# Bidirectional bond percolation model for the
spread of information in financial markets

Stefano Chiaradonna and Nicolas Lanchier This work is partially supported by
NSF grant CNS-2000792.

###### Abstract

Information is a key component in determining the price of an asset in
financial markets, and the main objective of this paper is to study the spread
of information in this context. The network of interactions in financial
markets is modeled using a Galton-Watson tree where vertices represent the
traders and where two traders are connected by an edge if one of the two
traders sells the asset to the other trader. The information starts from a
given vertex and spreads through the edges of the graph going independently
from seller to buyer with probability $p$ and from buyer to seller with
probability $q$. In particular, the set of traders who are aware of the
information is a (bidirectional) bond percolation cluster on the Galton-Watson
tree. Using some conditioning techniques and a partition of the cluster of
open edges into subtrees, we compute explicitly the first and second moments
of the cluster size, i.e., the random number of traders who learn about the
information. We also prove exponential decay of the diameter of the cluster in
the subcritical phase.

††footnotetext: AMS 2000 subject classifications: Primary 60K35 Keywords and
phrases: Bond percolation, Galton-Watson tree, financial market, information

##  1 Introduction

A wealth of models have been developed to study financial markets, defined as
_any places or systems that provide buyers and sellers the means to trade
financial instruments_ [9]. Financial markets consist in particular of systems
where buyers and sellers interact. One class of models of financial markets
that have gained attraction are the ones based on percolation theory, the
branch of probability concerned with the statistical properties of random
clusters on graphs [3, 6]. Popular examples of percolation models used to
study financial markets are the Cont-Bouchaud market model and the Ising
model. The Cont-Bouchaud market model uses percolation on the $d$-dimensional
lattice, where each occupied site represents a trader [12]. Traders can be
partitioned into clusters of the same opinion, and the price variations of the
asset is controlled by the size of these clusters [13]. Similarly, the Ising
model is used to study the interactions taking place in financial markets due
to its ability to describe the competition between social imitation of the
model’s individual elements and the impact of private information [11].
Due to the presence of interactions between buyers and sellers, a trader’s
decision is influenced by the news received from others within the market
[15]. In particular, information has become an important factor in determining
the price of an asset. This relates to the concept of price efficiency,
defined as _the degree to which prices reflect all available information in
terms of speed and accuracy_ [10]. The value of markets depends on the amount
of information about prices, which reveals whether an investor should buy or
sell [2]. The disagreements among investors about the expected prices due to
diverse investor information is a common feature in most financial markets
[1], and these differences in the distribution of private information may
explain why informational efficiency can vary significantly [4].
The literature about the impact of information on financial markets is
copious. Edmans et al. [5] argued that decisions depend not only on the total
amount of information in prices but also on the source of the information.
Albagli et al. [1] developed a model of asset pricing to demonstrate that the
heterogeneity of information and its aggregation within a financial market are
core forces in determining asset prices. Corgnet et al. [4] proposed that the
concentration of private information across the investors is the determinant
of information efficiency. Lv and Wu [8] investigated the effect of margin
trading on information content and price-adjustment speed of price efficiency.
In particular, developing and studying mathematical models for the spread of
information in financial markets is of primary importance.
Galton-Watson random tree. Before describing how the information spreads, the
first step is to explain the topological structure of a financial market and
how the traders interact. Our model starts with an Initial Public Offering
(IPO), which is _when a company first sells its shares to the public_ [14],
and the financial market consists of buyers and sellers that consider this IPO
as their only asset of interest. The IPO sells its shares to some buyers who
then sell their shares to other buyers, and so on, and we assume that
potential buyers only purchase from a single seller and that sellers are not
currently interested in repurchasing any shares of the asset. Therefore, the
only potential buyers are those who have not already possessed the shares. In
particular, thinking of each trader as a vertex and drawing an edge between
two traders if one of the two traders sells the shares to the other trader
results in a graph with no cycles, meaning that the network of interactions
among traders exhibits a tree structure. To also account for the randomness
inherent to a financial market, we assume that the network of interactions is
a Galton-Watson tree: the graph starts from the IPO (the root of the tree) and
the random number of buyers of each seller (the number of edges starting from
each vertex going away from the root) follows a fixed distribution.
Bidirectional bond percolation. As previously mentioned, information is a key
component in the pricing of an asset. To model the spread of information
within the financial market, we assume that a trader in the already formed
random tree has new information about the price of the shares. In order to
price the shares accurately, this trader and any other traders who learn the
information in the future gather more information by inquiring their buyers
and/or their seller so that the information spreads through the edges of the
graph with some fixed probabilities. Due to the asymmetry in the seller-buyer
relationship, we assume that the information spreads from a seller to each of
her buyers independently with probability $p$ while it spreads from a buyer to
her seller with probability $q$. In mathematical terms, the resulting set of
traders who learn the information is an open cluster in a (bidirectional) bond
percolation process on the Galton-Watson tree where edges are independently
open with probability $p$ in the seller-buyer direction going away from the
root and with probability $q$ in the buyer-seller direction going towards the
root.
The main objective of this paper is to study some of the statistical
properties of the size distribution of the open cluster, i.e., the number of
traders who learn about the information. More precisely, we compute explicitly
the first and second moments of the cluster size, and prove exponential decay
of the diameter of the cluster in the subcritical phase.

##  2 Model description and main results

To define the model rigorously, let $\mathbb{T}=(V,E)$ be a realization of the
Galton-Watson tree with offspring distribution $(p_{k})$, meaning that,
independently for each vertex, there are $k$ edges starting from this vertex
and going away from the root of the tree with probability $p_{k}$. In the
terminology of branching processes, two vertices connected by an edge are
called the parent and the offspring, with the parent being the vertex closer
to the root. In the context of financial markets, the parent plays the role of
the seller while the offspring plays the role of the buyer. To ensure survival
of the tree and avoid trivialities, we assume that $p_{0}=0$. To state our
results later, we let $\mu$ and $\sigma^{2}$ be respectively the mean and the
variance of the number of offspring per individual:

$\mu=\sum_{k=1}^{\infty}\,kp_{k}\quad\hbox{and}\quad\sigma^{2}=\sum_{k=1}^{\infty}\,(k-\mu)^{2}p_{k}.$

To model the spread of information within the financial market, we use
bidirectional bond percolation on the Galton-Watson tree. More precisely, we
let $p,q\in(0,1)$ and assume that each of the edges of the tree is
independently open

$\begin{array}[]{rcl}\hbox{with probability\leavevmode\nobreak\ $p$}&\hbox{in
the direction parent $\to$ offspring}\vspace*{4pt}\\\ \hbox{with
probability\leavevmode\nobreak\ $q$}&\hbox{in the direction offspring $\to$
parent}\end{array}$

In other words, each edge is identified to two arrows. The arrow going away
from the root is open with probability $p$ whereas the arrow going towards the
root is open with probability $q$. Injecting the information at vertex $x$,
which represents the trader who first learns about the information, we are
interested in the size $S$ of the cluster of open edges $\mathscr{C}$ starting
at $x$, i.e.,

$S=\operatorname{card\,}(\mathscr{C})\quad\hbox{where}\quad\mathscr{C}=\\{y\in
V:\hbox{there is a directed open path\leavevmode\nobreak\ $x\to y$}\\}.$

In the terminology of percolation theory, we assume that a fluid injected at
vertex $x$ can only spread through the open edges so the vertices in cluster
$\mathscr{C}$ are called wet vertices. In the context of financial markets,
vertex $x$ represents the source of the information while the wet vertices are
the traders who are aware of the information.
First moment. To begin with, we look at the first moment of the cluster size
$S$, which represents the mean number of traders who are aware of the
information. Although we model the financial market using an infinite Galton-
Watson tree, we study the first moment on the truncated Galton-Watson tree
with radius $R$, defined as the subgraph of $\mathbb{T}$ induced by the
vertices at distance at most $R$ from the root of the tree, in order to also
extend a result in [7] concerned with the symmetric case $p=q$. By monotone
convergence, the first moment on the infinite tree can be deduced by simply
taking the limit as $R\to\infty$. Partitioning the set of wet vertices into
subtrees and using independence, we obtain the following theorem.

###### Theorem 1

– The conditional first moment on the tree with radius $R$ given that the
information starts at distance $r$ from the root is equal to

$E_{r}(S)=\frac{1}{1-\mu
p}\bigg{(}1+q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}(1-p)-(\mu
p)^{R-r+1}\bigg{(}\frac{1-pq(1+(\mu-1)(\mu pq)^{r})}{1-\mu
pq}\bigg{)}\bigg{)}.$

It directly follows from the theorem that, in the supercritical phase $\mu
p>1$, the first moment of the cluster size on the infinite Galton-Watson tree
is infinite. In contrast, in the subcritical phase $\mu p<1$, the first moment
on the infinite tree reduces to

$E_{r}(S)=\frac{1}{1-\mu
p}\bigg{(}1+q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}(1-p)\bigg{)}.$

In addition, setting $p=q$ in the theorem gives

$E_{r}(S)=\frac{1}{1-\mu p}\bigg{(}1+p(1-p^{r})-(\mu
p)^{R-r+1}\bigg{(}\frac{1-p^{2}(1+(\mu-1)(\mu p^{2})^{r})}{1-\mu
p^{2}}\bigg{)}\bigg{)},$

which is exactly the expression found in [7, Theorem 4], but we point out
that, even though our result extends the result in [7] to the asymmetric case,
our approach leads to a much shorter and more elegant proof. More precisely,
the proof in [7] relies on a tedious combinatorial argument that consists in
counting the number of open paths of a given length starting from the source
of the information whereas our proof consists in finding the wet vertices
along the path going from the source of the information to the root of the
tree and then partitioning the cluster of wet vertices into (disjoint)
subtrees starting from each of these vertices.
Second moment. We now study the second moment of the cluster size on the
infinite tree. It follows from Theorem 1 that, on the infinite Galton-Watson
tree, all the higher moments are infinite in the supercritical phase so we
assume that $\mu p<1$. The key to computing the second moment is independence.
More precisely, writing again the cluster of wet vertices as a disjoint union
of subtrees, the second moment can be computed explicitly using that the sizes
of these subtrees are independent and identically distributed random
variables, which gives the following result.

###### Theorem 2

– Let $\mu p<1$. Then, the conditional second moment on the infinite tree
given that the information starts at distance $r$ from the root is equal to

$\begin{array}[]{rcl}\displaystyle E_{r}(S^{2})&=&\displaystyle\frac{1}{(1-\mu
p)^{2}}\ \bigg{(}1+\frac{p(1-p)\mu+p^{2}\sigma^{2}}{1-\mu
p}\bigg{)}\vspace*{8pt}\\\ &&\hskip 5.0pt+\
\displaystyle\bigg{(}1+\frac{2(1+(\mu-1)p)}{1-\mu
p}+\frac{2(\mu-1)p+p(1-p)(\mu-1)+p^{2}\sigma^{2}+(\mu-1)^{2}p^{2}}{(1-\mu
p)^{2}}\vspace*{8pt}\\\ &&\hskip 100.0pt+\
\displaystyle\frac{(p(1-p)\mu+p^{2}\sigma^{2})(\mu-1)p}{(1-\mu
p)^{3}}\bigg{)}\ q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}\vspace*{8pt}\\\ &&\hskip
5.0pt+\ \displaystyle\bigg{(}1+\frac{(\mu-1)p}{1-\mu p}\bigg{)}^{2}\
\frac{2q^{2}(1-rq^{r-1}+(r-1)q^{r})}{(1-q)^{2}}.\end{array}$

Combining Theorems 1 and 2 implies that, in the subcritical phase, the tail
distribution of the size of the cluster of wet vertices has a quadratic decay:
there exists $C=C(\mu p)<\infty$ such that

$P(S>n)\leq C/n^{2}\quad\hbox{for all}\quad n>0.$

Indeed, according to the theorems, both the first and second moments are
finite when $\mu p<1$, therefore it follows from Chebyshev’s inequality that

$P(S>n)\leq
P(|S-E(S)|>|n-E(S)|)\leq\frac{\operatorname{Var}(S)}{(n-E(S))^{2}}\leq\frac{E(S^{2})}{(n-E(S))^{2}}$

for all $n>E(S)$, which shows quadratic decay.

Exponential decay of the diameter. Another quantity of interest which also
accounts for the geometry of the set of wet vertices is the diameter of the
cluster $\mathscr{C}$ defined as the maximum graph distance between any two
wet vertices:

$\operatorname{diam}(\mathscr{C})=\max\,\\{d(x,y):x,y\in\mathscr{C}\\}.$

In this case, studying the spread of information from a dynamical point of
view starting from the highest wet vertex and moving one generation down the
tree at each time step, we can prove an exponential decay. More precisely, we
have the following theorem.

###### Theorem 3

– Let $\mu p<1$. Then, the conditional probability that the diameter is larger
than $2n$ given that the information starts at distance $r$ from the root is

$P_{r}(\operatorname{diam}(\mathscr{C})\geq 2n)\leq\frac{1-(q/\mu
p)^{r+1}}{1-(q/\mu p)}\ (\mu p)^{n}\quad\hbox{for all}\quad n>r.$

The theorem indeed implies that, in the subcritical phase $\mu p<1$, the tail
distribution of the diameter of the cluster of wet vertices, i.e., the maximum
distance between any two traders who learn about the information, decays
exponentially.

##  3 Partition into disjoint subtrees

To get ready for the proofs of Theorems 1 and 2 in the next two sections, we
first explain and study the partition of the set of wet vertices into disjoint
subtrees. More precisely, we first study the distribution of the random number
of subtrees and compute the first and second moment of the size of these
subtrees. From now on, we assume that the information starts at a vertex $x$
with $d(0,x)=r$. By spherical symmetry, the specific choice of $x$ is
unimportant as long as the vertex is at distance $r$ from the root. There is a
unique directed path

$x_{0}=0\to x_{1}\to x_{2}\to\cdots\to x_{r-1}\to x_{r}=x$

of length $r$ going from the root to vertex $x$ and we let

$D=\max\,\\{i=0,1,\ldots,r:x_{r-i}\ \hbox{is wet}\\}.$

This is the distance between the source of the information and the highest wet
vertex, and we refer the reader to Figure 1 for a picture. The following lemma
gives preliminary results about the random variable $D$ that will be useful
later to prove the theorems.

$x_{0}$$x_{1}$$x_{2}$$S(x_{1}\setminus x_{2})$$S(x_{2}\setminus
x_{3})$$S(x_{3})$$D=2$$x_{3}=x$$p$$q$

Figure 1: Picture of the partition into disjoint subtrees used to prove
Theorems 1 and 2. The black vertices represent the set of wet vertices while
the white vertices are not wet. In our example, the information starts from
$x_{3}$ and spreads up to $x_{1}$, which results in a partition of the cluster
of wet vertices into three disjoint subtrees. Starting from the source of the
information, the numbers of wet vertices in the subtrees are 3, 2 and 5,
respectively.

###### Lemma 4

– We have

$E(D)=q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}\quad\hbox{and}\quad
E(D(D-1))=\frac{2q^{2}(1-rq^{r-1}+(r-1)q^{r})}{(1-q)^{2}}.$

###### Proof.

Because the information spreads towards the root of the random tree with
probability $q$ independently through each of the edges, we have

$P(D=k)=q^{k}(1-q)\quad\hbox{for}\ k=0,1,\ldots,r-1,\quad\hbox{and}\quad
P(D=r)=q^{r}.$

In particular, using that

$\sum_{k=1}^{r-1}\,kx^{k-1}=\frac{\partial}{\partial
x}\bigg{(}\sum_{k=0}^{r-1}x^{k}\bigg{)}=\frac{\partial}{\partial
x}\bigg{(}\frac{1-x^{r}}{1-x}\bigg{)}=\frac{1-rx^{r-1}+(r-1)x^{r}}{(1-x)^{2}},$

we deduce that the first moment is given by

$\begin{array}[]{rcl}E(D)&=&\displaystyle\sum_{k=0}^{r-1}\,kq^{k}(1-q)+rq^{r}=q(1-q)\sum_{k=1}^{r-1}\,kq^{k-1}+rq^{r}\vspace*{4pt}\\\
&=&\displaystyle
q(1-q)\bigg{(}\frac{1-rq^{r-1}+(r-1)q^{r}}{(1-q)^{2}}\bigg{)}+rq^{r}=q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}.\end{array}$

Similarly, we have

$\begin{array}[]{rcl}\displaystyle\sum_{k=2}^{r-1}\,k(k-1)x^{k-2}&=&\displaystyle\frac{\partial^{2}}{\partial
x^{2}}\bigg{(}\sum_{k=0}^{r-1}x^{k}\bigg{)}=\displaystyle\frac{\partial}{\partial
x}\bigg{(}\frac{1-rx^{r-1}+(r-1)x^{r}}{(1-x)^{2}}\bigg{)}\vspace*{8pt}\\\
&=&\displaystyle\frac{2-r(r-1)x^{r-2}+2r(r-2)x^{r-1}-(r-1)(r-2)x^{r}}{(1-x)^{3}}\end{array}$

from which it follows that

$\begin{array}[]{rcl}\displaystyle
E(D(D-1))&=&\displaystyle\sum_{k=0}^{r-1}\,k(k-1)q^{k}(1-q)+r(r-1)q^{r}\vspace*{4pt}\\\
&=&\displaystyle q^{2}(1-q)\
\frac{2-r(r-1)q^{r-2}+2r(r-2)q^{r-1}-(r-1)(r-2)q^{r}}{(1-q)^{3}}\vspace*{4pt}\\\
&&\hskip 50.0pt+\ \displaystyle q^{2}\
\frac{r(r-1)q^{r-2}-2r(r-1)q^{r-1}+r(r-1)q^{r}}{(1-q)^{2}}\vspace*{4pt}\\\
&=&\displaystyle\frac{2q^{2}(1-rq^{r-1}+(r-1)q^{r})}{(1-q)^{2}}.\end{array}$

This completes the proof. ∎

Next, we define the random variables

$\begin{array}[]{rcl}S(x_{j})&=&\hbox{number of wet vertices in the subtree
starting at\leavevmode\nobreak\ $x_{j}$}\vspace*{4pt}\\\ S(x_{j}\setminus
x_{j+1})&=&\hbox{number of wet vertices in the subtree starting
at\leavevmode\nobreak\ $x_{j}$ but}\\\ &&\hbox{excluding the subtree starting
at\leavevmode\nobreak\ $x_{j+1}$.}\end{array}$ (1)

For instance, in the realization shown in Figure 1, we have

$S(x_{3})=3,\quad S(x_{2}\setminus x_{3})=2,\quad S(x_{1}\setminus
x_{2})=5,\quad S(x_{0}\setminus x_{1})=0.$

To estimate the first and second moments of $S$ later, we now compute the
first and second moments of the random variables in (1). To do this, let
$\xi_{i}=\operatorname{Bernoulli\,}(p)$ be independent, and let $Y$ be the
offspring distribution, i.e., the random variable describing the random number
of edges starting from a given vertex and going away from the root. In
particular,

$X_{+}=\xi_{1}+\xi_{2}+\cdots+\xi_{Y}\quad\hbox{and}\quad
X_{-}=\xi_{1}+\xi_{2}+\cdots+\xi_{Y-1}$

are the random variables describing the number of wet offspring of a given
vertex and the number of wet offspring of a given vertex excluding a given
offspring, respectively. The next lemma gives the first and second moments of
the first random variable in (1).

###### Lemma 5

– Let $\nu_{+}=E(X_{+})$ and $\Sigma_{+}^{2}=\operatorname{Var}(X_{+})$. Then,

$\begin{array}[]{rcl}E(S(x_{j}))&=&\displaystyle\frac{1-\nu_{+}^{R-j+1}}{1-\nu_{+}}\vspace*{4pt}\\\
E(S(x_{j})^{2})&=&\displaystyle\frac{\Sigma_{+}^{2}}{(1-\nu_{+})^{2}}\
\bigg{(}\frac{1-\nu_{+}^{2(R-j)+1}}{1-\nu_{+}}-(2(R-j)+1)\nu_{+}^{R-j}\bigg{)}+\bigg{(}\frac{1-\nu_{+}^{R-j+1}}{1-\nu_{+}}\bigg{)}^{2}.\end{array}$

For a detailed proof of this lemma, we refer to [7, Lemmas 8 and 9]. We now
compute the first and second moments of the second set of random variables in
(1).

###### Lemma 6

– Let $\nu_{-}=E(X_{-})$ and $\Sigma_{-}^{2}=\operatorname{Var}(X_{-})$. Then,

$\begin{array}[]{rcl}E(S(x_{j}\setminus
x_{j+1}))&=&1+\nu_{-}E(S(x_{j+1}))\vspace*{4pt}\\\ E(S(x_{j}\setminus
x_{j+1})^{2})&=&1+2\nu_{-}E(S(x_{j+1}))+\nu_{-}E(S(x_{j+1})^{2})+(\Sigma_{-}^{2}+\nu_{-}^{2}-\nu_{-})E(S(x_{j+1}))^{2}.\end{array}$

###### Proof.

Letting $y_{1},y_{2},\ldots,y_{X}$ be the wet offspring of $x_{j}$ other than
$x_{j+1}$,

$S(x_{j}\setminus x_{j+1})=1+\sum_{n=1}^{X}\,S(y_{n})\quad\hbox{and}\quad
X\overset{d}{=}X_{-}.$ (2)

In particular, conditioning on $X$, we get

$E(S(x_{j}\setminus
x_{j+1}))=E\bigg{(}E\bigg{(}1+\sum_{n=1}^{X}\,S(y_{n})\,\Big{|}\,X\bigg{)}\bigg{)}=1+E(X)E(S(y_{n}))=1+\nu_{-}E(S(x_{j+1})).$

Taking the square in (2)

$\begin{array}[]{rcl}S(x_{j}\setminus x_{j+1})^{2}&=&\displaystyle
1+2\,\sum_{n=1}^{X}S(y_{n})+\bigg{(}\sum_{n=1}^{X}S(y_{n})\bigg{)}^{2}\vspace*{4pt}\\\
&=&\displaystyle
1+2\,\sum_{n=1}^{X}S(y_{n})+\sum_{n=1}^{X}S(y_{n})^{2}+\sum_{n\neq
m}S(y_{n})S(y_{m})\end{array}$

then conditioning on $X$ and using independence,

$\begin{array}[]{rcl}E(S(x_{j}\setminus x_{j+1})^{2})&=&E(E(S(x_{j}\setminus
x_{j+1})^{2}\,|\,X))\vspace*{4pt}\\\
&=&1+2E(X)E(S(y_{n}))+E(X)E(S(y_{n})^{2})+E(X(X-1))E(S(y_{n}))^{2}\vspace*{4pt}\\\
&=&1+2\nu_{-}E(S(x_{j+1}))+\nu_{-}E(S(x_{j+1})^{2})+(\Sigma_{-}^{2}+\nu_{-}^{2}-\nu_{-})E(S(x_{j+1}))^{2}.\end{array}$

This completes the proof. ∎

##  4 Proof of Theorem 1 (first moment)

Throughout this section, we assume that the network of interactions consists
of the truncated Galton-Watson tree with radius $R$ obtained by removing from
the infinite tree all the vertices at distance more than $R$ from the root. To
prove Theorem 1, the first step is to compute the mean of the random variables
$X_{+}$ and $X_{-}$ defined above, which is done in the next lemma.

###### Lemma 7

– For $\nu_{\pm}$ defined as in Lemmas 5 and 6, we have $\nu_{+}=\mu p$ and
$\nu_{-}=(\mu-1)p$.

###### Proof.

Recalling the definition of $X_{\pm}$ and $Y$, and conditioning on $Y$,

$\begin{array}[]{rcl}\nu_{+}&=&E(X_{+})=E(E(X_{+}\,|\,Y))=E(E(\xi_{1}+\cdots+\xi_{Y}\,|\,Y))=E(Y)E(\xi_{n})=\mu
p\vspace*{4pt}\\\
\nu_{-}&=&E(X_{-})=E(E(X_{-}\,|\,Y))=E(E(\xi_{1}+\cdots+\xi_{Y-1}\,|\,Y))=E(Y-1)E(\xi_{n})=(\mu-1)p.\end{array}$

This completes the proof. ∎

Using the previous lemmas, we are now ready to prove Theorem 1. To begin with,
observe that, on the event $D=k$, the total number of wet vertices can be
written as

$S=S(x_{r})+\sum_{i=r-k}^{r-1}S(x_{i}\setminus
x_{i+1})=S(x_{r})+\sum_{i=1}^{k}\,S(x_{r-i}\setminus x_{r-i+1}).$ (3)

In particular, conditioning on $D$ and using Lemma 6, we get

$\begin{array}[]{rcl}E_{r}(S)&=&\displaystyle\sum_{k=0}^{r}\,E(S\,|\,D=k)\,P(D=k)\vspace*{4pt}\\\
&=&\displaystyle\sum_{k=0}^{r}\,\bigg{(}E(S(x_{r}))+\sum_{i=1}^{k}\,E(S(x_{r-i}\setminus
x_{r-i+1}))\bigg{)}P(D=k)\vspace*{4pt}\\\
&=&\displaystyle\sum_{k=0}^{r}\,\bigg{(}E(S(x_{r}))+\sum_{i=0}^{k-1}\
(1+\nu_{-}E(S(x_{r-i}))\bigg{)}P(D=k)\vspace*{4pt}\\\ &=&\displaystyle
E(S(x_{r}))+E(D)+\nu_{-}\ \sum_{k=1}^{r}\
\sum_{i=0}^{k-1}\,E(S(x_{r-i}))\,P(D=k).\end{array}$ (4)

Exchanging the two sums, and using Lemma 5, we obtain

$\begin{array}[]{l}\displaystyle\sum_{k=1}^{r}\
\sum_{i=0}^{k-1}\,E(S(x_{r-i}))\,P(D=k)=\displaystyle\sum_{i=0}^{r-1}\
\sum_{k=i+1}^{r}\,E(S(x_{r-i}))\,P(D=k)\vspace*{4pt}\\\ \hskip
40.0pt=\displaystyle\sum_{i=0}^{r-1}\,E(S(x_{r-i}))\,P(D>i)=\displaystyle\sum_{i=0}^{r-1}\,q^{i+1}\,E(S(x_{r-i}))\vspace*{8pt}\\\
\hskip
40.0pt=\displaystyle\sum_{i=0}^{r-1}\,q^{i+1}\bigg{(}\frac{1-\nu_{+}^{R-r+i+1}}{1-\nu_{+}}\bigg{)}=\displaystyle\frac{1}{1-\nu_{+}}\bigg{(}q\
\sum_{i=0}^{r-1}\,q^{i}-q\,\nu_{+}^{R-r+1}\
\sum_{i=0}^{r-1}\,(q\nu_{+})^{i}\bigg{)}\vspace*{8pt}\\\ \hskip
40.0pt=\displaystyle\frac{q}{1-\nu_{+}}\bigg{(}\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}-\nu_{+}^{R-r+1}\bigg{(}\frac{1-(q\nu_{+})^{r}}{1-q\nu_{+}}\bigg{)}\bigg{)}.\end{array}$
(5)

Combining (4) and (5), and using Lemmas 4 and 5, we deduce that

$\begin{array}[]{rcl}E_{r}(S)&=&\displaystyle\frac{1-\nu_{+}^{R-r+1}}{1-\nu_{+}}+q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}+\frac{q\nu_{-}}{1-\nu_{+}}\bigg{(}\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}-\nu_{+}^{R-r+1}\bigg{(}\frac{1-(q\nu_{+})^{r}}{1-q\nu_{+}}\bigg{)}\bigg{)}\vspace*{8pt}\\\
&=&\displaystyle\frac{1-\nu_{+}^{R-r+1}}{1-\nu_{+}}+q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}\bigg{(}\frac{1-\nu_{+}+\nu_{-}}{1-\nu_{+}}\bigg{)}+\frac{q\nu_{-}}{1-\nu_{+}}\bigg{(}-\nu_{+}^{R-r+1}\bigg{(}\frac{1-(q\nu_{+})^{r}}{1-q\nu_{+}}\bigg{)}\bigg{)}\vspace*{8pt}\\\
&=&\displaystyle\frac{1}{1-\nu_{+}}\bigg{(}1+q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}(1-\nu_{+}+\nu_{-})-\nu_{+}^{R-r+1}\bigg{(}1+q\nu_{-}\bigg{(}\frac{1-(q\nu_{+})^{r}}{1-q\nu_{+}}\bigg{)}\bigg{)}\bigg{)}.\end{array}$

Using Lemma 7 also gives $1-\nu_{+}+\nu_{-}=1-p$ and

$1+q\nu_{-}\bigg{(}\frac{1-(q\nu_{+})^{r}}{1-q\nu_{+}}\bigg{)}=\frac{1-pq(1+(\mu-1)(\mu
pq)^{r})}{1-\mu pq}.$

In conclusion,

$E_{r}(S)=\frac{1}{1-\mu
p}\bigg{(}1+q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}(1-p)-(\mu
p)^{R-r+1}\bigg{(}\frac{1-pq(1+(\mu-1)(\mu pq)^{r})}{1-\mu
pq}\bigg{)}\bigg{)},$

which completes the proof of Theorem 1.

##  5 Proof of Theorem 2 (second moment)

Throughout this section, we assume that the network of interactions consists
of the infinite Galton-Watson tree $\mathbb{T}$. To prove Theorem 2, we follow
the same strategy as for Theorem 1 and first compute the variance of the
random variables $X_{+}$ and $X_{-}$, which is done in the next lemma.

###### Lemma 8

– For $\Sigma_{\pm}^{2}$ defined as in Lemmas 5 and 6, we have

$\Sigma_{+}^{2}=p(1-p)\mu+p^{2}\sigma^{2}\quad\hbox{and}\quad\Sigma_{-}^{2}=p(1-p)(\mu-1)+p^{2}\sigma^{2}.$

###### Proof.

Conditioning on $Y$ and using the law of total variance, we get

$\begin{array}[]{rcl}\Sigma_{+}^{2}=\operatorname{Var}(X_{+})&=&E(\operatorname{Var}(X_{+}\,|\,Y))+\operatorname{Var}(E(X_{+}\,|\,Y))\vspace*{4pt}\\\
&=&E(p(1-p)Y)+\operatorname{Var}(pY)=p(1-p)\mu+p^{2}\sigma^{2}.\end{array}$

Similarly, for the variance of $X_{-}$,

$\begin{array}[]{rcl}\Sigma_{-}^{2}=\operatorname{Var}(X_{-})&=&E(\operatorname{Var}(X_{-}\,|\,Y))+\operatorname{Var}(E(X_{-}\,|\,Y))\vspace*{4pt}\\\
&=&E(p(1-p)(Y-1))+\operatorname{Var}(p(Y-1))=p(1-p)(\mu-1)+p^{2}\sigma^{2}.\end{array}$

This completes the proof. ∎

We are now ready to prove Theorem 2. Taking the square in (3)

$\begin{array}[]{rcl}S^{2}&=&\displaystyle S(x_{r})^{2}+2S(x_{r})\
\sum_{i=1}^{k}\,S(x_{r-i}\setminus
x_{r-i+1})+\bigg{(}\sum_{i=1}^{k}\,S(x_{r-i}\setminus
x_{r-i+1})\bigg{)}^{2}\vspace*{4pt}\\\ &=&\displaystyle
S(x_{r})^{2}+2S(x_{r})\ \sum_{i=1}^{k}\,S(x_{r-i}\setminus
x_{r-i+1})\vspace*{0pt}\\\ &&\hskip 25.0pt+\
\displaystyle\sum_{i=1}^{k}\,S(x_{r-i}\setminus x_{r-i+1})^{2}+\sum_{i\neq
j}\,S(x_{r-i}\setminus x_{r-i+1})S(x_{r-j}\setminus x_{r-j+1})\end{array}$

then conditioning on the event $D=k$ and using independence of the random
variables in (1) (because they represent the number of wet vertices in
disjoint subtrees), we obtain

$\begin{array}[]{rcl}\displaystyle
E_{r}(S^{2})=\sum_{k=0}^{r}\bigg{(}E(S(x_{r})^{2})&+&\displaystyle
2E(S(x_{r}))\ \sum_{i=1}^{k}\,E(S(x_{r-i}\setminus
x_{r-i+1}))\vspace*{-4pt}\\\
&+&\displaystyle\sum_{i=1}^{k}\,E(S(x_{r-i}\setminus
x_{r-i+1})^{2})\vspace*{0pt}\\\ &+&\displaystyle\sum_{i\neq
j}\,E(S(x_{r-i}\setminus x_{r-i+1}))\,E(S(x_{r-j}\setminus
x_{r-j+1}))\bigg{)}P(D=k).\end{array}$ (6)

Using the monotone convergence theorem, taking the limit as $R\to\infty$ in
(and simplifying) the expressions in Lemma 5, we deduce that, on the infinite
Galton-Watson tree,

$E(S(x_{r}))=\frac{1}{1-\nu_{+}}\quad\hbox{and}\quad
E(S(x_{r})^{2})=\frac{1}{(1-\nu_{+})^{2}}\
\bigg{(}1+\frac{\Sigma_{+}^{2}}{1-\nu_{+}}\bigg{)}.$ (7)

This, together with Lemma 6, implies that

$\begin{array}[]{rcl}E(S(x_{r-i}\setminus x_{r-i+1}))&=&\displaystyle
1+\frac{\nu_{-}}{1-\nu_{+}}\vspace*{4pt}\\\ E(S(x_{r-i}\setminus
x_{r-i+1})^{2})&=&\displaystyle
1+\frac{2\nu_{-}}{1-\nu_{+}}+\frac{\nu_{-}}{(1-\nu_{+})^{2}}\
\bigg{(}1+\frac{\Sigma_{+}^{2}}{1-\nu_{+}}\bigg{)}+\frac{\Sigma_{-}^{2}+\nu_{-}^{2}-\nu_{-}}{(1-\nu_{+})^{2}}\vspace*{8pt}\\\
&=&\displaystyle
1+\frac{2\nu_{-}}{1-\nu_{+}}+\frac{\Sigma_{-}^{2}+\nu_{-}^{2}}{(1-\nu_{+})^{2}}+\frac{\Sigma_{+}^{2}\nu_{-}}{(1-\nu_{+})^{3}}.\end{array}$
(8)

Using that, in the limit as $R\to\infty$, the terms in the two sums over $i$
and the terms in the sum over $i\neq j$ in equation (6) are constant, and
using (7) and (8), we obtain

$\begin{array}[]{rcl}\displaystyle
E_{r}(S^{2})&=&\displaystyle\frac{1}{(1-\nu_{+})^{2}}\
\bigg{(}1+\frac{\Sigma_{+}^{2}}{1-\nu_{+}}\bigg{)}+\frac{2}{1-\nu_{+}}\bigg{(}1+\frac{\nu_{-}}{1-\nu_{+}}\bigg{)}E(D)\vspace*{8pt}\\\
&&\hskip 5.0pt+\
\displaystyle\bigg{(}1+\frac{2\nu_{-}}{1-\nu_{+}}+\frac{\Sigma_{-}^{2}+\nu_{-}^{2}}{(1-\nu_{+})^{2}}+\frac{\Sigma_{+}^{2}\nu_{-}}{(1-\nu_{+})^{3}}\bigg{)}E(D)+\bigg{(}1+\frac{\nu_{-}}{1-\nu_{+}}\bigg{)}^{2}E(D(D-1))\vspace*{8pt}\\\
&=&\displaystyle\frac{1}{(1-\nu_{+})^{2}}\
\bigg{(}1+\frac{\Sigma_{+}^{2}}{1-\nu_{+}}\bigg{)}+\displaystyle\bigg{(}1+\frac{2(1+\nu_{-})}{1-\nu_{+}}+\frac{2\nu_{-}+\Sigma_{-}^{2}+\nu_{-}^{2}}{(1-\nu_{+})^{2}}+\frac{\Sigma_{+}^{2}\nu_{-}}{(1-\nu_{+})^{3}}\bigg{)}E(D)\vspace*{8pt}\\\
&&\hskip 5.0pt+\
\displaystyle\bigg{(}1+\frac{\nu_{-}}{1-\nu_{+}}\bigg{)}^{2}E(D(D-1)).\end{array}$

Then, using Lemma 4, we get

$\begin{array}[]{rcl}\displaystyle
E_{r}(S^{2})&=&\displaystyle\frac{1}{(1-\nu_{+})^{2}}\
\bigg{(}1+\frac{\Sigma_{+}^{2}}{1-\nu_{+}}\bigg{)}\vspace*{8pt}\\\ &&\hskip
5.0pt+\
\displaystyle\bigg{(}1+\frac{2(1+\nu_{-})}{1-\nu_{+}}+\frac{2\nu_{-}+\Sigma_{-}^{2}+\nu_{-}^{2}}{(1-\nu_{+})^{2}}+\frac{\Sigma_{+}^{2}\nu_{-}}{(1-\nu_{+})^{3}}\bigg{)}\
q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}\vspace*{8pt}\\\ &&\hskip 5.0pt+\
\displaystyle\bigg{(}1+\frac{\nu_{-}}{1-\nu_{+}}\bigg{)}^{2}\
\frac{2q^{2}(1-rq^{r-1}+(r-1)q^{r})}{(1-q)^{2}}.\end{array}$

and finally Lemmas 7 and 8,

$\begin{array}[]{rcl}\displaystyle E_{r}(S^{2})&=&\displaystyle\frac{1}{(1-\mu
p)^{2}}\ \bigg{(}1+\frac{p(1-p)\mu+p^{2}\sigma^{2}}{1-\mu
p}\bigg{)}\vspace*{8pt}\\\ &&\hskip 5.0pt+\
\displaystyle\bigg{(}1+\frac{2(1+(\mu-1)p)}{1-\mu
p}+\frac{2(\mu-1)p+p(1-p)(\mu-1)+p^{2}\sigma^{2}+(\mu-1)^{2}p^{2}}{(1-\mu
p)^{2}}\vspace*{8pt}\\\ &&\hskip 100.0pt+\
\displaystyle\frac{(p(1-p)\mu+p^{2}\sigma^{2})(\mu-1)p}{(1-\mu
p)^{3}}\bigg{)}\ q\bigg{(}\frac{1-q^{r}}{1-q}\bigg{)}\vspace*{8pt}\\\ &&\hskip
5.0pt+\ \displaystyle\bigg{(}1+\frac{(\mu-1)p}{1-\mu p}\bigg{)}^{2}\
\frac{2q^{2}(1-rq^{r-1}+(r-1)q^{r})}{(1-q)^{2}},\end{array}$

which completes the proof of Theorem 2.

##  6 Proof of Theorem 3 (exponential decay)

To prove exponential decay of the diameter of the cluster of wet vertices, the
idea is to study the process that keeps track of the random number of wet
vertices at distance $n$ from the highest wet vertex in the tree. More
precisely, on the event $D=k$, we let

$X_{n}=\operatorname{card\,}(\mathscr{C}_{n})\quad\hbox{where}\quad\mathscr{C}_{n}=\\{x\in\mathscr{C}:d(x,x_{r-k})=n\\}.$

Recall that $x_{r-k}$ is the unique vertex along the path connecting the
source of the information and the root of the tree that is at distance $k$ of
the source of the information. The next lemma shows that, in the subcritical
phase $\mu p<1$, the expected value of the process decays exponentially.

###### Lemma 9

– Given that the information starts at distance $r$ from the root,

$E_{r}(X_{n+1})=\left\\{\begin{array}[]{ccl}\mu
p\,E(X_{n})+(1-p)\,q^{n+1}&\hbox{for}&n<r\vspace*{2pt}\\\ \mu
p\,E(X_{n})&\hbox{for}&n\geq r.\end{array}\right.$

###### Proof.

Recall that each vertex produces $\mu$ offspring in average and that each of
the offspring of a wet vertex is wet independently with probability $p$, from
which it follows that each wet vertex has $\mu p$ wet offspring in average.
Now, given $D=k$, the process is conditioned so that

$x_{r-k}\in\mathscr{C}_{0},\quad x_{r-k+1}\in\mathscr{C}_{1},\quad\ldots\quad
x_{r-1}\in\mathscr{C}_{k-1}\quad\hbox{and}\quad x_{r}\in\mathscr{C}_{k}$

are wet therefore, until generation $k-1$, the vertices in $\mathscr{C}_{n}$
all have $\mu p$ wet offspring in average except for vertex $x_{r-k+n}$ that
has $(\mu-1)p+1$ wet offspring in average. This implies that

$E_{r}(X_{n+1}\,|\,X_{n},D=k)=\left\\{\begin{array}[]{ccl}\mu
p(X_{n}-1)+(\mu-1)p+1&\hbox{for}&n<k\vspace*{2pt}\\\ \mu
pX_{n}&\hbox{for}&n\geq k.\end{array}\right.$

Conditioning on the random variable $D$, we deduce that

$\begin{array}[]{rcl}E_{r}(X_{n+1}\,|\,X_{n})&=&\displaystyle\sum_{k=0}^{\infty}\,E_{r}(X_{n+1}\,|\,X_{n},D=k)P_{r}(D=k)\vspace*{4pt}\\\
&=&\displaystyle\sum_{k=0}^{n}\ (\mu
pX_{n})\,P_{r}(D=k)+\sum_{k=n+1}^{\infty}(\mu
p(X_{n}-1)+(\mu-1)p+1)\,P_{r}(D=k)\vspace*{4pt}\\\
&=&\displaystyle\sum_{k=0}^{n}\ (\mu
pX_{n})\,P_{r}(D=k)+\sum_{k=n+1}^{\infty}(\mu
pX_{n}+1-p)\,P_{r}(D=k)\vspace*{4pt}\\\ &=&\displaystyle\sum_{k=0}^{\infty}\
(\mu pX_{n})\,P_{r}(D=k)+(1-p)\sum_{k=n+1}^{\infty}P_{r}(D=k)=\mu
pX_{n}+(1-p)P_{r}(D>n).\end{array}$

Recalling the probability mass function of $D$, we deduce that

$E_{r}(X_{n+1})=E(E(X_{n+1}\,|\,X_{n}))=\left\\{\begin{array}[]{ccl}\mu
p\,E(X_{n})+(1-p)\,q^{n+1}&\hbox{for}&n<r\vspace*{2pt}\\\ \mu
p\,E(X_{n})&\hbox{for}&n\geq r.\end{array}\right.$

This completes the proof. ∎

It follows from the lemma that, for all $n\leq r$,

$\begin{array}[]{rcl}E_{r}(X_{n})&=&(\mu
p)\,E_{r}(X_{n-1})+(1-p)\,q^{n}\vspace*{4pt}\\\ &=&(\mu
p)^{2}E_{r}(X_{n-2})+(\mu p)(1-p)\,q^{n-1}+(1-p)\,q^{n}\vspace*{4pt}\\\
&=&(\mu p)^{3}E_{r}(X_{n-3})+(\mu p)^{2}(1-p)\,q^{n-2}+(\mu
p)(1-p)\,q^{n-1}+(1-p)\,q^{n}\vspace*{4pt}\\\ &=&(\mu
p)^{n}E_{r}(X_{0})+(1-p)((\mu p)^{n-1}\,q+(\mu p)^{n-2}\,q^{2}+\cdots+(\mu
p)\,q^{n-1}+q^{n})\vspace*{4pt}\\\ &\leq&(\mu p)^{n}E_{r}(X_{0})+(\mu
p)^{n-1}\,q+(\mu p)^{n-2}\,q^{2}+\cdots+(\mu p)\,q^{n-1}+q^{n}.\end{array}$

Then, using that $E_{r}(X_{0})=1$, we get

$E_{r}(X_{n})\leq\sum_{k=0}^{n}\,(\mu p)^{n-k}q^{k}=(\mu
p)^{n}\,\sum_{k=0}^{n}\bigg{(}\frac{q}{\mu p}\bigg{)}^{k}=\frac{1-(q/\mu
p)^{n+1}}{1-(q/\mu p)}\ (\mu p)^{n}\quad\hbox{for all}\quad n\leq r.$

Observing also that, for all $n>r$,

$E_{r}(X_{n})\leq(\mu p)\,E_{r}(X_{n-1})\leq(\mu
p)^{2}E_{r}(X_{n-2})\leq\cdots\leq(\mu p)^{n-r}E_{r}(X_{r})$

we conclude that, for all $n>r$, the diameter exceeds $2n$ with probability

$\begin{array}[]{rcl}P_{r}(\operatorname{diam}(\mathscr{C})\geq
2n)&\leq&\displaystyle
P_{r}(X_{n}>0)=\sum_{k=1}^{\infty}\,P_{r}(X_{n}=k)\leq\sum_{k=1}^{\infty}\,k\,P_{r}(X_{n}=k)=E_{r}(X_{n})\vspace*{8pt}\\\
&\leq&\displaystyle(\mu p)^{n-r}E_{r}(X_{r})\leq(\mu p)^{n-r}\ \frac{1-(q/\mu
p)^{r+1}}{1-(q/\mu p)}\ (\mu p)^{r}=\displaystyle\frac{1-(q/\mu
p)^{r+1}}{1-(q/\mu p)}\ (\mu p)^{n}.\end{array}$

This completes the proof of Theorem 3.

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Yuanqi<EMAIL_ADDRESS>Quan<EMAIL_ADDRESS>Hu
<EMAIL_ADDRESS>S. Kevin<EMAIL_ADDRESS>George Mason
University Institute of Computing Technology, Chinese Academy of Sciences
Where is the disease?

# Where is the disease? Semi-supervised pseudo-normality synthesis from an
abnormal image

###### Abstract

Pseudo-normality synthesis, which computationally generates a pseudo-normal
image from an abnormal one (e.g., with lesions), is critical in many
perspectives, from lesion detection, data augmentation to clinical surgery
suggestion. However, it is challenging to generate high-quality pseudo-normal
images in the absence of the lesion information. Thus, expensive lesion
segmentation data have been introduced to provide lesion information for the
generative models and improve the quality of the synthetic images. In this
paper, we aim to alleviate the need of a large amount of lesion segmentation
data when generating pseudo-normal images. We propose a Semi-supervised
Medical Image generative LEarning network (SMILE) which not only utilizes
limited medical images with segmentation masks, but also leverages massive
medical images without segmentation masks to generate realistic pseudo-normal
images. Specifically, our framework consists of four primary components,
including a pseudo-normality generator, a lesion segmentor, a confidence
enhancer and a pseudo-abnormality reconstructor. We adopt the idea of
adversarial training between the generator and the segmentor, and between the
reconstructor and the segmentor to produce realistic pseudo-normal and pseudo-
abnormal images. Furthermore, to reduce the number of required lesion
segmentation data, we propose a confidence enhancer, which maximizes the
confidence score of the segmentation prediction and enables our framework to
learn in semi-supervised learning. To stabilize the model training, a step-by-
step adversarial training scheme is also introduced. Extensive experiments
show that our model outperforms the best state-of-the-art model by up to 6%
for data augmentation task and 3% in generating high-quality images. Moreover,
the proposed semi-supervised learning achieves comparable medical image
synthesis quality with supervised learning model, using only 50% of
segmentation data.

##  1 Introduction

With the rapid development of deep learning and its successful applications in
computer vision, natural language processing, scientific discovery, etc [Deng
et al.(2009)Deng, Dong, Socher, Li, Li, and Fei-Fei, Devlin et
al.(2018)Devlin, Chang, Lee, and Toutanova, Graves et al.(2013)Graves,
Mohamed, and Hinton], more and more researchers attempt to assist the clinical
diagnosis with the deep learning algorithms in many tasks, e.g. segmentation
[Andermatt et al.(2018)Andermatt, Horváth, Pezold, and Cattin, Bowles et
al.(2017)Bowles, Qin, Guerrero, Gunn, Hammers, Dickie, Hernández, Wardlaw, and
Rueckert, Milletari et al.(2016)Milletari, Navab, and Ahmadi, Sun et
al.(2020)Sun, Wang, Huang, Ding, Greenspan, and Paisley, Ye et al.(2013)Ye,
Zikic, Glocker, Criminisi, and Konukoglu], lesion detection [Alex et
al.(2017)Alex, KP, Chennamsetty, and Krishnamurthi, Chen and Konukoglu(2018),
Tsunoda et al.(2014)Tsunoda, Moribe, Orii, Kawano, and Maeda] and so on [Shen
et al.(2017)Shen, Wu, and Suk]. Despite promising results in some tasks, there
are still several main challenges left unsolved. (1) Most of the deep learning
models are highly data-driven which require a large amount of labeled data to
achieve satisfying performance. However, the labels are especially expensive
for medical image data because it takes plenty of time for experts to
carefully annotate the data [Liu et al.(2021)Liu, Han, Du, Zhu, Li, Gu, Xiao,
Li, Zhao, Xiao, et al.]. (2) Deep learning algorithms often serve as a black
box and are hard to assist with doctors’ diagnosis.

Figure 1: Visualization of abnormal, pseudo-normal, pseudo-abnormal images.

Recently, many researchers have found synthetic images generated by deep
generative models, especially generative adversarial networks (GANs), can
greatly improve performance of many tasks in both natural images and medical
images [Perez and Wang(2017), Goodfellow et al.(2020)Goodfellow, Pouget-
Abadie, Mirza, Xu, Warde-Farley, Ozair, Courville, and Bengio, Han et
al.(2018)Han, Hayashi, Rundo, Araki, Shimoda, Muramatsu, Furukawa, Mauri, and
Nakayama]. At the same time, the high-quality synthetic images generated by
deep generative models can assist with doctors’ diagnosis. For example,
doctors can localize the lesions by comparing abnormal images with synthetic
pseudo-normal images. Thus, generating medical images with great quantity and
quality has become a necessary and urgent research topic.

The idea of pseudo-normality synthesis is to generate pseudo-normal images
(i.e. without lesion) from real abnormal images (i.e. with lesion). The
generated pseudo-normal images are important in two aspects. (1) The synthetic
pseudo-normal images can assist doctors with diagnosis by comparing the
abnormal and pseudo-normal images [Tsunoda et al.(2014)Tsunoda, Moribe, Orii,
Kawano, and Maeda]. (2) The synthetic pseudo-normal images can be utilized as
a data augmentation technique to improve performance of various downstream
tasks, (e.g. lesion segmentation, lesion detection, etc.) [Alex et
al.(2017)Alex, KP, Chennamsetty, and Krishnamurthi, Andermatt et
al.(2018)Andermatt, Horváth, Pezold, and Cattin, Bowles et al.(2017)Bowles,
Qin, Guerrero, Gunn, Hammers, Dickie, Hernández, Wardlaw, and Rueckert, Chen
and Konukoglu(2018), Milletari et al.(2016)Milletari, Navab, and Ahmadi, Sun
et al.(2020)Sun, Wang, Huang, Ding, Greenspan, and Paisley, Tsunoda et
al.(2014)Tsunoda, Moribe, Orii, Kawano, and Maeda, Ye et al.(2013)Ye, Zikic,
Glocker, Criminisi, and Konukoglu]. However, in most of the cases, the paired
abnormal and normal images are unavailable for training such a generative
network to synthesize pseudo-normal images from abnormal images. Thus,
previous works [Baumgartner et al.(2018)Baumgartner, Koch, Can Tezcan, Xi Ang,
and Konukoglu, Sun et al.(2020)Sun, Wang, Huang, Ding, Greenspan, and Paisley,
Xia et al.(2020)Xia, Chartsias, and Tsaftaris, Yunlong et al.(2020)Yunlong,
Xin, Liyan, Yihong, Yue, Xinghao, Xiaoqing, and Yizhou] leverage unpaired
normal and abnormal images to train generative networks (variants of GAN-based
models) to generative pseudo-normal images from abnormal images. Nevertheless,
they have several limitations: (1) low-quality of the generated pseudo-normal
images in the absence of the segmentation labels, (2) high-cost of the
required segmentation labels, (3) the dual generation problem (pseudo-abnormal
image synthesis) is not yet explored.

To solve the above limitations, we propose a Semi-supervised Medical Image
generative LEarning network (SMILE), which can leverage a small set of images
with segmentation masks and a large set of images without segmentation masks
to achieve robust pseudo-normality synthesis from abnormal ones. To be
specific, we propose a confidence enhancer component which maximizes the
confidence score of the predicted segmentation labels in our semi-supervised
setting. For the third limitation, we incorporate a new component in our
framework for pseudo-abnormal image synthesis. Thus, our model is capable of
generating both pseudo-normal and pseudo-abnormal images, as shown in Fig. 1.
Our contributions are summarized as follows:

  * •

We propose a semi-supervised generative modeling framework which uses
confidence enhancer to leverage both labeled and unlabeled data to generate
synthetic images.

  * •

We propose a confidence enhancement technique which enforces maximization of
the certainty over the predicted segmentation mask and is especially helpful
in the absence of ground-truth segmentation labels.

  * •

Extensive experiments show that our model outperforms the best of the state-
of-the-art methods by up to 3% in generating high-quality images and 6% for
data augmentation task. When using only 50% of labeled data, the proposed
model is able to achieve comparable pseudo-normality synthesis results to
fully supervised-methods.

##  2 Related Works

###  2.1 Pseudo-normality Synthesis

Pseudo-normality synthesis is important in various perspectives, including
clinical suggestion for doctors and data augmentation for many downstream
tasks [Andermatt et al.(2018)Andermatt, Horváth, Pezold, and Cattin, Bowles et
al.(2017)Bowles, Qin, Guerrero, Gunn, Hammers, Dickie, Hernández, Wardlaw, and
Rueckert, Sun et al.(2020)Sun, Wang, Huang, Ding, Greenspan, and Paisley, Ye
et al.(2013)Ye, Zikic, Glocker, Criminisi, and Konukoglu, Zhu et al.(2017)Zhu,
Park, Isola, and Efros]. Thanks to the satisfying visual effects of the
synthetic images, generative adversarial nets have been dominating in the
pseudo-normality synthesis task. In the previous work, [Sun et al.(2020)Sun,
Wang, Huang, Ding, Greenspan, and Paisley] utilizes anomaly detection, which
treats lesions as outliers, to guide the generation process. In a later work
[Xia et al.(2020)Xia, Chartsias, and Tsaftaris], they introduce the idea of
adversarial training, which co-trains a predictor to make binary predictions
over the synthetic images. The predictor is expected to detect whether the
synthetic images are normal (i.e. 0) or abnormal (i.e. 1). The generator is
expected to generate realistic pseudo-normality images which fool the
predictor to believe that the synthetic pseudo-normality images are real.
Additionally, a reconstructor is also introduced to reconstruct back the
pseudo-abnormality images taking the segmentation mask and the pseudo-
normality images as input. In the most recent work [Yunlong et
al.(2020)Yunlong, Xin, Liyan, Yihong, Yue, Xinghao, Xiaoqing, and Yizhou],
they extend the idea of adversarial training and co-train a segmentor which
provides more detailed information about the lesion than the predictor used in
[Xia et al.(2020)Xia, Chartsias, and Tsaftaris]. Besides GAN-like
architectures, in work [Chen and Konukoglu(2018)], they adopt a variational
auto-encoders (VAE) to learn the distribution of the normal images and
generate pseudo-normality images based on the learnt constrained latent
representations. However, they suffer from the poor reconstruction visual
effects of VAE as well as hard to eliminate small lesions.

###  2.2 Deep Generative Models for Medical Image Synthesis

To alleviate the need of the large amount of data required by deep learning
models, deep generative models have been proposed firstly to generate
synthetic data for effective data augmentation [Goodfellow et
al.(2020)Goodfellow, Pouget-Abadie, Mirza, Xu, Warde-Farley, Ozair, Courville,
and Bengio]. In medical image analysis, deep generative models are also
utilized for data augmentation and anonymization [Shin et al.(2018)Shin,
Tenenholtz, Rogers, Schwarz, Senjem, Gunter, Andriole, and Michalski, Han et
al.(2019)Han, Rundo, Araki, Nagano, Furukawa, Mauri, Nakayama, and Hayashi].
In the previous work, various downstream tasks have been utilized to guide the
generator to produce high-quality synthetic images, such as anomaly detection,
lesion detection, lesion segmentation, etc. For example, [Sun et al.(2020)Sun,
Wang, Huang, Ding, Greenspan, and Paisley] utilizes anomaly detection, which
treats lesions as anomalies, and learn a generator which removes the lesion
from the original images. However, since there is no ground-truth for the
pseudo-abnormal images, the labels are obtained by simply removing everything
within the contour of the brains from the images. Obviously, the assumption is
too strong for an abnormal image which should maintain the structures and
tissues within the brains. Later work in [Xia et al.(2020)Xia, Chartsias, and
Tsaftaris, Yunlong et al.(2020)Yunlong, Xin, Liyan, Yihong, Yue, Xinghao,
Xiaoqing, and Yizhou] eliminate the assumption by providing more information
about the lesions in the images (e.g. binary masks).

###  2.3 Generative Adversarial Learning in Medical Image Analysis

Generative Adversarial Nets (GAN) [Goodfellow et al.(2020)Goodfellow, Pouget-
Abadie, Mirza, Xu, Warde-Farley, Ozair, Courville, and Bengio] have been
proposed to generate realistic images and achieved satisfying results in
natural images [Zhu et al.(2017)Zhu, Park, Isola, and Efros]. The fundamental
mechanism of the GAN-based approach is to co-train a generator and a
discriminator together. The generator is expected to generate realistic images
which fool the discriminator to believe the synthetic images are real, while
the discriminator is expected to distinguish the real and synthetic images.
Later, the idea of adversarial training which one model attempts to fool the
other model by generating realistic images is largely adopted in several
different forms. For example, conditional GAN [Mirza and Osindero(2014)]
utilizes the idea of adversarial training to make the generator learn to
generate images of different classes. In medical image analysis, the idea of
adversarial training is also largely utilized [Han et al.(2018)Han, Hayashi,
Rundo, Araki, Shimoda, Muramatsu, Furukawa, Mauri, and Nakayama, Yi et
al.(2019)Yi, Walia, and Babyn]. Specifically, [Chen et al.(2017)Chen,
Papandreou, Kokkinos, Murphy, and Yuille, Tack et al.(2018)Tack, Mukhopadhyay,
and Zachow, Xue et al.(2018)Xue, Xu, Zhang, Long, and Huang] have shown that
adversarial training can improve the performance of various medical image
analysis tasks, such as segmentation, detection, etc.

Symbol | Description
---|---
$G$ | pseudo-normality generator
$S$ | lesion segmentor
$R$ | pseudo-abnormality reconstructor
$F$ | confidence enhancer
$I$ | input image w/ segmentation label
$L$ | segmentation label
$P_{L}$ | generated pseudo-label
$U$ | unlabeled input image
$M$ | predicted segmentation mask
$P_{n}$ | generated pseudo-normality image
$P_{ab}$ | reconstructed pseudo-abnomality image
Table 1: List of notations.

##  3 Methodology

The overall framework includes three main parts (1) pseudo-normality
generation, (2) adversarial segmentation and (3) pseudo-abnormality
generation. The confidence enhancement is for the semi-supervised learning
setting. The list of notations used throughout this section is first
introduced in Table 1. Then, all the three parts are illustrated in the
following sections.

Figure 2: (a) Pseudo-normality Generation Adversarial Training (b) Pseudo-
abnormality Reconstruction (c) Pseudo-abnormality Reconstruction Adversarial
Training (d) Semi-supervised Confidence Enhancement

###  3.1 Pseudo-normality Generation

We adopt a U-Net [Ronneberger et al.(2015)Ronneberger, Fischer, and Brox]
model, which is a widely used image segmentation architecture, to build our
generator because the architecture allows different resolutions of information
to flow through and skip connection to pass message along, which both are
critical for the generation task. The loss function of the generator consists
of two parts, the first part is a pixel-wise measurement on the quality of the
generated pseudo-normality image, and the second part measures how good the
generated pseudo-normality image is by the prediction from the discriminaotr.
Therefore, the loss of the generator is defined as follows,

$L_{gen}=L_{mse}((1-M)\odot
I,\>(1-M)\odot\mathbf{G}(I))+L_{ce}(\mathbf{S}(\mathbf{G}(I)),\>M)$ (1)

where $M$ is the segmentation mask of the expected normal image (i.e. zeros
matrix), $L_{mse}$ refers to a pixel-wise $L_{2}$ loss and $L_{ce}$ refers to
the cross-entropy loss. The pixel-wise generation $L_{2}$ loss enforces the
generator to maintain the identity of the no-lesion part. Notably, in order to
keep the model consistent and aware of the presence or absence of the lesion,
we also train the model to generate pseudo-normality image with the input real
normal image. In this case, the model is expected to output the original
image.

###  3.2 Adversarial Segmentation

In terms of the segmentor, we also adopt a U-Net architecture due to its
excellent performance in segmentation task [Ronneberger et
al.(2015)Ronneberger, Fischer, and Brox] and a softmax function in the end to
produce segmentation masks. The main objective for the segmentor is to
recognize the lesion behind the generated pseudo-normality images and
recognize the fake-lesion in the generated pseudo-abnormality images, as:

$\begin{split}L_{seg}=&L_{ce}(\mathbf{S}(\mathbf{G}(I)),\;M)+L_{ce}(\mathbf{S}(P_{ab}),\;M)\end{split}$
(2)

where $M$ is the segmentation mask and $P_{ab}$ is the reconstructed pseudo-
abnormality image.

###  3.3 Pseudo-abnormality Generation

To fully utilize the segmentation labels to provide more supervised signals to
the model as well as enrich the dataset better during data argumentation, we
introduce a reconstructor at the end of our model to learn to reconstruct the
input image with the segmentation mask and the pseudo-normality image. As in
the generator, we adopt a similar U-Net structure, except with a two-channel
input. The objective function is to minimize the pixel-wise loss between the
reconstructed image and the original input image.

$P_{ab}=\mathbf{R}(\mathbf{G}(I),\>M)\>\>L_{recons}=L_{mse}(P_{ab},\>I)$ (3)

###  3.4 Semi-supervised Objective

In our semi-supervised training setting, the input image is with unknown type,
either normal or abnormal. The input image $U$ first goes through the
segmenter. Two following things will happen: (1) the image trains the
segmentor without label by enhancing the confidence of the segmentor (as in
Eq. 4), (2) the segmentor assigns a pseudo-label to the image. After that, the
training is same as it is in the supervised setting, with the assigned pseudo-
label. In this setting, the segmentor is trained to be more confident about
the segmentation mask, which in turns provides a more precise pseudo-label and
leads to a better solution for both the segmentor and the generator. The loss
function for the confidence enhancement of the segmentor is:

$P_{L}=\begin{cases}1,&\text{if}\ M>\epsilon\\\
0,&\text{otherwise}\end{cases}$ (4) $L_{semi}=L_{ce}(S(U),P_{L})$ (5)

where $\epsilon$ is a desired confidence threshold, which we set as $0.9$ in
our model. It is also recommended to dynamically update $\epsilon$ by the
average confidence value every epoch. In this way, the confidence of the
segmentor with respect to the accuracy of the predicted segmentation mask is
maximized. It is especially useful under the semi-supervised learning which no
ground-truth is present to guide the training of the framework.

###  3.5 Training Scheme

We adopt a step-by-step pre-training and fine-tuning training scheme as shown
in Fig. 2, We first train $G$ and $S$ in an adversarial setting. Then, we
train $R$ by fixing $S$ and $G$. Next, we train $R$ and $S$ again in an
adversarial setting. Finally, we fine-tune the framework with $G$, $S$, $R$
together. We adopt a similar training scheme in the semi-supervised setting,
except the segmentation mask is predicted from the segmentor $S$.

##  4 Experiments

###  4.1 Dataset

We consider the publicly available dataset, Multimodal Brain Tumor
Segmentation Challenge 2019 (BraTS19) [Bakas et al.(2017)Bakas, Akbari,
Sotiras, Bilello, Rozycki, Kirby, Freymann, Farahani, and Davatzikos, Menze et
al.(2014)Menze, Jakab, Bauer, Kalpathy-Cramer, Farahani, Kirby, Burren, Porz,
Slotboom, Wiest, et al.] to evaluate our framework. We take the training set
of the BraTS19 challenge dataset. There are 259 GBM (i.e. glioblastoma) and 76
LGG (i.e. lower-grade glioma) volumes of magnetic resonance imaging (MRI) in
the dataset. Each of them is skull-stripped, interpolated to an isotropic
spacing of $1mm^{3}$ and co-registered to the same anatomical template. In
every volume, there are four different modalities (i.e. T1, T2, T1c and Flair)
with 240x240 slice available. We take the T2 modality of the GBM and split
train/validation/test set by 130/104/25 volumes, respectively. We clip the
intensities to [0, $V_{99.5}$] of each volume, where $V_{99.5}$ accounts for
top 99.5% pixel values of the volume.

Figure 3: Visualization of the randomly selected abnormal and synthetic
pseudo-normality and pseudo-abnormality images.

###  4.2 Comparison Methods

We consider four related frameworks for the supervised learning task of
pseudo-normality synthesis. The models are introduced below. We take the
baseline implementation from their open-source codes. VA-GAN[Baumgartner et
al.(2018)Baumgartner, Koch, Can Tezcan, Xi Ang, and Konukoglu] uses a
generator which can predict an additive maps and then generate a pseudo-
normality images and a discriminator which attempts to distinguish between the
generated images and the real images. Additionally, they aim to learn a least
required additive map to achieve the realistic pseudo-normality image
generation by minimizing a $L_{1}$ loss. ANT-GAN[Sun et al.(2020)Sun, Wang,
Huang, Ding, Greenspan, and Paisley] is a variant of a CycleGAN which it
enforces the consistency loss between the pseudo-normality and pseudo-
abnormality images. Compared with our approach, they do not intend adversarial
training in the network, while they only use the pixel-wise $L_{2}$ normality
consistency loss. PHS-GAN[Xia et al.(2020)Xia, Chartsias, and Tsaftaris]
introduces adversarial training with a discriminator to distinguish between
normal and abnormal images, while GVS[Yunlong et al.(2020)Yunlong, Xin, Liyan,
Yihong, Yue, Xinghao, Xiaoqing, and Yizhou] takes segmentation label as input
and co-train a discriminator to segment the lesion in the normal and abnormal
images.

###  4.3 Evaluation Metrics

For quantitative evaluation, we select two metrics proposed in [Xia et
al.(2020)Xia, Chartsias, and Tsaftaris], to evaluate the healthiness and
identity, in which healthiness refers to the effects of the erased tumor,
while identity refers to the maintenance of the original structure.
Healthiness ($h$) measures how normal the generated pseudo-normality image is.
The evaluation function of $h$ is defined as:

$h=1-\frac{E_{x_{p}\sim P}[N(\textbf{S}_{pred}(\textbf{G}(I))]}{E_{x_{p}\sim
P}[N(\textbf{S}_{pred}(I))]}$ (6)

Identity ($iD$) measures the structure maintenance of the generated pseudo-
normality image. The evaluation function of $iD$ is defined as:

$iD=MS-SSIM[(1-M)\odot\textbf{G}(I),(1-M)\odot I]$ (7)

where $MS-SSIM$ stands for multiscale structural similarity [Wang et
al.(2003)Wang, Simoncelli, and Bovik].

###  4.4 Experiment Details

The framework is trained with an Adam Optimizer. The batch size is set as 8
and the learning rate is set as 1e-4. The model is trained 10 epochs for each
phase of the training loop. All the experiments are conducted on a 64-bit
machine with two NVIDIA GPUs (TITAN RTX, 1770MHz, 24GB GDDR6).

##  5 Discussion

Methods | identity $iD\uparrow$ | healthiness $h\uparrow$
---|---|---
VA-GAN[Baumgartner et al.(2018)Baumgartner, Koch, Can Tezcan, Xi Ang, and Konukoglu] | 0.809 | 0.605
PHS-GAN[Xia et al.(2020)Xia, Chartsias, and Tsaftaris] | 0.960 | 0.731
ANT-GAN[Sun et al.(2020)Sun, Wang, Huang, Ding, Greenspan, and Paisley] | 0.969 | 0.747
GVS[Yunlong et al.(2020)Yunlong, Xin, Liyan, Yihong, Yue, Xinghao, Xiaoqing, and Yizhou] | 0.997 | 0.792
SMILE-50 | 0.978 | 0.653
SMILE-75 | 0.987 | 0.810
SMILE-sup | 1.000 | 0.822
Table 2: Quantitative evaluation on identity and healthiness metric of our
proposed SMILE framework and the other baseline models.

###  5.1 Evaluation on Generation Performance

In Figure 3, we visualize randomly selected triplets of (abnormal images,
pseudo-normal images, pseudo-abnormal images). In Table 2, we evaluate the
$iD$ and $h$ evaluation metrics of our proposed model as well as the baseline
models. In Table 2, it is clear to observe that our model outperforms other
models in both evaluation metrics. Notably, GVS ranks the second, even though
the framework is very simple. That demonstrates the adversarial training with
segmentation task is important to provide detailed information (e.g. shape,
location) about the lesion and further improve the quality of the synthetic
images. The performance of ANT-GAN and PHS-GAN are comparable and way better
than VA-GAN, which illustrates that adversarial training could help improve
the quality of the synthetic images. In summary, utilizing segmentation task
as an adversarial training task could greatly improve the quality of the
synthetic images. Rather, the additional pseudo-abnormality reconstructor
further improve the quality of the synthetic pseudo-normal images by 2%. For
the semi-supervised setting, we use only 75% segmentation-labeled data, along
with 25% unlabeled data to train out network, and achieve comparable results
with the fully-supervised learning setting. In practice, we do not involve any
extra unlabeled data. Rather, we remove the label of the 25% labeled data from
the same training set to produce the unlabeled data, thanks to the proposed
confidence maximization technique, which continues to train the model, even
when the labels are missing. Besides, we evaluate how the amount of labeled
data influence our semi-supervised learning performance in table 2 (a). It is
worth noting that the labeled data we use is at least 50% of the whole labeled
training dataset because the framework crashes with only few labeled data and
we leave it as a future study.

###  5.2 Evaluation on Data Augmentation

We evaluate the effectiveness of the method when working as a data
augmentation task performance by providing synthetic pseudo-normality and
pseudo-abnormality data for the downstream lesion segmentation tasks. (Note
that we do not apply traditional data augmentation (e.g. rotation, crop, etc.)
here.) We show the performance (Dice score) of the models in Table 3. To be
specific, we set up the experiments by utilizing our training dataset to
generate the data-augmented dataset, which we generate two types of data (i.e.
pseudo-normality, pseudo-abnormality) with the same amount as in the training
dataset. We compare the performance of each different type of augmented-
dataset, namely, without augmentation (None), with pseudo-normality only (PN
or GVS [Yunlong et al.(2020)Yunlong, Xin, Liyan, Yihong, Yue, Xinghao,
Xiaoqing, and Yizhou]), with pseudo-abnormality only (PA) and with both
pseudo-normality and -abnormality (Both). Note, the setting PN is the baseline
model, GVS [Yunlong et al.(2020)Yunlong, Xin, Liyan, Yihong, Yue, Xinghao,
Xiaoqing, and Yizhou], which produces the best performance over the current
state-of-the-art models. For each of the data-augmented experiment, adding
pseudo-images means incorporating the synthetic data into the training data.
The result shows that both the generated pseudo-normality and pseudo-
abnormality data can improve the performance of the downstream task (i.e.
segmentation) when used for data augmentation.

Methods | $Dice\uparrow$
---|---
None | 0.72
Pseudo-normality (GVS) | 0.76
Pseudo-abnormality | 0.77
Both | 0.82
Table 3: Quantitative evaluation on dice score of the downstream lesion
segmentation task.

##  6 Conclusion

In this paper, we proposed a Semi-supervised Medical Image generative LEaning
(SMILE) framework to generate pseudo-normality and pseudo-abnormality images
to assist in medical image analysis tasks through eye screening and data
augmentation. A confidence enhancement technique is introduced for semi-
supervised generative learning. Extensive experiment results suggest that our
proposed SMILE model can generate images with better quality and support
better data augmentation than the state-of-the-art models. We plan to study
more data-efficient ways (e.g. self-supervised learning) for generative
learning in medical images in the future.

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