare a singlelayer Erdős-Rényi random graph with mean out-degree
z to a multiplex Erdős-Rényi random graph with mean
out-degree z/M in each of the M layers.
The Jacobian matrix J M -ER for an M -layer directedErdős-Rényi random graph with edge density hli i in each�������� ������ �� ����� ��� ���� = �Q0���� -����� ������� ������
� = � ��������� �����
� = � ��������� ������
� = � ��������� ������
� = � ��������� ������

25

20

15

10

5

0
0.05

0.10

0.15

0.20

0.25

0.30

0.35

��������� �� �� ��� ���� ������ �����

FIG. 5. Distinguishing more types of seniority reduces thesize of the cascade region. In other words, with more types ofseniority, the expected size of a default cascade is negligibly
small for more pairs ofPthresholds R1 and expected numberof loans per bank z = M
i=1 hli i. Here we consider splitting a
directed Erdős-Rényi network into M ∈ {1, 2, 3, 4} many independent, identically distributed layers, each with the samemean degree hli i = z/M .

Notice in Fig. 5 that the cascade region shrinks with
the number M of seniority levels, meaning that the system is less vulnerable to large cascades of default. (Bycontrast, in the model in [13], the cascade region grows
with M .) For some capital-to-interbank-asset ratios R1 ,the cascade region is even eliminated as M grows (e.g.,
R1 = 0.3 for M = 1).
The reason the cascade region shrinks with M is thatmaking a loan more senior makes it less significant early
in a cascade. For instance, if M = 1, then a defaulted
node reduces the assets of all of its in-neighbors, whereas
if M > 1, then only the in-neighbors in layer 1 lose an
asset, so less contagion occurs. The decline in importanceof more senior layers is captured by the multiplicative
Qi−1
factor k=1 pk,borrowin Eq. (11c) and by the exponential
0

9debts than junior debts. Such disparity between individual interests and system-wide stability is the “regulator’s
dilemma” [34].
The seniority of debt is just one dimension of multiplexity in financial networks. Other dimensions of debt
include the maturity of loans and whether loans are collateralized [8, 9]. Furthermore, other important interactions include the trade of financial assets [35] and common exposures (i.e., multiple institutions holding thesame financial assets) [9, 36]. We hope our multiplex
model will stimulate further research on these many typesof multiplexity in financial markets.

factor in Eq. (13). In short, distinguishing more types ofseniority makes more interbank debts insignificant (early
in a crisis); hence a systemic crisis is less likely.
This result hinges on the assumption that banks’buffers against insolvency R1 remain constant as thenumber of seniority levels (M ) grows. It is possible that if
there were more seniority levels, then banks would erode
their buffers against insolvency, R1 , thereby pushing thesystem back toward the cascade region, where bankruptcies likely spread widely.

IV.

CONCLUSION AND DISCUSSIONACKNOWLEDGMENTS

We model seniority of debt using a multiplex network,
and we study the likelihood of bankruptcy spreading
widely using a generalized cascade condition. The model
has interesting tradeoffs. Whereas self-interested banks
would prefer to make their loans at the most senior level,a mixture of junior and senior debts optimally reduces
systemic risk, in that it minimizes the set of network
densities such that cascades are likely large. For two
types of seniority, we show that the optimal multiplex
networks have on the order of 50% to 100% more senior

C.D.B. is supported by the James S. McDonnell Postdoctoral Fellowship in Studying Complex Systems. T.K.
is supported by KAKENHI 25780203 and 24243044. The
authors thank the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) for support and hospitality during the programme Systemic Risk: Mathematical
Modelling and Interdisciplinary Approaches, where workon this paper was undertaken.

Appendix A: Thresholds of default for M levels of seniority

Here we consider the general case of M ≥ 1 levels of seniority of interbank debt (rather than just two, junior andsenior), and we label the levels of seniority by α ∈ {1, 2, . . . , M }. We choose the convention that debts of seniority α
are more senior than debts of seniority α0 whenever α > α0 .
Recall from Fig. 1 the stylized balance sheet of a bank: a bank’s assets consist of external assets e and lα many
interbank loans of seniority α ∈ {1, 2, . . . , M }, while its liabilities consist of external liabilities d and bα many interbank
liabilities of seniority α ∈ {1, 2, . . . , M }. We assume that the external liabilities d (which could include, for example,
demand deposits) are more senior than debts to other financial institutions. The equity w is the difference between
assets and liabilities,
MX

w =e−d+(lα − bα ).

α=1PM
Now suppose that this bank has some interbank assets (i.e., α=1 lα > 0) and that it suffers losses on mα of itsPMlα many interbank assets (for α ∈ {1, 2, . . . , M }), so that its interbank assets now have total value α=1 (lα − mα ).
Then exactly one of the following events occurs:
PM• The bank is solvent if and only if the capital buffer w can absorb the losses, i.e., w ≥ α=1 mα .
• The bank is in default at layer α0 ∈ {1, 2, . . . , M } but not in any layer ≥ α0 + 1 if and only if
w+

0
αX
−1

bα <

α=1

M
X

0

mα ≤ w +

α=1

α
X

bα .

(A1)

α=1

• The bank is in complete default, meaning that it cannot pay off any of its interbank debts nor any of its liabilities
to external creditors, if and only ifMX
α=1mα > w +

M
X
α=1

bα .

10
Here, we focus on cascades in the interbank network; for this purpose, we do not need to distinguish between default
at the most senior level (M ) from complete default because both have the same effect on the banks that had lent to
the defaulted bank.
PM
Rearranging the strict inequality in Eq. (A1) and dividing by the total lending α=1 lα > 0 gives the response
function in Eq. (10).

Appendix B: Relationship with “multistage complex contagion”

Our model is related to the “multistage complex contagion” model of Melnik et al. [29]. In their model, nodes
exist in one of three states: inactive, active, and hyper-active. Their hyper-active nodes are active nodes with greater
influence on their neighbors. To see the commonality between the models, consider our model with every node
having the same junior threshold RJ and with the multiplex network consisting of two identical layers (i.e., all themultiplex edges are “fully overlapped” [17, 32]). Then every neighbor in junior-default (respectively, in senior-default)
contributes mJ + mS = 1 (respectively, mJ + mS = 2) in the numerator of the response function. Thus, this model
is equivalent to a special case of the model in [29] in which (1) the “bonus influence parameter” β (which captures
the extra influence of hyper-active nodes) takes the value 1; (2) the network is a directed single-layer graph; and (3)the thresholds for activation and hyper-activation are RJ and RJ + bJ /(lJ + lS ) = RJ + bJ /(2lJ ), respectively. To
model real financial systems, however, edges should be only partially overlapped, as in the percolation models studiedin [17, 32]. Extending our multilevel contagion model to such a partially overlapped case is an important avenue for
future research.

Appendix C: Derivation of the first-order cascade condition

Now we obtain a simple expression that approximately captures for which parameters a vanishingly small seed leads
to a cascade that results in a finite fraction of banks in junior- or senior-default. Note that if the threshold RJ > 0,kthen (0, 0) is a fixed point of the recursion (4) [because Bm(0) ≡ 1m=0 and Fα (~l, ~b, ~0) ≡ 0 provided that RJ > 0]. The
first-order cascade condition is the linear instability of Eq. (4) at this fixed point at the origin (φJt , φSt ) = (0, 0) in the
double limit φJ0 → 0 and φS0 → 0. This linear instability provides a sufficient condition for the numbers of junior- andsenior-defaulted banks to grow (at least initially). The Jacobian matrix of the right-hand side of Eq. (4) is
JJJ JJSJ =(C1)
JSJ JSS
E[lJ (11>RJ (lJ +lS ) − 10>RJ )]E[lS (11>RJ (lJ +lS ) − 10>RJ )]=E[lJ (11−bJ >RJ (lJ +lS ) − 1−bJ >RJ (lJ +lS ) )] E[lS (11−bJ >RJ (lJ +lS ) − 1−bJ >RJ (lJ +lS ) )](C2)
(The expectations E are over the degree distributions pJ,loan, pS,loan, pJ,borrow.) Assume that RJ > 0, so that thelJ
lSbJ
factors 10>RJ in the top row of (C2) vanish. Also, note that bJ ≥ 0 and that RJ > 0 and lJ + lS ≥ 0, so thefactors 1−bJ >RJ (lJ +lS ) in the bottom row of Eq. (C2) vanish. Finally, the other indicator in the bottom row of
Eq. (C2), namely 11−bJ >RJ (lJ +lS ) , simplifies to 1bJ =0 11>RJ (lJ +lS ) because bJ ∈ {0, 1, 2, . .. }. Because each node’s
in- and out-degrees (lJ , bJ , lS , bS ) are all independent, we can factor the expectation E lα 1bJ =0 11>RJ (lJ +lS ) =E lα 11>RJ (lJ +lS ) for both α ∈ {J, S} (i.e., for the bottom-left and bottom-right entries of Eq. (C2)). It
pJ,borrow0
follows that
E[lJ 11>RJ (lJ +lS ) ]E[lS 11>RJ (lJ +lS ) ]J =(C3)
E[lJ 11>RJ (lJ +lS ) ] pJ,borrow
E[lS 11>RJ (lJ +lS ) ]pJ,borrow
0
0




1
0
0
1
+ E[lS 11>RJ (lJ +lS ) ].
(C4)= E[lJ 11>RJ (lJ +lS ) ] J,borrowp0
0
0 pJ,borrow0
Note that det J = 0. Thus, from Eq. (6), the largest eigenvalue of the Jacobian is λmax (J ) = trJ = JJJ + JSS .
It should be pointed out that this equality, λmax (J ) = trJ , does not necessarily hold true in other variants ofmultiplex contagion models. For example, in multiplex models with (independent) undirected networks and/or an
exogenous senior-default threshold RS , one would generally have λmax (J ) 6= trJ . A more detailed explanation is
given in Appendix E.

11Appendix D: Comparison between the first-order cascade condition with the fixed point of the recursion
equations and with numerical simulations

Figure 6 shows that the first-order cascade condition [the parameters satisfying Eq. (6), which are enclosed by thesolid blue line in Fig. 6] agrees closely with the fixed point (φJ∞ , φS∞ ) of the recursion equations (4) [Fig. 6(a) and 6(b)]and agrees reasonably closely with the results of numerical simulations [Fig. 6(c) and 6(d)]. In the recursion equations
and simulations, we use the same initial condition, with φJ0 = φS0 = 5 × 10−4 . The threshold for junior default is
RJ = 0.18, as in Fig. 3.
Notice in Fig. 6 that when a large cascade occurs, many nodes are in junior-default, but many nodes are insenior-default if and only if, loosely speaking, there are many more senior loans than junior ones. [Look at the dark
region in the left-hand sides of Figs. 6(b) and 6(d).] In practice, we are most concerned with whether the number ofbankruptcies of any type is large. Thus, our definition of the optimal seniority ratio in Sec. II C considered only themultiplex cascade region and not the fractions φJ∞ and φS∞ .
Appendix E: Multilevel cascades under alternative settings

In Appendix C, we claimed that the equality λmax (J ) = trJ , which is proved in the main text for our model, does
not necessarily hold true in other variants of the multilevel cascade model. In what follows, we substantiate this claimby examining two variants of the 2-layer multiplex model.

1.

Exogenous senior thresholds

In the model described in the main text, the threshold of senior-default, RS , depends on the threshold of juniordefault, RJ , and on the in-degree of the node on the junior layer, bJ . In this subsection, we consider an alternativemodel of multilevel cascades in which the threshold of senior-default is an exogenous parameter. Let the (locallytreelike) graph hang down like a tree from a root node chosen uniformly at random. The recursion equation for the
probability of default of a node t hops above the leaves of the tree is
X

(α) J
α
φ̂α
(φ̂t , φ̂St ) ≡ φ̂αt+1 = ĝ
0 + (1 − φ̂0 )pJ,outpS,outlJlS

lJ +lS ≥1

lJ
lS
X
X

lJ
lS
Bm
(φ̂Jt )Bm(φ̂St )F̂α (~l, m)
~
J
S

(E1)

mJ =0 mS =0

for α ∈ {J, S}, where pJ,out
and pS,out
denote the out-degree distributions on the junior and senior layers, respectively.
lJlS~
The response functions are now F̂α (l, m)~ = 1mJ +mS >Rα (lJ +lS ) for α ∈ {J, S}, where 0 < RJ ≤ RS ≤ 1.
The first-order cascade condition is thus given by λmax (Jˆ) > 1, where the entries of Jˆ are E[lj 11>Ri (lJ +lS ) ] for(i, j) ∈ {J, S}2 . Notice that the equality det Jˆ = 0 does not necessarily hold if RJ < RS . It follows that λmax (Jˆ) is
not in general equal to trJˆ.

2.

Undirected edges

Next, consider a multilevel cascade model in which edges are undirected. Again, let the graph hang down like atree from a root node chosen uniformly at random. Consider a node located t hops above the leaves of the tree.
The probability that this node is in α-default due to its children, conditioned on its parent in layer α ∈ {J, S} being
solvent at layer α, is given by the recursion equations
Jθ̄t+1= θ̄0J + (1 − θ̄0J )X
lJ +lS ≥1

Sθ̄t+1= θ̄0S + (1 − θ̄0S )X
lJ +lS ≥1lJ
lS
X
pJlJ lJ S Xp
B lJ (θ̄J )B lS (θ̄S )F̄J (~l, m),~hlJ i lS m =0 m =0 mJ t mS t

(E2a)

lJ
lS
X
pSlS lS X
B lJ (θ̄J )B lS (θ̄S )F̄S (~l, m),~
hlS i m =0 m =0 mJ t mS t

(E2b)

J

pJlJ

J

S

S

where one uses Eq. (E2a) [respectively, Eq. (E2b)] if the node’s parent is a neighbor in layer J (respectively, layer S),
and pαlα is the degree distribution on layer α. Because the graph is unweighted, and because this node lies at the end12���� ������ (ϕ�∞ � ϕ�∞ ) �� ��� ��������� ���������
(�)��������� �������� λ��� () > �������-���� ��������  �� > �8������-���� ��������  �� > �
6

ϕ�∞
���

4

���
���
���

2

�

�������� ������ �� ������ ����� ��� ���� = 〈�� 〉�������� ������ �� ������ ����� ��� ���� = 〈�� 〉(�)0��������� �������� λ��� () > �������-���� ��������  �� > �8������-���� ��������  �� > �
6

ϕ�∞
���

4

���
���
���

2

�

0
0

2

4

6

8

0

�������� ������ �� ������ ����� ��� ���� = 〈�� 〉

2

4

6

8

�������� ������ �� ������ ����� ��� ���� = 〈�� 〉��������� �����������

8(�)��������� �������� λ��� () > �������-���� ��������  �� > �������-���� ��������  �� > �

6��������� ϕ�
���

4

���
���
���

2

�

0

0

2

4

6

�������� ������ �� ������ ����� ��� ���� = 〈�� 〉8�������� ������ �� ������ ����� ��� ���� = 〈�� 〉�������� ������ �� ������ ����� ��� ���� = 〈�� 〉(�)8��������� �������� λ��� () > �������-���� ��������  �� > �������-���� ��������  �� > �

6��������� ϕ�
���

4

���
���
���

2

�

0

0

2

4

6

8

�������� ������ �� ������ ����� ��� ���� = 〈�� 〉FIG. 6. (Color online) Here we compare the junior-only, senior-only and multiplex cascade conditions [Eqs. (6), (7), and (8),respectively] with the fixed points of the recursion equations (4) [in panels (a) and (b)] and with numerical simulations on
random graphs with 104 nodes [in panels (c) and (d)]. The junior-default threshold of every node is RJ = 0.18 (as in Fig. 3).
The axes are the edge densities hlJ i and hlS i (i.e, the average numbers of junior and senior loans per bank). The solid blue,
dashed orange, and dot-dashed green lines surround the multiplex, junior-only, and senior-only cascade regions [i.e., the sets
of values of (hlJ i, hlS i) that satisfy the inequalities (6), (7), and (8), respectively]. In panels (a) and (b), the grayscale in the
JS−4background shows the fixed points φJ∞ and φS. In∞ of the recursion (4), begun from the initial condition φ0 = φ0 = 5 × 10
panels (c) and (d), the grayscale in the background shows the fraction of nodes in junior- and senior-default, respectively, in
numerical simulations with 104 nodes and 5 banks initially in senior-default, averaged over 75 simulations. In the upper-leftpart of all four panels, where most loans are senior, the fractions of nodes (φJ∞ , φS∞ ) in junior- and senior-default at the endof the cascade are both ≈ 1. By contrast, in the lower-right part, where most loans are junior, the fraction φJ∞ of nodes injunior-default is large [φJ∞ ≈ 1; panels (a) and (c)], whereas the fraction φS∞ of nodes in senior-default is ≈ 0.4 [see the light-grayregion surrounded by the dashed orange line in panels (b) and (d)].

13of an α-type edge chosen uniformly at random, its degree in layer α is the excess degree distribution, pα
lα lα /hlα i. The
response functions are(S
1 if mlJJ +m+lS > RJ ,(E3a)
F̄J (~l, m)~ =0 otherwise
(
S −lJ
> RJ
1 if mJ l+m~
J +lSF̄S (l, m)
~ =
.
(E3b)
0 otherwise
Because the graphs are undirected, the term −bJ in FS in the original model [Eq. (2b)] is now −lJ in Eq. (E3b).
The corresponding Jacobian matrix is
EJ [(lJ − 1)11>RJ (lJ +lS ) ]EJ [lS 11>RJ (lJ +lS ) ]
¯
J =
,
(E4)
pJ0 ES [lJ 11>RJ (lJ +lS ) ] pJ0 ES [(lS − 1)11>RJ (lJ +lS ) ]
pJ lJ

pS lS

where the expectations EJ and ES are over the joint probability distributions hllJJ i pSlS and pJlJ hllSS i , respectively. It
follows that det J¯ is generally nonzero, and therefore λmax (J¯) = trJ¯ does not necessarily hold.
REFERENCES[1] P. Gai and S. Kapadia, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, 2401
(2010).
[2] E. Nier, J. Yang, T. Yorulmazer, and A. Alentorn, Journal of Economic Dynamics and Control 31, 2033 (2007).
[3] T. Kobayashi, Economics Letters 124, 113 (2014).
[4] P. Gai, A. Haldane, and S. Kapadia, Journal of Monetary Economics 58, 453 (2011).
[5] C. Upper, Journal of Financial Stability 7, 111 (2011).
[6] R. M. May and N. Arinaminpathy, Journal of the Royal Society Interface 7, 823 (2010).
[7] T. R. Hurd, D. Cellai, S. Melnik, and Q. Shao, (2013), arXiv:1310.6873.
[8] L. Bargigli, G. di Iasio, L. Infante, F. Lillo, and F. Pierobon, (2013), arXiv:1311.4798.
[9] M. Montagna and C. Kok, Kiel Working Papers no.1873 (2013).
[10] M. Elliott, B. Golub, and M. O. Jackson, American Economic Review 104, 3115 (2014).
[11] M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter, Journal of Complex Networks 2, 203(2014).
[12] S. Boccaletti, G. Bianconi, R. Criado, C. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, andM. Zanin, Physics Reports 544, 1 (2014).
[13] C. D. Brummitt, K.-M. Lee, and K.-I. Goh, Physical Review E 85, 045102(R) (2012).
[14] O. Yağan and V. Gligor, Physical Review E 86, 036103 (2012).
[15] K.-M. Lee, C. D. Brummitt, and K.-I. Goh, Physical Review E 90 (2014).
[16] B. Min, S. D. Yi, K.-M. Lee, and K.-I. Goh, Physical Review E 89, 042811 (2014).
[17] D. Cellai, E. López, J. Zhou, J. Gleeson, and G. Bianconi, Physical Review E 88, 052811 (2013).
[18] D. J. Watts, Proceedings of the National Academy of Sciences 99, 5766 (2002).
[19] J. Gleeson and D. Cahalane, Physical Review E 75, 56103 (2007).
[20] L. C. G. Rogers and L. A. M. Veraart, Management Science 59, 882 (2013).
[21] H. Elsinger, Oesterreichische Nationalbank Working Paper 156 (2009).
[22] C. Gourieroux, J. C. Heam, and A. Monfort, Journal of Banking and Finance 37, 5261 (2013).
[23] L. Eisenberg and T. Noe, Management Science 47, 236 (2001).
[24] J. P. Gleeson, Physical Review E 77, 46117 (2008).
[25] D. Centola, V. Eguı́luz, and M. W. Macy, Physica A: Statistical Mechanics and its Applications 374, 449 (2007).
[26] J. Payne, P. Dodds, and M. Eppstein, Physical Review E 80 (2009).
[27] J. Payne, K. Harris, and P. Dodds, Physical Review E 84 (2011).
[28] R. R. Liu, W. X. Wang, Y. C. Lai, and B. H. Wang, Physical Review E , 026110 (2012).
[29] S. Melnik, J. A. Ward, J. P. Gleeson, and M. A. Porter, Chaos: An Interdisciplinary Journal of Nonlinear Science 23,013124 (2013).
[30] R. Cont, A. Moussa, and E. B. Santos, in Handbook on Systemic Risk, edited by J.-P. Fouque and J. A. Langsam(Cambridge University Press, New York, 2013).
[31] M. Granovetter, American Journal of Sociology 83, 1420 (1978).

14[32] G. Bianconi, Physical Review E 87, 062806 (2013).
[33] Data on leverage ratios of banks was accessed at http://www.bankregdata.com/allHMmet.asp?met=LEV on January 22,
2015. The source of the data is the Federal Financial Institutions Examination Council (FFIEC), available at https://cdr.ffiec.gov/public/.
[34] N. Beale, D. G. Rand, H. Battey, K. Croxson, R. M. May, and M. A. Nowak, Proceedings of the National Academy of
Sciences 108, 12647 (2011).
[35] C. D. Brummitt, R. Sethi, and D. J. Watts, PloS one 9, e104219 (2014).
[36] F. Caccioli, J. D. Farmer, N. Foti, and D. Rockmore, Journal of Economic Dynamics and Control 51, 50 (2015).