# Rough statistical convergence of double sequences in probabilistic normed
spaces

1Rahul Mondal, 2Nesar Hossain  1Department of Mathematics, Vivekananda
Satavarshiki Mahavidyalaya, Manikpara, Jhargram -721513, West Bengal, India.
2Department of Mathematics, The University of Burdwan, Burdwan - 713104, West
Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we have defined rough convergence and rough statistical
convergence of double sequences in probabilistic normed spaces which is more
generalized version than the rough statistical convergence of double sequences
in normed linear spaces. Also, we have defined rough statistical cluster
points of double sequences and then, investigated some important results
associated with the set of rough statistical limits of double sequences in
these spaces. Moreover, in the same spaces, we have proved an important
relation between the set of all rough statistical cluster points and rough
statistical limits under certain condition.

###### Key words and phrases:

Probabilistic normed space; rough statistical convergence of double sequences;
rough statistical cluster points of double sequences.

###### 2020 Mathematics Subject Classification:

40A35, 40A05, 40A99, 26E50, 40G99

##  1\. Introduction

In $1951$, the concept of usual convergence of real sequences was extended to
statistical convergence of real sequences based on the natural density of a
set by Fast [9] and Steinhaus [30] independently. Later on, this idea has been
studied in different directions and in different spaces by many authors as in
[6, 7, 8, 11, 12, 20, 21, 25, 32, 34] and many more.

In $2001$, Phu [27] has initially introduced the concept of rough convergence
of sequences in finite dimensional normed linear spaces which is basically a
generalization of usual convergence and, in the same paper he has investigated
that $r$-limit set is bounded, closed, convex and many more interesting
results and later on, this concept has been extended to infinite dimensional
normed linear spaces [29]. Also, He [28] has defined the notion of rough
continuity of linear operators. Later, Ayter [3] extended this notion to rough
statistical convergence based on natural density of a set. Malik and Maity
[23, 24] has defined rough convergence and rough statistical convergence of
double sequences in normed linear spaces. After that, the research work on
this concept is still being carried out in different directions as in [4, 13,
14, 18, 26] and many references therein.

In $1942$, Menger [19] first proposed the concept of statistical metric space,
now called probabilistic metric space, which is an interesting and important
generalization of the notion of metric space. This concept, later on, was
studied by Schweizer and Sklar [33]. Combining the idea of statistical metric
space and normed linear space, Šerstnev [31] introduced the idea of
probabilistic normed space. In $1993$ Alsina et al. gave a new definition of
probabilistic normed space whic is basically a special case of the definition
of Šerstnev. Recently, Antal et al. [5] defined the notion of rough
convergence and rough statistical convergence in probabilistic normed spaces.
In this space, we have presented the notion of rough statistical convergence
of double sequences and investigated some interesting results associated with
the sets of rough statistical cluster points and rough statistical limits of
double sequences.

##  2\. Preliminaries

Throughout the paper $\mathbb{N}$ and $\mathbb{R}$ denote the set of positive
integers and set of reals respectively. First we recall some basic definitions
and notations.

######  Definition 2.1.

[33] A triangular norm, briefly $t$-norm, is a binary operation on $[0,1]$
which is continuous, commutative, associative, non decreasing and has $1$ as
unit element, i.e., it is the continuous mapping
$\diamond:[0,1]\times[0,1]\rightarrow[0,1]$ such that for all
$a,b,c,d\in[0,1]$:

  1. (1)

$a\diamond 1=a$;

  2. (2)

$a\diamond b=b\diamond a$;

  3. (3)

$a\diamond b\geq c\diamond d$ whenever $a\geq c$ and $b\geq d$;

  4. (4)

$a\diamond(b\diamond c)=(a\diamond b)\diamond c$.

######  Example 2.1.

[15] The following are the examples of $t$-norms:

  1. (1)

$x\diamond y=min\\{x,y\\}$;

  2. (2)

$x\diamond y=x.y$;

  3. (3)

$x\diamond y=max\\{x+y-1,0\\}$. This $t$-norm is known as Lukasiewicz
$t$-norm.

######  Definition 2.2.

[10] A function $f:\mathbb{R}\rightarrow\mathbb{R}_{0}^{+}$ is said to be a
distribution function if it is non decreasing and left continuous with
$\inf_{t\in\mathbb{R}}f(t)=0$ and $\sup_{t\in\mathbb{R}}f(t)=1$. We denote $D$
as the set of all distribution functions.

######  Definition 2.3.

[10] A triplet $(X,\vartheta,\diamond)$ is called a probabilistic normed space
(shortly PNS) if $X$ is a real vector space, $\nu$ is a mapping from $X$ into
$D$ (for $x\in X,t\in\mathbb{(}R)$, $\vartheta(x;t)$ is the value of the
distribution function $\vartheta(x)$ at $t$) and $\diamond$ is a $t$-norm
satisfying the following conditions:

  1. (1)

$\vartheta(x;0)=0$;

  2. (2)

$\vartheta(x;t)=1$, $\forall\ t>0$ iff $x=\theta$, $\theta$ being the zero
element of $X$;

  3. (3)

$\vartheta(\alpha x;t)=\vartheta(x;\frac{t}{|\alpha|})$, $\forall\
\alpha\in\mathbb{R}\setminus\\{0\\}$ and $\forall\ t>0$;

  4. (4)

$\vartheta(x+y;s+t)\geq\vartheta(x;t)\diamond\vartheta(y;s)$, $\forall\ x,y\in
X$ and $\forall\ s,t\in\mathbb{R}_{0}^{+}$.

######  Example 2.2.

[2] For a real normed space $(X,\left\lVert\cdot\right\rVert)$, we define the
probabilistic norm $\vartheta$ for $x\in X,t\in\mathbb{R}$ as
$\vartheta(x;t)=\frac{t}{t+\left\lVert x\right\rVert}$. Then
$(X,\vartheta,\diamond)$ is a PNS under the $t$-norm $\diamond$ defined by
$x\diamond y=\min\\{x,y\\}$. Also,
$x_{n}\xrightarrow{\left\lVert\cdot\right\rVert}\xi$ if and only if
$x_{n}\xrightarrow{\vartheta}\xi$.

######  Definition 2.4.

[2] Let $(X,\vartheta,\diamond)$ be a PNS. For $r>0$, the open ball
$B(x,\lambda;r)$ with center $x\in X$ and radius $\lambda\in(0,1)$ is the set

$B(x,\lambda;r)=\\{y\in X:\vartheta(y-x;r)>1-\lambda\\}.$

Similarly, the closed ball is the set $\overline{B(x,\lambda;r)}=\\{y\in
X:\vartheta(y-x;r)\geq 1-\lambda\\}$

######  Definition 2.5.

[17] Let $\\{x_{mn}\\}$ be a double sequence in a PNS
$(X,\vartheta,\diamond)$. Then $\\{x_{mn}\\}$ is said to be convergent to
$\xi\in X$ with respect to the probabilistic norm $\vartheta$ if for every
$\varepsilon>0$ and $\lambda\in(0,1)$, there exists a positive integer $n_{0}$
such that $\vartheta(x_{mn}-\xi;\varepsilon)>1-\lambda$ whenever $m,n\geq
n_{0}$. In this case we write $\vartheta_{2}\text{-}\lim x_{mn}=\xi$ or
$x_{mn}\xrightarrow{\vartheta_{2}}\xi$.

######  Definition 2.6.

Let $K\subset\mathbb{N}$. Then the natural density $\delta(K)$ of $K$ is
defined by

$\delta(K)=\lim_{n\to\infty}\frac{1}{n}|\\{k\leq n:k\in K\\}|,$

provided the limit exists.

It is clear that if $K$ is finite then $\delta(K)=0$.

######  Definition 2.7.

[5] Let $\\{x_{n}\\}_{n\in\mathbb{N}}$ be a sequence in an PNS
$(X,\vartheta,\diamond)$. Then $\\{x_{n}\\}_{n\in\mathbb{N}}$ is said to be
rough convergent to $\xi\in X$ with respect to the probabilistic norm
$\vartheta$ if for every $\varepsilon>0$, $\lambda\in(0,1)$ and some non
negative number $r$ there exists $n_{0}\in\mathbb{N}$ such that
$\vartheta(x_{n}-\xi;r+\varepsilon)>1-\lambda$ for all $n>n_{0}$. In this case
we write $r_{\vartheta}\text{-}\lim_{n\to\infty}x_{n}=\xi\ \text{or}\
x_{n}\xrightarrow{r_{\vartheta}}\xi$ and $\xi$ is called $r_{\vartheta}$-limit
of $\\{x_{n}\\}_{n\in\mathbb{N}}$.

######  Definition 2.8.

[5] Let $\\{x_{n}\\}_{n\in\mathbb{N}}$ be a sequence in an PNS
$(X,\vartheta,\star)$. Then $\\{x_{n}\\}_{n\in\mathbb{N}}$ is said to be rough
statistically convergent to $\xi\in X$ with respect to the probabilistic norm
$\vartheta$ if for every $\varepsilon>0$ and $\lambda\in(0,1)$ and some non
negative number $r$,
$\delta(\\{n\in\mathbb{N}:\vartheta(x_{n}-\xi;r+\varepsilon)\leq
1-\lambda\\})=0$. In this case we write
$r\text{-}St_{\vartheta}\text{-}\lim_{n\to\infty}x_{n}=\xi\ \text{or}\
x_{n}\xrightarrow{r-St_{\vartheta}}\xi$.

######  Definition 2.9.

[21] The double natural density of the set
$K\subseteq\mathbb{N}\times\mathbb{N}$ is defined by

$\delta_{2}(K)=\lim_{m,n\to\infty}\frac{|\\{(i,j)\in K:i\leq m\ \text{and}\
j\leq n\\}|}{mn}$

where $|\\{(i,j)\in K:i\leq m\ \text{and}\ j\leq n\\}|$ denotes the number of
elements of $K$ not exceeding $m$ and $n$ respectively. It can be observed
that if $K$ is finite, then $\delta_{2}(K)=0$. Also, if $A\subseteq B$, then
$\delta_{2}(A)\leq\delta_{2}(B)$.

######  Definition 2.10.

[17] Let $\\{x_{mn}\\}$ be a double sequence in a PNS
$(X,\vartheta,\diamond)$. Then $\\{x_{mn}\\}$ is said to be statistically
convergent to $\xi\in X$ with respect to the probabilistic norm $\vartheta$ if
for every $\varepsilon>0$ and $\lambda\in(0,1)$, $K=\\{(m,n),m\leq i,n\leq
j:\vartheta(x_{mn}-\xi;\varepsilon)\leq 1-\lambda\\}$ has double natural
density zero, that is, if $K(i,j)$ become the numbers of $(m,n)$ in $K$:

$\lim_{i,j}\frac{K(i,j)}{ij}=0.$

In this case we write $st_{2}^{\vartheta}\text{-}\lim x_{mn}=\xi$ or
$x_{mn}\xrightarrow{st_{2}^{\vartheta}}\xi$.

######  Definition 2.11.

[24] A subsequence $x^{\prime}=\\{x_{j_{p}k_{q}}\\}$ of a double sequence
$\\{x_{jk}\\}$ is called a dense subsequence , if
$\delta_{2}(\\{(j_{p}k_{q})\in\mathbb{N}\times\mathbb{N}:p,q\in\mathbb{N}\\})=1$.

##  3\. Main Results

First we define rough convergence and rough statistical convergence of double
sequences in probabilistic normed spaces.

######  Definition 3.1.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$ and
$r$ be a non negative real number. Then $\\{x_{mn}\\}$ is said to be rough
convergent to $\beta\in X$ with respect to the probabilistic norm $\vartheta$
if for every $\varepsilon>0$, $\lambda\in(0,1)$ there exists
$n_{0}\in\mathbb{N}$ such that
$\vartheta(x_{mn}-\beta;r+\varepsilon)>1-\lambda$ for all $m,n\geq n_{0}$. In
this case $\beta$ is called $r_{2}^{\vartheta}$-limit of $\\{x_{mn}\\}$ and we
write $x_{mn}\xrightarrow{r_{2}^{\vartheta}}\beta$.

######  Definition 3.2.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$ and
$r$ be a non negative real number. Then $\\{x_{mn}\\}$ is said to be rough
statistical convergent to $\beta\in X$ with respect to the probabilistic norm
$\vartheta$ if for every $\varepsilon>0$, $\lambda\in(0,1)$,
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\beta;r+\varepsilon)\leq
1-\lambda\\})=0$. In this case $\beta$ is called
$r\text{-}st_{2}^{\vartheta}$-limit of $\\{x_{mn}\\}$ and we write
$r\text{-}st_{2}^{\vartheta}\text{-}\lim_{m,n\to\infty}x_{mn}=\beta$ or
$x_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\beta$.

######  Remark 3.1.

$(a)$ If we put $r=0$ in Definition 3.1, then the notion of rough convergence
of a double sequence with respect to the probabilistic norm $\vartheta$
coincides with notion of ordinary convergence of the double sequence with
respect to the probabilistic norm $\vartheta$.
$(b)$ From Definition 3.1, it is clear that $r_{2}^{\vartheta}$-limit of a
double sequence may not be unique. So, we denote
$LIM_{x_{mn}}^{r_{\vartheta}}$ to mean the set of all
$r_{2}^{\vartheta}$-limit of $\\{x_{mn}\\}$ with respect to the probabilistic
norm $\vartheta$.
$(c)$ If we put $r=0$ in Definition 3.2, then the notion of rough statistical
convergence of a double sequence with respect to the probabilistic norm
$\vartheta$ coincides with statistical convergence of the double sequence with
respect to the probabilistic norm $\vartheta$. So, our whole discussion is on
the fact $r>0$.
$(d)$ From Definition 3.2, it is clear that
$r\text{-}st_{2}^{\vartheta}$-limit of a double sequence may not be unique.
So, we denote $st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ to mean the set of
all $r\text{-}st_{2}^{\vartheta}$-limit of $\\{x_{mn}\\}$ with respect to the
probabilistic norm $\vartheta$.

The sequence $\\{x_{mn}\\}$ is said to be $r_{2}^{\vartheta}$-convergent if
$LIM_{x_{mn}}^{r_{\vartheta}}\neq\emptyset$. But, if the sequence is unbounded
with respect to the probabilistic norm $\vartheta$ then
$LIM_{x_{mn}}^{r_{\vartheta}}=\emptyset$ although in this case
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\neq\emptyset$ may be happened
which has been shown in the following example.

######  Example 3.1.

Let $(X,\left\lVert\cdot\right\rVert)$ be a real normed linear space and let
$\vartheta(x;t)=\frac{t}{t+\left\lVert x\right\rVert}$ for $x\in X$ and $t>0$.
Then $(X,\vartheta,\diamond)$ is a PNS under the $t$-norm $\diamond$ defined
by $x\diamond y=\min\\{x,y\\}$. For all $m,n\in\mathbb{N}$, we define a
sequence $\\{x_{mn}\\}$ by $x_{mn}=\begin{cases}(-1)^{m+n},\ m,n\neq i^{2}\
(i\in\mathbb{N})\\\ mn,\ \text{otherwise}\end{cases}$. Then, we have
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}=\begin{cases}\emptyset,\ r<1\\\
[1-r,r-1],\ \text{otherwise}\end{cases}$ and
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}=\emptyset$ when $r=0$. Also,
$LIM_{x_{mn}}^{r_{\vartheta}}=\emptyset$ for any $r\geq 0$.

######  Remark 3.2.

From Example 3.1, we have
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\neq\emptyset$ does not imply
$LIM_{x_{mn}}^{r_{\vartheta}}\neq\emptyset$. But,
$LIM_{x_{mn}}^{r_{\vartheta}}\neq\emptyset$ always implies
$LIM_{x_{mn}}^{r_{\vartheta}}\neq\emptyset$ as
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\ \text{either}\ m\
\text{or}\ n\ \text{runs over finite subsets of}\ \mathbb{N}\\})=0$. So,
$LIM_{x_{mn}}^{r_{\vartheta}}\subset
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$.

######  Example 3.2.

We take the PNS in Example 3.1 and define the double sequence $\\{x_{mn}\\}$
by $x_{mn}=\begin{cases}mn,\ m,n=i^{2}\ (i\in\mathbb{N})\\\ 0,\
\text{otherwise}\end{cases}$. Then,
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}=[-r,r]$. Now, if we consider a
subsequence $\\{x_{m_{j}n_{k}}\\}$ of $\\{x_{mn}\\}$ such that
$m_{j}=j^{2},n_{k}=k^{2}$, $j,k\in\mathbb{N}$, then
$st_{2}^{\vartheta}\text{-}LIM_{x_{m_{j}n_{k}}}^{r}=\emptyset$.

######  Remark 3.3.

From Example 3.2, for any subsequence of a double sequence we not not conclude
that $st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\subseteq
st_{2}^{\vartheta}\text{-}LIM_{x_{m_{j}n_{k}}}^{r}$.

But, this inclusion may be hold under certain condition which has been given
in the following theorem.

######  Theorem 3.1.

Let $\\{x_{m_{j}n_{k}}\\}$ be a dense subsequence of $\\{x_{mn}\\}$ in a PNS
$(X,\vartheta,\diamond)$. Then
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\subseteq
st_{2}^{\vartheta}\text{-}LIM_{x_{m_{j}n_{k}}}^{r}$.

###### Proof.

The proof is obvious. So, we omit details. ∎

######  Definition 3.3.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$.
Then $\\{x_{mn}\\}$ is said to be statistically bounded with respect to the
probabilistic norm $\vartheta$ if for every $\lambda\in(0,1)$ there exists a
positive real number $G$ such that
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn};G)\leq
1-\lambda\\})=0$.

######  Theorem 3.2.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$.
Then $\\{x_{mn}\\}$ is statistically bounded if and only if
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\neq\emptyset$ for some $r>0$.

###### Proof.

First suppose that $\\{x_{mn}\\}$ is statistically bounded. Then for every
$\lambda\in(0,1)$ there exists a positive real number $G$ such that
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn};G)\leq
1-\lambda\\})=0$. Now, let
$M=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn};G)\leq 1-\lambda\\}$
and $\theta$ be the zero element in $X$. Now for $m,n\in M^{c}$ we have
$\vartheta(x_{mn}-\theta;r+G)\geq\vartheta(x_{mn};G)\diamond\vartheta(\theta;r)>(1-\lambda)\diamond
1=1-\lambda$. This gives $\theta\in
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ and consequently
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\neq\emptyset$.

Conversely suppose that
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\neq\emptyset$. Let $\xi\in
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\neq\emptyset$. Then for every
$\varepsilon>0$ and $\lambda\in(0,1)$,
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\xi;r+\varepsilon)\leq
1-\lambda\\})=0$. Therefore almost all $x_{mn}$ are contained in some ball
with center $\xi$. This shows that $\\{x_{mn}\\}$ is statistically bounded.
This completes the proof. ∎

Now we give the algebraic characterization of rough statistically convergent
double sequences in probabilistic normed spaces.

######  Theorem 3.3.

Let $\\{x_{mn}\\}$ and $\\{y_{mn}\\}$ be double sequences in a PNS
$(X,\vartheta,\diamond)$. Then for some $r>0$ the following statements hold:

  1. (1)

If $x_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\beta$ and
$y_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\eta$ then
$x_{mn}+y_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\beta+\eta$

  2. (2)

If $x_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\beta$ and $\alpha(\neq
0)\in\mathbb{R}$ then $\alpha
x_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\alpha\beta$.

###### Proof.

Let $\\{x_{mn}\\}$ and $\\{y_{mn}\\}$ be double sequences in PNS
$(X,\vartheta,\diamond)$ and $r>0$.

  1. (1)

Let $x_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\beta$ and
$y_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\eta$. Let $\varepsilon>0$. Now, for
a given $\lambda\in(0,1)$, choose $s\in(0,1)$ such that
$(1-s)\diamond(1-s)>1-\lambda$. So, $\delta_{2}(A)=0$ and $\delta_{2}(B)=0$
where
$A=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\beta;\frac{r+\varepsilon}{2})\leq
1-s\\}$ and
$B=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(y_{mn}-\eta;\frac{r+\varepsilon}{2})\leq
1-s\\}$. Now for $(i,j)\in A^{c}\cap B^{c}$, we have
$\vartheta(x_{ij}+y_{ij}-(\beta+\eta);r+\varepsilon)\geq\vartheta(x_{ij}-\beta;\frac{r+\varepsilon}{2})\diamond(y_{ij}-\eta;\frac{r+\varepsilon}{2})>(1-s)\diamond(1-s)>1-\lambda$,
i.e. $A^{c}\cap
B^{c}\subset\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}+y_{ij}-(\beta+\eta);r+\varepsilon)>1-\lambda\\}$.
Therefore
$\delta_{2}(\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}+y_{ij}-(\beta+\eta);r+\varepsilon)\leq
1-\lambda\\})=0$, which gives
$x_{mn}+y_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\beta+\eta$.

  2. (2)

Since $x_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\beta$ and $\alpha\neq 0$, then
for every $\varepsilon>0$ and $\lambda\in(0,1)$,
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\beta;\frac{r+\varepsilon}{|\alpha|})\leq
1-\lambda\\})=0$ i.e.,
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(\alpha
x_{mn}-\alpha\beta;r+\varepsilon)\leq 1-\lambda\\})=0$, which gives $\alpha
x_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\alpha\beta$. This completes the
proof.

∎

We will discuss on some topological and geometrical properties of the set
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$.

######  Theorem 3.4.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$.
Then the set $st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ is closed.

###### Proof.

If $st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}=\emptyset$ then we have nothing
to prove. So, let $st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\neq\emptyset$.
Suppose that $\\{y_{mn}\\}$ is a double sequence in
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ such that
$y_{mn}\xrightarrow{\vartheta_{2}}\beta$. For a given $\lambda\in(0,1)$ choose
$s\in(0,1)$ such that $(1-s)\diamond(1-s)>1-\lambda$. Then for every
$\varepsilon>0$ there exists $n_{0}\in\mathbb{N}$, such that
$\vartheta(y_{mn}-\beta;\frac{\varepsilon}{2})>1-s$ for all $m,n>n_{0}$.
Suppose $i,j>n_{0}$. Then $\vartheta(y_{ij}-\beta;\frac{\varepsilon}{2})>1-s$.
Again, since $\\{y_{ij}\\}\in st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$,
then $\delta_{2}(P)=0$ where
$P=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-y_{ij};r+\frac{\varepsilon}{2})\leq
1-s\\}$. Now for $(s,t)\in P^{c}$, we have
$\vartheta(x_{st}-\beta;r+\varepsilon)\geq\vartheta(x_{st}-y_{ij};r+\frac{\varepsilon}{2})\diamond\vartheta(y_{ij}-\beta;\frac{\varepsilon}{2})>(1-s)\diamond(1-s)>1-\lambda$.
Therefore
$\\{(s,t)\in\mathbb{(}N)\times\mathbb{N}:\vartheta(x_{st}-\beta;r+\varepsilon)\leq
1-\lambda\\}\subset P$. Since $\delta_{2}(P)=0$, therefore
$\delta_{2}(\\{(s,t)\in\mathbb{(}N)\times\mathbb{N}:\vartheta(x_{st}-\beta;r+\varepsilon)\leq
1-\lambda\\})=0$. Consequently $\beta\in
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$. So,
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ is closed. This completes the
proof. ∎

######  Theorem 3.5.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$.
Then the set $st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ is convex for some
$r>0$.

###### Proof.

Let $\xi_{1},\xi_{2}\in st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ and
$t\in(0,1)$. Suppose $\lambda\in(0,1)$. Choose $s\in(0,1)$ such that
$(1-s)\diamond(1-s)>1-\lambda$. Then $\delta_{2}(A)=0$ and $\delta_{2}(B)=0$
where
$A=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\xi_{1};\frac{r+\varepsilon}{2(1-t)})\leq
1-s\\}$ and
$B=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\xi_{2};\frac{r+\varepsilon}{2t})\leq
1-s\\}$. Now for $i,j\in A^{c}\cap B^{c}$ we have
$\vartheta(x_{ij}-[(1-t)\xi_{1}+t\xi_{2}];r+\varepsilon)\geq\vartheta((1-t)(x_{ij}-\xi_{1});\frac{r+\varepsilon}{2})\diamond\vartheta(t(x_{ij}-\xi_{2});\frac{r+\varepsilon}{2})=\vartheta(x_{ij}-\xi_{1};\frac{r+\varepsilon}{2(1-t)})\diamond\vartheta(x_{ij}-\xi_{2};\frac{r+\varepsilon}{2t})>(1-s)\diamond(1-s)>1-\lambda$.
Therefore
$\\{(i,j)\in\mathbb{N}\times{N}:\vartheta(x_{ij}-[(1-t)\xi_{1}+t\xi_{2}];r+\varepsilon)\leq
1-\lambda\\}\subset A\cup B$. Hence
$\delta_{2}(\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-[(1-t)\xi_{1}+t\xi_{2}];r+\varepsilon)\leq
1-\lambda\\})=0$, i.e. $(1-t)\xi_{1}+t\xi_{2}\in
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$. Therefore
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ is convex. This completes the
proof. ∎

######  Theorem 3.6.

A double sequence $\\{x_{mn}\\}$ in a PNS $(X,\vartheta,\diamond)$ is rough
statistically convergent to $\xi\in X$ with respect to the probabilistic norm
$\vartheta$ for some $r>0$ if there exists a double sequence $\\{y_{mn}\\}$ in
$X$ such that $st_{2}^{\vartheta}\text{-}\lim y_{mn}=\xi$ and for every
$\lambda\in(0,1)$, $\vartheta(x_{mn}-y_{mn};r)>1-\lambda$ for all
$m,n\in\mathbb{N}$.

###### Proof.

Let $\varepsilon>0$ be given. For a given $\lambda\in(0,1)$, choose
$s\in(0,1)$ such that $(1-s)\diamond(1-s)>1-\lambda$. Suppose that
$st_{2}^{\vartheta}\text{-}\lim y_{mn}=\xi$ and
$\vartheta(x_{mn}-y_{mn};r)>1-\lambda$ for all $m,n\in\mathbb{N}$. Then
$\delta_{2}(A)=0$ where
$A=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(y_{mn}-\xi;\varepsilon)\leq
1-s\\}$. Now for $(i,j)\in A^{c}$, we have
$\vartheta(x_{ij}-\xi;r+\varepsilon)\geq\vartheta(x_{ij}-y_{ij};r)\diamond\vartheta(y_{ij}-\xi;r)>(1-s)\diamond(1-s)>1-\lambda$.
Therefore
$\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-\xi;r+\varepsilon)\leq
1-\lambda\\}\subset A$. Hence
$\delta_{2}(\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-\xi;r+\varepsilon)\leq
1-\lambda\\})=0$. Consequently $x_{mn}\xrightarrow{r-st_{2}^{\vartheta}}\xi$.
This completes the proof. ∎

######  Theorem 3.7.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$.
Then there do not exist $x_{1},x_{2}\in
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ for some $r>0$ and every
$\lambda\in(0,1)$ such that $\vartheta(x_{1}-x_{2};mr)\leq 1-\lambda$ for
$m(\in\mathbb{R})>2$.

###### Proof.

On contrary, we assume that there exist $x_{1},x_{2}\in
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$ for which
$\vartheta(x_{1}-x_{2};mr)\leq 1-\lambda$ for $m(\in\mathbb{R})>2$. Let
$\varepsilon>0$ be given. For a given $\lambda\in(0,1)$, choose $s\in(0,1)$
such that $(1-s)\diamond(1-s)>1-\lambda$. Then
$P=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-x_{1};r+\frac{\varepsilon}{2})\leq
1-s\\}$ and
$Q=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-x_{2};r+\frac{\varepsilon}{2})\leq
1-s\\}$ have double natural density zero. Now for $(i,j)\in P^{c}\cap Q^{c}$,
we have
$\vartheta(x_{1}-x_{2};2r+\varepsilon)\geq\vartheta(x_{ij}-x_{1};r+\frac{\varepsilon}{2})\diamond\vartheta(x_{ij}-x_{2};r+\frac{\varepsilon}{2})>(1-s)\diamond(1-s)>1-\lambda$.
Therefore,

$\vartheta(x_{1}-x_{2};2r+\varepsilon)>1-\lambda$ (3.1)

Now, if we choose $\varepsilon=mr-2r,m>2$, in Equation 3.1 then we have
$\vartheta(x_{1}-x_{2};mr)>1-\lambda,m>2$, which is a contradiction. This
completes the proof. ∎

######  Definition 3.4.

(c.f. [16]) Let $\\{x_{mn}\\}$ be a double sequence in a PNS
$(X,\vartheta,\diamond)$. Then a point $\xi\in X$ is said to be statistical
cluster point of $\\{x_{mn}\\}$ with respect to the probabilistic norm
$\vartheta$ if for every $\varepsilon>0$ and $\lambda\in(0,1)$,
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\xi;\varepsilon)>1-\lambda\\})>0$.

We denote $\Lambda_{(x_{mn})}(st_{2}^{\vartheta})$ to mean ordinary
statistical cluster points of $\\{x_{mn}\\}$ with respect to the probabilistic
norm $\vartheta$.

######  Definition 3.5.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$.
Then a point $\xi\in X$ is said to be rough statistical cluster point of
$\\{x_{mn}\\}$ with respect to the probabilistic norm $\vartheta$ if for every
$\varepsilon>0$, $\lambda\in(0,1)$ and some $r>0$,
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\xi;r+\varepsilon)>1-\lambda\\})>0$.
The set of all rough statistical cluster points of $\\{x_{mn}\\}$ is denoted
as $\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$.

######  Remark 3.4.

If $r=0$, then
$\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})=\Lambda_{(x_{mn})}(st_{2}^{\vartheta})$.

######  Theorem 3.8.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$.
Then, the set $\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$ is closed for some
$r>0$.

###### Proof.

If $\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})=\emptyset$ then we have nothing
to prove. So, let $\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})\neq\emptyset$.
Suppose that $\\{\omega_{mn}\\}$ is a double sequence in
$\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$ such that
$\omega_{mn}\xrightarrow{\vartheta_{2}}\zeta$. Now for given
$\lambda\in(0,1)$, choose $s\in(0,1)$ such that
$(1-s)\diamond(1-s)>1-\lambda$. Then for every $\varepsilon>0$ there exists
$k_{\varepsilon}\in\mathbb{N}$ such that
$\vartheta(\omega_{mn}-\zeta;\frac{\varepsilon}{2})>1-s$ for all $m,n\geq
k_{\varepsilon}$. Now choose $m_{0},n_{0}\in\mathbb{N}$ such that
$m_{0},n_{0}>k_{\varepsilon}$. Then
$\vartheta(\omega_{m_{0}n_{0}}-\zeta;\frac{\varepsilon}{2})>1-s$. Again, since
$\\{\omega_{m_{0}n_{0}}\\}\in\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$, then
$\delta_{2}(K)>0$ where
$K=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\omega_{m_{0}n_{0}};r+\frac{\varepsilon}{2})>1-s\\}$.
Now, for $(i,j)\in K$, we have
$\vartheta(x_{ij}-\zeta;r+\varepsilon)\geq\vartheta(x_{ij}-\omega_{m_{0}n_{0}};r+\frac{\varepsilon}{2})\diamond\vartheta(\omega_{m_{0}n_{0}}-\zeta;\frac{\varepsilon}{2})>(1-s)\diamond(1-s)>1-\lambda$.
Therefore
$\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-\zeta;r+\varepsilon)>1-\lambda\\}\subset
K$. Since $\delta_{2}(K)>0$,
$\delta_{2}(\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-\zeta;r+\varepsilon)>1-\lambda\\})>0$.
So, $\zeta\in\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$. Hence
$\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$ is closed. ∎

######  Theorem 3.9.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$ and
$r>0$. Then, for an arbitrary $\zeta\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})$
and $\lambda\in(0,1)$, $\vartheta(\gamma-\zeta;r)>1-\lambda$ for all
$\gamma\in\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$.

###### Proof.

For given $\lambda\in(0,1)$, choose $s\in(0,1)$ such that
$(1-s)\diamond(1-s)>1-\lambda$. Since
$\zeta\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})$, then for every
$\varepsilon>0$, $\delta_{2}(M)>0$ where
$M=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\zeta;\varepsilon)>1-s\\}$.
Now, we will show that if $\gamma\in X$ satisfying
$\vartheta(\gamma-\zeta;r)>1-s$ then
$\gamma\in\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$. Now, for $(i,j)\in M$
we have
$\vartheta(x_{ij}-\gamma;r+\varepsilon)\geq\vartheta(x_{ij}-\zeta;\varepsilon)\diamond\vartheta(\gamma-\zeta;r)>(1-s)\diamond(1-s)>1-\lambda$.
Therefore
$M\subset\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-\gamma;r+\varepsilon)>1-\lambda\\}$.
Hence
$\delta_{2}(\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-\gamma;r+\varepsilon)>1-\lambda\\})>0$,
Consequently, $\gamma\in\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$. This
completes the proof. ∎

######  Theorem 3.10.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$ and
$r>0$. Then for $\lambda\in(0,1)$ and fixed $x_{0}\in X$,
$\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})=\bigcup_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}$.

###### Proof.

For $\lambda\in(0,1)$, choose $s\in(0,1)$ such that
$(1-s)\diamond(1-s)>1-\lambda$. Let
$\gamma\in\bigcup_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}$.
Then there exists a $x_{0}\in\Lambda_{(x_{mn})}$ such that
$\vartheta(x_{0}-\gamma;r)>1-s$. Since $x_{0}\in\Lambda_{(x_{mn})}$, then for
every $\varepsilon>0$, $\delta_{2}(Z)>0$ where
$Z=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-x_{0};\varepsilon)>1-s\\}$.
Now for $(i,j)\in Z$, we have
$\vartheta(x_{ij}-\gamma;r+\varepsilon)\geq\vartheta(x_{ij}-x_{0};\varepsilon)\diamond\vartheta(x_{0}-\gamma;r)>(1-s)\diamond(1-s)>1-\lambda$.
Therefore
$Z\subset\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-\gamma;r+\varepsilon)>1-\lambda\\}$.
Since $\delta_{2}(Z)>0$,
$\delta_{2}(\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-\gamma;r+\varepsilon)>1-\lambda\\})>0$.
Hence $\gamma\in\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$ and so,
$\bigcup_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}\subset\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$.

conversely suppose that $\gamma\in\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})$.
Now, we show that
$\gamma\in\bigcup_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}$.
If possible, let
$\gamma\notin\bigcup_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}$.
Then for every $x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})$,
$\vartheta(\gamma-x_{0};r)<1-\lambda$, which contradicts the fact of Theorem
3.9. Hence
$\gamma\in\bigcup_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}$.
Therefore
$\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})\subset\bigcup_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}$.
This completes the proof. ∎

######  Theorem 3.11.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$.
Then for some $r>0$ and any $\lambda\in(0,1)$, the following statements hold:

  1. (1)

If $x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})$, then
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\subseteq\overline{B(x_{0},\lambda,r)}$.

  2. (2)

$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}=\bigcap_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}=\\{y_{0}\in
X:\Lambda_{(x_{mn})}(st_{2}^{\vartheta})\subseteq\overline{B(y_{0},\lambda,r)}\\}$

###### Proof.

Suppose $\\{x_{mn}\\}$ is a double sequence in a PNS $(X,\vartheta,\diamond)$
and $r>0$.

  1. (1)

Now, for a given $\lambda\in(0,1)$, choose $s\in(0,1)$ such that
$(1-s)\diamond(1-s)>1-\lambda$. Let $\beta\in
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$. Then for every $\varepsilon>0$,
we have $\delta_{2}(K_{1})=0$ and $\delta_{2}(k_{2})>0$ where
$K_{1}=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\beta;r+\varepsilon)\leq
1-s\\}$ and
$K_{2}=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-x_{0};\varepsilon)>1-s\\}$.
Now for $(i,j)\in K_{1}^{c}\cap K_{2}$, we have $\vartheta(\beta-
x_{0};r)\geq\vartheta(x_{ij}-\beta;r+\varepsilon)\diamond\vartheta(x_{ij}-x_{0};\varepsilon)>(1-s)\diamond(1-s)>1-\lambda$.
Therefore $\beta\in\overline{B(x_{0},\lambda,r)}$. Hence
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\subseteq\overline{B(x_{0},\lambda,r)}$.

  2. (2)

By the previous part, we have
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\subseteq\bigcap_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}$.
Let
$\gamma\in\bigcap_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}$.
Then, $\vartheta(\gamma-x_{0};r)\geq 1-\lambda$ for all
$x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})$ and hence
$\Lambda_{(x_{mn})}(st_{2}^{\vartheta})\subseteq\overline{B(\gamma,\lambda,r)}$,
i.e.,
$\bigcap_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}\subseteq\\{y_{0}\in
X:\Lambda_{(x_{mn})}(st_{2}^{\vartheta})\subseteq\overline{B(y_{0},\lambda,r)}\\}$.
Further, suppose that $\gamma\notin
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$. Then for $\varepsilon>0$,
$\delta_{2}(\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\gamma;r+\varepsilon)\leq
1-\lambda\\})\neq 0$, which gives that there exists a statistical cluster
point $x_{0}$ of $\\{x_{mn}\\}$ for which $\vartheta(\gamma-
x_{0};r+\varepsilon)\leq 1-\lambda$. Therefore
$\Lambda_{(x_{mn})}(st_{2}^{\vartheta})\nsubseteq\overline{B(\gamma,\lambda,r)}$
and $\gamma\notin\\{y_{0}\in
X:\Lambda_{(x_{mn})}(st_{2}^{\vartheta})\subseteq\overline{B(y_{0},\lambda,r)}\\}$.
Therefore $\\{y_{0}\in
X:\Lambda_{(x_{mn})}(st_{2}^{\vartheta})\subseteq\overline{B(y_{0},\lambda,r)}\\}\subseteq
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$. Therefore,
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}=\bigcap_{x_{0}\in\Lambda_{(x_{mn})}(st_{2}^{\vartheta})}\overline{B(x_{0},\lambda,r)}=\\{y_{0}\in
X:\Lambda_{(x_{mn})}(st_{2}^{\vartheta})\subseteq\overline{B(y_{0},\lambda,r)}\\}$.
This completes the proof.

∎

######  Theorem 3.12.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$ such
that $x_{mn}\xrightarrow{st_{2}^{\vartheta}}\zeta$. Then, there exists
$\lambda\in(0,1)$ such that
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}=\overline{B(\zeta,\lambda,r)}$ for
some $r>0$.

###### Proof.

For given $\lambda\in(0,1)$, choose $s\in(0,1)$ such that
$(1-s)\diamond(1-s)>1-\lambda$. Since
$x_{mn}\xrightarrow{st_{2}^{\vartheta}}\zeta$, then for every $\varepsilon>0$,
the set
$Y=\\{(m,n)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{mn}-\zeta;\varepsilon)\leq
1-s\\}$ has the double natural density zero. Now, let
$y_{*}\in\overline{B(\zeta,s,r)}$. Then $\vartheta(y_{*}-\zeta;r)\geq 1-s$.
Now, for $(i,j)\in Y^{c}$,
$\vartheta(x_{ij}-y_{*};r+\varepsilon)\geq\vartheta(x_{ij}-\zeta;\varepsilon)\diamond\vartheta(\zeta-
y_{*};r)>(1-s)\diamond(1-s)>1-\lambda$. Therefore
$\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-y_{*};r+\varepsilon)\leq
1-\lambda\\}\subseteq Y$. Since $\delta_{2}(Y)=0$,
$\delta_{2}(\\{(i,j)\in\mathbb{N}\times\mathbb{N}:\vartheta(x_{ij}-y_{*};r+\varepsilon)\leq
1-\lambda\\})=0$. Consequently, $y_{*}\in
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$. Hence
$\overline{B(\zeta,\lambda,r)}\subseteq
st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$. Again, since
$x_{mn}\xrightarrow{st_{2}^{\vartheta}}\zeta$,
$\Lambda_{(x_{mn})}(st_{2}^{\vartheta})=\\{\zeta\\}$ and consequently, from
Theorem 3.11 we have
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}\subseteq\overline{B(\zeta,\lambda,r)}$.
Hence
$st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}=\overline{B(\zeta,\lambda,r)}$.
This completes the proof. . ∎

######  Theorem 3.13.

Let $\\{x_{mn}\\}$ be a double sequence in a PNS $(X,\vartheta,\diamond)$ such
that $x_{mn}\xrightarrow{st_{2}^{\vartheta}}\eta$. Then
$\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})=st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$
for some $r>0$.

###### Proof.

Since $x_{mn}\xrightarrow{st_{2}^{\vartheta}}\eta$,
$\Lambda_{(x_{mn})}(st_{2}^{\vartheta})=\\{\eta\\}$. By Theorem 3.10, for
$\lambda\in(0,1)$,
$\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})=\overline{B(\eta,\lambda,r)}$.
Again, from Theorem 3.12,
$\overline{B(\eta,\lambda,r)}=st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$.
Therefore,
$\Lambda_{(x_{mn})}^{r}(st_{2}^{\vartheta})=st_{2}^{\vartheta}\text{-}LIM_{x_{mn}}^{r}$.
This completes the proof. ∎

### Authors’ contributions

The authors have contributed equally and significantly in writing this paper.
Both the authors read and approved the final version of the manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Acknowledgments

The second author is grateful to The Council of Scientific and Industrial
Research (CSIR), HRDG, India, for the grant of Senior Research Fellowship
during the preparation of this paper.

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#  Dynamics of entanglement in the
transverse Ising model

Zhe Chang111e-mail<EMAIL_ADDRESS>and Ning Wu222e-mail:
<EMAIL_ADDRESS>

Institute of High Energy Physics
Chinese Academy of Sciences
P. O. Box 918(4), 100049 Beijing, China

###### Abstract

We study the evolution of nearest-neighbor entanglement in the one dimensional
Ising model with an external transverse field. The system is initialized as
the so called “thermal ground state” of the pure Ising model. We analyze
properties of generation of entanglement for different regions of external
transverse fields. We find that the derivation of the time at which the
entanglement reaches its first maximum with respect to the reciprocal
transverse field has a minimum at the critical point. This is a new indicator
of quantum phase transition.

PACS numbers: 03.67.Bg, 75.10.Pq

##  1 Introduction

Entanglement has been recognized as an important resource for quantum
information and computation [1], and has been the subject of intense research
over the past few years [2]. Recently, there has been extensive analysis of
entanglement in quantum spin models [3]. Various models have been considered
for entanglement generation, and their static [4, 5] as well as dynamical
properties [6, 7, 8] have been investigated. The reason for this is that, many
of these models can be realized in cold atomic gas in an optical lattice [9,
10].

A special one among these models is the anisotropic XY model. It can be solved
exactly by means of the Jordan-Wigner transformation [11]. A lot of works have
been done on the dynamics of entanglement in this model. In Ref.[6], the time
evolution of initial Bell states was studied. Ref.[7] investigated the
dynamics of entanglement of a system prepared in a thermal equilibrium state,
and the system starts evolving after the transverse field was turned off.

In this paper, we analyze a spin-1/2 transverse Ising chain with a particular
separable initial state, the so-called “thermal ground state” [4] for the case
of zero field. We study the generation of nearest-neighbor entanglement as a
function of the external transverse field, and find the time at which the
entanglement reaches its first maximum. It turns out that the derivation of
this time with respect to the reciprocal transverse field has a minimum at the
critical point. This is a new indicator of quantum phase transition.

##  2 Static correlations

The anisotropic XY model is described by the Hamiltonian

$\displaystyle
H=-\frac{\lambda}{4}\sum^{N}_{i=1}[(1+\gamma)\sigma^{x}_{i}\sigma^{x}_{i+1}+(1-\gamma)\sigma^{y}_{i}\sigma^{y}_{i+1}]-\frac{1}{2}\sum^{N}_{i=1}\sigma^{z}_{i}~{},$
(1)

where $\sigma^{\alpha}_{i}$ are the Pauli matrix of the spin at site $i$ and
we assume periodic boundary conditions. We have neglected a common factor and
absorbed the external field into the reciprocal field $\lambda$. The
anisotropy parameter $\gamma$ connects the quantum Ising model for $\gamma=1$.
In the range $0<\gamma\leq 1$, the model belongs to the Ising universality
class and undergoes a quantum phase transition at the critical point
$\lambda_{c}=1$. We will consider the case of $\gamma=1$ in this paper, namely
the transverse Ising model

$\displaystyle
H=-\frac{1}{2}\sum^{N}_{i=1}(\lambda\sigma^{x}_{i}\sigma^{x}_{i+1}+\sigma^{z}_{i})~{}.$
(2)

The Hamiltonian defined in Eq.(2) can be mapped into a one-dimensional
spinless fermion system with creation and annihilation operators
$c^{\dagger}_{i}$ and $c_{i}$ via the Jordan-Wigner transformation
$\frac{\sigma^{x}_{i}-i\sigma^{y}_{i}}{2}=\prod_{j<i}(1-2c^{\dagger}_{j}c_{j})c_{i},~{}\frac{\sigma^{x}_{i}+i\sigma^{y}_{i}}{2}=c^{\dagger}_{i}\prod_{j<i}(1-2c^{\dagger}_{j}c_{j}),~{}\frac{\sigma^{z}_{i}}{2}=c^{\dagger}_{i}c_{i}-\frac{1}{2}$.
By making use of the transformation

$\displaystyle\eta_{k}=\frac{1}{\sqrt{N}}\sum_{l}e^{ikl}(\alpha_{k}c_{l}+i\beta_{k}c^{\dagger}_{l})$
(3)

further, with $\alpha_{k}=\frac{\Lambda_{k}-(1+\lambda\cos
k)}{\sqrt{2[\Lambda^{2}_{k}-(1+\lambda\cos
k)\Lambda_{k}]}},~{}\beta_{k}=\frac{\lambda\sin
k}{\sqrt{2[\Lambda^{2}_{k}-(1+\lambda\cos k)\Lambda_{k}]}}$, we get the
fermionic Hamiltonian finally

$\displaystyle
H=\sum_{k}\Lambda_{k}\left(\eta^{\dagger}_{k}\eta_{k}-\frac{1}{2}\right)~{},$
(4)

where the spectrum is $\Lambda_{k}=\sqrt{1+\lambda^{2}+2\lambda\cos k}$.

The evolution of the system is governed by Eq.(2). In the Heisenberg picture,
the time evolution of the spinless fermion operator $c_{i}(t)$ is [6]

$\displaystyle c_{i}(t)=\frac{1}{N}\sum_{k,l}[\cos k(l-i)A(k,t)c_{l}+\sin
k(l-i)B(k,t)c^{\dagger}_{l}]~{},$ (5)

where $A(k,t)=e^{i\Lambda_{k}t}-2i\beta^{2}_{k}\sin\Lambda_{k}t$,
$B(k,t)=-2i\alpha_{k}\beta_{k}\sin\Lambda_{k}t$.

The dynamics of the system also depends on the initial state. We choose the
“thermal ground state” $\rho_{0}=\frac{1}{2}(|N^{+}\rangle\langle
N^{+}|+|N^{-}\rangle\langle N^{-}|)$ of the pure Ising model (no transverse
field is applied) as the initial state. Here
$|N^{+}\rangle=|\rightarrow\rangle_{1}...|\rightarrow\rangle_{N}$, and
$|N^{-}\rangle=|\leftarrow\rangle_{1}...|\leftarrow\rangle_{N}$ are the two
degenerate ground states of the Ising model with all spins pointing to the
positive (negative) $x$ direction. Note that $|N^{+}\rangle$ and
$|N^{-}\rangle$ are the ground states of the Hamiltonian (2) with
$\lambda\to\infty$. When $\lambda$ is finite, they are even not eigenstates of
the Hamiltonian. Thus, the evolution from this initial state must be
nontrivial for finite $\lambda$. Physically, this can be viewed as an
instantaneous, i.e., idealized sudden quench [8] in the reciprocal field
$\lambda_{1}\to\lambda_{2}$ with $\lambda_{1}\to\infty$ and
$\lambda_{2}=\lambda$.

Due to the translational symmetry, we need only to consider averages of the
form $\langle c_{1}...\rangle_{0}$. Namely, we choose site-1 as “the first
site”, where $\langle...\rangle_{0}$ denote the average over the initial state
$\rho_{0}$. We rewrite $|N^{\pm}\rangle$ in the fermionic representation,

$\displaystyle|N^{\pm}\rangle=\left(\frac{1}{\sqrt{2}}\right)^{N}(1\pm
c^{\dagger}_{1})(1\pm c^{\dagger}_{2})...(1\pm
c^{\dagger}_{N})|0_{1},0_{2},...,0_{N}\rangle~{},$ (6)

where $|0_{1},0_{2},...,0_{N}\rangle$ denotes the vacuum of the fermions, or
the state for all spins down. Here we have to pay attention to the order of
operators in this equation. We write $(1\pm c^{\dagger}_{1})$ before operators
on all other sites in order to be compatible with the correlation $\langle
c_{1}...\rangle_{0}$. In other words, if another site $i$ is chosen as “the
first site”, we should write the state as
$|N^{\pm}\rangle=(\frac{1}{\sqrt{2}})^{N}(1\pm c^{\dagger}_{i})(1\pm
c^{\dagger}_{i+1})...(1\pm
c^{\dagger}_{i-1})|0_{i},0_{i+1},...,0_{i-1}\rangle$ to calculate averages
like $\langle c_{i}...\rangle_{0}$, due to the periodic boundary conditions.

The only single-site averages are $\langle c_{1}\rangle_{0}$ and $\langle
c^{\dagger}_{1}c_{1}\rangle_{0}$ (or their complex conjugate). It is easy to
see that $\langle N^{+}|c_{1}|N^{+}\rangle=\frac{1}{2}=-\langle
N^{-}|c_{1}|N^{-}\rangle$, so $\langle c_{1}\rangle_{0}=0$. As to $\langle
c^{\dagger}_{1}c_{1}\rangle_{0}$, we have $\langle
N^{+}|c^{\dagger}_{1}c_{1}|N^{+}\rangle=\frac{1}{2}\langle
0_{1}|(1+c_{1})c^{\dagger}_{1}c_{1}(1+c^{\dagger}_{1})|0_{1}\rangle=\frac{1}{2}$.
Similarly, $\langle N^{-}|c^{\dagger}_{1}c_{1}|N^{-}\rangle=\frac{1}{2}$. So
that, $\langle c^{\dagger}_{1}c_{1}\rangle_{0}=\frac{1}{2}$. We can also see
this point from $0=\langle\frac{\sigma^{z}_{1}}{2}\rangle_{0}=\langle
c^{\dagger}_{1}c_{1}-\frac{1}{2}\rangle_{0}$.

Next we consider two-point correlations of the form $\langle
c_{1}c_{l}\rangle_{0},~{}l\geq 2$. For $l=2$, we have

$\displaystyle\langle N^{+}|c_{1}c_{2}|N^{+}\rangle$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)^{2}\langle
0_{1},0_{2}|(1+c_{2})(1+c_{1})c_{1}c_{2}(1+c^{\dagger}_{1})(1+c^{\dagger}_{2})|0_{1},0_{2}\rangle$
$\displaystyle=$ $\displaystyle\left(\frac{1}{2}\right)^{2}\langle
0_{1}|(1+c_{1})c_{1}(1-c^{\dagger}_{1})|0_{1}\rangle\cdot\langle
0_{2}|(1+c_{2})c_{2}(1+c^{\dagger}_{2})|0_{2}\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{4}~{}.$

Similarly $\langle N^{-}|c_{1}c_{2}|N^{-}\rangle=-\frac{1}{4}$. So that
$\langle c_{1}c_{2}\rangle_{0}=-\frac{1}{4}$. We can show that $\langle
c^{\dagger}_{1}c_{2}\rangle_{0}=\frac{1}{4}$ in the same manner. For $l>2$,
when $c_{l}$ or $c^{\dagger}_{l}$ crosses $(1\pm c^{\dagger}_{2})$ to its
right side, this will make it be $(1\mp c^{\dagger}_{2})$. But $\langle
0_{2}|(1\pm c_{2})(1\mp c^{\dagger}_{2})|0_{2}\rangle=0$. So we have obtained
all the two-point correlations

$\displaystyle\langle
c^{\dagger}_{i}c_{j}\rangle_{0}=\frac{1}{4}(\delta_{i+1,j}+\delta_{i-1,j}+2\delta_{i,j}),$
$\displaystyle\langle
c_{i}c_{j}\rangle_{0}=\frac{1}{4}(-\delta_{i+1,j}+\delta_{i-1,j}),$
$\displaystyle\langle
c^{\dagger}_{i}c^{\dagger}_{j}\rangle_{0}=\frac{1}{4}(\delta_{i+1,j}-\delta_{i-1,j}).$
(7)

By the same means one can check that the averages involving three fermion
operators all vanish.

##  3 Reduced density matrix and evolution of concurrence

Making use of the periodic boundary conditions, we can focus on the reduced
density matrix $\rho^{12}(t)$ at time $t$ of the first and second spins. In
the
$\\{|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle\\}$
basis, the matrix elements reads,

$\displaystyle\rho^{12}_{11}(t)$ $\displaystyle=$
$\displaystyle\langle\uparrow\uparrow|\rho^{12}(t)|\uparrow\uparrow\rangle$
$\displaystyle=$
$\displaystyle\langle\uparrow\uparrow|\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2}|\uparrow\uparrow\rangle+\langle\uparrow\downarrow|\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2}|\uparrow\downarrow\rangle$
$\displaystyle+\langle\downarrow\uparrow|\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2}|\downarrow\uparrow\rangle+\langle\downarrow\downarrow|\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2}|\downarrow\downarrow\rangle$
$\displaystyle=$ $\displaystyle
tr_{12}(\rho^{12}(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2})$ $\displaystyle=$
$\displaystyle tr(\rho(t)S^{+}_{1}S^{-}_{1}S^{+}_{2}S^{-}_{2})$
$\displaystyle=$ $\displaystyle\langle
c^{\dagger}_{1}c_{1}c^{\dagger}_{2}c_{2}\rangle_{t}.$

Other matrix elements can be obtained similarly. Thus, we have

$\displaystyle\rho^{12}(t)=\left(\begin{array}[]{cccc}\langle
c^{\dagger}_{1}c_{1}c^{\dagger}_{2}c_{2}\rangle_{t}&-\langle
c^{\dagger}_{1}c_{1}c_{2}\rangle_{t}&\langle
c_{2}^{\dagger}c_{2}c_{1}\rangle_{t}&\langle c_{2}c_{1}\rangle_{t}\\\ \\\
-\langle c^{\dagger}_{2}c^{\dagger}_{1}c_{1}\rangle_{t}&\langle
c^{\dagger}_{1}c_{1}c_{2}c^{\dagger}_{2}\rangle_{t}&-\langle
c_{1}c^{\dagger}_{2}\rangle_{t}&\langle
c_{1}c_{2}c^{\dagger}_{2}\rangle_{t}\\\ \\\ \langle
c^{\dagger}_{1}c_{2}^{\dagger}c_{2}\rangle_{t}&-\langle
c_{2}c_{1}^{\dagger}\rangle_{t}&\langle
c_{1}c^{\dagger}_{1}c^{\dagger}_{2}c_{2}\rangle_{t}&\langle
c_{1}c^{\dagger}_{1}c_{2}\rangle_{t}\\\ \\\ \langle
c^{\dagger}_{1}c^{\dagger}_{2}\rangle_{t}&\langle
c_{2}c^{\dagger}_{2}c^{\dagger}_{1}\rangle_{t}&\langle
c^{\dagger}_{2}c_{1}c^{\dagger}_{1}\rangle_{t}&\langle
c_{1}c^{\dagger}_{1}c_{2}c^{\dagger}_{2}\rangle_{t}\\\ \end{array}\right),$
(15)

All the elements can be equally evaluated in the Heisenberg picture, e.g.,
$\rho^{12}_{11}(t)=\langle
c^{\dagger}_{1}c_{1}c^{\dagger}_{2}c_{2}\rangle_{t}=\langle
c^{\dagger}_{1}(t)c_{1}(t)c^{\dagger}_{2}(t)c_{2}(t)\rangle_{0}$. We have
mentioned that all the static averages involving three fermion operators equal
zero. From Eq.(5), the fermion operators in the Heisenberg picture are merely
some linear superpositions of the same set of operators in the Schrodinger
picture. So all matrix elements involving three operators in Eq.(8) also
vanish. Using Wick’s theorem, one gets all non-vanishing elements of
$\rho^{12}(t)$ as following

$\displaystyle\rho^{12}_{11}(t)=\langle
c^{\dagger}_{1}(t)c_{1}(t)\rangle_{0}\langle
c^{\dagger}_{2}(t)c_{2}(t)\rangle_{0}-\langle
c^{\dagger}_{1}(t)c^{\dagger}_{2}(t)\rangle_{0}\langle
c_{1}(t)c_{2}(t)\rangle_{0}-\langle c^{\dagger}_{1}(t)c_{2}(t)\rangle\langle
c^{\dagger}_{2}(t)c_{1}(t)\rangle_{0},$
$\displaystyle\rho^{12}_{22}(t)=\rho^{12}_{33}(t)=\langle
c^{\dagger}_{1}(t)c_{1}(t)\rangle_{0}-\rho^{12}_{11}(t),$
$\displaystyle\rho^{12}_{44}(t)=1-\rho^{12}_{11}(t)-2\rho^{12}_{22}(t),$
$\displaystyle\rho^{12}_{14}(t)=\rho^{12*}_{41}(t)=\langle
c_{2}(t)c_{1}(t)\rangle_{0},$
$\displaystyle\rho^{12}_{23}(t)=\rho^{12*}_{32}(t)=-\langle
c_{1}(t)c^{\dagger}_{2}(t)\rangle_{0}.$ (16)

We see that if we can calculate $\langle
c^{\dagger}_{1}(t)c_{1}(t)\rangle_{0},~{}\langle
c^{\dagger}_{1}(t)c_{2}(t)\rangle_{0}$ and $\langle
c_{1}(t)c_{2}(t)\rangle_{0}$, then the reduced density matrix is totally
determined.

Using Eq.(5) and (7), we find, in the thermodynamic limit, that

$\displaystyle\langle c^{\dagger}_{1}(t)c_{1}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int^{\pi}_{-\pi}dk\sin
k(A^{*}(k)-A(k))(B(k)-B^{*}(k))+\frac{1}{4\pi}\int^{\pi}_{-\pi}dk(|A(k)|^{2}+|B(k)|^{2})$
$\displaystyle+\frac{1}{4\pi}\int^{\pi}_{-\pi}dk\cos k(|A(k)|^{2}-|B(k)|^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\pi}_{0}dk\frac{\lambda\sin^{2}k\sin^{2}\Lambda_{k}t}{\Lambda^{2}_{k}}+\frac{1}{2},$
$\displaystyle\langle c^{\dagger}_{1}(t)c_{2}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\int^{\pi}_{-\pi}dk(1+\cos
2k)(|A(k)|^{2}-|B(k)|^{2})+\frac{1}{4\pi}\int^{\pi}_{-\pi}dk\cos
k(|A(k)|^{2}+|B(k)|^{2})$ $\displaystyle+\frac{1}{8\pi}\int^{\pi}_{-\pi}dk\sin
2k[A(k)-A^{*}(k)]B^{*}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\pi}_{0}dk\frac{\lambda\cos
k\sin^{2}k\sin^{2}\Lambda_{k}t}{\Lambda^{2}_{k}}+\frac{1}{4},$
$\displaystyle\langle c_{1}(t)c_{2}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int^{\pi}_{-\pi}dk\sin
2kA(k)B(k)+\frac{1}{8\pi}\int^{\pi}_{-\pi}dk[B(k)^{2}-A(k)^{2}](1-\cos 2k)$
(17) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int^{\pi}_{0}dk\sin^{2}k[\frac{i\sin
2\Lambda_{k}t}{\Lambda_{k}}-\cos
2\Lambda_{k}t-2\frac{\lambda\sin^{2}\Lambda_{k}t}{\Lambda_{k}^{2}}(\cos
k+\lambda)]$

Note that $\langle c^{\dagger}_{1}(t)c_{2}(t)\rangle$ is indeed real, so that
we have $\rho^{12}_{23}(t)=\rho^{12}_{32}(t)$.

For bipartite entanglement, a commonly used measure for arbitrary states of
two qubits is the so-called concurrence [12]. The concurrence is defined as

$\displaystyle C(t)$ $\displaystyle=$
$\displaystyle\rm{max}\\{0,2\lambda_{max}(t)-tr\sqrt{\rho^{12}(t)\tilde{\rho}^{12}(t)}\\},$
(18) $\displaystyle\tilde{\rho}^{12}(t)$ $\displaystyle=$
$\displaystyle\sigma_{y}\otimes\sigma_{y}\rho^{12*}(t)\sigma_{y}\otimes\sigma_{y},$
(19)

where $\lambda_{max}$ is the largest eigenvalue of the matrix
$\sqrt{\rho^{12}(t)\tilde{\rho}^{12}(t)}$.

In the present case,

$\displaystyle\rho^{12}(t)=\left(\begin{array}[]{cccc}\rho^{12}_{11}(t)&0&0&\rho^{12}_{14}(t)\\\
\\\ 0&\rho^{12}_{22}(t)&\rho^{12}_{23}(t)&0\\\ \\\
0&\rho^{12}_{23}(t)&\rho^{12}_{22}(t)&0\\\ \\\
\rho^{12*}_{14}(t)&0&0&\rho^{12}_{44}(t)\\\ \end{array}\right),$ (27)

the four eigenvalues of $\sqrt{\rho^{12}(t)\tilde{\rho}^{12}(t)}$ are of the
form

$\displaystyle\lambda_{1}(t)$ $\displaystyle=$
$\displaystyle||\rho^{12}_{14}(t)|+\sqrt{\rho^{12}_{11}(t)\rho^{12}_{44}(t)}|,$
$\displaystyle\lambda_{2}(t)$ $\displaystyle=$
$\displaystyle||\rho^{12}_{14}(t)|-\sqrt{\rho^{12}_{11}(t)\rho^{12}_{44}(t)}|,$
$\displaystyle\lambda_{3}(t)$ $\displaystyle=$
$\displaystyle|\rho^{12}_{22}(t)+\rho^{12}_{23}(t)|,$
$\displaystyle\lambda_{4}(t)$ $\displaystyle=$
$\displaystyle|\rho^{12}_{22}(t)-\rho^{12}_{23}(t)|.$ (28)

Figure 1: The evolution of bipartite entanglement for the initial state being
the “thermal ground state” of the Ising model.

In Fig.1, we give a plot of the evolution of entanglement as a function of
time $t$ and parameter $\lambda$. At the beginning, the reduced density matrix
is

$\displaystyle\rho^{12}(0)=\frac{1}{4}\left(\begin{array}[]{cccc}1&0&0&1\\\
0&1&1&0\\\ 0&1&1&0\\\ 1&0&0&1\\\ \end{array}\right).$ (33)

So that the concurrence equals zero. At any later fixed instant, the
entanglement as a function of $\lambda$ may have different behaviors in
different time intervals. For times before about $t=2.5$ (Fig.2 (a)), the
entanglement maintains its magnitude for large $\lambda$ with slow
oscillations. As $\lambda$ increases, the height of each peak deceases
gradually, and finally vanishes as $\lambda\to\infty$. This is easy to
understand. For $\lambda\to\infty$, the Hamiltonian reduces to the pure Ising
model, and the initial state is just the eigenstate of the Ising model. Thus,
only a phase factor contributes to the times evolving state, which does not
change the entanglement. Physically, this corresponds to
$\lambda_{1}=\lambda_{2}=\infty$, hence no quench takes place. For the narrow
band $2.47<t<2.69$, there is no entanglement at all for all $\lambda$. For
times greater than around $t=3.0$ (Fig.2 (b)), the entanglement emerges
abruptly, but only has non-vanishing values for very short interval of
$\lambda$. As the time increases, the generations of entanglement are
different from each other for different $\lambda$. For $\lambda$ is less than
about $\lambda=0.8$ (Fig.3 (a)), the concurrence will encounter more than one
maximums. In a time region between the first two maximums, no entanglement
appear. Although weak oscillations emerge after this region, the entanglement
can last for a fairly long time with a not too small magnitude. For
$0.80<\lambda<1.07$, there is only one maximum as the time increases. After a
short time, the entanglement vanishes. For $\lambda>1$ (Fig. 3 (b)), strong
oscillations emerge. The entanglement reaches its maximum fast, then suddenly
decreases to zero. Such a behavior will repeat again at later times, and
finally the entanglement vanishes completely.

Figure 2: (a): The concurrence at different (b): The concurrence at different
times
times $t=0.5,1.0,1.5,2.0$ as a function of $\lambda$, $t=4.0,5.0,6.0,7.0$ as a
function of $\lambda$,
in the region of $t<2.5$, in the region of $t>3.0$.

Figure 3: (a): The dynamics of entanglement (b): The dynamics of entanglement
for
for different reciprocal fields $\lambda=0.2,0.4,0.6,0.8$, different
reciprocal fields $\lambda=1.5,3.0$,
in the region of $\lambda<0.8$, in the region of $\lambda>1.07$.  Figure 4:
The time at which the entanglement reaches its first maximum and its
derivative.

For any $\lambda$, there is a time $T_{m}(\lambda)$ at which the entanglement
reaches its first maximum (Fig.4). This time is a monotonically decreasing
function of $\lambda$. Taking its derivative with respect to $\lambda$, we
find that the derivative $\frac{dT_{m}(\lambda)}{d\lambda}$ has a minimum at
the critical point $\lambda=1$. This interesting property maybe an indicator
of the quantum phase transition at the critical point.

##  4 Brief discussions and conclusions

In this paper, we have studied the dynamics of the transverse Ising chain
prepared in the “thermal ground state” of the pure Ising model. For different
values of the reciprocal field, the evolution of the entanglement has dramatic
distinctions. We also obtained the times at which the entanglement reaches its
first maximum, which is a monotonically decreasing function of the parameter
$\lambda$. The derivative of this quantity with respect to the reciprocal
field gets its minimum at the critical field. This is a new indicator of the
quantum phase transition. Let us finally remark that the initial state we
choose is simple to be prepared experimentally. One can first cool the pure
Ising system down to near absolute zero, then turn on some desired transverse
field at $t=0$ to generate the wanted entanglement.

## Acknowledgments

We would like to thank X. Li for useful discussions. The work was supported
partially by the NSF of China under Grant No. 10875129.

## References

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#  FLGo: A Fully Customizable Federated Learning Platform

Zheng Wang1,2 Xiaoliang Fan1,2∗ Zhaopeng Peng1,2 Xueheng Li1,2 Ziqi Yang1,2
Mingkuan Feng1,2 Zhicheng Yang1,2 Xiao Liu3 Cheng Wang1,2 1 Fujian Key
Laboratory of Sensing and Computing for Smart Cities, School of Informatics,
Xiamen University, Xiamen, China
2 Key Laboratory of Multimedia Trusted Perception and Efficient Computing,
Ministry of Education of China, Xiamen University, Xiamen, China
3 School of Information Technology, Deakin University, Geelong, Australia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Federated learning (FL) has found numerous applications in healthcare,
finance, and IoT scenarios. Many existing FL frameworks offer a range of
benchmarks to evaluate the performance of FL under realistic conditions.
However, the process of customizing simulations to accommodate application-
specific settings, data heterogeneity, and system heterogeneity typically
remains unnecessarily complicated. This creates significant hurdles for
traditional ML researchers in exploring the usage of FL, while also
compromising the shareability of codes across FL frameworks. To address this
issue, we propose a novel lightweight FL platform called FLGo, to facilitate
cross-application FL studies with a high degree of shareability. Our platform
offers 40+ benchmarks, 20+ algorithms, and 2 system simulators as out-of-the-
box plugins. We also provide user-friendly APIs for quickly customizing new
plugins that can be readily shared and reused for improved reproducibility.
Finally, we develop a range of experimental tools, including parallel
acceleration, experiment tracker and analyzer, and parameters auto-tuning.
FLGo is maintained at flgo-xmu.github.io.

##  1 Introduction

Figure 1: FLGo implements benchmarks, simulators, and algorithms as independent plugins, where developers can download plugins and share locally customized plugins through a plugin library. | TFF | LEAF | PySyft | FedML | FederatedScope | FedScale | Flower | FLGo
---|---|---|---|---|---|---|---|---
Data Heterogeneity | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
System Heterogeneity | 0 | $\times$ | $\times$ | 0 | 1 | 1 | 0 | 1
High-Level API | 0 | $\times$ | 0 | 1 | 1 | 1 | 1 | 1
Multi-Architecture | $\times$ | $\times$ | $\times$ | 1 | 1 | 0 | 0 | 1
Asynchrounous | $\times$ | $\times$ | $\times$ | 0 | 1 | $\times$ | $\times$ | 1
Experiment Manager | $\times$ | $\times$ | $\times$ | 1 | 1 | 0 | $\times$ | 1
Behavior Customization | $\times$ | $\times$ | $\times$ | 1 | 1 | 0 | 0 | 1
Benchmark Customization | $\times$ | $\times$ | $\times$ | 0 | $\times$ | $\times$ | $\times$ | 1
Heterogeneity Customization | $\times$ | $\times$ | $\times$ | 0 | 0 | 0 | 0 | 1
Table 1: Comparison on FL Frameworks. 0 represents partially implemented, 1
represents fully implemented, $\times$ represents not implemented.

Federated learning (FL) has attracted wide attention from both academics and
industries Kairouz et al. (2019); He et al. (2020); Li et al. (2020). Its
characteristic of privacy protection has made it a popular choice for data
security compliance applications, including medicine Liu et al. (2021a),
finance Long et al. (2021), internet of things Nguyen et al. (2021), etc.
Researchers have proposed numerous FL algorithms to address both data Wang et
al. (2021); Li et al. (2020); Karimireddy et al. (2020) and system
heterogeneityLi et al. (2020); Wang et al. (2020, 2022). However, the
effectiveness of these algorithms is limited Li et al. (2022), where each
algorithm can only bring non-trivial improvement, or even be worse than FedAvg
in specific contexts Li et al. (2022). Therefore, it is essential for
engineers to conduct quantitative evaluations on available FL algorithms
before deploying them.

Recently, several platforms were proposed to enable researchers to fairly
evaluate FL algorithms by providing multiple benchmarks. FedScaleLai et al.
(2022) uses systematic datasets to simulate realistic mobile-device settings
and real-world federated datasets, while FederatedScopeXie et al. (2023)
employs an architecture driven by events to customize the behaviors of
different participants. However, the complexity involved in implementing
customized settings, such as different applications, heterogeneous data, and
system heterogeneity, is commonly overlooked. This substantial limitation
poses a significant barrier for machine learning researchers seeking to employ
FL frameworks in practical scenarios. Additionally, the high complexity of
customization further harms the code shareability due to the absence of
standardized guidelines for customization, even when employing the same FL
framework. In short, addressing these issues is crucial in order to develop an
adaptable FL framework that facilitates cross-application FL studies with a
high degree of shareability.

To tackle these problems, we develop FLGo, a novel lightweight FL platform
that streamlines cross-application FL research with a high degree of
shareability. By leveraging the plugin library, developers can easily
distribute their customization as plugins, as illustrated in Fig.1. Our main
contributions are summerized as follows:

  * •

We propose a novel FL platform, FLGo, to support customization with high
shareability, where benchmarks, algorithms and simulators are implemented as
plugins enable the fast development of cross-application FL scenarios.

  * •

We provide 40+ comprehensive FL benchmarks, 20+ algorithms, and 2 system
simulators in FLGo. We also develop useful tools to support various
experimental purposes including parallel acceleration, experiment tracker and
analyzer, and parameters auto-tuning.

  * •

We conduct comprehensive experiments on 6 datasets to evaluate the
functionality of FLGo platform by customizing various data and system
heterogeneity settings.

##  2 Related Works

##### Existing frameworks for FL research.

Extensive federated frameworks were developed to better support FL studies in
both academic and industrial scenarios He et al. (2020); Liu et al. (2021b).
LEAF Caldas et al. (2018) provides several open-source federated benchmarks.
TFF Bonawitz et al. (2019) focuses on the application of large-scale mobile
devices and PySyft Ryffel et al. (2018) focuses on deep learning for data
security. FedML He et al. (2020) facilitates equitable comparison of
algorithms through the provision of benchmarks, multiple topologies, and
diverse computing paradigms. FedScale Lai et al. (2022) provides real-world
datasets to evaluate the statistical efficiency and system efficiency and
improve the efficiency under large scalability. FederatedScope Xie et al.
(2023) employs an event-driven architecture to enable arbitrary customization
on parties’ behaviors. Flower Beutel et al. (2020) supports different backends
and is also scalable. However, the challenge of evaluating FL algorithms for
customized applications, data heterogeneity, and system heterogeneity is often
overlooked. This issue presents a natural divide between conventional machine
learning developers and the FL community. By addressing this challenge and
enabling greater compatibility and shareability, FL can become more accessible
to a wider range of developers, further promoting its adoption and
advancement.

##### Existing frameworks are inadequate.

Although great efforts have been made in developing FL community, these
frameworks failed to 1) fast convert any ML task into a federated benchmark to
attract more researchers in the traditional ML community, and 2) provide high-
level APIs for customization on heterogeneity. To simulate more realistic and
complex scenarios when adopting FL, we propose FLGo. We make a comparison
between FLGo and other frameworks to distinguish the advantages of FLGo in
Table.1.

##  3 FLGo Features

###  3.1 Overview

Figure 2: The main workflow in FLGo. Category | Task | Scenario | Datasets
---|---|---|---
CV | Classification | Horizontal & Vertical |  | CIFAR10, CIFAR100, MNIST, FashionMNIST,
---
FEMNIST, EMNIST, SVHN
Detection | Horizontal | Coco, VOC
Segmentation | Horizontal | Coco, SBDataset
NLP | Classification | Horizontal | Sentiment140, AG_NEWS, sst2
Text Prediction | Horizontal | Shakespeare, Reddit
Translation | Horizontal | Multi30k
Graph | Node Classification | Horizontal | Cora, Citeseer, Pubmed
Link Prediction | Horizontal | Cora, Citeseer, Pubmed
Graph Classification | Horizontal | Enzymes, Mutag
Rec | Rating Prediction | Horizontal&Vertical | Ciao, Movielens, Epinions, Filmtrust, Douban
Series | Time series forecasting | Horizontal | Electricity, Exchange Rate
Tabular | Classification | Horizontal | Adult, Bank Marketing, Heart Disease
Synthetic | Regression | Horizontal | Synthetic, DistributedQP
Table 2: An overview of benchmarks in FLGo. Name | Feature Skew | Label Skew | Concept Drift | Concept Shift | Quantity Skew
---|---|---|---|---|---
IIDPartitioner McMahan et al. (2017) |  |  |  |  | $\checkmark$
DiversityPartitioner McMahan et al. (2017) | $\checkmark$ | $\checkmark$ |  |  |
DirichletPartitioner Hsu et al. (2019) | $\checkmark$ | $\checkmark$ |  |  | $\checkmark$
GaussianPerturbationPartitioner Li et al. (2022) | $\checkmark$ |  |  |  | $\checkmark$
IDPartitioner | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
VerticalPartitioner | $\checkmark$ |  |  |  | $\checkmark$
NodeLouvainPartitioner Zhang et al. (2021) | $\checkmark$ |  |  |  |
Table 3: Partitioners for data heterogeneity in FLGo.

The workflow in FLGo framework is as shown in Fig.2. First, the blue benchmark
plugin generates a static federated task on the disk, which describes how the
data is distributed among participants. Second, FLGo system will load the
model and the stored task to construct the simulation. Then, the red algorithm
plugin will optimize the model under the virtual environment created by the
green simulator plugin. Finally, the logger will trace the running-time
variables and save them as a record on the disk for further analysis.

###  3.2 Benchmark Module

####  3.2.1 Comprehensive Benchmarks

We have provided 40+ out-of-the-box FL benchmarks in Table.2. The details of
each dataset are on the website111https://flgo-xmu.github.io. These benchmarks
enable FLGo to support a wide range of studies through three characteristics:

  * •

Cross-Category. We implement the templates of benchmark plugins for different
types of input data including images, text, graphs, series, and tables.

  * •

Cross-Application. Tasks of different applications like finance, healthcare,
and IoT scenarios are integrated.

  * •

Cross-Architecture. Benchmark plugins can also support FL with different
architectures like horizontal FL, vertical FL, decentralized FL, and
hierarchical FL.

####  3.2.2 Customization

/*Code.1 import flgo flgo.gen_benchmark(name, config, target_path, data_type,
task_type) We design a general paradigm to ease the customization of various
benchmarks. In addition, we provide templates of benchmark plugins for
different categories of input data and different types of tasks. By using the
templates, developers can integrate their customized benchmarks with only one
line of code as shown in Code.1.

Figure 3: FLGo simulates synchronous and asynchronous scenarios by using the
global clock and the client-state machine, where each client’s state can shift
either as time passes or when a particular condition is reached (e.g., being
selected by the center).

####  3.2.3 Paritioner for Data Heterogeneity

To support studies in FL with different types of data heterogeneity
categorized by Kairouz et al. (2019), we create reusable data partitioners to
simulate real-world local data distributions. In this way, developers can
arbitrarily combine the benchmark plugins and partitioners together to design
the data heterogeneity. We also provide APIs for customizing new partitioners.
We list the data heterogeneity currently supported by FLGo in Table.3.

###  3.3 Simulator Module

Existing frameworks Xie et al. (2023); Lai et al. (2022) provided realizations
of particular system heterogeneity as benchmarks, but they fail to support
APIs for customization on the system heterogeneity. Since both different types
and degrees of system heterogeneity have non-trivial impacts on the training
effectiveness Lai et al. (2022); Wang et al. (2022), it’s essential to allow
developers to customize the system heterogeneity. To this end, we first
conclude four types of system heterogeneity that were investigated in previous
FL studies:

  * •

Availability. If a client is unavailable, the server cannot select it to join
the model training Wang et al. (2022).

  * •

Responsiveness. Responsiveness describes the length of the period for the
server to wait for the responses of participants Chai et al. (2020).

  * •

Completeness. The client model updates may be incompleteLi et al. (2020); Wang
et al. (2020).

  * •

Connectivity. Some participants may lose connection for a long period Jiang et
al. (2023) even if they had been selected to join.

According to the aforementioned four types of system heterogeneity, we design
2 simulators, where one is based on synthetic data and another one is based on
real-world data Lai et al. (2022). We first develop a client-state machine and
a global virtual clock to simulate heterogeneous systems. For example in
Fig.3, it takes client $i$ 3 units of time to finish local training at the
first round, and the length of time units that are spent by client $u$ is 4.
Based on the global clock and the client-state machine, we then design easy-
to-use APIs for developers to directly define how the state will shift for
each participant at each moment or each round. In this way, the four types of
system heterogeneity can be easily customized.

###  3.4 Algorithm Module

In FLGo, each algorithm plugin is described by parties with their behaviors.
We conclude the main features of the algorithm module of FLGo in the following
subsections.

####  3.4.1 Time-based asynchronism

Most of the previous studies in asynchronous FL are based on round-wise
asynchronism, which cannot be fairly compared with the synchronous FL methods
since the time cost of each round is overlooked Xie et al. (2019); Nguyen et
al. (2022). To this end, we implement the asynchronous FL algorithms based on
the global virtual clock, where the developers should consider the behaviors
of parties at each time unit when realizing asynchronous algorithms. For
example, the server of fully asynchronous FL in Fig.3 samples participants
once they become available (e.g. client $j$ at the 13th time unit) in round
$3$, and receives the locally trained model from client $k$ at the same time
unit. As a comparison, the synchronous FL server in rounds 1 and 2 will wait
for the slowest participant at each iteration. In this way, the algorithms in
syncFL and asyncFL are comparable under the same metric of time, which enables
developers to search for the best suitable strategy under more realistic
settings.

####  3.4.2 Flexible Communication

To enable different communication behaviors of parties, we follow the gRPC
protocol to model the communication processes. Every time a party sends a
message to a remote one, it will receive a response that is returned by a
corresponding remote function. The transferred information is organized as
key-value pairs for high flexibility. By registering the message handler to
the specific message, developers can arbitrarily customize the communication
process. For example in above Code.2, a party can send a specific message
’forward’ to ask another remote party for partial activations by registering
this message to the action of forward computing. /*Code.2 import
flgo.algorithm.vflbase

class ActiveParty(vflbase.ActiveParty): def iterate(self): … # prepare the
sending package pkg = self.set_message( mtype=’forward’, package=) # request
activations from the party acts = self.communicate_with( party.id, pkg)[’res’]

class PassiveParty(vflbase.PassiveParty): def initialize(self): # register the
action to ’forward’ self.actions = ’forward’: self.my_forward

# define the action def my_forward(self, package): … return ’res’:activations

####  3.4.3 Cross-Architecture

To ease the development of algorithms with different architectures (i.e.
horizontal FL McMahan et al. (2017), vertical FL Liu et al. (2022),
decentralized FLBeltrán et al. (2023), and hierarchical FL Liu et al. (2019)),
we respectively conclude general paradigms for each type of architecture based
on previous works. By using these paradigms, developers can easily customize
their algorithms by only writing a few codes to replace the corresponding
parts in the paradigm.

Figure 4: The acceleration by two levels of parallelism: (left) runner-level,
and (right) participant-level.

###  3.5 Experiment Tools

| Data Heterogeneity | System Heterogeneity | Asynchronous
---|---|---|---
Dataset | Cifar 10 | Cora | Shakespeare | Fashion Mnist | Reddit | SVHN
Task |  | Image
---
Classification
| Node
---
Classification
| Next-Character
---
Prediction
| Image
---
Classification
| Next-word
---
Prediction
| Image
---
Classification
Model |  | Two-layer
---
CNN
| Two-layer
---
GCN
| Stacked
---
LSTM
| Two-layer
---
CNN
| Stacked
---
LSTM
| Two-layer
---
CNN
Table 4: Experimental datasets and corresponding models.

####  3.5.1 Parallel Acceleration

We provide two-level parallels in FLGo to accelerate the FL training process
in simulations. The first one is the runner-level, where each runner
$r=(\mathcal{A},\theta,G,\mathcal{T},s)$ denotes using the algorithm
$\mathcal{A}$ to optimize the global model $\theta$ with the hyperparameters
$G$ on the federated task $\mathcal{T}$ under the environment created by the
simulator $s$. The runner-level parallelism enables developers to use multiple
GPUs to save efficiency, where devices will be automatically scheduled by the
device scheduler module to avoid out-of-memory errors. A runner queue is used
to promise that all the runners will successfully finish their tasks, whereas
the losers will be automatically put back in the queue again. The second-level
parallelism is the participant-level, where each participant denotes a virtual
client of each runner. The participant-level parallelism accelerates the
training process by reducing the wall-clock time of iterative local training
of all the participants. The acceleration results are shown in Fig.4. The left
part in Fig.4 shows that increasing the runner-level parallelism can well
reduce the time cost as the number of runners grows. The right part in Fig.4
shows that increasing the number of processes can also save the training
efficiency, especially when the cost of data/model transmission across
different devices is relatively smaller than the local training processes.

| Cifar10 - CNN |  |  | Cora - GCN |  |  | Shakespeare - LSTM |  |
---|---|---|---|---|---|---|---|---|---
Data Heterogeneity | IID | Imbalance | Dir(0.3) | IID | Imbalance | Louv. | IID | Imbalance | Div.
fedavg | 0.8216 | 0.8303 | 0.7355 | 0.9059 | 0.9022 | 0.7693 | 0.4457 | 0.4424 | 0.3340
fedprox | 0.8206 | 0.8207 | 0.7099 | 0.8911 | 0.9022 | 0.7638 | 0.4443 | 0.4432 | 0.2934
scaffold | 0.8551 | 0.8474 | 0.7492 | 0.9151 | 0.9022 | 0.7638 | 0.4304 | 0.4309 | 0.2775
Table 5: Comparison on model testing accuracy of different methods across
datasets and heterogeneity settings.

####  3.5.2 Experiment Tracker and Analyzer

We realize two modules to help do experiments. First, we use the logger to
track the running-time variables of interest without writing intrusive codes,
and we preserve APIs to customize loggers for different experimental purposes.
The logger module is also compatible with popular ML logging tools like wandb
Biewald (2020). The running-time variables of interest will finally be stored
on the disk as records. Second, we develop the analyzer to load the
experimental records from the disk and help make further analyses on them. We
also provide basic tools to visualize the records and generate tables in a
customizable manner.

####  3.5.3 Parameters Auto-Tuning

We realize automatically tuning hyper-parameters based on the aforementioned
parallelism. By using this module, developers can input the grid of hyper-
parameters and specify the GPUs to find the group of hyper-parameters that
achieves the optimal performance on the validation dataset.

##  4 Experiments

In this section, we conduct experiments to show how FLGo facilitates FL
studies by customization from three aspects: data heterogeneity, system
heterogeneity, and asynchronism. The meanings of heterogeneity notions are in
the appendix.

###  4.1 Experimental Settings

####  4.1.1 Datasets and Models

The datasets and corresponding models are concluded in Tabel.4. For data
heterogeneity experiments, we respectively conduct I.I.D, quantity skew, and
non-I.I.D scenarios for three datasets. For CIFAR10, we use I.I.D, Imbalance
(0.5), and Dirichlet (0.3) data partition. For Cora, we employ IID, Imbalance,
and Louvain data partition. For the Shakespeare dataset, we use IID,
Imbalance, and Diversity (0.2) data partition. For system heterogeneity and
asynchronism experiments, we use I.I.D and Dirichlet (1.0) data partition.

####  4.1.2 Baselines

For data heterogeneity experiments, the baselines are FedAvgMcMahan et al.
(2017), FedProxLi et al. (2020), and ScaffoldKarimireddy et al. (2020). For
system heterogeneity experiments, the baselines are FedAvgMcMahan et al.
(2017), FedNovaWang et al. (2020), and FedProxLi et al. (2020). For
asynchronism experiments, the baselines are synchronous algorithms
FedAvgMcMahan et al. (2017), FedProxLi et al. (2020) and the asynchronous
algorithm FedAsyncXie et al. (2019).

####  4.1.3 Hyparameters

We tune all the methods by grid search. The hyperparameters for the CIFAR10
and Cora datasets: the number of rounds to be 1000, the learning rate to be
0.1, proportion to be 1.0,epoch $E\in$$\left\\{1,3,5\right\\}$, and batch size
to be 50. For the Shakespeare dataset, we set rounds to 100, learning rate
$\eta\in$$\left\\{0.1,0.3,0.8\right\\}$, proportion to be 0.1, epoch
$E\in$$\left\\{5,10\right\\}$, and batch size to be 64. For Fashion Mnist and
SVHN dataset, the number of rounds to be 1000, the learning rate to be 0.1,
epoch to be 5, and batch size to be 50. For Reddit dataset, the number of
rounds to be 1000, the learning rate, epoch and batch size to be 1, 1 and 50,
respectively.

####  4.1.4 Implementation

Experiments are run on a 64 GB-RAM Ubuntu 18.04.6 server with Intel(R) Xeon(R)
CPU E5-2630 v4 @ 2.20GHz, 4 NVidia(R) 2080Ti GPUs, and PyTorch 1.12.0.

###  4.2 Data Heterogeneity

We compare the performance of three state-of-the-art FL algorithms toward the
non-IID problem in Table.5. The algorithm’s performance varies significantly
with different datasets and distribution settings. For CIFAR10, Scaffold
achieves the best performance under all the data distributions. For Cora, all
the methods achieve similar model performance. Scaffold achieves the best
results on the IID distribution, while FedAvg performs the best on the Louvain
distribution. For Shakespeare, the best results lie in FedAvg and FedProx,
where Scaffold is the worst one across all the data distributions. These
results suggest that there is no method that always outperforms other methods
across datasets and heterogeneity settings, which is consistent with the
observations in Li et al. (2022). In addition, the imbalance setting won’t
have an obvious impact on the results of all these methods. The non-IID
settings cause performance reduction for all the methods on different datasets
compared to the IID settings.

###  4.3 System Heterogeneity

We respectively investigate the impact of 4 different types of system
heterogeneity on three FL algorithms in FashionMNIST and Reddit datasets:
FedAvgMcMahan et al. (2017), FedNovaWang et al. (2020) and FedProxLi et al.
(2020). Fig.5(a) suggests that the heterogeneity of client availability has a
non-trivial negative impact on the performance of all the methods. Especially,
when data heterogeneity meets availability heterogeneity, the performance is
further worsened than the IID scenario on both two datasets. Fig.5(b) shows
the impact of heterogeneity in client completeness. When clients are not able
to complete all the local training epochs, the training efficiency is
correspondingly degraded. Although FedNova is claimed to be robust to varying
completeness, its performance is slightly worse than FedAvg and FedProx in
FashionMNIST, which may be due to the over-enlarging of short model updates.
Fig.5(c) shows the impact of heterogeneity in client connectivity. We use the
virtual time as the horizontal axis instead of the communication round since
the server will wait for another unit of time once there exist dropping-out
clients. The results show that less connectivity will increase the time cost
of the training process. For FashionMNIST, the connectivity has only a trivial
impact on the model performance. For Reddit, less connectivity causes
performance degradation in the non-IID setting. Fig.5(d) shows the impact of
heterogeneity in client responsiveness. This heterogeneity will only influence
the time cost of training since the server will wait for all the clients. The
results show that a larger variance of client latencies leads to larger time
costs with the same client latency mean. This is consistent with the Cask
Effect where the slowest client will dominate the time costs for synchronous
algorithms.

(a) The impact of client availability

(b) The impact of client completeness

(c) The impact of client connectivity

(d) The impact of client responsiveness

(e) Asynchronism

Figure 5: The subfigures in the first 4 rows conduct the impact of system
heterogeneity, where each row’s heterogeneity is of: (a) availability; (b)
completeness; (c) connectivity; and (d) responsiveness. For each row, in the
order from left to right: IID-Fashion, Dirichlet-Fashion, IID-Reddit, and
Dirichlet-Reddit. (e) compares synchronous and asynchronous strategies in the
simulation of the complex combinations of different heterogeneity.

###  4.4 Asynchronism

We create a complex combination of different heterogeneity to customize the
environment. First, we make the probability of each client follow the
distribution $lognormal(0,-log(0.1))$. We use the same completeness
heterogeneity setting in Li et al. (2020). We make the clients have the same
drop probability of $0.5$, and we set the response latencies of clients to
follow a log-normal distribution with $(mean,var.)=(200,50)$. The results in
Fig.5(e) show that asynchronous strategies can significantly reduce time costs
when carrying out the same times of aggregation. Under the IID setting,
FedAsync reduces the time cost to achieve the same model performance against
other baselines. Under the non-IID setting, the performance of FedAsync is
worse than FedAvg and FedProx after the same amounts of aggregations, which
suggests that the effectiveness of aggregation for asynchronous strategies can
be further improved.

##  5 Conclusion

In this work, we presented a novel lightweight FL platform, FLGo, to
facilitate cross-application FL studies with a high ability of shareability .
FLGo offers 40+ benchmarks, 20+ algorithms, and 2 system simulators as out-of-
the-box plugins. We also developed a range of experimental tools for various
experiment purposes. Comprehensive experiments were conducted to verify the
ability of customization of FLGo, under various data and system heterogeneity
settings.

## Acknowledgments

The research was supported by Natural Science Foundation of China (62272403,
61872306).

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