# Financial fire sales as continuous-state complex contagion

Tomokatsu Onaga  Frontier Research Institute for Interdisciplinary Sciences,
Graduate School of Information Sciences,
Tohoku University, Sendai, Japan  Fabio Caccioli  Department of Computer
Science, University College London, London, UK
Systemic Risk Centre, London School of Economics and Political Science,
London, UK
London Mathematical Laboratory, London, UK  Teruyoshi Kobayashi
<EMAIL_ADDRESS>Department of Economics,
Center for Computational Social Science, Kobe University, Kobe, Japan

###### Abstract

Trading activities in financial systems create various channels through which
systemic risk can propagate. An important contagion channel is financial fire
sales, where a bank failure causes asset prices to fall due to asset
liquidation, which in turn drives further bank defaults, triggering the next
rounds of liquidation. This process can be considered as complex contagion,
yet it cannot be modeled using the conventional binary-state contagion models
because there is a continuum of states representing asset prices. Here, we
develop a threshold model of continuous-state cascades in which the states of
each node are represented by real values. We show that the solution of a
multi-state contagion model, for which the continuous states are discretized,
accurately replicates the simulated continuous state distribution as long as
the number of states is moderately large. This discretization approach allows
us to exploit the power of approximate master equations (AME) to trace the
trajectory of the fraction of defaulted banks and obtain the distribution of
asset prices that characterize the dynamics of fire sales on asset-bank
bipartite networks. We examine the accuracy of the proposed method using real
data on asset-holding relationships in exchange-traded funds (ETFs).

##  I introduction

Financial markets are complex systems in which many economic agents, such as
individuals, firms, financial institutions, and central banks, are
interconnected at the global scale through trading activities. The networks of
money and credit flows have been expanding as more market participants have
access to a greater variety of financial assets, bonds, and other securities.
On one hand, this increases the efficiency of the market. On the other hand,
the past two decades made it clear that increased connectivity between market
participants leads to higher risk of systemic failures in the global financial
market: Once a part of the financial system malfunctions, interconnections can
facilitate a chain reaction that can spread globally [1, 2, 3], for instance
in the form of fire sales of financial assets [4, 5, 6], or of cascades of
defaults among banks with mutual exposures [7, 8, 9, 10, 11].

In models of financial contagion, nodes can be financial institutions,
individual traders, or financial assets, depending on which aspects of
financial phenomena one would examine. For example, in cascades of bank
defaults due to counterparty default risk, nodes are banks and edges are
lending and borrowing relationships in an interbank market [7, 8, 12, 13, 10,
11, 14]. In cascades of financial fire sales, nodes are financial assets and
financial institutions, which form a bipartite network defined by asset-
holding relationships, i.e., overlapping portfolios [15, 5, 6, 16].

However, full analytical solutions for these models are difficult to obtain
since the states of nodes in financial networks are generally non-binary
unlike the standard threshold models [17, 18, 19, 7, 10, 20, 21, 22, 23, 24].
While the state of a bank can for instance be well captured by a binary
variable that denotes whether the bank is solvent or insolvent, the state of
an asset (e.g., a stock price) is a continuous variable, since in principle
its price can take any non-negative real value. Financial fire sales are thus
continuous-state complex contagion processes in which a group of nodes are
characterized by real-valued states, while the others by binary states.

Here we develop a threshold model of continuous-state contagion in which the
states of each node are represented by real values. Compared to the
conventional binary-state cascade models, the difficulty is that the
distribution of nodes’ states is generally given as a density function as
opposed to a probability mass function. We therefore employ a discretization
approach in which we approximate a set of real-valued states by a finite
number of discretized states.

As a tool for the calculation of spreading dynamics, a _message-passing
method_ has been used in the analysis of binary-state cascades [18, 19, 23,
24]. While the message-passing method is highly accurate in describing binary-
state spreading processes [23], it can be imprecise in models with more than
two states, i.e., _multistate_ cascade models [25, 26]. We address this
problem by employing a more general _approximate master equations_ (AMEs)
approach [27, 28, 26], in which the states of neighboring nodes are not
necessarily independent unlike the conventional mean-field approaches. We show
that the AME solution well explains the distribution of continuous states in
the generalized threshold model as long as the number of discretized states is
moderately large.

By extending the continuous-state threshold model, we develop an analytical
framework to study the dynamics of financial fire sales due to overlapping
portfolios. In the model, _bank nodes_ take binary states (i.e., solvent or
insolvent) while the states of _asset nodes_ (i.e., asset prices) are
represented by real values. The analysis reveals that the discrete-price
(i.e., discrete-state) equilibrium obtained by the AME method converges to the
continuous-price (i.e., continuous-state) equilibrium as one increases the
number of discretized states. This suggests that the behavior of continuous
asset prices in the propagation of fire sales, which has been examined only
numerically in previous studies [15, 6, 16], can be analyzed analytically
within a framework of multistate cascade models.

##  II Continuous-state complex contagion

###  II.1 A threshold model with continuous states

We first present a model of continuous-state contagion in which the state of
each node is represented by a real value $s\in[0,1]$. We will then consider a
model of financial fire sales with both binary and continuous states in
section III.

The model we consider in this section is a generalized version of the binary-
state Watts’ threshold model [17]; in addition to state 0 (i.e., fully
inactive) and state 1 (i.e., fully active), there is a spectrum of
intermediate states between them (i.e., partially active).

Let us focus on a node with degree $k$. We will denote by
$\bm{\mathbf{s}}=(s_{1},\ldots,s_{k})^{\top}$ the vector containing the states
of the node’s neighbors. While there can be various specifications of
continuous-state cascades, we consider a generalized fractional threshold rule
in which the state of each node is affected by the fraction of its “active
neighbors” $m/k$, as it is assumed in many variants of the discrete-state
threshold model [17, 29, 25, 30, 20, 10, 31, 32, 23, 22, 33]. For a given
threshold $\theta\in[0,1)$, the response function
$\mathcal{F}_{\bm{\mathbf{s}}}$ is generically given as:

$\displaystyle\mathcal{F}_{\bm{\mathbf{s}}}=\begin{cases}f\left(\frac{m}{k}\right)\;{\text{
if }}\;\frac{m}{k}>\theta,\\\ 0\;\text{ otherwise},\end{cases}$ (1)

where $m$ denotes the number of “active” neighbors whose states are positive:
$m=|\\{j:s_{j}>0,s_{j}\in\bm{\mathbf{s}}\\}|$. $f$ is a continuous non-
decreasing function on $(\theta,1]$ such that $f(\cdot)\in[0,1]$. That is,
$\mathcal{F}_{\bm{\mathbf{s}}}$ gives the state of a node, taking the
neighbors’ states $\bm{\mathbf{s}}$ as input. Note that the standard Watts
model is recovered when $f(m/k)=1$ for $m/k>\theta$, corresponding to the most
“progressive” response function. In Appendix A, we examine a more general
nonlinear form of $f(m/k)$.

We also consider an alternative response function that depends on the sum of
neighbors’ states:

$\displaystyle\widetilde{\mathcal{F}}_{\bm{\mathbf{s}}}=\begin{cases}f\left(\frac{\sum_{j}s_{j}}{k}\right)\;{\text{
if }}\;\frac{\sum_{j}s_{j}}{k}>\theta,\\\ 0\;\text{ otherwise}.\end{cases}$
(2)

It should be noted that the influence of seed nodes will generally be weakened
when the response function is given by
$\widetilde{\mathcal{F}}_{\bm{\mathbf{s}}}$ rather than
$\mathcal{F}_{\bm{\mathbf{s}}}$. For example, consider a tree in which each
node (excluding the seed node) has $k$ edges, and all nodes are initially in
state $0$. Suppose that $f(\sum_{j}s_{j}/k)=\sum_{j}s_{j}/k$, and the state of
the seed node shifts from $0$ to $1$. Since the seed node is one of the $k$
neighbors for the child nodes, the states of the child nodes become $1/k$ if
$\theta<1/k$. Then, the states of the grandchild nodes become $1/k^{2}$ if
$\theta<1/k^{2}$, the great-grandchild nodes’ state become $1/k^{3}$ if
$\theta<1/k^{3}$, and so on. While the actual propagation process can be more
complicated due to repercussion of peer effects, the spread of influence with
response function $\widetilde{\mathcal{F}}_{\bm{\mathbf{s}}}$ is more likely
to decay than that obtained with the response function
$\mathcal{F}_{\bm{\mathbf{s}}}$.

###  II.2 Multistate dynamical process as an approximation

In the continuous-state contagion model, it is difficult to describe the
collective dynamics in an analytical manner because the shares of nodes in
each state are represented as a continuous density function in the limit of an
infinitely large network size. While the conventional mean-field/message-
passing methods could be applied to a class of multi-stage contagion models
with several nodal states, the accuracy of the approximation methods easily
deteriorates when there are more than two states [25].

To address this problem, we introduce a discretization approach using an
approximate master equation (AME) method [27, 28, 26]. To apply the AME
method, we transform the continuous-state model into a discrete multistate
model by discretizing the range of continuous states into $n$ grids. Let
$\mathbf{m}\equiv(m_{0},m_{1},\ldots,m_{n-1})^{\top}$ be the profile of
neighbors’ states, where $m_{\ell}$ denotes the number of neighbors in
(discretized) state $\ell\in\\{0,\ldots,n-1\\}$. For a node with degree
$k\in\\{1,\ldots,k_{\rm max}\\}$, we have $\sum_{\ell=0}^{n-1}m_{\ell}=k$.

Transitions in the states of nodes are captured by the response function
$F_{\mathbf{m}}(i\rightarrow j)$, the rate at which a node in state $i$
changes its state to $j$. For a given sequence of thresholds
$\Theta=\\{\theta_{0},\theta_{1},\ldots,\theta_{n-1}\\}$, $F_{\mathbf{m}}$ is
specified as

$F_{\mathbf{m}}(i\rightarrow j)=\begin{cases}1\;\text{ if
}\theta_{j-1}<P(\mathbf{m})\leq\theta_{j}\;\;\\\ 0\;\text{
otherwise}\end{cases}$ (3)

for $1\leq j\leq n-1$, and

$F_{\mathbf{m}}(i\rightarrow 0)=\begin{cases}1\;\text{ if
}P(\mathbf{m})\leq\theta_{0}\;\;\\\ 0\;\text{ otherwise},\end{cases}$ (4)

where $P(\mathbf{m})$ denotes the “peer pressure”, i.e., the influence from
neighbors, and $\theta_{j-1}$ and $\theta_{j}$ are the thresholds between
which the node is in state $j$.

To approximate the continuous-state model, the peer-pressure corresponding to
the response function $\mathcal{F}_{\bm{\mathbf{s}}}$ (Eq. 1) is given as

$P(\mathbf{m})=f\left(\frac{\sum_{\ell=1}^{n-1}m_{\ell}}{k}\right).$ (5)

Note that the multistate model reduces to a binary-state model when $n=2$,
$0<\theta_{0}<1$ and $\theta_{1}=1$. We specify the sequence of threshold
values $\Theta$ such that the responsiveness $f(m/k)$ in the continuous model
is evenly spaced on $[0,1]$, where $f(\theta_{j})-f(\theta_{j-1})=1/(n-1)$ for
all $j=1,\ldots,n-1$. A detailed description of the specifications of
threshold values is provided in Appendix A and Fig. 9. When the response
function is given by $\widetilde{\mathcal{F}}_{\bm{\mathbf{s}}}$ (Eq. 2), the
peer pressure for the discretized model leads to

$\displaystyle P(\mathbf{m})=f\left(\frac{1}{k}\frac{\sum_{\ell=1}^{n-1}\ell
m_{\ell}}{n-1}\right),$ (6)

where $\ell/(n-1)\in[0,1]$ is the normalized state of neighbors in state
$\ell$. In the following, we specify the functional form of $f$ as
$f(u)=\left(\frac{u-\theta}{1-\theta}\right)^{\eta}$ with $\eta\geq 0$, where
$\eta=0$ corresponds to a linear threshold model (see Appendix A for details).

Let $x_{k,{\bf m}}^{i}(t)$ be the fraction of $k$-degree nodes in state $i$
that face neighbors’ profile $\bf m$ at time $t$. Employing the AME formalism,
we solve the following differential equations for the dynamics of $x_{k,{\bf
m}}^{i}(t)$ [26]:

$\displaystyle\frac{dx^{i}_{k,\mathbf{m}}}{dt}$ $\displaystyle=$
$\displaystyle-\sum_{j>i}F_{\mathbf{m}}(i\rightarrow
j)x^{i}_{k,\mathbf{m}}+\sum_{j<i}F_{\mathbf{m}}(j\rightarrow
i)x^{j}_{k,\mathbf{m}}$ (7)
$\displaystyle-\sum_{l=0}^{n-1}\sum_{\ell^{\prime}>\ell}m_{\ell}\beta^{i}(\ell\rightarrow\ell^{\prime})x^{i}_{k,\mathbf{m}}$
$\displaystyle+\sum_{l=0}^{n-1}\sum_{\ell^{\prime}<\ell}(m_{\ell^{\prime}}+1)\beta^{i}(\ell^{\prime}\rightarrow\ell)x^{i}_{k,\mathbf{m}-\mathbf{e}_{\ell}+\mathbf{e}_{\ell^{\prime}}},$

where ${\bf e}_{\ell}$ denotes the $n\times 1$ vector that contains $1$ for
the $\ell$-th element and $0$ for the other elements.
$\beta^{i}(\ell\to\ell^{\prime})$ denotes the probability that a neighbor of a
node being in state $i$ changes its state from $\ell$ to $\ell^{\prime}$:

$\beta^{i}(\ell\rightarrow\ell^{\prime})=\frac{\langle\sum_{|\mathbf{m}|=k}m_{i}x^{\ell}_{k,\mathbf{m}}(t)F_{\mathbf{m}}(\ell\rightarrow\ell^{\prime})\rangle_{k}}{\langle\sum_{|\mathbf{m}|=k}m_{i}x^{\ell}_{k,\mathbf{m}}(t)\rangle_{k}},$
(8)

where $\langle\cdot\rangle_{k}$ denotes the average over degree $k$ with
degree distribution $p_{k}$. The first (resp. second) term of Eq. (7) captures
the rate at which the states of a $k$-degree node in the $(i,{\bf m})$ class
(resp. $(j,{\bf m})$ class) shifts from $i$ to $j(\neq i)$ (resp. $j$ to
$i(\neq j)$) in an infinitesimal time interval $dt$. It should be noted that
Eq. (7) describes the dynamics under _asynchronous update_ in which only a
fraction $dt$ of nodes can change their states in response to their neighbor
profiles in a small time interval [18, 25]. The third term denotes the rate at
which the neighbors’ state profile will be different from ${\bf m}$. The forth
terms indicates the rate at which the neighbors’ profile newly becomes ${\bf
m}$. The expression ${\bf m}-{\bf e}_{\ell}+{\bf e}_{\ell^{\prime}}$
represents the neighbor profile that has $m_{\ell^{\prime}}+1$ in the
$\ell^{\prime}$-th element and $m_{\ell}-1$ in the $\ell$-th element. For $k$
ranging from $0$ to $k_{\rm max}$, the total number of differential equations
leads to $n\sum_{k=0}^{k_{\rm max}}\binom{k+n-1}{k}$ [26].

Figure 1: State transition in the continuous- and discrete-state threshold
models. (a) Visualization of node states at $t=3$ in different models. The
network structure and the seed nodes are common for the three cases. (b)
Comparison between the AME solution and simulation results in the discrete-
state model with $n=6$. (c) Distribution of states at a given time. In the
AME, the fraction of nodes in discretized state $i\in\\{0,\ldots,n-1\\}$ is
given by $\rho_{i}$ (blue bar). For the continuous-state model, density
$\pi(s)$ for state $s\in[0,1]$ is calculated by the kernel density estimation
with bandwidth $0.1$ based on numerical simulations (black line). The index of
discretized states is normalized by $n-1$ so that the states are distributed
on $[0,1]$. Seed fraction is determined such that the fraction $0.05$ of nodes
are in the “most-active” state (i.e., $i=n-1$ or $s=1$) while $0.95$ of nodes
are in state 0. For panels b and c, we run 40 simulations on Erdős-Rényi
networks with $N=3000$, $z=3$, $\theta=0$, and $\theta_{j}=j/(n-1)$.

###  II.3 Validation

To examine the accuracy of the discrete approximation, we calculate the
average fraction of active nodes in a discretized state $i$ at time $t$,
denoted by $\rho_{i}(t)$, using the solutions of the AMEs (7): where
$\sum_{|\mathbf{m}|=k}$ denotes the sum over all combinations of $\mathbf{m}$
such that $\sum_{\ell=0}^{n-1}m_{\ell}=k$. On the other hand, the states of
nodes in the continuous-state model are represented by $\pi(s)$ on
$s\in[0,1]$, where $\pi(s)$ is a density of nodes in state $s$. We consider
Erdős-Rényi random networks in which the degree distribution follows a Poisson
distribution with mean $z$. We first consider a simple responsiveness given as
$f(m/k)=m/k$ (i.e., $\eta=1$, $\theta=0$), which is a linear map of the
fractional influence $m/k$. The corresponding thresholds are given by
$\theta_{j}=j/(n-1)$ for $j=0,\ldots,n-1$. Results for more general nonlinear
functional forms are presented in Appendix A. For the calculation of the AME
solution, we use the MATLAB code developed by [26, 34].

While the states of nodes are $0$ or $1$ when $n=2$ (Fig. 1a, _left_), there
are many nodes whose states are in between $0$ and $1$ when $n>2$ once the
state index is normalized by $n-1$ (Fig. 1a, _middle_). In the discrete-state
model with $n=6$, the node states are distributed as heterogeneously as in the
continuous-state model (Fig. 1a, _middle_ and _right_). On the other hand, the
steady-state fraction of nodes in state 1 in the binary-state approximation
(i.e., $n=2$) is too large, compared to that of the continuous-state model
(Fig. S1 in Supplemental Material (SM)).

As shown in previous studies, the AME method is highly accurate in predicting
the dynamical path of the fractions of each state in a class of discrete-state
contagion models [26] (Figs. 1b and S2). It is thus sufficient to compare
$\rho_{i}$ calculated by the AMEs with the density function $\pi(s)$ obtained
via numerical simulations of the continuous-state model. Indeed, the
probability mass function for $\rho_{i}$ gives an accurate approximation to
the continuous density function $\pi(s)$ (Fig. 1c). In Figs. S1 and S3 in SM,
we show that the discrete approximation with $n=6$ still maintains its
accuracy when $\theta>0$ and the responsiveness is more “progressive” (i.e.,
$f$ is concave, Fig. 9a) or “conservative” (i.e., $f$ is convex, Fig. 9c). As
expected, the probability mass function converges to the continuous density
function $\pi$ as one increases the number of states $n$ (Fig. S1).

On the other hand, when the response function is given by
$\widetilde{\mathcal{F}}_{\bm{\mathbf{s}}}$, the AME method does not
necessarily provide a good approximation. The AME method works well when
$\eta<1$ and $f$ is concave, where the threshold values are skewed toward the
origin as illustrated in Fig. 9a, but otherwise it may incorrectly predict the
shares of each state (Fig. S4). This suggests that for the continuous model to
be well approximated, the discretized thresholds need to be sufficiently
densely distributed near $\theta(=\theta_{0})$.

##  III A model of financial fire sales

Figure 2: Schematic of fire sales through overlapping portfolios. (a) Asset-
bank bipartite network. The asset prices are assumed to be $1$ in the initial
state. (b) Cascade of fire sales initiated by a bank failure. The asset held
by a defaulted bank will be liquidated and its price falls to 0.6 (top). Then,
the price fall could cause other banks to fail due to capital loss (bottom),
triggering further asset liquidations.

In this section, we construct a model of financial fire sales as a dynamical
process on bipartite networks, where _asset nodes_ take continuous states
(i.e., price levels) while _bank nodes_ take binary states $\\{0,1\\}$ (i.e.,
solvent or insolvent). The asset nodes and bank nodes are connected with each
other by undirected and unweighted edges, while there is no edge within the
same group of nodes, forming a bipartite network. The structure of the
bipartite network represents the asset holdings, or portfolios, of banks (Fig.
2a). For simplicity, the initial price of each asset is set to $1$ throughout
the analysis.

The dynamic of contagion works as follows: A bank defaults when the value of
its assets falls below a given threshold. In turn, the value of an asset
depends on how many of the banks connected to the asset have defaulted (the
higher the fraction of defaults, the lower the value of the asset). We will
now describe in details the model for the case of continuous-price and its
multistate approximation.

###  III.1 Continuous-price model

We first describe a continuous-price model in which asset prices are given by
real values in $[0,1]$. To specify the correspondence between asset prices and
the fraction of defaulted banks, we consider a simple continuous price
function of the form [4]:

$\displaystyle
p\left(\frac{n_{1}}{h}\right)=\left[1-\left(\frac{n_{1}}{h}\right)^{\alpha}\right]^{\frac{1}{\alpha}},\;\;\alpha>0,$
(9)

where $h$ denotes the degree of an asset node, and $n_{1}$ is the number of
bank nodes in state 1 (i.e., insolvent banks). The price function (9) captures
the responsiveness of asset prices to liquidation events; the more banks fail,
the lower the price of the assets held by the failed banks since those assets
would be liquidated (i.e., sold in the market) (Fig. 2b, _top_). Eq. (9) has
some desirable properties: $p^{\prime}<0$, $p(0)=1$ and $p(1)=0$ (Fig. 3).
That is, the price of an asset is $1$ if there are no failed banks and $0$ if
all banks that hold the asset fail. The elasticity of asset price to the
fraction of failed banks, defined by $-d\log p/d\log(n_{1}/h)$, is given by
$x^{\alpha}/(1-x^{\alpha})$, where $x\equiv n_{1}/h$. We assume $\alpha<1$ to
focus on situations in which thresholds are skewed toward the origin, since
otherwise the discretization approach would not work well as explained in the
previous section.

Figure 3: Discretization of asset prices. $n=6$ and $\alpha=0.6$.

The total loss of bank $b$ due to price falls, denoted by $L_{b}$, is then
given by

$\displaystyle L_{b}=\sum_{a}A_{ba}(1-p_{a}),$ (10)

where $A=(A_{ba})$ denotes the adjacency matrix: $A_{ba}=1$ if bank $b$ holds
asset $a$, and $A_{ba}=0$ otherwise. $p_{a}\in[0,1]$ is the price of asset
$a$. Given the loss $L_{b}$, the state of bank $b$ shifts according to the
following threshold rule:

$\displaystyle\mathcal{F}(0\to 1)$ $\displaystyle=\begin{cases}1&\text{if
}\;\;\frac{L_{b}}{k}>\theta^{\rm bank},\\\ 0&\text{otherwise},\end{cases}$
(11)

where $\theta^{\rm bank}\in(0,1)$ denotes the threshold of the fractional loss
above which the bank goes bankrupt (Fig. 2b, _bottom_) [7, 8, 14]. The
threshold $\theta^{\rm bank}$ can also be interpreted as the capital-asset
ratio in the balance sheet of a bank, where a bank is forced to sell assets if
the total loss $L_{b}$ exceeds the amount of capital $k\theta^{\rm bank}$ [4,
5, 6, 14]. Since we assume that there is no possibility of recovery in the
state of banks [35, 8, 9], we have $\mathcal{F}(1\to 1)=1$.

###  III.2 Multistate contagion on bipartite networks

Now we develop a model of multistate contagion as an approximation to the
continuous-price model. Let $k$ denote the degree of a bank node, and let
$\ell\in\\{0,\ldots,n-1\\}$ denote a state of an asset, where $\ell/(n-1)$
represents the size of price fall for the assets in state $\ell$. The prices
of the assets held by a bank are encoded in vector
$\mathbf{m}=(m_{0},m_{1},\ldots,m_{n-1})^{\top}$, where $m_{\ell}$ denotes the
number of assets in state $\ell$, and
$|\mathbf{m}|=\sum_{\ell=0}^{n-1}m_{\ell}=k$. The states of banks holding an
asset are encoded in vector $\mathbf{n}=(n_{0},n_{1})^{\top}$, where $n_{0}$
and $n_{1}$ denote the numbers of solvent and insolvent banks, respectively,
and $|\mathbf{n}|=n_{0}+n_{1}=h$.

####  III.2.1 Bank nodes

A bank node takes one of the binary states $0$ (solvent) or $1$ (insolvent),
depending on the prices of the assets that the bank has. The state-transition
rate for bank nodes is given by

$\displaystyle F_{\mathbf{m}}(0\to 1)$ $\displaystyle=\begin{cases}1&\text{if
}\;\;\frac{L(\mathbf{m})}{k}>\theta^{\rm bank},\\\
0&\text{otherwise},\end{cases}$ (12) $\displaystyle\text{where
}\;\;\;L(\mathbf{m})$ $\displaystyle=\frac{\sum_{\ell=1}^{n-1}\ell
m_{\ell}}{n-1}\in[0,k].$ (13)

$L(\mathbf{m})$ represents the total loss that a bank incurs due to a decline
in asset prices. The threshold condition (12) corresponds to Eq. (11) in the
continuous-price model. Note that the loss from an asset price, $\ell/(n-1)$,
takes (discrete) values between $0$ and $1$, so that $0\leq L(\mathbf{m})\leq
k$ and $0\leq L(\mathbf{m})/k\leq 1$.

####  III.2.2 Asset nodes

The state-transition rate for asset nodes facing a neighbor profile
$\mathbf{n}$ is given by

$G_{\mathbf{n}}(\ell\rightarrow\ell^{\prime})=\begin{cases}1&\text{if
}\;\theta_{\ell^{\prime}-1}^{\rm asset}<P^{\rm
asset}(\mathbf{n})\leq\theta_{\ell^{\prime}}^{\rm asset}\\\
0&\text{otherwise}\end{cases},$ (14)

for $1\leq\ell^{\prime}\leq n-1$, where $P^{\rm asset}(\mathbf{n})=n_{1}/h$ is
the fraction of insolvent banks. $\theta_{\ell^{\prime}-1}^{\rm asset}$ and
$\theta_{\ell^{\prime}}^{\rm asset}$ are the threshold values between which
the asset is in state $\ell^{\prime}$. The threshold condition (14) indicates
that an asset price changes in a discrete manner in accordance with the
fraction of insolvent banks among all the asset holders. For
$\ell^{\prime}=0$, we have

$G_{\mathbf{n}}(\ell\rightarrow 0)=\begin{cases}1&\text{if }\;P^{\rm
asset}(\mathbf{n})\leq\theta_{0}^{\rm asset}\\\ 0&\text{otherwise}\end{cases}$
(15)

This suggests that when $\theta_{0}^{\rm asset}=0$, an asset price will remain
unchanged if and only if there are no failed banks that hold the asset. Note
that when $n=2$, the model reduces to a model of binary-state contagion on
bipartite networks [36, 37].

By implementing a discretization such that the price of assets in state-$\ell$
is equal to $1-\ell/(n-1)$, the threshold values are given by

$\theta_{\ell}^{\rm
asset}=\left[1-\left(1-\frac{\ell}{n-1}\right)^{\alpha}\right]^{\frac{1}{\alpha}},\;\ell=0,\ldots,n-1.$
(16)

Note that $\theta_{0}^{\rm asset}=0$ and $\theta_{n-1}^{\rm asset}=1$ (Fig.
3). Clearly, the set of threshold values in the limit of $n\to\infty$ leads to
$\\{\theta:\theta\in[0,1]\\}$, in which case the asset prices are determined
by the continuous price function (9). Thus, the continuous-price model is also
interpreted as a continuous-threshold model as long as the price function is
continuous.

It should be noted that due to the discretization of thresholds, the
multistate cascade process only provides an approximation to the “true”
dynamics that would be observed if the asset price would follow the original
price function (9). Nonetheless, we will show in section IV.1 that the
simulated cascade dynamics obtained with the continuous price function can be
well replicated in the discrete multistate model for a moderately large $n$.

###  III.3 Approximate master equations

Let $x^{i}_{k,\mathbf{m}}(t)$ denote the fraction of $k$-degree banks that are
in state $i$ and face asset-price profile $\mathbf{m}$ at time $t$. For each
degree $k$, we have $\sum_{l=0}^{n-1}m_{\ell}=k$. The expected fraction of
banks in state $i$ at time $t$ is given by

$\displaystyle\rho_{i}^{\rm
bank}(t)=\left\langle\sum_{|\mathbf{m}|=k}x^{i}_{k,\mathbf{m}}(t)\right\rangle\stackengine{11.5pt}{}{$\\!\scriptstyle
k$}{U}{l}{F}{F}{L},$ (17)

where $\langle\cdot\rangle_{k}$ denotes the average over $k$ with degree
distribution $p_{k}^{\rm bank}$.

For asset nodes, we define $y^{\ell}_{h,\mathbf{n}}(t)$ as the fraction of
$h$-degree assets that are in state $\ell$ and face asset holders’ profile
$\mathbf{n}$. The expected fraction of assets in state $\ell$ at time $t$
leads to

$\rho_{\ell}^{\rm
asset}=\left\langle\sum_{|\mathbf{n}|=h}y^{\ell}_{h,\mathbf{n}}(t)\right\rangle\stackengine{11.5pt}{}{{\hbox{\\!\scriptstyle
h}}}{U}{l}{F}{F}{L},$ (18)

where $\langle\cdot\rangle_{h}$ denotes the average over $h$ with degree
distribution $p^{\rm asset}_{h}$.

By employing the AME formalism, the dynamics of $x^{i}_{k,\mathbf{m}}$ and
$y^{l}_{h,\mathbf{n}}$ are respectively described by the following
differential equations:

$\displaystyle\frac{dx^{i}_{k,\mathbf{m}}}{dt}$ $\displaystyle=$
$\displaystyle-\delta_{i0}F_{\mathbf{m}}(0\rightarrow
1)x^{0}_{k,\mathbf{m}}+\delta_{i1}F_{\mathbf{m}}(0\rightarrow
1)x^{0}_{k,\mathbf{m}}$ (19)
$\displaystyle-\sum_{\ell=0}^{n-1}\sum_{\ell^{\prime}>\ell}m_{\ell}\beta^{i}(\ell\rightarrow\ell^{\prime})x^{i}_{k,\mathbf{m}}$
$\displaystyle+\sum_{\ell=0}^{n-1}\sum_{\ell^{\prime}<\ell}(m_{\ell^{\prime}}+1)\beta^{i}(\ell^{\prime}\rightarrow\ell)x^{i}_{k,\mathbf{m}-\mathbf{e}_{\ell}+\mathbf{e}_{\ell^{\prime}}},$
$\displaystyle\frac{dy^{\ell}_{h,\mathbf{n}}}{dt}$ $\displaystyle=$
$\displaystyle-\sum_{\ell^{\prime}>\ell}G_{\mathbf{n}}(\ell\rightarrow\ell^{\prime})y^{\ell}_{h,\mathbf{n}}+\sum_{\ell^{\prime}<\ell}G_{\mathbf{n}}(\ell^{\prime}\rightarrow\ell)y^{\ell^{\prime}}_{h,\mathbf{n}}$
(20) $\displaystyle-n_{0}\gamma^{\ell}(0\rightarrow 1)y^{\ell}_{h,\mathbf{n}}$
$\displaystyle+(n_{0}+1)\gamma^{\ell}(0\rightarrow
1)y^{\ell}_{h,\mathbf{n}-\mathbf{e}_{1}+\mathbf{e}_{0}},$

where $\delta_{ij}$ is the Kronecker delta. The link transition rates
$\beta^{i}(\ell\rightarrow\ell^{\prime})$ and $\gamma^{\ell}(i\rightarrow j)$
are given by

$\displaystyle\beta^{i}(\ell\rightarrow\ell^{\prime})$
$\displaystyle=\frac{\langle\sum_{|\mathbf{n}|=h}n_{i}y^{\ell}_{h,\mathbf{n}}(t)G_{\mathbf{n}}(\ell\rightarrow\ell^{\prime})\rangle_{h}}{\langle\sum_{|\mathbf{n}|=h}n_{i}y^{\ell}_{h,\mathbf{n}}(t)\rangle_{h}},$
(21) $\displaystyle\gamma^{\ell}(i\rightarrow j)$
$\displaystyle=\frac{\langle\sum_{|\mathbf{m}|=k}m_{\ell}x^{i}_{k,\mathbf{m}}(t)F_{\mathbf{m}}(i\rightarrow
j)\rangle_{k}}{\langle\sum_{|\mathbf{m}|=k}m_{\ell}x^{i}_{k,\mathbf{m}}(t)\rangle_{k}}.$
(22)

Eqs. (19) and (20) are obtained in the same way as the standard AMEs (Eq. 7).
Eq. (19) describes the dynamics of bank states while Eq. (20) represents the
dynamics of the share of state-$\ell$ assets. Since there is no edge within
the same group, the states of neighbors for bank nodes (resp. asset nodes)
correspond to the price of assets that the bank has (resp. the solvency of the
asset-holding banks). $\beta^{i}(\ell\rightarrow\ell^{\prime})$ thus denotes
the probability of an asset held by a state-$i$ bank changing its state from
$\ell$ to $\ell^{\prime}$. Analogously, $\gamma^{\ell}(i\rightarrow j)$
denotes the chance that the state of a bank holding state-$\ell$ assets shifts
from $i$ to $j$. Combining Eqs. (19) and (20), the total number of
differential equations to be solved is
$2\sum_{k=0}^{k_{\max}}\binom{k+1}{k}+n\sum_{h=0}^{h_{\max}}\binom{n+h-1}{h}$.

Figure 4: Comparison between theory and simulation. (a) Dynamical paths of
$\rho_{1}^{\rm bank}(t)$ obtained by the AME method (lines) and simulation
(symbols). We set $z_{\rm bank}=4$, $N=5000$, and $M=5000$. (b) Steady-state
fraction of insolvent banks against the bank degree. (c) The average fraction
of assets in each state. (d) Distribution of asset prices calculated by the
AME (blue bar) and numerical simulations in the continuous-price model (black
line). Density of the continuous prices is obtained by the kernel density
estimation with bandwidth $0.1$. Unless otherwise noted, we set
$k_{\max}=\min\\{k^{\prime}:\sum_{k=0}^{k^{\prime}}z_{\rm
bank}^{k}e^{-z}/k!\geq 0.99\\}$, $\rho_{1}^{\rm bank}(0)=10^{-3}$,
$\alpha=0.6$, $\theta^{\rm bank}=0.2$, $N=10^{4}$, $M=10^{4}$, $T=60$ and
$dt=0.01$. The average is taken over 40 runs.

##  IV Results

###  IV.1 Convergence to the continuous-state equilibrium

Now we calculate the average size of default cascades $\rho_{1}^{\rm bank}(t)$
based on Eqs. (19) and (20), assuming that banks and asses are connected
uniformly at random; bank degrees follow a Poisson distribution with mean
$z_{\rm bank}$, and a bank with degree $k$ is connected to an asset with
probability $k/M$, where $M$ is the total number of asset nodes. The number of
edges is given by $Nz_{\rm bank}$, where $N$ is the number of banks. Since the
presence of multiedges can be ignored as long as $z_{\rm bank}/M$ is
sufficiently small, the degree distribution for assets is also Poissonian with
mean $Nz_{\rm bank}/M$.

Fig 4a shows that simulated paths of the average fraction of insolvent banks
are well replicated by the corresponding AME solutions. We find that a global
cascade can occur only in a certain range of network connectivity captured by
$z_{\rm bank}$ (Fig. 4b), which is consistent with many studies on binary-
state cascades [17, 7, 10, 20, 14]. Note that the cascade region gradually
shrinks as the number of states $n$ increases since the discretized threshold
values $\\{\theta_{\ell}^{\rm asset}\\}$ tend to rise on average (Fig. 3). In
particular, when $n=6$, the the steady-state fraction $\rho_{1}^{\rm bank}(T)$
calculated by the AME (black solid in Fig. 4a and b) well matches the
corresponding steady state in the continuous-state model (purple cross),
indicating that $\rho_{1}^{\rm bank}(t)$ converges to the continuous
equilibrium as $n$ increases. That is, the solution to the continuous-state
model can be accurately replicated with a finite set of discretized states as
long as $n$ is moderately large. Our multistate model thus provides an
analytical foundation to the previously proposed numerical models of financial
fire sales with continuous prices [15, 5, 14, 16].

In addition to the average fraction of defaulted banks, the share of assets in
each state and even the distribution of continuous asset prices are
approximated by the AME when the number of states is moderately large (Fig. 4c
and d). The figures show that the price distribution drastically changes from
$\rho_{0}^{\rm asset}\approx 1$ to $\rho_{n-1}^{\rm asset}\approx 1$ around
$t=12$, after which most of the asset prices drop to $0$.

Figure 5: Cascade region obtained by the AMEs and simulated continuous-price
model. (a) $\alpha$ vs. $z_{\rm bank}$, and (b) $\alpha$ vs. $\theta^{\rm
bank}$. We set $n=6$ for the AME method. See Fig. 4 for the other parameter
values.

Fig. 5 presents a comparison of the cascade regions obtained by theory and
simulation under different parameter combinations. First, the larger the
elasticity parameter $\alpha$, the smaller the cascade region. Generally, the
average of discretized threshold values increases as $\alpha$ rises, thereby
shrinking the cascade region (Eq. 16). Second, the effect of network
connectivity $z_{\rm bank}$ is twofold, as is well known in the literature on
threshold cascades; a network needs to be moderately sparse for a global
cascade to occur since there must be routes through which influence spreads
and, at the same, the average degree needs to be sufficiently low so that
influence from a neighbor would not be diluted [17, 7].

###  IV.2 Preventing cascades through policy intervention

In this section, we explore how cascades of fire sales could be avoided by
intervention. Here, we consider policy interventions in which the
government/financial authority may operate in the market to reduce the
probability of asset prices declining or to prevent banks from going bankrupt.
In practice, the former type of policies has been implemented extensively
through massive asset-purchasing programs introduced in response to the global
financial crisis of 2007–2009 [38, 39, 40]. Along with this, defaulting banks
have been occasionally rescued by the governments’ capital injection (e.g.,
purchasing the banks’ stocks) over the past decades [41, 42, 43]. Within our
analytical framework, each of these policies may be represented as an
alteration to the state-transition probabilities in $G_{\mathbf{n}}$ or
$F_{\mathbf{m}}$.

To analyze the effect of an “asset-purchasing policy,” we introduce a
probability $q_{\rm asset}$ that an asset price does not change even if a
sufficient fraction of its asset holders newly default. That is, with
probability $1-q_{\rm asset}$, the state of an asset shifts as is specified by
Eq. (14), and with probability $q_{\rm asset}$, the asset’s state is kept
unchanged regardless of the state of the asset holders. Analogously, we can
also consider another policy tool that could prevent banks from being
defaulted. Such a “bank-rescue policy” is represented by a probability $q_{\rm
bank}$ that the state of a bank is kept unchanged regardless of the asset
prices that the bank has (see Appendix B for the specification of the modified
transition probabilities). Clearly, setting $q_{\rm asset}=q_{\rm bank}=0$
recovers the baseline model in which there is no policy intervention.

Figure 6: Effect of policy intervention on the fraction of defaulted banks.
(a) Asset-purchasing policy and (b) bank-rescue policy. $q_{\rm asset}$ and
$q_{\rm bank}$ respectively represent the probabilities that the price of an
asset and the state of a bank are kept unchanged due to policy intervention.

We find that both policies have a similar effect on cascades (Fig. 6); the
possibility of policy intervention delays the onset of fire sales while the
terminal state attained around $t\approx 100$ is unaffected (i.e.,
$\rho_{1}^{\rm bank}\approx 1$). In the process of propagation, the cascade
region naturally shrinks as $q_{\rm asset}$ increases at a given point in time
(Fig. S5). This suggests that a probabilistic policy intervention may delay
the propagation of fire sales, but eventually the undesirable steady state
will be attained.

###  IV.3 Application to the ETF market

Figure 7: Bipartite network of ETF portfolios. (a) Schematic of the bipartite
network between funds and assets. For instance, the ETF labeled “IXN” holds
the stocks of Apple, Microsoft, and TSMC. (b) Visualization of the actual ETF
network, where the number of funds $N$ is $212$ (light blue), and the number
of assets $M$ is $810$ (pale red). (c) Complementary cumulative distribution
function (CCDF) for the asset degrees.  Figure 8: Comparison between theory
and simulation for the US ETF portfolios. $\rho_{1}^{\rm ETF}$ and
$\theta^{\rm ETF}$ correspond to $\rho_{1}^{\rm bank}$ and $\theta^{\rm bank}$
in the baseline model, respectively. Simulated results for a continuous-price
model are compared with the corresponding AME solutions with $n=6$. (a)
Dynamical path of the fraction of collapsed ETFs and (b) the cascade region.
The initial seed fractions $\rho_{1}^{\rm ETF}(0)$ are set at $0.05$ and $1/N$
in panels a and b, respectively. In panel a, we set $\alpha=0.6$ and
$\theta^{\rm ETF}=0.04$. In panel b, the color denotes the average over 40
runs for $T=20$.

In this section we examine the possibility of a global cascade occurring in a
real-world financial network. We use data on the asset portfolios of exchange-
traded funds (ETFs) listed in exchanges in the US (e.g., NASDAQ, NYSE Arca).
The ETF data are collected from the website of Monex Inc. [44]. We consider a
bipartite network in which there are two types of nodes, ETFs (i.e., fund
nodes) and stocks (i.e., asset nodes). A fund node and an asset node are
connected by an undirected and unweighted edge if the asset is held by the ETF
(Fig. 7a). The network data is available in Github at [45]. Due to the limited
availability of portfolio information, we use the top 10 assets for each ETF,
in terms of the share in the portfolio, in constructing the bipartite network.
To capture systemic risk in the ETF market, rather than the risk of each
individual asset, we combine several individual stocks that apparently belong
to the same group (e.g., BHP Group Ltd and BHP Group Plc). Therefore, each
fund node is connected to at most ten asset nodes (i.e., $k\leq 10$ for all
fund nodes). We focus on the largest connected component, where the numbers of
the fund nodes and the asset nodes are $N=212$ and $M=810$, respectively (Fig.
7b). We find that the asset degrees are quite heterogeneous and the degree
distribution is heavy-tailed (Fig. 7c).

In the ETF network, fund nodes correspond to bank nodes in the baseline model,
so we assume that an ETF will be liquidated (i.e., collapsed) if its loss
exceeds a certain fraction $\theta^{\rm ETF}$, in the same way that a bank
defaults according to the threshold condition (12). Fig. 8 compares the
solution of the AME and the corresponding simulation results in the
continuous-price model. In the AME method, we use the actual degree
distributions of the ETF network in calculating the fraction of collapsed
ETFs, denoted by $\rho_{1}^{\rm ETF}$. The dynamical path of $\rho_{1}^{\rm
ETF}(t)$ suggested by the AME method well matches the simulated values (Fig.
8a) while the network structure is not random as assumed in the baseline
model.

On the other hand, since the AME method is developed for sufficiently large
networks, there are necessarily some discrepancy between the AME solution and
simulation results stemming from the finite-size effect. The finite-size
effect is also visible in Fig. 8b, which shows the cascade region for
different values of $\alpha$ and $\theta^{\rm ETF}$. Nevertheless, simulated
cascades in the continuous-price model are well predicted by the discrete-
state AME method with $n=6$.

##  V Conclusion

We develop an analytical framework for the continuous-state model of complex
contagion which can be extended to describe the dynamics of financial fire
sales with continuous prices. While the discretized model involves a limited
number of states, its AME solution well matches the continuous-state
equilibrium as long as the number of possible states is moderately large. The
analytical tractability allows us to examine various properties, such as the
size of cascade region and the effects of policy interventions, without
running massive numerical simulations.

There are still some issues to be addressed in future research. First, while
the model is based on a system of differential equations, the cascade
condition is not obtained in an analytical manner as is done in many binary-
state models [17, 18, 19, 10, 20]. We in fact found that the maximum
eigenvalue of the Jacobian of the differential equations is not necessarily
informative in predicting the cascade region. Arguably, this is because the
system is highly nonlinear so that a local linearization around the initial
point may not accurately capture the stability of the system. Second, since
our method describes the collective dynamics that would materialize on
average, it is not possible to theoretically identify important nodes in the
network. In practice, however, there is a substantial need for identifying
systemically important banks in terms of their possible impact on financial
cascades. Finally, the idea that a discrete multistate contagion model can
provide an approximation of continuous-state contagion could also be utilized
in other contexts, such as opinion dynamics and cascades of load [46, 47, 48].
We hope that our work will stimulate further research in these directions.

## Acknowledgments

T. K. acknowledges financial support from JSPS KAKENHI 19H01506, 20H05633 and
22H00827. T. O. acknowledges financial support from JSPS KAKENHI 19K14618 and
19H01506. We would like to thank Tatsuro Kawamoto and Yoshitaka Ogisu for
constructing the ETF data set, and James Gleeson and Takehisa Hasegawa for
valuable comments.

##  Appendix A Discretization of threshold values

In the continuous-state threshold model with $\theta\geq 0$, the response
function can be expressed as:

$\displaystyle\mathcal{F}_{\bm{\mathbf{s}}}=\begin{cases}\left(\frac{m/k-\theta}{1-\theta}\right)^{\eta}\;{\text{
if }}\;\frac{m}{k}>\theta\\\ 0\;\text{ otherwise},\end{cases}$ (23)

where $f(m/k)=\left(\frac{m/k-\theta}{1-\theta}\right)^{\eta}$ represents the
responsiveness to the neighbors’ states. Note that $f$ satisfies $f(\theta)=0$
and $f(1)=1$. $\eta\geq 0$ captures the strength parameter; when $\eta<1$, the
response is “progressive” in that the slope of $f(m/k)$ is larger than that in
the case of $\eta=1$ when $m/k$ is close to $\theta$ (Fig. 9a and b), while
the response becomes “conservative” when $\eta>1$ (Fig. 9c). Note that the
Watts model of binary-state cascades corresponds to the most progressive
response function where $\eta=0$.

Figure 9:  Specification of threshold values $\theta_{j}$ for $n=5$. The
responsiveness in the continuous response function is given by
$f(m/k)=\left(\frac{m/k-\theta}{1-\theta}\right)^{\eta}$ (Eq. 23). For given
$\eta$ and $\theta$, discretized thresholds are determined by Eq. (24), where
each interval size for the responsiveness is given by
$f(\theta_{j})-f(\theta_{j-1})=\frac{1}{n-1}$ and is common for all
$j=1,\ldots,n-1$. $\theta(=\theta_{0})$ is set at $0.1$, while the values of
$\eta$ are (a) 0.5, (b) 1, and (c) 2. The distribution of $\\{\theta_{j}\\}$
is skewed to the right as $\eta$ rises, meaning that the response to the
neighbors’ states becomes more conservative.

In the discrete-state model, the threshold values corresponding to the above
response function are given by

$\displaystyle\theta_{j}$
$\displaystyle=\theta+(1-\theta)\left(\frac{j}{n-1}\right)^{\frac{1}{\eta}},$
$\displaystyle\;\;j=0,\ldots,n-1,\;0<\eta<\infty,$ (24)

where $\theta$ is the threshold given in Eq. (23) and is equal to
$\theta_{0}$. This specification ensures that each interval size between two
adjacent responsiveness values is common:
$f(\theta_{j})-f(\theta_{j-1})=\frac{1}{n-1}$, for all $j=1,\ldots,n-1$. Fig.
9 illustrates three cases: progressive, neutral and conservative responses.

We can also consider an alternative nonlinear response function based on the
sum of neighbors’ states:

$\displaystyle\widetilde{\mathcal{F}}_{\bm{\mathbf{s}}}=\begin{cases}\left(\frac{\sum_{j}s_{j}/k-\theta}{1-\theta}\right)^{\eta}\;{\text{
if }}\;\frac{\sum_{j}s_{j}}{k}>\theta\\\ 0\;\text{ otherwise}.\end{cases}$
(25)

Fig. S4 reveals that the dynamical paths predicted by the AME method well
match the simulated ones when $\eta<1$. This suggests that the AME method
provides a good approximation when $\eta<1$, in which case threshold values
are skewed to the left. When $\eta\geq 1$, the AME method still predicts that
a cascade occurs while simulation results do not (Fig. S4, _middle_ and
_bottom_).

##  Appendix B Transition probabilities in the presence of policy intervention

In the presence of an asset-purchasing policy, the transition matrix
$G_{\mathbf{n}}$ in the baseline model (Eq.(14)) is modified since the policy
alters the probability of an asset price declining. For
$\ell,\ell^{\prime}\geq 1$, we have

$\displaystyle
G_{\mathbf{n}}(\ell\rightarrow\ell^{\prime})=\begin{cases}q_{\rm
asset}&\text{if }\;\ell\notin\\{\ell:\theta_{\ell-1}^{\rm asset}<P^{\rm
asset}(\mathbf{n})\leq\theta_{\ell}^{\rm asset}\\}\text{ and
}\ell^{\prime}=\ell\\\ 1-q_{\rm asset}&\text{if
}\;\ell\notin\\{\ell:\theta_{\ell-1}^{\rm asset}<P^{\rm
asset}(\mathbf{n})\leq\theta_{\ell}^{\rm asset}\\}\text{ and
}\ell^{\prime}\in\\{\ell^{\prime}:\theta_{\ell^{\prime}-1}^{\rm asset}<P^{\rm
asset}(\mathbf{n})\leq\theta_{\ell^{\prime}}^{\rm asset}\\}\\\ 1&\text{if
}\;\ell\in\\{\ell:\theta_{\ell-1}^{\rm asset}<P^{\rm
asset}(\mathbf{n})\leq\theta_{\ell}^{\rm asset}\\}\text{ and
}\ell^{\prime}=\ell\\\ 0&\text{otherwise}\end{cases}.$ (26)

For $\ell=0$ or $\ell^{\prime}=0$, we have

$\displaystyle G_{\mathbf{n}}(0\rightarrow\ell^{\prime})=\begin{cases}q_{\rm
asset}&\text{if }\;P^{\rm asset}(\mathbf{n})>\theta_{0}^{\rm asset}\text{ and
}\ell^{\prime}=0\\\ 1-q_{\rm asset}&\text{if
}\;\ell^{\prime}\in\\{\ell^{\prime}:\theta_{\ell^{\prime}-1}^{\rm
asset}<P^{\rm asset}(\mathbf{n})\leq\theta_{\ell^{\prime}}^{\rm
asset}\\}\text{ and }\ell^{\prime}\geq 1\\\ 0&\text{otherwise}\end{cases},$
(27) $\displaystyle G_{\mathbf{n}}(\ell\rightarrow 0)$
$\displaystyle=\begin{cases}1&\text{if }\;P^{\rm
asset}(\mathbf{n})\leq\theta_{0}^{\rm asset}\\\
0&\text{otherwise}\end{cases}.$ (28)

The probability $q_{\rm asset}$ thus represents the chance that an asset-
purchasing policy is implemented, where setting $q_{\rm asset}=0$ recovers the
baseline model without policy intervention.

In the presence of a bank-rescue policy, the transition matrix
$F_{\mathbf{m}}$ is given by

$\displaystyle F_{\mathbf{m}}(0\to 1)$ $\displaystyle=\begin{cases}1-q_{\rm
bank}&\text{if }\;\;\frac{L(\mathbf{m})}{k}>\theta^{\rm bank},\\\
0&\text{otherwise},\end{cases}$ (29) $\displaystyle F_{\mathbf{m}}(0\to 0)$
$\displaystyle=\begin{cases}q_{\rm bank}&\text{if
}\;\;\frac{L(\mathbf{m})}{k}>\theta^{\rm bank},\\\
0&\text{otherwise},\end{cases}$ (30)

where $q_{\rm bank}$ denotes the chance that the government rescues the
defaulting bank. Clearly, $q_{\rm bank}=0$ recovers the baseline model in
which there is no policy intervention.

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Supplemental Material

“Financial fire sales as continuous-state complex contagion”

Tomokatsu Onaga, Fabio Caccioli and Teruyoshi Kobayashi

Figure S1:  State distribution in the discrete- and continuous threshold
models for different number of states $n$: response function (23). In the
discretized model, we use the AMEs to calculate the fraction of nodes in state
$i\in\\{0,\ldots,n-1\\}$, denoted by $\rho_{i}$ (blue bar). In the continuous-
state model, the density $\pi(s)$ for state $s\in[0,1]$ is calculated by the
kernel density estimation with bandwidth $0.1$ based on numerical simulations
(black line). The index of discretized states is normalized by $n-1$ so that
the states are distributed on $[0,1]$. Seed fraction is determined such that
the fraction $0.05$ of nodes are in the “most-active” state (i.e., $i=n-1$ or
$s=1$) while $0.95$ of nodes are in state 0. We run 40 simulations on Erdős-
Rényi networks with $N=3000$, $z=3$, $\theta=0.18$, and $\eta=1$, where
$\theta_{j}=\theta+(1-\theta)j/(n-1)$. Clearly, the larger the number of
states, the better the matching between the simulation in the continuous-state
model and its discrete approximation. Figure S2: Dynamical path of the share
of each state for different responsiveness $\eta$. The responsiveness is (a)
progressive $(\eta<1)$, (b) neutral $(\eta=1)$, and (c) conservative
$(\eta>1)$. We run 40 simulations on Erdős-Rényi networks with $N=3000$,
$z=3$, $n=6$, $\theta=0.18$, where $\theta_{j}$ is given by Eq. (24). See the
caption of Fig. S1 for the seed size.  Figure S3: State distribution for
different responsiveness $\eta$: response function (23). We run 40 simulations
on Erdős-Rényi networks with $N=3000$, $z=3$, $n=6$, $\theta=0$, where
$\theta_{j}=\left(\frac{j}{n-1}\right)^{\frac{1}{\eta}}$. See the caption of
Fig. S1 for the seed size. $\eta>1$ and $\eta<1$ respectively indicate
conservative and progressive responses, while $\eta=1$ is neutral.  Figure S4:
State distribution for different responsiveness $\eta$: response function
(25). We run 40 simulations on Erdős-Rényi networks with $N=3000$, $z=3$,
$n=6$, $\theta=0$, where
$\theta_{j}=\left(\frac{j}{n-1}\right)^{\frac{1}{\eta}}$. See the caption of
Fig. S1 for the seed size.

Figure S5: Cascade region under different degrees of policy intervention:
asset-purchasing policy. We set $T=60$. See the caption of Fig. 4 for the
other parameter values.

Figure S6: Cascade region under different degrees of policy intervention:
bank-rescue policy. We set $T=60$. See the caption of Fig. 4 for the other
parameter values.