# Normal edge ideals of edge-weighted graphs

Thanh Vu  Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam
<EMAIL_ADDRESS>and  Guangjun Zhu  School of Mathematical Sciences, Soochow
University, Suzhou, Jiangsu, 215006, P.R. China<EMAIL_ADDRESS>
###### Abstract.

We classify all normal edge ideals of edge-weighted graphs.

###### Key words and phrases:

Integral closure; normal ideal; edge-weighted graph

###### 2010 Mathematics Subject Classification:

13B22, 13F65, 90C05

##  1\. Introduction

Let $R=\mathrm{k}[x_{1},\ldots,x_{n}]$ be a standard graded polynomial ring
over a field $\mathrm{k}$. In [MVZ], Minh, Vu, and Zhu classified all
integrally closed and normal edge ideals of weighted oriented graphs. In this
work, we use the results developed in [MVZ] to classify all normal edge ideals
of edge-weighted graphs. Recall that a non-zero homogeneous ideal $I$ of $R$
is normal if $I^{t}=\overline{I^{t}}$ for all natural exponents $t\geq 1$,
where $\overline{I^{t}}$ denotes the integral closure of $I^{t}$. We refer to
[SH] for criteria for the normality of general homogeneous ideals. Even when
$I$ is a monomial ideal, determining the normality of $I$ is a difficult
problem and has a deep connection to the theory of linear and integer
programming [DV, HL, HT, Sch1, Sch2].

Let us now recall the notion of edge ideals of edge-weighted graphs introduced
by Paulsen and Sather-Wagstaff [PS]. Let $G$ be a simple graph on the vertex
set $V(G)=[n]=\\{1,\ldots,n\\}$ and the edge set $E(G)$. Assume that
$\mathbf{w}:E(G)\rightarrow\operatorname{\mathbb{Z}}_{>0}$ is a weight
function on the edges of $G$. The edge ideal of the edge-weighted graph
$(G,\mathbf{w})$ is defined by

$I(G,\mathbf{w})=\big{(}(x_{i}x_{j})^{\mathbf{w}(e)}\mid e=\\{i,j\\}\in
E(G)\big{)}\subseteq S=\mathrm{k}[x_{1},\ldots,x_{n}].$

If $\mathbf{w}(e)=1$ for all edge $e\in E(G)$, then $I(G,\mathbf{w})=I(G)$ is
the usual edge ideal of $G$. For simplicity of notation, we call an edge-
weighted graph $(G,\mathbf{w})$ integrally closed (normal) if its edge ideal
is. Ohsugi and Hibi [OH] and Simis, Vasconcelos, and Villarreal [SVV] computed
the normalization of the edge ring of a simple graph. One can then deduce that
the edge ideal of a simple graph $G$ is normal if and only if $G$ satisfies
the odd cycle conditions. In terms of forbidden subgraphs, it says that $I(G)$
is normal if and only if $G$ does not have the disjoint union of two odd
cycles as an induced subgraph. A squarefree monomial ideal is integrally
closed. In particular, edge ideals of simple graphs are integrally closed.
Nonetheless, most edge-weighted graphs are not integrally closed. Duan, Zhu,
Cui, and Li [DZCL] proved that $I(G,\mathbf{w})$ is integrally closed if and
only if $(G,\mathbf{w})$ does not have the following configurations as induced
weighted subgraphs, where the weights are all nontrivial, i.e., have weights
greater than $1$.

ab$F_{1}$abc$F_{2}$ab$F_{3}$

Figure 1. Forbidden configurations for integrally closed edge-weighted graphs

The normality of $I(G,\mathbf{w})$ imposes further restrictions on
$(G,\mathbf{w})$. In the following figure, a triangle with a dashed line
represents an induced odd cycle. When we do not specify the weight of the
edge, we mean the weight could be any arbitrary positive integer. In
particular, $F_{4}$ represents the disjoint union of an odd cycle with a
nontrivial edge. $F_{5}$ represents two odd cycles with possibly only
nontrivial edges connecting them.

a$F_{4}$a$F_{5}$

Figure 2. Further forbidden configurations for normal edge-weighted graphs

######  Theorem 1.1.

Let $(G,\mathbf{w})$ be an arbitrary edge-weighted graph. Then
$I(G,\mathbf{w})$ is normal if and only if $(G,\mathbf{w})$ does not have
$F_{1},\ldots,F_{5}$ as an induced edge-weighted subgraph.

Let us outline the idea of the proof of Theorem 1.1. First, for each
configuration $F_{1},\ldots,F_{5}$, we give an explicit monomial in the
integral closure of a certain power of $I(G,\mathbf{w})$ but not in that power
itself. Now, assume that $(G,\mathbf{w})$ does not have $F_{1},\ldots,F_{5}$
as an induced edge-weighted subgraph. We follow the steps below to prove the
normality of $I(G,\mathbf{w})$ by induction on the number of vertices and the
number of edges of $G$.

  1. (1)

Reduce to the case where $G$ is connected and $(G,\mathbf{w})$ has no leaf
edges with trivial weights.

  2. (2)

Prove that dropping a nontrivial edge of $(G,\mathbf{w})$ results in a graph
that does not have $F_{1},\ldots,F_{5}$ as an induced edge-weighted subgraph.

  3. (3)

Reduce to the case where $G$ has only one nontrivial edge.

  4. (4)

Reduce to the case where $G$ has no even cycles. In other words, $G$ is
compact in the sense of Wang and Lu [WL].

  5. (5)

Establish the normality for each connected compact graph having only one
nontrivial edge and no $F_{4},F_{5}$ as induced edge-weighted subgraphs.

We would like to note that the steps are similar to those in [MVZ], though the
combinatorics of the edge-weighted graphs and weighted oriented graphs are
subtly different. Furthermore, the proofs in both cases are quite technical
with many details so we decide to separate this work from [MVZ] to better
present the results.

##  2\. Integrally closed edge ideals of edge-weighted graphs

Throughout the paper, we denote by $S=\mathrm{k}[x_{1},\ldots,x_{n}]$ a
standard graded polynomial ring over an arbitrary field $\mathrm{k}$. We first
recall the definition and some properties of the integral closure of monomial
ideals; see [MV] for more details. We then give a simple proof of a result of
[DZCL], which classifies integrally closed edge-weighted graphs.

######  Definition 2.1.

Let $I$ be a monomial ideal of $S$. The exponent set of $I$ is
$E(I)=\\{\mathbf{a}\in\operatorname{\mathbb{N}}^{n}\mid x^{\mathbf{a}}\in
I\\}$. The Newton polyhedron of $I$, denoted by $\operatorname{NP}(I)$, is the
convex hull of the exponent set of $I$ in $\operatorname{\mathbb{R}}^{n}$.

The following result [Ei, Exercise 4.23] is standard.

######  Lemma 2.2.

Let $I$ be a monomial ideal of $S$. The integral closure of $I$ is a monomial
ideal with exponent set
$E(\overline{I})=\operatorname{NP}(I)\cap\operatorname{\mathbb{Z}}^{n}$.

For
$\mathbf{a}=(a_{1},\ldots,a_{n}),\mathbf{b}=(b_{1},\ldots,b_{n})\in\operatorname{\mathbb{R}}^{n}$,
the notation $\mathbf{a}\geq\mathbf{b}$ means that $a_{j}\geq b_{j}$ for all
$j=1,\ldots,n$.

######  Lemma 2.3.

Let $I$ be a monomial ideal and $\mathbf{a}\in\operatorname{\mathbb{N}}^{n}$
be an exponent. Then $x^{\mathbf{a}}\in\overline{I}$ if and only if there
exist nonnegative real numbers $c_{j}$ and exponents $\mathbf{b}_{j}$ of
$E(I)$ corresponding to minimal generators of $I$ for $j=1,\ldots,s$ such that
$\sum_{j=1}^{s}c_{j}\geq 1$ and
$\mathbf{a}\geq\sum_{j=1}^{s}c_{j}\mathbf{b}_{j}$. In particular, when
$x^{\mathbf{a}}$ is a minimal generator of $\overline{I}$, we can choose
$c_{j}$ such that $\sum_{j=1}^{s}c_{j}=1$.

###### Proof.

The conclusion follows from Lemma 2.2 and the fact that for exponents
$\mathbf{a},\mathbf{b}\in\operatorname{\mathbb{N}}^{n}$ and a monomial ideal
$I$, $\mathbf{a}\geq\mathbf{b}$ and $x^{\mathbf{b}}\in I$ implies that
$x^{\mathbf{a}}\in I$. ∎

For a monomial $f$ in $S$, the support of $f$, denoted by
$\operatorname{supp}(f)$, is the set of all indices $i\in[n]$ such that
$x_{i}\mid f$. For a monomial ideal $J$ of $S$ and a subset $V$ of $[n]$, the
restriction of $J$ to $V$, denoted by $J_{V}$, is

$J_{V}=(f\mid f\text{ is a minimal generator of }J\text{ such that
}\operatorname{supp}f\subseteq V).$

We have

######  Corollary 2.4.

Let $I$ be a monomial ideal. Then

$\overline{I_{V}}=(\overline{I})_{V}.$

We also fix the following notation throughout the paper. Let $G$ be a finite
simple graph on the vertex set $V(G)=[n]$ and the edge set $E(G)$. For a
vertex $i\in V(G)$, the neighborhood of $i$ is defined by

$N_{G}(i)=\\{j\in V(G)\mid\\{i,j\\}\in E(G)\\}.$

The degree of a vertex $i$ is the cardinality of its neighborhood. A vertex of
degree $1$ is called a leaf of $G$. Let
$\mathbf{w}:E(G)\to\operatorname{\mathbb{Z}}_{+}$ be a weight function on the
edges of $G$. An edge $e\in E(G)$ is called a nontrivial edge if
$\mathbf{w}(e)>1$.

We now prove that the configurations $F_{1},F_{2},F_{3}$ give rise to non-
integrally closed edge-weighted graphs. We denote by
$\mathbf{e}_{1},\ldots,\mathbf{e}_{n}$ the canonical basis of
$\operatorname{\mathbb{R}}^{n}$.

######  Lemma 2.5.

Assume that $I(G,\mathbf{w})=(x_{1}^{a}x_{2}^{a},x_{2}^{b}x_{3}^{b})$ with
$a,b>1$. Then, $I(G,\mathbf{w})$ is not integrally closed.

###### Proof.

We may assume that $a\leq b$. We have $\mathbf{u}=(a,a,0)$,
$\mathbf{v}=(0,b,b)\in\operatorname{NP}(I(G,\mathbf{w}))$. Let

$\mathbf{w}=\frac{1}{2}(\mathbf{u}+\mathbf{v})=\left(\frac{a}{2},\frac{a+b}{2},\frac{b}{2}\right).$

By Lemma 2.3 and the assumption that $a,b>1$, we deduce that
$x_{1}^{a-1}x_{2}^{b}x_{3}^{b-1}\in\overline{I(G,\mathbf{w})}$. The conclusion
follows. ∎

######  Lemma 2.6.

Assume that
$I(G,\mathbf{w})=(x_{1}^{a}x_{2}^{a},x_{2}^{b}x_{3}^{b},x_{1}^{c}x_{3}^{c})$
with $a,b,c>1$. Then, $I(G,\mathbf{w})$ is not integrally closed.

###### Proof.

We may assume that $a\leq b\leq c$. We have $\mathbf{u}=(a,a,0)$,
$\mathbf{v}=(0,b,b)$ and
$\mathbf{w}=(c,0,c)\in\operatorname{NP}(I(G,\mathbf{w}))$. Let

$\mathbf{x}=\frac{1}{2}(\mathbf{u}+\mathbf{v})=\left(\frac{a}{2},\frac{a+b}{2},\frac{b}{2}\right).$

Since $a,b\geq 2$, by Lemma 2.3, we deduce that
$f=x_{1}^{a-1}x_{2}^{b}x_{3}^{b-1}\in\overline{I(G,\mathbf{w})}$. Since $c\geq
a,b$, $f\notin I(G,\mathbf{w})$. The conclusion follows. ∎

######  Lemma 2.7.

Assume that $I(G,\mathbf{w})=(x_{1}^{a}x_{2}^{a},x_{3}^{b}x_{4}^{b})$ with
$a,b>1$. Then, $I(G,\mathbf{w})$ is not integrally closed.

###### Proof.

We have $\mathbf{u}=(a,a,0,0)$ and
$\mathbf{v}=(0,0,b,b)\in\operatorname{NP}(I(G,\mathbf{w}))$. Let

$\mathbf{w}=\frac{1}{2}(\mathbf{u}+\mathbf{v})=\left(\frac{a}{2},\frac{a}{2},\frac{b}{2},\frac{b}{2}\right).$

Since $a,b>1$, by Lemma 2.3, we deduce that
$x_{1}^{a-1}x_{2}^{a-1}x_{3}^{b-1}x_{4}^{b-1}\in\overline{I(G,\mathbf{w})}$.
The conclusion follows. ∎

######  Theorem 2.8.

Let $(G,\mathbf{w})$ be an edge-weighted graph. Then $I(G,\mathbf{w})$ is
integrally closed if and only if $G$ does not have $F_{1},F_{2},F_{3}$ as an
induced edge-weighted subgraph.

###### Proof.

By Lemmas 2.5, 2.6, and 2.7, we may assume that $(G,\mathbf{w})$ does not have
$F_{1},F_{2},F_{3}$ as induced edge-weighted subgraphs. We need to prove that
$\overline{I(G,\mathbf{w})}=I(G,\mathbf{w})$. We prove this by induction on
$|V(G)|$. The base case where $|V(G)|\leq 2$ is clear. Hence, we may assume
that $|V(G)|\geq 3$. Let $f=x^{\mathbf{a}}$ be a minimal generator of
$\overline{I(G,\mathbf{w})}$. By induction and Corollary 2.4, we may assume
that $G$ does not have isolated vertices and $\operatorname{supp}f=[n]$.

Since $n\geq 3$ and $(G,\mathbf{w})$ has no $F_{1},F_{2},F_{3}$ as an induced
weighted subgraph, we deduce that $(G,\mathbf{w})$ must have an edge with
trivial weight. Let $\\{i,j\\}$ be an edge of $G$ with trivial weight. Then
$x_{i}x_{j}\in I(G,\mathbf{w})$. Since $\operatorname{supp}f=[n]$, $f$ is
divisible by $x_{i}x_{j}$. Hence, $f\in I(G,\mathbf{w})$. The conclusion
follows. ∎

##  3\. Normal edge ideals of edge-weighted graphs

In this section, we prove Theorem 1.1. First, we have some simple lemmas.

######  Lemma 3.1.

Let $I\subseteq S$ be a monomial ideal and
$\mathbf{a}\in\operatorname{\mathbb{N}}^{n}$ be an exponent. Then
$x^{\mathbf{a}}\in\overline{I^{k}}$ if and only if there exist nonnegative
real numbers $c_{j}$ and exponents $\mathbf{b}_{j}$ of $E(I)$ for
$j=1,\ldots,s$ such that $\sum_{j=1}^{s}c_{j}\geq k$ and
$\mathbf{a}\geq\sum_{j=1}^{s}c_{j}\mathbf{b}_{j}$. In particular, when
$x^{\mathbf{a}}$ is a minimal generator of $\overline{I^{k}}$, we can choose
$c_{j}$ such that $\sum_{j=1}^{s}c_{j}=k$.

###### Proof.

The conclusion follows from Lemma 2.2, Lemma 2.3, and the fact that a vector
$\mathbf{v}\in\operatorname{\mathbb{R}}^{n}$ belongs to
$\operatorname{NP}(I^{k})$ if and only if
$\frac{1}{k}\mathbf{v}\in\operatorname{NP}(I)$. ∎

######  Lemma 3.2.

Let $I\subseteq S$ be a monomial ideal and $V\subseteq[n]$. Assume that $I$ is
normal. Then $I_{V}$ is normal.

###### Proof.

The conclusion follows from Corollary 2.4 and the fact that
$(I^{k})_{V}=(I_{V})^{k}$. ∎

We now prove that the configurations $F_{4}$ and $F_{5}$ give rise to non-
normal edge-weighted graphs.

######  Lemma 3.3.

Assume that $(G,\mathbf{w})$ is isomorphic to the configuration $F_{4}$. Then,
$I(G,\mathbf{w})$ is not normal.

###### Proof.

By Lemma 2.7, we may assume that all edges of the odd cycle $C$ in $G$ have
trivial weights. In other words, we may assume that

$I=I(G,\mathbf{w})=(x_{1}x_{2},\ldots,x_{2k-2}x_{2k-1},x_{1}x_{2k-1})+(x_{2k+1}^{a}x_{2k}^{a}),$

where $k,a$ are integers at least $2$. We will prove that $I^{k}$ is not
integrally closed. Let

$\mathbf{u}_{1}=\mathbf{e}_{1}+\mathbf{e}_{2},\ldots,\mathbf{u}_{2k-2}=\mathbf{e}_{2k-2}+\mathbf{e}_{2k-1},\mathbf{u}_{2k-1}=\mathbf{e}_{2k-1}+\mathbf{e}_{1},\mathbf{u}_{2k}=a\mathbf{e}_{2k}+a\mathbf{e}_{2k+1}$

be the vectors in the Newton polyhedron of $I$ corresponding to minimal
generators of $I$. We have

$\mathbf{v}=\frac{1}{2}(\mathbf{u}_{1}+\cdots+\mathbf{u}_{2k})=(1,\ldots,1,\frac{a}{2},\frac{a}{2}).$

Since $a\geq 2$, by Lemma 3.1, we deduce that $f=x_{1}\cdots
x_{2k-1}x_{2k}^{a-1}x_{2k+1}^{a-1}\in\overline{I^{k}}.$ Since there is no edge
between $j$ and $2k$ or $2k+1$ for any $j\in\\{1,\ldots,2k-1\\}$, we deduce
that $f\notin I^{k}$. The conclusion follows. ∎

######  Lemma 3.4.

Assume that $(G,\mathbf{w})$ is isomorphic to the configuration $F_{5}$. Then
$I(G,\mathbf{w})$ is not normal.

###### Proof.

By Lemmas 2.5 and 2.7, we may assume that all edges on the two odd cycles have
trivial weights. In other words, we may assume that

$\displaystyle I=I(D,\mathbf{w})$
$\displaystyle=(x_{1}x_{2},\ldots,x_{2k-2}x_{2k-1},x_{1}x_{2k-1})$
$\displaystyle+(x_{2k}x_{2k+1},\ldots,x_{2k+2l-1}x_{2k+2l},x_{2k}x_{2k+2l})+((x_{i}x_{j})^{w_{ij}}\mid\\{i,j\\}\in
W),$

where $k\geq 2$, $l$ are positive integers and $W\subseteq E(G)$ is a possibly
empty collection of edges $\\{i,j\\}$ with $i\in\\{1,\ldots,2k-1\\}$ and
$j\in\\{2k,\ldots,2k+2l\\}$, and $w_{ij}>1$ for all $\\{i,j\\}\in W$. Let

$\displaystyle\mathbf{u}_{1}$
$\displaystyle=\mathbf{e}_{1}+\mathbf{e}_{2},\ldots,\mathbf{u}_{2k-1}=\mathbf{e}_{2k-1}+\mathbf{e}_{1},$
$\displaystyle\mathbf{u}_{2k}$
$\displaystyle=\mathbf{e}_{2k}+\mathbf{e}_{2k+1},\ldots,\mathbf{u}_{2k+2l}=\mathbf{e}_{2k}+\mathbf{e}_{2k+2l}$

be the vectors in the Newton polyhedron of $I$ corresponding to minimal
generators of $I$. We have

$\mathbf{v}=\frac{1}{2}(\mathbf{u}_{1}+\cdots+\mathbf{u}_{2k+2l})=(1,\ldots,1,1,\ldots,1).$

By Lemma 3.1, we deduce that $f=x_{1}\cdots x_{2k+2l}\in\overline{I^{k+l}}.$
Since every possible edge connecting the two odd cycles has nontrivial weight,
we deduce that $f\notin I^{k+l}$. The conclusion follows. ∎

The following three lemmas are analogous to the corresponding lemmas in [MVZ]
and allow us to achieve the first two reduction steps outlined in the
Introduction.

######  Lemma 3.5.

Assume that $G=G_{1}\cup G_{2}$ is the disjoint union of two subgraphs $G_{1}$
and $G_{2}$ where $G_{2}$ is bipartite. Let $\mathbf{w}$ be a weight function
on $E(G)$ such that $\mathbf{w}(e)=1$ for all $e\in E(G_{2})$. Denote by
$\mathbf{w}_{1}$ the induced weight function of $\mathbf{w}$ on $E(G_{1})$.
Then $I(G,\mathbf{w})$ is normal if and only if $I(G_{1},\mathbf{w}_{1})$ is
normal.

###### Proof.

The conclusion follows from Lemma 3.2 and [MT, Theorem 2.1]. See also [BH,
Corollary 2.8]. ∎

The following result is a consequence of [ANKRQ, Theorem 2.1]. We give an
alternative proof using our approach.

######  Lemma 3.6.

Assume that $v$ is a leaf of $G$ and $\mathbf{w}(\\{u,v\\})=1$, where $u$ is
the unique neighbor of $v$. Let $G^{\prime}=G\backslash v$ be the induced
subgraph of $G$ on $V(G)\backslash v$ and $\mathbf{w}^{\prime}$ be the induced
weight function of $\mathbf{w}$ on $E(G^{\prime})$. Then $I(G,\mathbf{w})$ is
normal if and only if $I(G^{\prime},\mathbf{w}^{\prime})$ is normal.

###### Proof.

By Lemma 3.2, it suffices to prove that if $I(G^{\prime},\mathbf{w}^{\prime})$
is normal then $I(G,\mathbf{w})$ is normal. For simplicity of notation, we
denote by $J=I(G^{\prime},\mathbf{w}^{\prime})$ and $K=I(G,\mathbf{w})$. We
prove by induction on $t$ that $K^{t}=\overline{K^{t}}$. The base case $t=1$
follows from Theorem 2.8. Let $f=x^{\mathbf{a}}\in\overline{K^{t}}$ be a
minimal generator. By induction on the number of vertices and Corollary 2.4,
we may assume that $\operatorname{supp}(f)=[n]$. By Lemma 3.1, there exist
nonnegative real numbers $\beta_{j}$ for $j=1,\ldots,s$ such that
$\sum_{1}^{s}\beta_{j}=t$ and

$\mathbf{a}\geq\mathbf{b}=\sum_{j=1}^{s}\beta_{j}\mathbf{b}_{j},$

where $\mathbf{b}_{j}=(b_{j,1},\ldots,b_{j,n})$ are exponents corresponding to
some minimal generators of $K$. We may assume that $v=n$, $u=n-1$ and
$\mathbf{b}_{s}=\mathbf{e}_{n-1}+\mathbf{e}_{n}$ be the exponent corresponding
to the generator $x_{n-1}x_{n}$ of $K$. By induction on $t$, we may assume
that $\beta_{j}<1$ for all $j=1,\ldots,s$. If $\beta_{s}=0$ then
$f\in\overline{J^{t}}$. By assumption, we deduce that $f\in J^{t}\subseteq
K^{t}$. Hence, we may assume that $\beta_{s}>0$. By the proof of [MV, Lemma
2.9], we deduce that $a_{n}=1$. We can further assume that
$\mathbf{b}_{u},\ldots,\mathbf{b}_{s-1}$ are all the other exponents such that
$b_{j,n-1}>0$. There are two cases.

Case 1. $\sum_{j=u}^{s-1}\beta_{j}<1-\beta_{s}$. In particular,
$\sum_{j=1}^{u-1}\beta_{j}>t-1$. Let
$\mathbf{a}^{\prime}=\mathbf{a}-\mathbf{b}_{s}$. Since $b_{j,n-1}=b_{j,n}=0$
for all $j=1,\ldots,u-1$, we have
$\mathbf{a}^{\prime}\geq\sum_{j=1}^{u-1}\beta_{j}\mathbf{b}_{j}$. By Lemma
3.1, we deduce that $x^{\mathbf{a}^{\prime}}\in\overline{J^{t-1}}=J^{t-1}$.
Hence, $x^{\mathbf{a}}=x^{\mathbf{a}^{\prime}}(x_{n-1}x_{n})\in K^{t}$.

Case 2. $\sum_{j=u}^{s-1}\beta_{j}>1-\beta_{s}$. Then there exist nonnegative
real numbers $\gamma_{j}$ for $j=u,\ldots,s-1$ such that
$\sum_{j=u}^{s-1}\gamma_{j}=1-\beta_{s}$. Let

$\mathbf{b}^{\prime}=\sum_{j=1}^{u-1}\beta_{j}\mathbf{b}_{j}+\sum_{j=u}^{s-1}(\beta_{j}-\gamma_{j})\beta_{j},\;\mathbf{b}^{{}^{\prime\prime}}=\sum_{j=u}^{s-1}\gamma_{j}\beta_{j}+\beta_{s}\mathbf{b}_{s}.$

Then we have $\mathbf{b}=\mathbf{b}^{\prime}+\mathbf{b}^{{}^{\prime\prime}}$
where $\mathbf{b}^{\prime}=(b^{\prime}_{1},\ldots,b^{\prime}_{n})$ with
$b^{\prime}_{n-1}\leq b_{n-1}-1$ and $b^{\prime}_{n}=0$. Let
$\mathbf{a}^{\prime}=\mathbf{a}-\mathbf{b}_{s}$. Then, we have
$\mathbf{a}^{\prime}\geq\mathbf{b}^{\prime}$. By Lemma 3.1, we deduce that
$x^{\mathbf{a}^{\prime}}\in\overline{J^{t-1}}=J^{t-1}$. Hence,
$x^{\mathbf{a}}=x^{\mathbf{a}^{\prime}}(x_{n-1}x_{n})\in K^{t}$. The
conclusion follows. ∎

In the sequel, when $(G,\mathbf{w})$ has an induced subgraph isomorphic to
$F_{i}$ for some $i=1,\ldots,5$ we say that $(G,\mathbf{w})$ has an
$F_{i}$-minor.

######  Lemma 3.7.

Let $(G,\mathbf{w})$ be an edge-weighted graph. Assume that $(G,\mathbf{w})$
has no $F_{i}$-minor for all $i=1,\ldots,5$ and $e\in E(G)$ is an edge of $G$
with $\mathbf{w}(e)>1$. Let $G^{\prime}=G\setminus e$, the subgraph of $G$
obtained by removing the edge $e$, and $\mathbf{w}^{\prime}$ be the induced
weight function on $E(G^{\prime})$. Then $(G^{\prime},\mathbf{w}^{\prime})$
has no $F_{i}$-minor for all $i=1,\ldots,5$.

###### Proof.

Assume by contradiction that $(G^{\prime},\mathbf{w}^{\prime})$ has an
$F_{i}$-minor for some $i=1,\ldots,5$. It is easy to see the following.

If $(G^{\prime},\mathbf{w}^{\prime})$ has an $F_{1}$-minor, then
$(G,\mathbf{w})$ has an $F_{1}$-minor or an $F_{2}$-minor.

If $(G^{\prime},\mathbf{w}^{\prime})$ has an $F_{2}$-minor, then
$(G,\mathbf{w})$ has an $F_{2}$-minor.

If $(G^{\prime},\mathbf{w}^{\prime})$ has an $F_{3}$-minor, then
$(G,\mathbf{w})$ has an $F_{1}$-minor or an $F_{3}$-minor.

If $(G^{\prime},\mathbf{w}^{\prime})$ has an $F_{4}$-minor, then
$(G,\mathbf{w})$ has an $F_{1}$-minor or an $F_{4}$-minor.

If $(G^{\prime},\mathbf{w}^{\prime})$ has an $F_{5}$-minor, then
$(G,\mathbf{w})$ has an $F_{5}$-minor.

In all cases, $(G,\mathbf{w})$ has an $F_{i}$-minor for some $i=1,\ldots,5$, a
contradiction. The conclusion follows. ∎

######  Lemma 3.8.

Let $G$ be an even cycle. We denote by $\mathbf{b}_{1},\ldots,\mathbf{b}_{2k}$
the exponent vectors corresponding to the minimal generators of $I(G)$. Let
$\beta_{1},\ldots,\beta_{2k}$ be positive real numbers. Then there exist
nonnegative real numbers $\gamma_{1},\ldots,\gamma_{2k}$ such that
$\sum_{j=1}^{2k}\gamma_{j}=\sum_{j=1}^{2k}\beta_{j}$, at least one of
$\gamma_{j}$ is zero and
$\sum_{j=1}^{2k}\gamma_{j}\mathbf{b}_{j}=\sum_{j=1}^{2k}\beta_{j}\mathbf{b}_{j}$.

###### Proof.

By the assumption, we deduce that

$\mathbf{b}_{1}+\mathbf{b}_{3}+\cdots+\mathbf{b}_{2k-1}=\mathbf{b}_{2}+\cdots+\mathbf{b}_{2k}.$

We may assume that $\beta_{2k}=\min\\{\beta_{2j}\mid j=1,\ldots,k\\}$. We have

$\displaystyle\sum_{j=1}^{2k}\beta_{j}\mathbf{b}_{j}$
$\displaystyle=\sum_{j=0}^{k-1}\beta_{2j+1}\mathbf{b}_{2j+1}+\sum_{j=1}^{k}\beta_{2j}\mathbf{b}_{2j}$
$\displaystyle=\sum_{j=0}^{k-1}(\beta_{2j+1}+\beta_{2k})\mathbf{b}_{2j+1}+\sum_{j=1}^{k-1}(\beta_{2j}-\beta_{2k})\mathbf{b}_{2j}.$

The conclusion follows. ∎

######  Lemma 3.9.

Let $(G,\mathbf{w})$ be an edge-weighted even cycle with exactly one
nontrivial edge. We denote by $\mathbf{b}_{1},\ldots,\mathbf{b}_{2k}$ the
exponent vectors corresponding to the minimal generators of $I(G)$. Let
$\beta_{1},\ldots,\beta_{2k}$ be positive real numbers. Then there exist
nonnegative real numbers $\gamma_{1},\ldots,\gamma_{2k}$ such that
$\sum_{j=1}^{2k}\gamma_{j}=\sum_{j=1}^{2k}\beta_{j}$, at least one of
$\gamma_{j}$ is zero and
$\sum_{j=1}^{2k}\gamma_{j}\mathbf{b}_{j}\leq\sum_{j=1}^{2k}\beta_{j}\mathbf{b}_{j}$.

###### Proof.

We may assume that $\mathbf{b}_{1}=a(\mathbf{e}_{1}+\mathbf{e}_{2})$ is the
exponent corresponding to the only nontrivial edge of $G$. Let
$\beta_{2j+1}=\min\\{\beta_{2l+1}\mid l=0,\ldots,k-1\\}$ and

$\gamma_{i}=\begin{cases}\beta_{i}+\beta_{2j+1}&\text{ if }i\text{ is
even},\\\ \beta_{i}-\beta_{2j+1}&\text{ if }i\text{ is odd}.\end{cases}$

Then $\sum_{i=1}^{2k}\gamma_{i}=\sum_{i=1}^{2k}\beta_{i}$, at least one of
$\gamma_{i}$ is zero. Furthermore,

$\sum_{i=1}^{2k}\gamma_{i}\mathbf{b}_{i}=\sum_{i=1}^{2k}\beta_{i}\mathbf{b}_{i}+\beta_{2j+1}\left(\frac{1}{a}-1\right)\mathbf{b}_{1}\leq\mathbf{b}.$

The conclusion follows. ∎

We now introduce the compact graphs defined by Wang and Lu [WL] and their
classification.

######  Definition 3.10.

A simple graph $G$ is compact if $G$ has no even cycles and satisfies the odd
cycle conditions.

The following classification of compact graphs is [WL, Theorem 2.12]. For
convenience in our applications, we use different terminology from that of
Wang and Lu. We call a finite collection of odd cycles sharing a common vertex
a bouquet of odd cycles, and the common vertex is called its stem. In
particular, an odd cycle itself is considered to be a special bouquet of odd
cycles.

######  Theorem 3.11.

A connected simple graph $G$ without leaves is compact if and only if it
belongs to one of the following classes of graphs.

  1. (1)

A bouquet of odd cycles.

  2. (2)

Two bouquets of odd cycles whose stems are joined by an edge and possibly
another path of even length.

  3. (3)

Three bouquets of odd cycles whose stems form a triangle.

######  Lemma 3.12.

Let $G$ be an edge-weighted connected compact graph and $\mathbf{w}$ be a
weight function on the edges of $G$. Assume that $(G,\mathbf{w})$ has only one
nontrivial weight and has no $F_{4},F_{5}$-minor. Then $I(G,\mathbf{w})$ is
normal.

###### Proof.

We prove by induction on the number of vertices of $G$. The base case where
$|V(G)|=2$ is clear. By Lemma 3.6, we may assume that $(G,\mathbf{w})$ has no
leaf edges with trivial weights. Let $e=\\{1,2\\}$ be the unique edge of $G$
such that $\mathbf{w}(e)=a>1$. Let $f=x^{\mathbf{a}}$ be a minimal generator
of $\overline{I^{t}}$ for some $t$. By Lemma 3.1, there exist nonnegative real
numbers $\beta_{1},\ldots,\beta_{s}$ and exponents
$\mathbf{b}_{1},\ldots,\mathbf{b}_{s}$ corresponding to minimal generators of
$I(G,\mathbf{w})$ such that

$\sum_{j=1}^{s}\beta_{j}=t\text{ and
}\mathbf{a}\geq\mathbf{b}=\sum_{j=1}^{s}\beta_{j}\mathbf{b}_{j}.$

We prove by induction on $t$ that $f\in I^{t}$. Since $(G,\mathbf{w})$ has no
$F_{4}$-minor, we deduce that $e$ contains a unique stem of $G$ and one of the
following must hold:

  1. (1)

$e$ is adjacent to an odd cycle in $G$.

  2. (2)

$e$ is connected to an odd cycle in $G$ by a trivial edge.

  3. (3)

$e$ is an edge of an odd cycle in $G$.

Furthermore, if (1) or (2) does not hold then $G$ contains a unique cycle.

Case 1. $e$ is adjacent to an odd cycle in $G$. Let $C=1,3,\ldots,2k$ be an
odd cycle in $G$. We may assume that
$\mathbf{b}_{1}=a(\mathbf{e}_{1}+\mathbf{e}_{2})$,
$\mathbf{b}_{2}=\mathbf{e}_{1}+e_{3},\mathbf{b}_{3}=\mathbf{e}_{3}+\mathbf{e}_{4},\ldots,\mathbf{b}_{2k}=\mathbf{e}_{1}+\mathbf{e}_{2k}$
are the exponents corresponding to the edge $e$ and the odd cycle $C$. If
$\beta_{1}=0$ then $f\in\overline{I(G\backslash e)^{t}}=I(G\backslash
e)^{t}\subseteq I(G,\mathbf{w})^{t}$. Hence, we may assume that $\beta_{1}>0$.
Let $\beta_{2j+1}=\min\\{\beta_{2l+1}\mid 1<l\leq k-1\\}.$ There are two
subcases.

Case 1.a. $\beta_{1}\leq\beta_{2j+1}$. Let

$\gamma_{i}=\begin{cases}\beta_{i}+\beta_{1}&\text{ if }i\text{ is even},\\\
\beta_{i}-\beta_{1}&\text{ if }i\text{ is odd}.\end{cases}$

Then we have $\sum_{i=1}^{2k}\gamma_{i}=\sum_{i=1}^{2k}\beta_{i}$ and

$\sum_{i=1}^{s}\gamma_{i}\mathbf{b}_{i}=\sum_{i=2}^{s}\beta_{i}\mathbf{b}_{i}+\beta_{1}2\mathbf{e}_{1}\leq\mathbf{b}.$

Hence, we may reduce to the case $\beta_{1}=0$ and the conclusion follows.

Case 1.b. $\beta_{1}>\beta_{2j+1}$. Let

$\gamma_{i}=\begin{cases}\beta_{i}+\beta_{2j+1}&\text{ if }i\text{ is
even},\\\ \beta_{i}-\beta_{2j+1}&\text{ if }i\text{ is odd}.\end{cases}$

Then we have $\sum_{i=1}^{s}\gamma_{i}=\sum_{i=1}^{s}\beta_{i}$ and

$\sum_{i=1}^{s}\gamma_{i}\mathbf{b}_{i}=\sum_{i=2}^{s}\beta_{i}\mathbf{b}_{i}+(\beta_{1}-\beta_{2j+1})\mathbf{b}_{1}+2\mathbf{e}_{1}\beta_{2j+1}\leq\mathbf{b}.$

Since $\gamma_{2j+1}=0$ and $G\backslash e_{2j+1}$ is compact, by induction on
the number of edges and Lemma 3.1, we deduce that

$f\in\overline{I(G\backslash e_{2j+1},\mathbf{w})^{t}}=I(G\backslash
e_{2j+1},\mathbf{w})^{t}\subseteq I(G,\mathbf{w})^{t}.$

Case 2. $e=\\{1,2\\}$ is connected to an odd cycle in $G$ by a trivial edge.
Since $(G,\mathbf{w})$ has no $F_{4}$-minor, we deduce that the vertices of
the connecting edge are the stems of $G$. In other words, we may assume that
$\\{1,3\\}\in E(G)$ and $3$ is a stem of a bouquet of odd cycles in $G$. In
particular, we have $C=3,\ldots,2k+1$ is an odd cycle of $G$. We assume that
$\mathbf{b}_{1},\ldots,\mathbf{b}_{2k+1}$ are the exponents corresponding to
the edges $\\{1,2\\},\\{1,3\\}$ and edges of $C$. Let
$\beta_{2j+1}=\min\\{\beta_{2l+1}\mid 1<l\leq k-1\\}.$ There are two subcases.

Case 2.a. $\beta_{1}\leq 2\beta_{2j+1}/a$. Let $\gamma=a\beta_{1}/2$,
$\gamma_{1}=0$ and

$\gamma_{i}=\begin{cases}\beta_{i}+\gamma&\text{ if }i>2\text{ is even},\\\
\beta_{i}-\gamma&\text{ if }i\text{ is odd}\\\ \beta_{i}+2\gamma&\text{ if
}i=2.\end{cases}$

Then we have
$\sum_{i=1}^{2k+1}\gamma_{i}=\sum_{i=1}^{2k+1}\beta_{i}+\gamma-\beta_{1}$.
Hence,

$\sum_{i=1}^{s}\gamma_{i}\geq\sum_{i=1}^{s}\beta_{i}\text{ and
}\sum_{i=1}^{s}\gamma_{i}\mathbf{b}_{i}=\sum_{i=1}^{s}\beta_{i}\mathbf{b}_{i}-a\beta_{1}\mathbf{e}_{2}\leq\mathbf{b}.$

Since $\gamma_{1}=0$, by Lemma 3.1, we deduce that $f\in\overline{I(G\setminus
e)^{t}}=I(G\backslash e)^{t}\subseteq I(G,\mathbf{w})^{t}$.

Case 2.b. $\beta_{1}>2\beta_{2j+1}/a$. Let

$\gamma_{i}=\begin{cases}\beta_{i}+\beta_{2j+1}&\text{ if }i>2\text{ is
even},\\\ \beta_{i}-\beta_{2j+1}&\text{ if }i>1\text{ is odd}\\\
\beta_{i}+2\beta_{2j+1}&\text{ if }i=2\\\ \beta_{1}-2\beta_{2j+1}/a&\text{ if
}i=1.\end{cases}$

Then we have

$\displaystyle\sum_{i=1}^{s}\gamma_{i}$
$\displaystyle=\sum_{i=1}^{s}\beta_{i}+\beta_{2j+1}\left(1-\frac{2}{a}\right)\geq\sum_{i=1}^{s}\beta_{i}\text{
and }$ $\displaystyle\sum_{i=1}^{s}\gamma_{i}\mathbf{b}_{i}$
$\displaystyle=\sum_{i=1}^{s}\beta_{i}\mathbf{b}_{i}-2\beta_{2j+1}\mathbf{e}_{2}\leq\mathbf{b}.$

Since $\gamma_{2j+1}=0$ and $G\backslash e_{2j+1}$ is compact, by induction on
the number of edges and Lemma 3.1, we deduce that

$f\in\overline{I(G\backslash e_{2j+1},\mathbf{w})^{t}}=I(G\backslash
e_{2j+1},\mathbf{w})^{t}\subseteq I(G,\mathbf{w})^{t}.$

Case 3. $e$ is an edge of an odd cycle and $G$ is the odd cycle itself. This
case follows from [DZCL, Theorem 4.9]. We give an alternative proof using our
approach. We denote by $\mathbf{b}_{1},\ldots,\mathbf{b}_{2k+1}$ be the
exponents corresponding to minimal generators of $I(G,\mathbf{w})$
corresponding to the edges of $C$ by clockwise order, where
$\mathbf{b}_{1}=a(\mathbf{e}_{1}+\mathbf{e}_{2})$ corresponds to $e$. In this
case, we have $s=2k+1$. If $\beta_{j}=0$ for some $j$ then we are done by
induction on the number of edges. Hence, we may assume that $\beta_{j}>0$ for
all $j$. Fix a pair of exponents $\mathbf{b}_{2j},\mathbf{b}_{2j+1}.$ Let
$\gamma$ be any positive numbers less than or equal to $\beta_{i}$ for all
$i\geq 2j$. Then we let $\gamma_{1}=\beta_{1}-\gamma/a$ and for all $i\geq
2j+1$, we set

$\gamma_{i}=\begin{cases}\beta_{i}+\gamma&\text{ if }i\text{ is odd},\\\
\beta_{i}-\gamma&\text{ if }i\text{ is even}.\end{cases}$

Then we have

$\displaystyle\sum_{i=1}^{2k+1}\gamma_{i}$
$\displaystyle=\sum_{i=1}^{2k+1}\beta_{i}+\gamma\left(1-\frac{1}{a}\right)\geq\sum_{i=1}^{2k+1}\beta_{i}\text{
and }$ $\displaystyle\sum_{i=1}^{2k+1}\gamma_{i}\mathbf{b}_{i}$
$\displaystyle=\sum_{i=1}^{2k+1}\beta_{i}\mathbf{b}_{i}-\gamma\mathbf{e}_{2}+\gamma\mathbf{e}_{2j+1}.$

Note that $b_{2j+1}=\beta_{2j}+\beta_{2+1}$. Hence, if we choose
$\gamma\leq\lceil\beta_{2j}+\beta_{2j+1}\rceil-(\beta_{2j}+\beta_{2j+1})$ we
deduce that we may replace $\beta_{i}$ by $\gamma_{i}$ and still get
$\mathbf{a}\geq\sum_{i=1}^{2k+1}\gamma_{i}\beta_{i}$. Now, we start clockwise
and apply these changes to get either
$\gamma_{2j}+\gamma_{2j+1}=\lceil\beta_{2j}+\beta_{2j+1}\rceil$ for all
$j=1,\ldots,k$ or $\gamma_{i}=0$ for some $i$. If $\gamma_{i}=0$ for some $i$
then we are done by induction on the number of edges. If
$\gamma_{2j}+\gamma_{2j+1}\geq 2$ then one of them is at least $1$ and we are
done by induction on $t$. Hence, we assume that $\gamma_{2j}+\gamma_{2j+1}=1$
for all $j=1,\ldots,k$. Since $\sum_{j=1}^{2k+1}\gamma_{j}\geq t$, we deduce
that $\gamma_{1}\geq t-k.$ By induction on $t$ we may then reduce to the case
that $\gamma_{1}=0$. That completes the proof of Case 3 and the lemma. ∎

We are now ready for the proof of the main theorem.

###### Proof of Theorem 1.1.

We assume that $\mathbf{w}$ is nontrivial. By Lemmas 2.5, 2.6, 2.7, 3.3, and
3.4, we may assume that $(G,\mathbf{w})$ has no $F_{i}$-minor for all
$i=1,\ldots,5$. We prove by induction on $|V(G)|$ and on $|E(G)|$ that
$I=I(G,\mathbf{w})$ is normal. For ease of reading, we divide the proof into
several steps.

Step 1. Reduction to the case where $G$ is connected. Assume that $G=G_{1}\cup
G_{2}$ is the disjoint union of two induced subgraphs $G_{1}$ and $G_{2}$.
Since $G$ has no $F_{3}$-minor, we deduce that one of the weight functions
$\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ induced on the edges of $G_{1}$ and
$G_{2}$ are trivial. We may assume that $\mathbf{w}_{1}$ is nontrivial and
$\mathbf{w}_{2}$ is trivial. Since $(G,\mathbf{w})$ has no $F_{4}$-minor, we
deduce that $G_{2}$ is bipartite. By Lemma 3.5 and induction on $|V(G)|$, we
may assume that $G$ is connected.

Step 2. Reduction to the case where $(G,\mathbf{w})$ has only one nontrivial
edge. Let $f=x^{\mathbf{a}}$ be a minimal generator of $\overline{I^{t}}$. By
Lemma 3.1, there exist nonnegative real numbers $\beta_{1},\ldots,\beta_{s}$
and exponents $\mathbf{b}_{1},\ldots,\mathbf{b}_{s}$ corresponding to minimal
generators of $I$ such that $\sum_{j=1}^{s}\beta_{j}=t$ and
$\mathbf{a}\geq\mathbf{b}=\sum_{j=1}^{s}\beta_{j}\mathbf{b}_{j}$. We may
assume that $\mathbf{b}_{1},\mathbf{b}_{2}$ are exponents in
$\operatorname{NP}(I)$ corresponding to edges $e_{1},e_{2}$ of
$(G,\mathbf{w})$ with nontrivial weights $\mathbf{w}(e_{1})=a$ and
$\mathbf{w}(e_{2})=b$. There are two cases.

Case 1. $e_{1}$ and $e_{2}$ have no common vertices. We may assume that
$e_{1}=\\{1,2\\}$ and $e_{2}=\\{3,4\\}$. Since $(G,\mathbf{w})$ has no
$F_{3}$-minor, there is a trivial edge in $(G,\mathbf{w})$ connecting $e_{1}$
and $e_{2}$. We may assume that $\\{2,3\\}\in E(G)$ is a trivial edge in
$(G,\mathbf{w})$ and $\mathbf{b}_{3}$ is the corresponding exponent in
$\operatorname{NP}(I)$. Note that,
$\mathbf{b}_{1}=a(\mathbf{e}_{1}+\mathbf{e}_{2}),\mathbf{b}_{2}=b(\mathbf{e}_{3}+\mathbf{e}_{4}),\mathbf{b}_{3}=\mathbf{e}_{2}+\mathbf{e}_{3}$.
We may further assume that $\beta_{1}a\leq\beta_{2}b$. Let $\gamma_{1}=0$,
$\gamma_{2}=(\beta_{2}b-\beta_{1}a)/b$, $\gamma_{3}=\beta_{3}+\beta_{1}a$.
Then

$\displaystyle\mathbf{c}$
$\displaystyle=\gamma_{1}\mathbf{b}_{1}+\gamma_{2}\mathbf{b}_{2}+\gamma_{3}\mathbf{b}_{3}=\beta_{2}\mathbf{b}_{2}-\beta_{1}a(\mathbf{e}_{3}+\mathbf{e}_{4})+(\beta_{3}+\beta_{1}a)(\mathbf{e}_{2}+\mathbf{e}_{3})$
$\displaystyle=\beta_{2}\mathbf{b}_{2}+\beta_{3}\mathbf{b}_{3}-\beta_{1}a\mathbf{e}_{4}+\beta_{1}a\mathbf{e}_{2}\leq\beta_{1}\mathbf{b}_{1}+\beta_{2}\mathbf{b}_{2}+\beta_{3}\mathbf{b}_{3}.$

Furthermore,

$\gamma_{1}+\gamma_{2}+\gamma_{3}=\beta_{2}+\beta_{3}+\beta_{1}(a-a/b)\geq\beta_{1}+\beta_{2}+\beta_{3}.$

Since $\gamma_{1}=0$, by Lemma 3.1, we deduce that
$x^{\mathbf{a}}\in\overline{I(G^{\prime},\mathbf{w}^{\prime})^{t}}$, where
$G^{\prime}=G\backslash e_{1}$ and $\mathbf{w}^{\prime}$ is the induced weight
function on the edges of $G^{\prime}$. By Lemma 3.7, we have that
$(G^{\prime},\mathbf{w}^{\prime})$ has no $F_{i}$-minors for all
$i=1,\ldots,5$. By induction on the number of edges, we deduce that
$x^{\mathbf{a}}\in I(G^{\prime},\mathbf{w}^{\prime})^{t}\subseteq I^{t}$.

Case 2. $e_{1}$ and $e_{2}$ have a common vertex. We may assume that
$e_{1}=\\{1,2\\}$ and $e_{2}=\\{1,3\\}$. Since $(G,\mathbf{w})$ has no
$F_{1}$-minor, we deduce that $e_{3}=\\{2,3\\}$ is an edge of $G$ with
$\mathbf{w}(e_{3})=1$. Note that,
$\mathbf{b}_{1}=a(\mathbf{e}_{1}+\mathbf{e}_{2}),\mathbf{b}_{2}=b(\mathbf{e}_{1}+\mathbf{e}_{3}),\mathbf{b}_{3}=\mathbf{e}_{2}+\mathbf{e}_{3}$.
As in case 1, we may further assume that $\beta_{1}a\leq\beta_{2}b$. Let
$\gamma_{1}=0$, $\gamma_{2}=(\beta_{2}b-\beta_{1}a)/b$,
$\gamma_{3}=\beta_{3}+\beta_{1}a$. Then

$\displaystyle\mathbf{c}$
$\displaystyle=\gamma_{1}\mathbf{b}_{1}+\gamma_{2}\mathbf{b}_{2}+\gamma_{3}\mathbf{b}_{3}=\beta_{2}\mathbf{b}_{2}-\beta_{1}a(\mathbf{e}_{1}+\mathbf{e}_{3})+(\beta_{3}+\beta_{1}a)(\mathbf{e}_{2}+\mathbf{e}_{3})$
$\displaystyle=\beta_{2}\mathbf{b}_{2}+\beta_{3}\mathbf{b}_{3}-\beta_{1}a\mathbf{e}_{1}+\beta_{1}a\mathbf{e}_{2}\leq\beta_{1}\mathbf{b}_{1}+\beta_{2}\mathbf{b}_{2}+\beta_{3}\mathbf{b}_{3}.$

We may now proceed as in Case 1 to obtain the reduction.

Step 3. Conclusion step. By Step 1, Step 2, and Lemmas 3.7, 3.8, 3.9, we may
assume that $G$ is compact with only one nontrivial edge. The conclusion
follows from Lemma 3.12. ∎

### Acknowledgments

Guangjun Zhu is supported by the Natural Science Foundation of Jiangsu
Province (No. BK20221353).

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# Lost in Transcription: Identifying and Quantifying the Accuracy Biases of
Automatic Speech Recognition Systems Against Disfluent Speech

Dena Mujtaba1, Nihar R. Mahapatra1, Megan Arney1, J. Scott Yaruss1
Hope Gerlach-Houck2, Caryn Herring3, and Jia Bin1
1Michigan State University  2Western Michigan University
3FRIENDS: The National Association of Young People Who Stutter
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Automatic speech recognition (ASR) systems, increasingly prevalent in
education, healthcare, employment, and mobile technology, face significant
challenges in inclusivity, particularly for the 80 million-strong global
community of people who stutter. These systems often fail to accurately
interpret speech patterns deviating from typical fluency, leading to critical
usability issues and misinterpretations. This study evaluates six leading
ASRs, analyzing their performance on both a real-world dataset of speech
samples from individuals who stutter and a synthetic dataset derived from the
widely-used LibriSpeech benchmark. The synthetic dataset, uniquely designed to
incorporate various stuttering events, enables an in-depth analysis of each
ASR’s handling of disfluent speech. Our comprehensive assessment includes
metrics such as word error rate (WER), character error rate (CER), and
semantic accuracy of the transcripts. The results reveal a consistent and
statistically significant accuracy bias across all ASRs against disfluent
speech, manifesting in significant syntactical and semantic inaccuracies in
transcriptions. These findings highlight a critical gap in current ASR
technologies, underscoring the need for effective bias mitigation strategies.
Addressing this bias is imperative not only to improve the technology’s
usability for people who stutter but also to ensure their equitable and
inclusive participation in the rapidly evolving digital landscape.

Lost in Transcription: Identifying and Quantifying the Accuracy Biases of
Automatic Speech Recognition Systems Against Disfluent Speech

##  1 Introduction

Figure 1: Transcript produced by an automatic speech recognition (ASR) model
from a speech sample of an individual who stutters, sourced from FluencyBank
Ratner and MacWhinney (2018). This figure illustrates a real-world scenario
where the individual is conducting an interview. Highlighted are both the
disfluencies characteristic of stuttered speech and the corresponding
transcription inaccuracies produced by the ASR.

###  1.1 Background and Motivation

Stuttering is a complex neurodevelopmental condition affecting over 80 million
people worldwide Wu (2023); Yairi and Ambrose (2013). This condition is
characterized by various involuntary speech disruptions or disfluencies
Johnson (1959); Ambrose and Yairi (1999), including repetitions of parts of
words (“li-li-like this”), prolongations of sounds (“lllllike this”), and
blocks (silent periods, “l—ike this”). While everyone may experience
disfluencies, in people who stutter, these are more pronounced and are
accompanied by a loss of control over speech Tichenor and Yaruss (2019).
Society’s negative perceptions of stuttered speech often leads to
discrimination and lower job satisfaction, as fluent speakers tend to be
preferred in employment settings Gerlach et al. (2018); Gonzales (2022);
Plexico et al. (2019).

In the post-COVID-19 era, characterized by the widespread adoption of voice-
activated artificial intelligence (voice AI), individuals who stutter are
confronted with heightened challenges, especially in employment contexts.
Automatic speech recognition (ASR) systems, the fundamental technology
underpinning voice AI and designed to transcribe spoken language into text
Prabhavalkar et al. (2023), have emerged as notable barriers in various
aspects of daily and professional life for this demographic. The broad
integration of ASR technology across sectors such as education, healthcare,
and employment, as well as in mobile devices and other applications
Prabhavalkar et al. (2023); Malik et al. (2021), has led to the inadvertent
marginalization of people who stutter. These systems frequently struggle to
accurately decode speech that differs from typical patterns Wheeler ; Wu
(2023); Lea et al. (2023), a fact exemplified in Fig. 1. In stark contrast,
ASRs demonstrate up to 95% word accuracy when processing standard speech Tobin
and Tomanek (2022). The ramifications of this disparity are profound, ranging
from difficulties in executing simple tasks, such as using automated phone
systems for prescription refills, to experiencing bias in AI-driven job
interviews Junior et al. (2020); Center (2019); Meyer (2018).

As the global voice recognition market is projected to nearly quintuple by
2030, reaching a valuation of US$59.62 billion Insights (2022), the prevalence
of AI-powered voice technologies, bolstered by the recent success of large
language models, is set to increase dramatically Horvitz (2016). This growth
suggests an impending escalation in bias against individuals who stutter.
Leading technology companies like Apple and Google have begun addressing
variations in speech, introducing alternative solutions such as typing
commands to voice assistants Wheeler . However, these measures, while helpful,
fall short of addressing the inherent bias in ASR accuracy. In particular,
initiatives like Google’s Project Euphonia focus on collecting data for
differing speech patterns, aiming to enhance ASR capabilities for
understanding people with speech impairments like dysarthria Tobin and Tomanek
(2022). However, these efforts often neglect the unique aspects of stuttered
speech. Given the significant variability of stuttering across different
individuals and situations Tichenor and Yaruss (2021), this oversight reveals
a critical gap. Unless ASRs are specifically tailored and refined to
accommodate the diverse expressions of stuttering, they are likely to continue
underperforming for individuals who stutter, exacerbating existing societal
biases.

###  1.2 Our Contributions

This paper examines the response of contemporary ASRs to disfluent speech
characteristic of stuttering, with a focus on identifying and quantifying
biases. While previous research has explored speech variations such as
dysarthria Shor et al. (2019); Moore et al. (2018) and ASR fairness in the
context of gender, age, or accent differences Feng et al. (2021); Liu et al.
(2022), biases specific to disfluent speech have remained largely unexplored.
The work of Lea et al. Lea et al. (2023) investigates interactions between
people who stutter and AI-based voice assistants, yet it stops short of
providing a quantitative analysis of biases.

Overall, the main contributions of this paper relative to previous work are
threefold: (1) Quantitative analysis of bias: We conduct a comprehensive
analysis of bias in both the word error rate (WER) and character error rate
(CER) of six popular ASRs, utilizing real speech data from people who stutter.
This approach offers a deeper understanding of how these systems perform with
speech patterns that differ from the norm. (2) Semantic accuracy evaluation:
We extend our analysis to the semantic accuracy of these ASRs, exploring how
well they preserve the meaning of disfluent speech, which is crucial for
effective communication. (3) Synthetic disfluent speech dataset: To address
the scarcity of stuttered speech datasets, we have developed a synthetic
disfluent speech dataset using novel text-to-speech and audio manipulation
techniques. This synthetic dataset, designed to complement real stuttered
speech samples, allows us to finely control the types and frequencies of
disfluencies in speech samples. This controlled environment enables a granular
examination of how various disfluencies and the inherent variability in
stuttering affect ASR performance. Our findings aim to inform bias mitigation
strategies in ASR development, ultimately aiding people who stutter in their
daily interactions with voice-driven technologies.

The remainder of the paper is organized as follows. In Section 2, we provide a
literature review of state-of-the-art ASRs, ASRs for people with speech
variations, and define fairness/bias measurement in the context of AI. Section
3 details our methodology, datasets, models, and metrics for studying bias.
Section 4 presents our findings and their analysis. Finally, Section 5
concludes with remarks and suggestions for future research directions.

##  2 Related Work

This section provides a brief literature review of three key areas to set the
stage for subsequent sections examining bias in ASRs: the evolution and
current state of ASR models, their adaptation for people with speech
differences, and the measurement of bias in AI models.

###  2.1 Automatic Speech Recognition

Recent advancements in ASR have been largely driven by the integration of deep
neural networks, leading to a significant reduction in WER compared to
classical statistical multi-stage models Prabhavalkar et al. (2023). These
neural network models adopt an end-to-end (E2E) approach, directly mapping
input acoustic features to their corresponding transcripts Prabhavalkar et al.
(2023). Key datasets like LibriSpeech Panayotov et al. (2015), encompassing
1,000 hours of diverse audiobook recordings, and TIMIT Garofolo (1983),
featuring recordings of 630 speakers across American English dialects, have
been pivotal in training and benchmarking ASRs. Among the notable ASR
architectures are the Conformer Gulati et al. (2020), wav2vec 2.0 Baevski et
al. (2020), and Whisper Radford et al. (2023), with Whisper demonstrating
remarkable zero-shot performance and the best WER at the time of this writing
Gulati et al. (2020); Baevski et al. (2020); Radford et al. (2023). However,
these models’ effectiveness on stuttered speech remains underexplored.
Comprehensive surveys on ASR systems are provided by Malik et al. Malik et al.
(2021) and on E2E ASRs by Prabhavalkar et al. Prabhavalkar et al. (2023).

###  2.2 Automatic Speech Recognition for People with Speech Differences

Efforts to enhance ASR performance for people with different speech patterns,
such as stuttering, have primarily focused on disfluency identification. This
involves classifying disfluency events to enhance ASR accuracy, either by
adjusting, ignoring, or removing these elements before they are processed by
an ASR, or by providing their location to the ASR Zayats et al. (2016);
Shonibare et al. (2022). Notable datasets supporting stuttering identification
include SEP-28K, KSoF, and LibriStutter, although they lack properly coded
transcripts for ASR training and evaluation Sheikh et al. (2022); Lea et al.
(2021); Kourkounakis et al. (2020). While there is limited research on
directly enhancing ASRs for stuttering, some progress has been made. Lea et
al. fine-tuned Apple’s Speech framework models with data from people who
stutter, observing a reduction in WER. They also applied disfluency refinement
and endpoint truncation techniques to improve accuracy Lea et al. (2023).
Mitra et al.’s work on tuning decoding parameters of ASRs also contributes to
this effort Mitra et al. (2021). Additionally, other research, such as those
by Tobin et al., Shor et al., and MacDonald et al., has focused on adapting
ASRs for various speech differences Tobin and Tomanek (2022); Shor et al.
(2019); MacDonald et al. (2021); Kim et al. (2008); Moore et al. (2020).

###  2.3 Fairness in Artificial Intelligence

Fairness in AI involves mitigating human and systemic biases that may arise
from factors like unrepresentative training data, inappropriate features used
during training, or biased target labels Mujtaba and Mahapatra (2019). The
concept of fairness in AI has evolved, with historical foundations in anti-
discrimination legislation and contemporary frameworks like the AI Risk
Management Framework by NIST aiming to enhance AI trustworthiness Commision ;
NIST . A fundamental fairness criterion in AI literature is “fairness through
unawareness” Kusner et al. (2017):

######  Definition 1.

Fairness Through Unawareness: An AI model $C$ is deemed fair under the
principle of ’Fairness Through Unawareness’ if it does not explicitly use
protected attributes $A$ in its decision-making process. This principle is
operationalized through a mapping $\hat{Y}{:}\;X\rightarrow Y$, where $X$
represents the set of observable features excluding any protected attributes
$A$, $\hat{Y}$ represents a model’s prediction, and $Y$ the intended outcome.

In this paper, $A$ represents the presence of disfluencies, $C$ is the ASR
model under evaluation, $X$ is the user’s speech input, and $Y$ is the
transcript produced by an ASR. Our study investigates the correlation between
the presence of disfluencies ($A$) and model error rates to evaluate ASR
fairness. This can be captured through the notion of “counterfactual fairness”
Kusner et al. (2017); Wachter et al. (2017):

######  Definition 2.

Counterfactual Fairness: An AI model $C$ is deemed counterfactually fair if
its predictions $\hat{Y}$ in the real world are the same as those in the
counterfactual world where an individuals’ protected attributes $A$ belonged
to a different group.

##  3 Methodology

To investigate accuracy biases in contemporary ASR systems against disfluent
speech, we have designed two experiments, each focusing on a different speech
dataset. The first dataset is well-known in the speech-language pathology
(SLP) community, and the second is widely used in ASR modeling and evaluation.
Our objective is to compare ASR performance on two versions of these datasets:
one with disfluencies and another identical in content but without
disfluencies.

Experiment 1 utilizes the FluencyBank dataset Ratner and MacWhinney (2018),
known for its collection of real stuttered speech samples with accompanying
transcripts (detailed in Sec. 3.2). We refer to this original dataset as
FluencyBank-Y (FB-Y), where “Y” indicates the presence of disfluencies. These
samples are transcribed using six different ASR models (Sec. 3.1), and the
accuracy of these transcriptions is measured using WER, CER, and BERTScore, a
metric for semantic similarity analysis (Sec. 3.4). For comparative analysis,
we generate a version of this dataset with apparent instances of stuttering or
disfluencies removed, named FluencyBank-N (FB-N), where “N” signifies the
absence of disfluencies, using text-to-speech (TTS) techniques (Sec. 3.3). We
then employ the Spearman correlation coefficient test to systematically
compare the performance of the ASRs on both the original FB-Y and the modified
FB-N datasets.

To complement Experiment 1, Experiment 2 utilizes the LibriSpeech dataset
Panayotov et al. (2015), frequently used in training and evaluating ASRs and
characterized by standard, non-disfluent speech samples. We refer to this
dataset as LibriSpeech-N or LS-N to convey the absence of disfluencies. To
assess the impact of disfluencies, we create a synthetic dataset by
introducing controlled disfluency events into the LibriSpeech dataset. This is
achieved through a combination of audio processing and TTS methods. This newly
modified dataset, denoted LibriSpeech-Y or LS-Y, where “Y” indicates the
presence of disfluencies, is then subjected to the same testing process as the
FluencyBank dataset in Experiment 1, using the same set of six ASR models.
This approach ensures a systematic analysis of bias within these ASR systems
across diverse speaking scenarios.

###  3.1 ASR Models Studied

We investigate a mix of commercially-available market ASRs and research-
oriented, open-source models, to uncover biases across diverse ASR systems.
The market ASRs include: (a) Google Cloud’s Speech-to-Text V1 Cloud , (b) IBM
Watson’s Speech-to-Text IBM , (c) Microsoft Azure Azure , and (d) RevAI Rev.AI
. It is important to note that the specific details regarding the training
processes of these commercial models are proprietary and not publicly
disclosed.

In addition to these market ASRs, our investigation extends to two open-
source, research-based ASR models, which are:

  * •

wav2vec 2.0 Baevski et al. (2020): This is an end-to-end ASR model, utilizing
connectionist temporal classification loss for training. For our analysis, we
employ the ‘wav2vec2-large-960h’ configuration, available from Meta on the
HuggingFace Library Facebook . This specific model was trained on a
substantial corpus of 960 hours of the LibriSpeech dataset.

  * •

Whisper Radford et al. (2023): Representing the current state-of-the-art in
ASR technology, Whisper is a transformer-based encoder-decoder model. It has
been pretrained on over 680k hours of multilingual data, leveraging semi-
supervised learning and multitasking approaches—including tasks like language
identification and voice activity detection. The primary aim of Whisper is to
achieve generalization across a variety of benchmarks. For our purposes, we
utilize the ‘whisper-1’ model configuration provided by OpenAI API, which is
the most up-to-date version at the time of this writing.

Figure 2: Histogram showing the frequency distribution of utterance durations
in seconds for each of the four speech datasets.

###  3.2 FluencyBank: Stuttered Speech Dataset

Our study utilizes the FluencyBank dataset Ratner and MacWhinney (2018) to
examine biases in real stuttered speech. FluencyBank comprises a series of
video recordings featuring individuals who stutter, captured while reading
passages and during interviews. Each video in this dataset is accompanied by a
transcript, which has been segmented into individual utterances—a sentence or
a fragment of a sentence (containing whole phrases) spoken by a single
individual.

Due to temporal misalignment and incorrect transcript errors, which were
identified by us and others Sheikh et al. (2022); Lea et al. (2021), we had
the dataset relabeled by trained SLPs with expertise in stuttering. They used
the standard Codes for the Human Analysis of Transcripts format MacWhinney
(2021), and added start and end timestamps for each utterance within a given
audio file, while ensuring inter-annotator agreement. The dataset utilized in
our analysis consists of 1,373 utterances, derived from 7 reading and 12
interview videos. The distribution of these utterances in terms of duration is
shown in Fig. 2, with the cumulative audio duration amounting to approximately
2.21 hours.

In addition to the original dataset (FB-Y), we produced a version with
apparent instances of stuttering or disfluencies removed, referred to as
FluencyBank-N or FB-N. This variant was created using TTS, which synthesized
speech from reference transcripts devoid of disfluencies. This approach,
detailed in Sec. 3.3, serves as a crucial component in our comparative
analysis of bias.

###  3.3 Synthetic Disfluent Speech Generation

To complement our analysis using the FluencyBank dataset, we produce a
synthetic dataset (LibriSpeech-Y or LS-Y) derived from LibriSpeech, an ASR
dataset featuring public domain audiobook recordings. We utilized the ‘clean’
sample from LibriSpeech’s test set, available through the HuggingFace library,
to ensure that the wav2vec 2.0 model was not previously exposed to this data
during training. However, for commercial models (i.e., those available from
Google and Microsoft), we do not have specific training data details and hence
can not exclude the possibility that LibriSpeech has been used in their
training, thus potentially boosting their performance in our analysis.

Our approach improves upon the LibriStutter dataset by FluentNet Kourkounakis
et al. (2020), which lacks diversity in disfluency events. LibriStutter’s
limitations include inadequate representation of the variability of
stuttering, such as inserting only one to two word repetitions. Moreover,
LibriStutter only uses audio processing methods for disfluency insertion. We
introduce a more diverse range of disfluency events Einarsdóttir and Ingham
(2005): interword blocks, intraword blocks, prolongations, word repetitions,
phrase repetitions, and interjections. While these do not cover all types of
disfluencies associated with stuttering, they serve to demonstrate key
variations for our bias study.

We introduce disfluencies into the speech samples using two methods: textual
disfluency insertion and audio disfluency insertion. For textual disfluency
insertion, we modify the transcript and then use a TTS model, specifically
SpeechT5 TTS Ao et al. (2022, 2021), to generate speech that naturally
includes the disfluency events. This process, supported by a set of pre-
trained speaker vectors Hollemans , allows us to simulate a diverse range of
voices. This technique is primarily used for word repetitions, phrase
repetitions, and interjections. The frequency and location of these
disfluencies within each sentence are selected randomly within predetermined
ranges: (1) for word repetition, we select between 1 to 3 words in each
utterance to repeat 1 to 4 additional times; (2) for phrase repetition, we
choose a phrase of 2 to 4 words and repeat it 1 to 3 times; (3) for
interjections, we randomly insert either “uh” or “um” in 1 to 4 locations and
repeat them 1 to 4 times.

For audio disfluency insertion, we directly modify the speech waveform to add
disfluencies that TTS cannot create, such as blocks and prolongations. We
employ a pre-trained wav2vec 2.0 model and PyTorch CTC forced alignment Zhang
and Hira  for precise alignment of words with their timestamps. This method
allows for the insertion of interword blocks, intraword blocks, and additional
word and phrase repetitions with parameters similar to those used in textual
disfluency insertion. The parameters for disfluency events applied in audio
processing are also randomly selected: (1) for interword blocks, we insert a
pause after 1 to 4 words with a duration of 1 to 3 seconds; (2) for intraword
blocks, we insert a pause into 1 to 3 words, each lasting between 1 to 3
seconds; (3) for prolongations, we extend the sound of 1 to 3 words at their
beginning, stretching it to 5 to 25 times its original length. Additionally,
word and phrase repetitions are applied following the same parameters as in
the textual disfluency insertion process.

Each disfluency is randomly applied, without replacement, to an equal number
of samples, amounting to 10% of the LibriSpeech dataset, ensuring a balanced
representation. The resulting dataset comprises approximately one hour of
speech samples, with 308 records in total. The duration of these samples, in
seconds, is depicted in Figure 2.

Table 1: Examples of disfluency events including word repetition (WR), phrase repetition (PR), prolongation (Prolong.), interword block (InterWB), intraword block (IntraWB), and interjection (Interject.). Words in <> brackets are added through textual disfluency insertion, while (pause) signifies the audio insertion of silence. Disfluency Event |  Example
---|---
WR |  How <are> are you?
PR |  How <are you> are you?
Interject. |  How are <um> you?
InterWB |  How are (pause) you?
IntraWB |  How a(pause)re you?
Prolong. |  How aaare you?

This approach allows for a comprehensive and nuanced analysis of ASR
performance across a range of disfluency types, thereby enriching our
understanding of bias in these systems.

###  3.4 ASR Evaluation Metrics

To effectively assess biases in ASR systems, we employ three key metrics, each
offering a unique lens on transcription accuracy and semantic fidelity.

The first metric is the word error rate (WER), a widely recognized standard in
ASR evaluations, defined as: ${\rm WER}=\frac{S+D+I}{S+D+C},$ where $S$, $D$,
$I$, and $C$ represent the number of word substitutions, word deletions, word
insertions, and correct words, respectively, in the ASR-produced transcript
relative to the reference transcript. WER provides a direct measure of
transcription accuracy at the word level. A lower WER, approaching zero, is
desirable, indicating higher word transcription accuracy.

In conjunction with WER, we analyze the character error rate (CER), which
evaluates transcription errors at the character level. Similar to WER but more
granular, CER offers insight into the accuracy of finer transcription details.
As with WER, a lower CER is indicative of better performance.

Our third metric, BERTScore Zhang et al. (2019), examines semantic similarity
between predicted and target sentences. This metric is crucial for cases where
a transcription error might involve just a single word, yet lead to a
significant change in meaning. For instance, incorrectly transcribing "not" in
a sentence could completely reverse its intended meaning, despite being a
single-word error. BERTScore assesses this aspect, with a score closer to 1
indicating greater semantic similarity between the predicted and target
sentences.

With BERTScore, we can observe the meaning behind each word. Given an ASR-
produced sentence $x=\left\langle x_{1},...,x_{k}\right\rangle$ and a
reference sentence
$\hat{x}=\left\langle\hat{x}_{1},...,\hat{x}_{l}\right\rangle$, where $x_{i}$
and $\hat{x}_{i}$ represent the $i^{th}$ token embeddings derived directly
from a pre-trained BERT model’s contextual embeddings, BERTScore computes the
cosine similarity of each token in $x$ with every token in $\hat{x}$ to obtain
an F1 score. We utilize embeddings from the ‘microsoft/deberta-large-mnli’
model He et al. (2021) alongside the re-scaled version of BERTScore, wherein
the range is within $[-1,1]$, where a 1 would be semantically identical
transcripts, and -1 opposite. We refer to this as $F_{BERT}$ Tobin et al.
(2022); Zhang et al. (2019) and it is defined as the following:

$F_{BERT}=2\frac{P_{BERT}\times R_{BERT}}{P_{BERT}+R_{BERT}},{\textrm{where}}$
(1) $R_{BERT}=\frac{1}{|x|}\sum_{x_{i}\in
x}\max_{\hat{x}_{j}\in\hat{x}}(x_{i}^{\textup{T}}\cdot\hat{x}_{j}),{\textrm{and}}$
(2) $P_{BERT}=\frac{1}{|\hat{x}|}\sum_{\hat{x}_{i}\in\hat{x}}\max_{x_{j}\in
x}(x_{i}^{\textup{T}}\cdot\hat{x}_{j}).$ (3)

We use the Spearman correlation coefficient to analyze the relationship
between ASR performance metrics across disfluent and non-disfluent speech
datasets. We opt for this non-parametric test because the performance
distributions are non-normal (as detailed in Appendix B.1), necessitating an
analysis approach that does not assume normality. The Spearman correlation
coefficient, defined as:

$r_{WER}=1-\frac{6\sum d_{i}^{2}}{n(n^{2}-1)},$ (4)

facilitates our understanding of variations in ASR model performance across
the two datasets. Here, $d_{i}$ is the difference in ranks between each
observation’s WER in disfluent and non-disfluent speech datasets, and $n$
represents the total number of observations or utterances. This approach helps
us quantify the impact of disfluent speech patterns, like stuttering, on ASR
accuracy. A value closer to 1 or -1 indicates a strong positive or negative
correlation, respectively, between the presence of disfluencies and the
difference in WER, suggesting a significant impact of disfluencies on ASR
performance. Conversely, a correlation closer to 0 implies a weaker
relationship. We perform similar computations for CER ($r_{CER}$) and
BERTScore ($r_{BERT}$), offering a multidimensional view of each ASR system’s
bias and accuracy.

Additionally, we incorporate the calculation of the p-value to determine the
statistical significance of the observed biases. The $p$-value quantifies the
probability of observing the results under the null hypothesis, which in this
context is the absence of bias. A $p$-value lower than a typical threshold
(e.g., 0.05) suggests that the observed bias is statistically significant and
not due to random chance. This metric is essential for substantiating the
reliability of our findings.

##  4 Results and Discussion

Our analysis, divided into two parts corresponding to each experiment in Sec.
3, reveals significant insights into ASR performance with real and synthetic
disfluent speech. The evaluation was conducted using ASR services as of March.
14, 2024, utilizing default API settings. For wav2vec 2.0, the analysis was
performed locally using an NVIDIA T4 GPU. Transcripts returned blank by models
due to audio processing challenges (e.g., due to noise or blocks) were
retained in our dataset. Prior to evaluation, all transcripts underwent
normalization using Whisper’s “BasicTextNormalizer” and
“EnglishNumberNormalizer” functions from the Transformers library HuggingFace
, ensuring consistent casing and punctuation removal. Any speech atmospherics
in output transcripts, such as laughing or indications of silence, were also
removed.

###  4.1 Experiment 1: Real Stuttered Speech

Table 2: Comparative analysis of ASR model performance metrics (WER, CER, and $F_{BERT}$) on FluencyBank datasets: ‘Y’ denotes the presence of stuttering (FB-Y), and ‘N’ denotes its absence (FB-N). The mean of scores across ASRs is given by $\mu$, and the standard deviation by $\sigma$. | WER | CER | $F_{BERT}$
---|---|---|---
| Y | N | Y | N | Y | N
Azure |  .168 |  .073 |  .117 |  .052 |  .805 |  .896
GCP |  .275 |  .069 |  .182 |  .050 |  .623 |  .920
IBM |  .476 |  .167 |  .407 |  .150 |  .463 |  .838
RevAI |  .263 |  .159 |  .187 |  .133 |  .586 |  .508
Whisp |  .121 |  .063 |  .082 |  .047 |  .870 |  .935
W2V2 |  .380 |  .101 |  .212 |  .061 |  .549 |  .865
$\mu$ |  .281 |  .105 |  .198 |  .082 |  .649 |  .827
$\sigma$ |  .120 |  .042 |  .103 |  .042 |  .143 |  .146
Table 3: Spearman correlation coefficients ($r$), all of which had $p<0.05$, indicating the statistical significance of the impact of disfluencies on $\bigtriangleup$WER, $\bigtriangleup$CER, and $\bigtriangleup$$F_{BERT}$. FB refers to comparisons between FB-Y and FB-N datasets, and LS to comparisons between synthetic (LS-Y) and original (LS-N) LibriSpeech datasets. “All” refers to the combination of all ASR predictions. |  $\bigtriangleup$WER |  $\bigtriangleup$CER |  $\bigtriangleup$$F_{BERT}$
---|---|---|---
| FB | LS | FB | LS | FB | LS
Azure |  .363 |  .656 |  .385 |  .700 |  -.33 |  -.62
GCP |  .552 |  .547 |  .559 |  .609 |  -.35 |  -.47
IBM |  .531 |  .616 |  .539 |  .690 |  -.52 |  -.49
RevAI |  .245 |  .557 |  .246 |  .611 |  -.21 |  -.50
Whisp |  .285 |  .501 |  .298 |  .528 |  -.24 |  -.48
W2V2 |  .563 |  .712 |  .562 |  .781 |  -.54 |  -.67
All |  .413 |  .566 |  .423 |  .623 |  -.38 |  -.50

The outcomes of ASR model performance on the FluencyBank dataset, with and
without disfluencies (denoted as FB-Y and FB-N, respectively), are presented
in Table 2. Our findings indicate a consistent pattern across all models:
lower WER, CER, and higher $F_{BERT}$ on the FB-N dataset, implying a systemic
bias against disfluent speech. Among the evaluated models, Whisper exhibited
the lowest WER and highest $F_{BERT}$. However, its performance on FB-Y was
markedly worse (about two times) compared to that on FB-N, where it achieved a
WER of just 6.3%. We also note that the RevAI model showed a higher $F_{BERT}$
for FB-Y than for FB-N, primarily due to blank transcript predictions. Upon
removing these from the analysis, the reported $F_{BERT}$ on FB-Y is 0.809 and
on FB-N is 0.933. This disparity in performance highlights a significant bias
in ASRs against disfluent speech.

Moreover, Fig. 3 illustrates the WER distribution for the wav2vec 2.0 model
across all datasets. The mean WER is noticeably higher for datasets with
disfluencies, highlighting a broader range and more outliers with elevated
WER.

###  4.2 Experiment 2: Synthetic Speech

In our second experiment, we evaluated the ASR models against our synthetic
disfluency dataset compared to the original LibriSpeech. The results (Table 4)
mirror those from Experiment 1, with higher WER and CER for the synthetic
disfluency dataset. This finding confirms the bias observed in Experiment 1,
not just for real disfluent speech but also for synthesized versions. The
Whisper model again showed superior performance, but with a significant drop
(WER of 3.1% vs. 19.8%) when disfluencies were introduced.

Table 4: Evaluation of ASR models on synthetic stuttering and original LibriSpeech datasets. Metrics include WER, CER, and $F_{BERT}$, with ‘Y’ indicating synthetic stuttering (LS-Y) and ‘N’ indicating the original dataset (LS-N). The mean of scores across ASRs is given by $\mu$, and the standard deviation by $\sigma$. | WER | CER | $F_{BERT}$
---|---|---|---
| Y | N | Y | N | Y | N
Azure |  .228 |  .037 |  .197 |  .018 |  .772 |  .957
GCP |  .268 |  .070 |  .217 |  .034 |  .744 |  .904
IBM |  .560 |  .191 |  .518 |  .127 |  .448 |  .757
RevAI |  .254 |  .062 |  .209 |  .031 |  .742 |  .901
Whisp |  .198 |  .031 |  .157 |  .015 |  .811 |  .969
W2V2 |  .259 |  .033 |  .219 |  .009 |  .718 |  .952
$\mu$ |  .295 |  .071 |  .253 |  .039 |  .706 |  .907
$\sigma$ |  .121 |  .056 |  .120 |  .040 |  .119 |  .072
Table 5: WER results for each ASR on the synthetic dataset, broken down by specific disfluency event types. These include audio-based word repetition (Aud-WR), audio-based phrase repetition (Aud-PR), prolongation (Prolong.), interword block (InterWB), intraword block (IntraWB), interjection (Interject.), text-TTS-based word repetition (Txt-WR), and text-TTS-based phrase repetition (Txt-PR). ‘Y’ denotes the presence of each disfluency event type, and ‘N’ indicates its absence. The mean of scores across ASRs is given by $\mu$, and the standard deviation by $\sigma$. | Aud-WR | Aud-PR | Prolong. | InterWB | IntraWB | Interject. | Txt-WR | Txt-PR
---|---|---|---|---|---|---|---|---
| Y | N | Y | N | Y | N | Y | N | Y | N | Y | N | Y | N | Y | N
Azure | .27 | .02 | .22 | .03 | .35 | .04 | .12 | .05 | .08 | .03 | .32 | .05 | .32 | .05 | .26 | .03
GCP | .33 | .05 | .24 | .06 | .53 | .08 | .11 | .08 | .08 | .05 | .38 | .07 | .36 | .09 | .28 | .10
IBM | .67 | .42 | .35 | .22 | .65 | .20 | .72 | .21 | .68 | .11 | .52 | .15 | .42 | .20 | .49 | .12
RevAI | .34 | .05 | .23 | .04 | .44 | .08 | .10 | .08 | .07 | .05 | .36 | .08 | .36 | .09 | .28 | .04
Whisp | .23 | .03 | .12 | .03 | .32 | .03 | .07 | .04 | .04 | .02 | .41 | .04 | .33 | .05 | .23 | .02
W2V2 | .33 | .02 | .21 | .03 | .55 | .03 | .11 | .03 | .09 | .03 | .34 | .04 | .32 | .04 | .26 | .05
$\mu$ | .36 | .10 | .23 | .07 | .47 | .08 | .20 | .08 | .17 | .05 | .39 | .07 | .35 | .09 | .30 | .06
$\sigma$ | .14 | .14 | .07 | .07 | .12 | .06 | .23 | .06 | .23 | .03 | .07 | .04 | .04 | .05 | .08 | .04

Further analysis on disfluency-event-type specific WER (Table 5) reveals that
certain disfluencies, such as word repetitions, prolongations, and
interjections, significantly increase WER. In contrast, intraword block and
interword block events showed a less pronounced impact. This suggests that
different disfluency types vary in their effect on ASR performance,
underscoring the need for more inclusive and diverse training data in ASRs.

The Spearman correlation coefficients and corresponding $p$-values (Table 3)
across all models and for different disfluencies (Appendix B.2) underscore
this finding. We find all correlation values in Table 3 are statistically
significant, with $p<0.05$, indicating that the presence of disfluencies
substantially impacts ASR accuracy. For example, prolongations and
interjections showed a high correlation with increased WER, highlighting
specific areas where ASRs struggle in handling disfluent speech. We also note
the correlation coefficients for $F_{BERT}$ present an inverse correlation,
where ASRs consistently had lower $F_{BERT}$ scores on FB-Y and LS-Y,
indicating poor semantic similarity.

Overall, our results indicate a clear bias in current ASR technology against
disfluent speech, both real and synthetic. This bias is evident across various
disfluency event types and is statistically significant, emphasizing the need
for targeted improvements in ASRs to accommodate speech variations.

Figure 3: Violin plot illustrating the range and distribution of WER for the
wav2vec 2.0 model on FluencyBank (FB) and synthetic (LS) datasets. The plot
highlights mean values and variability in WER scores.

##  5 Conclusion

This study represents a significant step in identifying and quantifying the
biases in ASR systems against individuals who stutter. By analyzing the
performance of six popular ASR models using two distinct datasets—FluencyBank
and a custom-generated synthetic dataset incorporating disfluency events from
LibriSpeech—we have uncovered substantial biases. Our findings reveal that all
six models demonstrate notably worse WER, CER, and BERTScore when processing
disfluent speech. This bias is consistent across both real and synthetic
datasets and is statistically significant.

Our research also delves into disfluency-event-specific biases, revealing that
the majority of disfluencies—six out of eight types—exhibit a statistically
significant WER bias across all models. This discovery is crucial in
understanding the nuances of ASR performance and the challenges faced by
people who stutter in real-world scenarios.

To address these challenges, there are a few approaches that can be
considered. First, incorporating speech from people who stutter in ASR
training; this is challenging due to the limited availability of relevant
data. Additionally challenging is the task of acquiring sufficient data that
encompasses samples from a diverse population to capture the variability in
speech associated with stuttering. However, with a strategic data collection
approach, this can address a primary limitation in ASR accessibility.
Secondly, obtaining feedback from people who stutter and benchmarking ASRs
before deployment are crucial steps towards accessibility. Rigorous testing on
disfluent speech can identify ASR deficiencies, allowing for targeted training
or post-processing to address them.

Looking forward, we also identify two promising directions for future
research. Firstly, there is a need to explore the impact of longer-form audio
on ASR biases. Current models often struggle with correctly segmenting and
transcribing longer audio samples containing disfluencies, leading to
significant transcription inaccuracies or even cessation of live
transcription. This aspect is particularly critical as it can severely limit
the practical usability of ASR technologies for individuals who stutter.

Secondly, incorporating human feedback from people who stutter represents an
important area of exploration. Current metrics like WER and semantic accuracy
might not fully capture the transcription preferences of individuals who
stutter. While our study assumes that the transcription of FluencyBank, with
stuttering indicators removed, accurately represents the intended speech, this
may not align with the preferences of those who stutter. They might desire
their speech patterns, such as interjections, prolongations, and other
disfluencies, to be reflected in the transcripts. Hence, engaging directly
with individuals who stutter to establish ground-truth labels could offer a
more authentic and respectful approach to ASR development.

In conclusion, our research is among the first to systematically examine and
highlight the biases in ASR systems towards disfluent speech. The implications
of these biases are far-reaching, potentially exacerbating disparities in
various domains, from healthcare to recruitment, where ASR technologies are
increasingly integrated. To pave the way for more inclusive and equitable ASR
technologies, bias mitigation strategies, including diversifying training data
with disfluent speech and incorporating feedback from the stuttering
community, are essential. Our work lays the foundation for this crucial
endeavor, opening avenues for future research and development in the field of
speech recognition.

## Limitations

We acknowledge three limitations of our work. First, our study focuses solely
on accuracy bias by ASR models transcribing speech in English, and people who
stutter speaking English. This does not cover the many multilingual ASR models
and people who stutter speaking other languages. Second, our synthetic dataset
does not encompass all types of disfluencies, and not all disfluencies (e.g.,
interjections) can be considered stuttering in all circumstances since they
depend upon the presence of a loss of control by the speaker while speaking
Tichenor and Yaruss (2019). Our intended use for this dataset is for studying
bias at a fine-grained disfluency level, with control over the frequency and
location of occurrence of disfluencies. Therefore, this dataset should not be
used for ASR model training as a bias mitigation measure, and should also not
be used to study bias unaccompanied by other datasets. Likewise, third, our
study method should not be used as the sole assessment of bias in ASRs. We
define bias by observing the transcript accuracy (i.e., WER, CER, and semantic
accuracy) difference with and without disfluencies. However, other metrics for
bias and other facets, such as age or gender, can affect ASR bias. Overall, we
seek to develop more inclusive ASRs, and have these limitations addressed in
future work.

## Acknowledgment

This material is based upon work supported by the U.S. National Science
Foundation under Grant Nos. 2235916 and 2345086.

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Table 6: Comparative analysis of similarity scores between our synthetic dataset LS-Y with disfluencies, and both FluencyBank datasets where ‘Y’ denotes the presence of disfluencies (FB-Y), and ‘N’ denotes its absence (FB-N). This is broken down by specific disfluency event types. These include audio-based word repetition (Aud-WR), audio-based phrase repetition (Aud-PR), prolongation (Prolong.), interword block (InterWB), intraword block (IntraWB), interjection (Interject.), text-TTS-based word repetition (Txt-WR), and text-TTS-based phrase repetition (Txt-PR). The mean of similarity scores is given by $\mu$, and the standard deviation by $\sigma$. “All’ is the average distance across all disfluency types. We observe across all disfluency types, our synthetic speech dataset is closer to resembling stuttered speech than non-stuttered speech. | FB-Y | FB-N
---|---|---
| $\mu$ | $\sigma$ | $\mu$ | $\sigma$
Aud-WR |  .659 |  .178 |  .614 |  .190
Aud-PR |  .589 |  .192 |  .524 |  .192
Prolong. |  .634 |  .178 |  .590 |  .190
InterWB |  .637 |  .187 |  .569 |  .192
IntraWB |  .546 |  .190 |  .472 |  .176
Interject. |  .667 |  .174 |  .635 |  .189
Txt-WR |  .648 |  .184 |  .597 |  .197
Txt-PR |  .598 |  .193 |  .531 |  .194
All |  .623 |  .189 |  .567 |  .197
Table 7: Spearman correlation coefficient ($r_{WER}$ denoted by $r$ below) and corresponding $p$-value results for each ASR model on the synthetic dataset, broken down by specific disfluency event types. These include audio-based word repetition (Aud-WR), audio-based phrase repetition (Aud-PR), prolongation (Prolong.), interword block (InterWB), intraword block (IntraWB), interjection (Interject.), text-TTS-based word repetition (Txt-WR), and text-TTS-based phrase repetition (Txt-PR). ‘Y’ denotes the presence of each disfluency event type, and ‘N’ indicates its absence. “All” refers to the combination of all ASR predictions. | Aud-WR | Aud-PR | Prolong. | InterWB | IntraWB | Interject. | Txt-WR | Txt-PR
---|---|---|---|---|---|---|---|---
| $r$ | $p$ | $r$ | $p$ | $r$ | $p$ | $r$ | $p$ | $r$ | $p$ | $r$ | $p$ | $r$ | $p$ | $r$ | $p$
Azure | .78 | .00 | .83 | .00 | .40 | .00 | .56 | .00 | .58 | .00 | .70 | .00 | .76 | .00 | .79 | .00
GCP | .72 | .00 | .81 | .00 | .45 | .00 | .18 | .11 | .27 | .01 | .71 | .00 | .69 | .00 | .62 | .00
IBM | .54 | .00 | .51 | .00 | .48 | .00 | .76 | .00 | .84 | .00 | .61 | .00 | .48 | .00 | .66 | .00
RevAI | .79 | .00 | .80 | .00 | .46 | .00 | .11 | .33 | .29 | .01 | .65 | .00 | .68 | .00 | .74 | .00
Whisp | .64 | .00 | .40 | .00 | .36 | .00 | .17 | .13 | .10 | .40 | .77 | .00 | .74 | .00 | .78 | .00
W2V2 | .82 | .00 | .83 | .00 | .71 | .00 | .48 | .00 | .66 | .00 | .78 | .00 | .78 | .00 | .72 | .00
All | .67 | .00 | .67 | .00 | .46 | .00 | .33 | .00 | .40 | .00 | .69 | .00 | .67 | .00 | .70 | .00

##  Appendix A Evaluation of Synthetic Disfluent Speech

To ensure that our synthetic disfluent speech dataset (LS-Y) presented in
Section 3.3 is comparable to real stuttered speech, we assess its similarity
to the FluencyBank dataset (FB-Y). It is important to note that our dataset’s
objective is not to mimic stuttering precisely; instead, it aims to provide a
comparable framework for evaluating ASR biases toward stuttered speech.

To compare, our evaluation procedure starts by computing the cosine similarity
between all spectrograms of audio samples in LS-Y and FB-Y. Following this, we
calculate the cosine similarity between all spectrograms of audio samples in
LS-Y and FB-N, our corresponding FluencyBank dataset without disfluencies.
This approach aims to determine whether our dataset exhibits greater
similarity to FB-Y, which contains stuttered speech, than to FB-N, which lacks
stuttering. Such a comparison helps gauge the resemblance of our dataset to
stuttered speech versus non-stuttered speech.

The effectiveness of this approach and our dataset is demonstrated through the
results presented in Table 6, which display the average and standard deviation
of similarities on a disfluency-specific level between LS-Y and both FB-Y and
FB-N datasets. We observe that our synthetic speech dataset exhibits a greater
similarity to real stuttered speech (FB-Y) than to non-stuttered speech (FB-N)
across all disfluency types. This finding is further supported by the fact
that our FB-N dataset was generated using voices through TTS (Sec. 3.3)
overlapping with our LS-Y dataset, possibly increasing the similarity between
LS-Y and FB-N. Therefore, overall, our synthetic stuttered speech generation
method proves to be an effective means of introducing disfluencies in non-
stuttered speech.

##  Appendix B Additional Experiment Results

###  B.1 ASR performance distributions

The distribution of WER, CER, and $F_{BERT}$ scores for all ASR models on real
and stuttered speech datasets are presented in Figure 4, Figure 5, and Figure
6, respectively. We observe that the mean is lower for each ASR when
disfluencies are present, specifically in LS-Y and FB-Y, in comparison to LS-N
and FB-N without perceived stuttering.

###  B.2 Correlation Coefficient Test Results

We test the correlation between degraded ASR accuracy and the presence of
disfluent speech using the Spearman rank correlation coefficient, as presented
in Table 3. Similarly, we also compute the individual correlation coefficients
and associated $p$-values for each disfluency type in our synthetic dataset
(LS-Y). These results are presented in Table 7. We find all correlation
coefficients, excluding those for interword blocks, statistically significant,
with $p<0.05$. In the case of interword blocks, we find a weak correlation,
$r_{WER}<0.20$, in the GCP, RevAI, and Whisper models without statistical
significance, whereas other ASR models have a strong correlation with
statistical significance. This is due to the difference in WER being quite
small when an interword block is present, as observed in Table 1, with only a
2% change for RevAI, and 3% change for GCP and Whisper. This is also seen for
the Whisper ASR on the intraword block, where a low correlation $r_{WER}=0.10$
is observed, as well as only a small decrease, of 2%, in WER. In comparison,
other disfluency types such as word repetitions, interjections, and phrase
repetitions show a high correlation with their presence in speech and a
decrease in WER. When combining predictions across ASR models, we also observe
a statistically significant correlation between all disfluency types, with
phrase repetitions having the strongest, and interword blocks having the
weakest.

Figure 4: Violin plots illustrating the range and distribution of WER for each
ASR model on FluencyBank (FB) and LibriSpeech (LS), both with disfluencies (Y)
and without disfluencies (N).

Figure 5: Violin plots illustrating the range and distribution of CER for each
ASR model on FluencyBank (FB) and LibriSpeech (LS), both with disfluencies (Y)
and without disfluencies (N).

Figure 6: Violin plots illustrating the range and distribution of $F_{BERT}$
for each ASR model on FluencyBank (FB) and LibriSpeech (LS), both with
disfluencies (Y) and without disfluencies (N).