# Recognizing number of communities and detecting community structures in
complex networks

Hongjue Wang
School of Systems Science
Beijing Normal University
Beijing, 100875, China
Tao Wang
School of Mathematics and Physics
North China Electric Power University
Baoding, HeBei, 071003,China

###### Abstract

Recognizing number of communities and detecting community structures of
complex network are discussed in this paper. As a visual and feasible
algorithm, block model has been successfully applied to detect community
structures in complex network. In order to measure the quality of the block
model, we first define an objective function $WQ$ value. For obtaining block
model $B$ of a network, GSA algorithm is applied to optimize $WQ$ with the
help of random keys. After executing processes $AO$ (Adding Ones) and $RO$
(Removing Ones) on block model $B$, the number of communities of a network can
be recognized distinctly. Furthermore, based on the advantage of block model
that its sort order of nodes is in correspondence with sort order of
communities, so a new fuzzy boundary algorithm for detecting community
structures is proposed and successfully applied to some representative
networks. Finally, experimental results demonstrate the feasibility of the
proposed algorithm.

Keywords: Community structures, Block model, Complex network, GSA algorithm

##  1 Introduction

In recent years, detecting community structure in complex networks has
attracted a growing attention of many researchers [1, 2]. Community structures
widely exist in many networked systems including friendship networks [1],
coauthor networks, protein-protein interaction network [3, 4], mobile phone
networks [5], WWW networks [6], transportation networks [7], online shopping
networks, animal relationship networks, etc. However, it is very difficult to
provide an accurate and recognized definition for community structure, due to
its idiosyncratic middle scale property and the complexity and heterogeneity
of these real networks in our society [8]. It is generally thought that
community structures are subgroups of a network with dense internal links and
sparse external links. As one of the most important tasks in complex networks
researches, detecting community structure can be utilized for researching
network topological structure, determining cause of group formation [9],
understanding network underlying laws [2, 10], learning network functions [11,
12] and discovering group evolution [12], etc.

A great deal of novel approaches and technologies have been proposed
successfully to discover communities in complex networks. Among these existing
methods, after the measurement of algorithms modularity was introduced by
Newman and Girvan [13] modularity optimization methods emerged with a large
number in a brief period [1, 4, 14, 15]. These methods include greedy
optimization [16] hierarchical clustering [17], extremal optimization [18],
spectral algorithm [19, 20], intelligent algorithm [21, 22, 23], mathematical
programming [24, 25, 26] and so on. However, modularity optimization methods
are NP-complete problem and modularity has the resolution limit problem due to
its definition and selection of null model. So many researchers improved and
extended modularity through great efforts, and some new algorithm measurements
were also proposed such as partition density [27], normalized mutual
information (NMI) [28], modularity density [29], local modularity [30], linked
matrix factorization [31]. For some other algorithms, such as core nodes based
methods [28, 32, 33], $k$-means methods [4, 34, 35, 36] $k$-rank methods [37],
hierarchical clustering methods [4] and spectral clustering [2] for more
communities, they can obtain satisfactory community structures, but they are
bound by the priori knowledge of the number of communities or a related
threshold value [1, 9, 11, 33, 37]. Identifying gaps between the eigenvalues
of Laplacian matrix [4] of a network or Clumpiness matrix [2] was often
applied to estimate the number of communities. For networks with indistinct
community boundaries and mixed communities, there are not distinct gaps
between the eigenvalues of their Laplacian matrixes or Clumpiness matrixes. In
order to detect communities precisely, Wei Tang et al. proposed a block model
approximation algorithm called Linked Matrix Factorization Algorithm [31].
This algorithm needs a block indicator matrix $S$ that is costly obtained by
calculating top $k$ eigenvectors of network adjacency matrix with maximum
eigenvalues.

Considering that complex calculation of eigenvectors of matrices exists in the
current block model approximation algorithms, a novel measurement WQ of block
model is proposed according to the property that most elements with a value of
one are around the leading diagonal in block model. Furthermore, GSA [38, 39]
and Random Keys algorithms [40] are employed to optimize objective function WQ
value in order to obtain the block model of complex network. For some
algorithms are bound by the number of communities, a method to recognize
number of communities is proposed using strong block of the network. The
proposed processes AO and RO are executed on block model and a strong block is
obtained, then the number of communities can be counted artificially from
strong block model. In order to detect communities in complex networks, a
novel fuzzy boundary algorithm is proposed according to the advantage that the
order of rows and columns of block model B corresponds with the order of
communities. Experimental results demonstrate the feasibility of the proposed
algorithm.

##  2 Preliminaries and definitions

A connected network with no node strength, link weight and direction is
denoted as $G=(V,E)$, where $V$ and $E$ are the set of nodes and links,
respectively. $\left|V\right|=n$ is the number of nodes and $\left|E\right|=m$
is the number of links of network $G$. The network $G$ can also be represented
by its adjacency matrix $A$, in which elements represent connections of
network nodes. When a link between nodes ${v_{i}}$ and ${v_{j}}$ exists, then
${a_{i,j}}=1$ and ${a_{i,j}}=0$ otherwise. The degree of node ${v_{i}}$ is
denoted as ${k_{i}}$ and ${k_{i}}=\sum\limits_{i}{{a_{i,j}}}$. Let
${G_{i}}({V_{i}},{E_{i}})$ represents a sub-network of a partition of $G$,
i.e., a community of network $G$.

###  2.1 Definition

Definition 1. $WQ$ value. We take adjacency matrix $A$ of network $G$ as a
table $\Delta$ with ${n^{2}}$ cells and the size of each cell is 1. Element
${a_{i,j}}$ is a cell of $\Delta$. If ${a_{i,j}}{\rm{=}}1$, the element is
called as 1-element. Then the distance between cell ${a_{i,j}}$ and leading
diagonal of matrix $A$ is
$\frac{{\sqrt{2}\left({\left|{i-j}\right|+1}\right)}}{2}$. So the $WQ$ value
of network $G$ is defined as the sum of all distances from 1-elements to main
diagonal divided by $2\sqrt{2}n(2n-1)$.

$WQ=\frac{1}{{4n(2n-1)}}\sum\limits_{1\leq i,j\leq
n}{\left({\left|{i-j}\right|+1}\right)}$ (1)

Clearly, $WQ\in\left({0,1}\right)$. In the following we will verify that less
$WQ$ value corresponds more distinct blocks and two isomorphic networks may
have different $WQ$ values.

Definition 2. Airtightness Value. Suppose the cell of element ${a_{i,j}}$ has
$r$ neighbor cells, then the airtightness value of element in ${a_{i,j}}$ is
defined as

$H_{i,j}^{A}=\frac{{\sum\limits_{i-1\leq k\leq i+1,j-1\leq l\leq
j+1}{{a_{k,l}}}}}{{r{\rm{+}}1}}$ (2)

Clearly, $H_{i,j}^{A}\in\left({0,1}\right)$ and the larger the airtightness
value, the tighter the network structure.

Definition 3. Block. For a network $G$ and its adjacency matrix $A$, we can
get its isomorphic network ${G^{\rm{{}^{\prime}}}}$ and corresponding
adjacency matrix ${A^{\rm{{}^{\prime}}}}$ by renumbered all nodes of $G$. And
we can obtain ${A^{\rm{{}^{\prime}}}}$ by swapping two rows and corresponding
two columns of adjacency matrix $A$ repeatedly. So the main idea of model
approximation algorithm can be visualized in Fig.1, where Fig.1(a) is heat map
of adjacency matrix $A$ and Fig.1(b) represents ${A^{\rm{{}^{\prime}}}}$ which
is called block structure or block model denoted as $B$. Then the block
${B_{i,j}}(i<j)$ of block model $B$ in this paper is defined as

${B_{i,j}}=\left[{\begin{array}[]{*{20}{c}}{{b_{i,i}}}&{{b_{i,i+1}}}&\cdots&{{b_{i,j}}}\\\
{{b_{i+1,i}}}&{{b_{i+1,i+1}}}&\cdots&{{b_{i+1,j}}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{{b_{j,i}}}&{{b_{j,i+1}}}&\cdots&{{b_{j,j}}}\end{array}}\right]$ (3)

where ${b_{i,j}}\in B$. Clearly, a block of network $G$ is a sub-matrix of its
block model $B$ and the sub-matrix is symmetric around the leading diagonal of
$B$. One block in $B$ is a community.

Definition 4. d-Strong Block. For an arbitrary threshold value
$d\in\left({0,1}\right)$, the block ${B_{i,j}}$ is a d-Strong Block if

$H_{k,l}^{{B_{i,j}}}>d,\forall k,l\in\left({1,j-i+1}\right)$ (4)

In this paper, if all blocks of a block model B satisfy equation (4), then $B$
is a d-Strong block model and denoted as $d-B$ in this paper.

Definition 5. d-Weak Block. For an arbitrary threshold value
$d\in\left({0,1}\right)$, the block ${B_{i,j}}$ is a d-Weak Block if

$\sum\nolimits_{k,l\in(1,j-i+1)}{H_{k,l}^{{B_{i,j}}}}>d$ (5)

Figure 1: Example network with non-overlapping communities. (a) Heat map of
original adjacency matrix with no blocks. (b) Block model of the network.
Clearly, there are four blocks which represent four communities.

###  2.2 Gravitational Search Algorithm (GSA)

Inspired by the law of gravity, a novel heuristic optimization algorithm,
namely Gravitational Search Algorithm (GSA) was proposed [38]. It has been
proved that GSA is superior to some existing intelligent algorithms including
real genetic algorithm, particle swarm optimization and central force
optimization. GSA has five main steps which are as follows:

Step 1: Initialize Agents. Randomly initialize the positions of $N$ agents as

${X_{i}}=\left({x_{i}^{1},\ldots x_{i}^{k},\ldots
x_{i}^{h}}\right),i=1,2,\ldots,N$ (6)

where $x_{i}^{k}$ is the position of the $i$th agent in the $k$th dimension
and $h$ is the dimension of an agent.

Step 2: Determine the best and worst fitness of agents. For minimization
problems, the best (denoted as $best(t)$) and worst (denoted as $worst(t)$)
fitness for all agents at the $t$th iteration are

$best(t)=\mathop{\min
fi{t_{j}}\left(t\right)}\limits_{j\in\left\\{{1,\cdots,N}\right\\}}$ (7)

$worst(t)=\mathop{\max
fi{t_{j}}\left(t\right)}\limits_{j\in\left\\{{1,\cdots,N}\right\\}}$ (8)

where $fi{t_{j}}(t)$ is the fitness value (object function value) of the $j$th
agent at the $t$th iteration.

Step 3: Calculate and update attribute values of agents.

(1) Gravitational constant $\Psi$. Gravitational constant $\Psi$ is a function
of iteration $t$ and defined as

$\Psi(t)={\Psi_{0}}\times{e^{-\alpha t/T}}$ (9)

where ${\Psi_{0}}$ is the initial value of gravitational constant, $T$ is the
number of iterations and $\alpha$ is a free parameter. According to
experiments and analysis in reference [38], we set ${\Psi_{0}}=100$ and
$\alpha=20$ in this paper.

(2) Masses of the agents. At the $t$th iteration, gravitational and inertia
masses of agents can be updated by the followings

$\left\\{\begin{array}[]{l}{M_{a,i}}(t)={M_{p,i}}(t)={M_{i,i}}(t)={M_{i}}(t)\\\
\\\
{m_{i}}\left(t\right)=\frac{{fi{t_{i}}(t)-worst(t)}}{{best(t)-worst(t)}}\\\
\\\
{M_{i}}(t)={m_{i}}(t)/(\sum\limits_{j=1}^{N}{{m_{j}}(t)})\end{array}\right.$
(10)

where ${M_{a,i}}(t)$ is the active gravitational mass and ${M_{p,i}}(t)$ is
the passive gravitational mass related to the $i$th agent. ${M_{i,i}}(t)$
represents inertial mass of the $i$th agent.

(3) Acceleration. At the $t$th iteration, the force acting on the $i$th agent
from the $j$th agent at the $k$th dimension is

$F_{i,j}^{k}(t)=\Psi(t)\frac{{{M_{p,i}}(t)\times{M_{a,j}}(t)}}{{{R_{i,j}}{{(t)}^{power}}+\varepsilon}}(x_{j}^{k}(t)-x_{i}^{k}(t))$
(11)

where ${R_{i,j}}(t)$ is the Euclidian distance between ${X_{i}}$ and
${X_{j}}$, $power$ is a positive integer, $\varepsilon$ is a very small
constant. Here we set $power=1$, $\varepsilon=eps$. $eps$ is machine zero in
the tool of MATLAB. Then the resultant force of the $i$th agent exerted from
other agents at the $k$th dimension is

$F_{i}^{k}(t)=\sum\limits_{j=1,j\neq i}^{N}{ran{d_{j}}\times F_{i,j}^{k}(t)}$
(12)

where $ran{d_{j}}$ is a random variable in the interval [0,1]. So the
acceleration of the $i$th agent at the $k$th dimension is

$a_{i}^{k}(t)=\frac{{F_{i}^{k}(t)}}{{{M_{i,i}}(t)}}$ (13)

Step 4: Calculate and update positions and velocities of agents. After
executing the above procedures, the velocities and positions of agents at the
$(t+1)$th iteration can be updated by the following equations

$\left\\{{\begin{array}[]{*{20}{c}}{u_{i}^{k}=ran{d_{i}}\times
u_{i}^{k}(t)+a_{i}^{k}(t)}\\\
{x_{i}^{k}(t+1)=x_{i}^{k}(t)+u_{i}^{k}(t+1)}\end{array}}\right.$ (14)

where $ran{d_{i}}$ is a uniform random variable in the interval [0,1],
$u_{i}^{k}(t)$ and $x_{i}^{k}(t)$ are the velocity and position of the $i$th
agents in the $k$th dimension at the $t$th iteration, respectively. We set
$u_{i}^{k}(1)=0$ for $i=1,2,\ldots,N$, $k=1,2,\ldots,h$.

Step 5: Repeat steps 2 to 4 until $t>T$.

##  3 Methods

###  3.1 Recognizing number of communities

For many community detection algorithms such as partitional clustering methods
[4], the number of communities should be given in advance and is often imposed
as stopping conditions of algorithms. Identifying gaps between eigenvalues of
Laplacian matrix of a network was often applied to estimate the number of
communities. However, for networks with indistinct community boundaries and
mixed communities, there are not distinct gaps between their Laplacian
matrixes or Clumpiness matrixes eigenvalues. So we creatively utilize $GSA$
algorithm to optimize the proposed $WQ$ value and eventually obtain network
block model, then the number of communities (that is strong or weak blocks)
can be recognized readily. In order to obtain network block model $B$, GSA
algorithm is employed to search the best sort order of nodes by swapping two
rows and corresponding two columns of adjacency matrix $A$ for each
circulation. As we all know, $A$ and $B$ have the same eigenvalues and
eigenvectors. So we can approximate network adjacency matrix $A$ as $A\approx
PB{P^{\rm T}}$, where $P$ corresponds to the top $k$ eigenvectors of $A$ with
maximum eigenvalues.

In order to apply GSA to develop the block models of a complex network, there
are three problems should be resolved primarily. Firstly, for the problem of
initializing agents in GSA, 50 agents are initialized as
${X_{i}}=\left({x_{i}^{1},\ldots x_{i}^{k},\ldots x_{i}^{n}}\right)$ for
$i=1,2,\ldots,50$, where $n$ is the number of nodes, $x_{i}^{k}$ is a random
integer, $x_{i}^{k}\in\\{1,2,\ldots,n\\}$ and $x_{i}^{k}\neq x_{i}^{s}$ for
$k\neq s$. Secondly, for calculating the fitness value in Step 2 of GSA, the
WQ value is utilized as the target function of minimization problem in the
optimization model. Thirdly, the best agent we want to obtain is a permutation
of integers, but the position of agents may be non-integer after updated in
Step 4 of GSA. So the random keys [40, 41] is applied to deal with this
problem. As an example of the idea of random keys, suppose
$x=\left({\begin{array}[]{*{20}{c}}{0.5,}&{1.2,}&{0.1,}&{0.3}\end{array}}\right)$
is the updated position of an agent with four dimensions in Step 4 of GSA.
After sorting elements of $x$ , we can get
${x^{{}^{\prime\prime}}}=\left({\begin{array}[]{*{20}{c}}{0.1,}&{0.3,}&{0.5,}&{1.2}\end{array}}\right)$.
According to the position of every element, the new position of agent $x$ is
set as
${x^{*}}=\left({\begin{array}[]{*{20}{c}}{3,}&{4,}&{1,}&2\end{array}}\right)$
eventually. For example, the first element of $x$ is 0.5, then it gets to the
third position in ${x^{{}^{\prime\prime}}}$, so the third dimension is set as
1 in ${x^{*}}$. Similarly, the second element 1.2 of $x$ goes to the fourth
position in ${x^{{}^{\prime\prime}}}$, so the fourth dimension of ${x^{*}}$ is
set as 2. Other elements of $x$ are processed analogically.

In order to recognize blocks more clearly, a two-steps strategy is proposed
including Adding Ones (AO) and Removing Ones (RO). AO is executed by assigning
element ${a_{i,j}}$ and all its neighbor cells to one when $H_{i,j}^{A}$ is
greater than a threshold value $d_{1}\in\left({0,1}\right)$. Whereas RO is
executed by assigning element ${a_{i,j}}$ and all its neighbor cells to zero
when $H_{i,j}^{A}$ is less than a threshold value
$d_{2}\in\left({0,1}\right)$. Then a block model with d-Strong blocks can be
obtained eventually.

Moreover, according to the special structure of block model, we calculate the
number of 1-elements along each diagonal (line) which parallel to minor
diagonal (including minor diagonal). For a $n\times n$ block model B, there
are $2n-2$ diagonals parallel to its minor diagonal and the first one is at
the top left of B, the last one at the bottom right. So the value range of
X-axis is $\left[{1,2n-1}\right]$, and the $(2k-1)$th line passes through the
element ${b_{k,k}}$, we can use this rule to calculate the label of a node
according to the label of parallel diagonal.

In this paper, hierarchical communities, overlapping communities and non-
overlapping communities are considered in un-weighted and undirected connected
networks. Figs.2-4 are block models of three typical networks obtained by GSA
algorithm. The RB125 network shown in Fig.2 is a popular synthetic
hierarchical and scale-free network [4] and it has 25 little communities or 5
large communities. The proposed three-star network shown in Fig.3 has 27 nodes
and can be divided into three communities. The networks ${G^{\rm{\\#}}}$ with
overlapping communities in Fig.4 and non-overlapping communities in Figs.1 are
GN benchmark networks, which have 128 nodes and four communities. Clearly, in
Fig.2(c), the block model of RB125 has 25 blocks or communities. In Fig.2(b)
and Fig.3(b), there are five ”cross” in block model of RB125 and three ”cross”
in block model of three-star network. As a matter of fact, the cross
represents community. So we can conclude that RB125 has five communities and
the three-star network has three communities. For GN benchmark networks in
Fig.4, the block model has four blocks with two pairs of overlapping blocks.
So the network has four communities and two overlapping communities in Fig.4.

Figure 2: RB125. (a) Visualization of RB125. There are 25 little communities
and 5 larger communities. (b) Block model of RB125 obtained by GSA algorithm.
Clearly, there are 25 little blocks and 5 ”cross” structures which represent 5
larger communities. (c) 0.25-B of RB125. This 0.25-Strong block model was
obtained by executing AO with $d=0.7$ and RO with $d=0.25$. (d) Number of
1-elements along each parallel line. This figure shows relation between the
label of parallel lines and number of 1-elements along such parallel lines,
where X-axis is the label of parallel lines and Y-axis is the number of
1-elements. The value range of X-axis is $\left[{1,249}\right]$. Figure 3:
Three-star network. (a) Visualization of three-star network with three
communities. (b) Block model of three-star network obtained by GSA algorithm.
Clearly, there are three ”cross” structures which represent three communities.
(c) 0.5-B of three-star network. This 0.5-Strong block model was obtained by
executing AO with $d=0.3$ and RO with $d=0.5$. (d) Number of 1-elements along
each parallel line. This figure shows relation between the label of parallel
lines and number of 1-elements along such parallel lines, where X-axis is the
label of parallel lines and Y-axis is the number of 1-elements. The value
range of X-axis is $\left[{1,53}\right]$. The three peaks mean three
communities. Figure 4:  Example network ${G^{\rm{\\#}}}$ with two overlapping
communities. (a) Heat map of original adjacency matrix with no blocks. Its WQ
value is 0.4274. (b) Block model of ${G^{\rm{\\#}}}$ with WQ = 0.4128 obtained
by GSA algorithm. Clearly, there are four blocks which represent four
communities. Two pairs of blocks are overlapping. (c) 0.2-B of network
${G^{\rm{\\#}}}$. This 0.2-Strong block model was obtained by executing AO
with $d=0.5$ and RO with $d=0.2$. (d) Number of 1-elements along each parallel
line. This figure shows relation between the label of parallel lines and
number of 1-elements along such parallel lines, where X-axis is the label of
parallel lines and Y-axis is the number of 1-elements. The value range of
X-axis is $\left[{1,255}\right]$. There are four small-scale peaks on two
large-scale peaks, and it was caused by two overlapping communities. So we can
consider this network has two large communities or four little communities
with two overlapping communities.

###  3.2 Detecting community structures

From the above discussion, we can obtain the block model of a network by the
GSA algorithm with the optimal WQ value proposed in this paper. Then the
communities can be detected by many existing algorithms such as $k$-means [4,
34, 35, 36], and hierarchical clustering [4]. Moreover, these algorithms need
to calculate node similarities and even some similarity indices which cost
vast time. So we present a novel fuzzy boundary algorithm that can directly
process the block model and utilize the proposed fuzzy boundary to detect
community structures.

A block model has a corking advantage that its sort order of nodes or rows and
columns is in correspondence with the sort order of communities, for example,
the first 5 nodes belong to the first community in Fig.2. So the communities
can be discovered by detecting boundaries between blocks. We can clearly see
that it is hard to observe the distinct boundary between blocks in Fig.2(c),
because of the attribution of some nodes could not be decided by our
eyeballing. So the set of nodes in a gap between two strong blocks is defined
as their fuzzy boundary denoted as ${F_{b}}$. For a strong block model B of a
network, we renumbered all nodes and blocks by their sequence from left to
right in the block model. So all nodes from left to right are ${v_{1}}$,
${v_{2}}$, ${v_{3}}$, $\cdots$, ${v_{n}}$ and all blocks are ${B_{1}}$,
${B_{2}}$, ${B_{3}}$,$\cdots$, ${B_{k}}$ which correspond to communities
${G_{1}}$, ${G_{2}}$, ${G_{3}}$, $\cdots$, ${G_{k}}$. For
$\forall{v_{l}}\in{F_{b}}$, $\exists{B_{i}},{B_{i+1}}$, and for
$\forall{v_{p}}\in{B_{i}}$ and $\forall{v_{q}}\in{B_{i{\rm{+}}1}}$ we can
obtain $p\leq l\leq q$.

After obtaining all ${F_{b}}$, the rest blocks which are separated by
${F_{b}}$ compose initial community structures
$\left\\{{{G_{1}},{G_{2}},\ldots,{G_{k}}}\right\\}$, where $k$ is the number
of blocks. In order to distribute nodes in ${F_{b}}$, the similarity between a
node and blocks (communities) is defined as the number of links between them.
Moreover, the nodes in ${F_{b}}$ belong to the adjacent block which has the
greatest similarity with them, and the nodes with identical similarity are
overlapping nodes. After each circulation of distributing nodes in ${F_{b}}$,
all similarities of remaining nodes should be updated until ${F_{b}}$ is empty
or all nodes in ${F_{b}}$ are overlapping nodes. We call this strategy of
detecting communities as fuzzy boundary algorithm and it can be described as
follows:

Input: Network $G=(V,E)$

Output: Communities $\left\\{{{G_{1}},{G_{2}},\ldots,{G_{k}}}\right\\}$

Step 1: Use GSA algorithm to optimize the WQ value of adjacency matrix $A$ to
get block model $B$.

Step 2: Obtain $d$-Strong block model $d$-B by applying AO and RO to $B$.

Step 3: Detect ${F_{b}}$ and initial community structures
$\left\\{{{G_{1}},{G_{2}},\ldots,{G_{k}}}\right\\}$ of $d$-B.

Step 4: Calculate similarities between nodes in ${F_{b}}$ and
$\left\\{{{G_{1}},{G_{2}},\ldots,{G_{k}}}\right\\}$, $i=1,2,\ldots,k$.

Step 5: Distribute nodes in ${F_{b}}$ to the adjacent communities and update
$\left\\{{{G_{1}},{G_{2}},\ldots,{G_{k}}}\right\\}$ and ${F_{b}}$.

Step 6: Repeat steps from Step 4 to Step 6 until ${F_{b}}=\emptyset$.

Step 7: Calculate similarities between nodes and communities, adjust
misallocated nodes.

Step 8: Output communities
$\left\\{{{G_{1}},{G_{2}},\ldots,{G_{k}}}\right\\}$.

###  3.3 Experimental results

The fuzzy boundary algorithm is applied to two representative networks
including Football team network [42] and pol-blog network [6], and
experimental results are compared with some existing methods by modularity Q
and NMI. Experimental results are shown in Figs.5-7. The Football team network
contains 115 nodes and 613 edges, nodes are college football teams and edges
are competition schedule of teams, and it has 12 communities originally. The
pol-blog network, with 1490 nodes and 19090 links, represents a set of web-
blogs of US politics. This network primitively has two communities including
conservative and liberal, and its maximal connected network with 1222 nodes
and 16714 links is discussed in this paper.

On Football team network, the block model B has 11 blocks shown in Fig.5, then
11 communities are discovered with WQ=0.1420 which is less than WQ= 0.4271 of
original adjacency matrix A. After executing AO with d=0.9 and RO with d=0.6,
block model 0.6-B is obtained, and its corresponding
${F_{b}}=\\{{v_{30}},{v_{35}},{v_{27}},{v_{37}},{v_{40}},{v_{7}},{v_{65}},{v_{107}},{v_{90}},{v_{2}},{v_{43}},{v_{81}},{v_{83}},{v_{64}},{v_{63}},{v_{77}},{v_{60}},{v_{114}},{v_{88}},$

${v_{59}},{v_{45}},{v_{58}},{v_{92}},{v_{49}},{v_{98}},{v_{4}},{v_{103}},{v_{12}},{v_{25}},{v_{29}},{v_{70}},{v_{54}},{v_{68}},{v_{91}},{v_{51}},{v_{1}},{v_{24}},{v_{8}}\\}$,
11 communities are detected and shown in Fig.6. The optimal modularity
Q=0.5621 and NMI=0.9043. Modularity of initial communities is Q = 0.5540. This
network is also discussed in [14], and its optimal modularity is Q = 0.546 and
NMI=0.6838 with 6 communities. In [43] their optimal modularity is Q=0.6005
and NMI=0.9095 with 11 communities.

From Fig.6, the proposed fuzzy boundary algorithm can successfully detect
communities ${G_{1}}$, ${G_{2}}$, ${G_{3}}$, ${G_{4}}$, ${G_{5}}$, ${G_{6}}$
and ${G_{9}}$ of original communities. Community ${G_{7}}$ is divided into two
communities, and nodes of communities ${G_{11}}$ and ${G_{12}}$ are
distributed to other communities. In football team network, the initial 12
communities are obtained by football teams from different geographical
location, and their links represent schedule of competition between teams. So
in initial community structures, several nodes such as ${v_{29}}$, ${v_{37}}$
and ${v_{111}}$ may have more intercommunity links than intra-community links.
Then experimental results may have some distinctions compared with initial
community structure. In general, the fuzzy boundary algorithm is feasible.

Figure 5: Football team network. (a) Visualization of original 12 communities
of football team network. Different colors represent different communities.
(b) Heat map of original adjacency matrix with no blocks. Its WQ value is
0.4271. (c) Block model of football team network obtained by GSA algorithm.
Its WQ value is 0.1420 which is less than WQ in (b). (d) 0.7-B of football
team network. This 0.7-Strong block model was obtained by executing AO with
$d=0.7$ and RO with $d=0.7$. (e) Number of 1-elements along each parallel
line. This figure shows relation between the label of parallel lines and
number of 1-elements along
such parallel lines, where X-axis is the label of parallel lines and Y-axis is
the number of 1-elements. The value range of X-axis is $\left[{1,229}\right]$.
There are 11 non-overlapping peaks and it means 11 non-overlapping
communities. According to gaps between peaks, ${F_{b}}$ can be obtained.
Figure 6:  Community structure of football team network detected by the
proposed fuzzy boundary algorithm. 11 communities are detected with WQ=0.1420,
modularity Q=0.5514 and NMI=0.8827.Nodes of the same color originally belong
to the same community.

Two communities are detected from pol-blog network shown in Fig.7, and the
optimal WQ value is 0.5803 which is less than 0.6804 of original adjacency
matrix A. After executing AO with d=0.2 and RO with d=0.9, 0.9-Strong block
model 0.9-B is obtained. Because there are two communities in pol-blog
network, the labels of nodes in ${F_{b}}$ correspond to the labels of parallel
lines from 1292th to 1450th, then ${F_{b}}$ can be obtained by transforming
parallel line label to node label. Eventually, we get two communities with
modularity Q=0.4052. This network was also discussed in [1] and [44], their
optimal modularity are Q=0.424236 and Q=0.4176, respectively. Actually, it can
be found from Fig.7(a) that two blocks exist in the heat map of original
adjacency matrix, so experimental results of the proposed fuzzy boundary
algorithm are reasonable.

Figure 7:  Pol-blog network. (a) Visualization of original two communities of
pol-blog network. (b) Heat map of original adjacency matrix with two blocks.
Its WQ value is 0.6804. (c) Block model of pol-blog network obtained by GSA
algorithm. It has two blocks and its WQ value is 0.5803 which is less than WQ
in (b). (d) 0.9-B of football team network. This 0.9-Strong block model was
obtained by executing AO with d=0.2 and RO with d=0.9. (e) Number of
1-elements along each parallel line. This figure shows relation between the
label of parallel lines and number of 1-elements along such parallel lines,
where X-axis is the label of parallel lines and Y-axis is the number of
1-elements. The value range of X-axis is $\left[{1,2443}\right]$.

##  4 Conclusion

In this paper, in order to measure the quality of block model, we first define
an objective function WQ value. For obtaining block model B of a network, GSA
algorithm is applied to optimize WQ with the help of random keys. After
executing AO and RO on block model B, the number of communities of a network
can be recognized distinctly. Furthermore, based on the advantage of block
model that its sort order of nodes or rows and columns is in correspondence
with sort order of communities, an algorithm for detecting community
structures called Fuzzy Boundary is proposed and successfully applied to some
representative networks. Finally, experimental results demonstrate the
feasibility of our algorithm. There are some rooms need to improve in the
future. Firstly, the threshold value d in AO and RO needs an in-depth study,
for example, AO may merge some block together with too small threshold and RO
may break a block into more with too large threshold. Secondly, for some
large-scale network, GSA has high time-complexity and the problem of trapping
in local optimum. So the experimental results in this paper may not be optimal
division but feasible. Thirdly, if the number of nodes in ${F_{b}}$ is
overmuch, cohesiveness of existing communities might be weakened. What s more,
other similarity indices between nodes and communities can be applied to
distribute nodes in ${F_{b}}$. Finally, we hope the algorithm presented here
can be expanded to other types of networks, such as weighted network, directed
network, bipartite network and dynamic network. We hope such improvements and
more applications of the proposed algorithm in the future.

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# Charge Radius of Neutron-deficient 54Ni and Symmetry Energy Constraints
Using the Difference in Mirror Pair Charge Radii

Skyy V. Pineda<EMAIL_ADDRESS>National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
Department of Chemistry, Michigan State University, East Lansing, Michigan
48824, USA  Kristian König  National Superconducting Cyclotron Laboratory,
Michigan State University, East Lansing, Michigan 48824, USA  Dominic M. Rossi
Institut für Kernphysik, Technische Universität Darmstadt, 64289 Darmstadt,
Germany  GSI Helmholtzzentrum für Schwerionenforschung mbH, Planckstr. 1,
64291 Darmstadt, Germany  B. Alex Brown  National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
Department of Physics and Astronomy, Michigan State University, East Lansing,
Michigan 48824, USA  Anthony Incorvati  National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
Department of Physics and Astronomy, Michigan State University, East Lansing,
Michigan 48824, USA  Jeremy Lantis  National Superconducting Cyclotron
Laboratory, Michigan State University, East Lansing, Michigan 48824, USA
Department of Chemistry, Michigan State University, East Lansing, Michigan
48824, USA  Kei Minamisono<EMAIL_ADDRESS>National Superconducting
Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824,
USA  Department of Physics and Astronomy, Michigan State University, East
Lansing, Michigan 48824, USA  Wilfried Nörtershäuser  Institut für Kernphysik,
Technische Universität Darmstadt, 64289 Darmstadt, Germany  Jorge Piekarewicz
Department of Physics, Florida State University, Tallahassee, Florida 32306,
USA  Robert Powel  National Superconducting Cyclotron Laboratory, Michigan
State University, East Lansing, Michigan 48824, USA  Department of Physics and
Astronomy, Michigan State University, East Lansing, Michigan 48824, USA  Felix
Sommer  Institut für Kernphysik, Technische Universität Darmstadt, 64289
Darmstadt, Germany

###### Abstract

The nuclear root-mean-square charge radius of 54Ni was determined with
collinear laser spectroscopy to be $R(^{54}$Ni) = 3.737 (3) fm. In conjunction
with the known radius of the mirror nucleus 54Fe, the difference of the charge
radii was extracted as $\Delta R_{\rm ch}$ = 0.049 (4) fm. Based on the
correlation between $\Delta R_{\rm ch}$ and the slope of the symmetry energy
at nuclear saturation density ($L$), we deduced $20\leq L\leq 70$ MeV. The
present result is consistent with the $L$ from the binary neutron star merger
GW170817, favoring a soft neutron matter EOS, and barely consistent with the
PREX-2 result within 1$\sigma$ error bands. Our result indicates the neutron-
skin thickness of 48Ca as 0.15 - 0.19 fm.

††preprint: APS/123-QED

## Introduction —

Knowledge of the slope of the symmetry energy $L$ in the nuclear equation of
state (EOS) is critical for the extrapolation to the higher densities [1] that
are required to predict the properties of both super-heavy nuclei and neutron
stars [2, 3, 4]. In the case of neutron stars, the “softness” or “stiffness”
of the EOS has a direct link to the neutron star radius [5]. Note that a stiff
EOS indicates that the pressure increases rapidly with increasing density.
Conceptually, the symmetry energy is closely related to the difference between
the energy per nucleon of pure neutron matter and symmetric nuclear matter.
Given that symmetric nuclear matter saturates, $L$ is proportional to the
pressure of pure neutron matter at nuclear saturation density $\rho_{0}$ [6].
Different parameterizations of Skyrme energy density functionals show dramatic
variations in the stiffness of the EOS [1], therefore making the
extrapolations to higher densities uncertain. The stiffness of the EOS in the
vicinity of $\rho_{0}$ is controlled by $L$, and although $L$ cannot be
directly determined through experiment, the neutron skin thickness $\Delta
R_{\rm np}$ of neutron rich nuclei is strongly correlated to $L$ [7, 8], which
may then be used to set boundaries on its value [6].

The lead radius experiments PREX-1 [9] and PREX-2 [10] provide a direct probe
of neutron densities via parity violating electron scattering. Given that the
weak charge of the neutron is much larger than that of the proton, it paves an
electroweak avenue to constrain the density dependence of the symmetry energy.
Other electromagnetic methods involve a correlation between the electric
dipole polarizability and the $\Delta R_{\rm np}$ [11, 12]. Such measurements
have been performed in 208Pb [13, 14], 48Ca [15], and in radioactive 68Ni
[16]. Besides terrestrial experiments, the binary neutron star merger GW170817
has placed important constraints on the EOS through the analysis of the tidal
polarizability (or deformability) [17]. Various studies have aimed to
translate the measurements on the neutron star merger into constraints on the
EOS of dense neutron matter. However, whether the EOS is soft or stiff—which
in turn translates into smaller or larger neutron star radii, respectively—is
still under debate [18, 17, 19, 20, 21, 22, 23, 24, 25, 26].

Another purely electromagnetic method to constrain $L$ has been introduced in
[6], where the $\Delta R_{\rm np}$ is deduced from the difference in charge
radii between a mirror pair. Assuming perfect charge symmetry, the neutron
radius of a given nucleus should be equal to the proton radius of the
corresponding mirror nucleus. The $\Delta R_{\rm np}$ can then be obtained
from the difference $\Delta R_{\rm ch}$ of the root-mean-square (rms) charge
radii $R_{\rm ch}$ of mirror nuclei [6, 27] as $\Delta R_{\rm np}=R_{\rm
ch}\left({}^{A}_{Z}X_{N}\right)-R_{\rm ch}\left({}^{A}_{N}Y_{Z}\right)=\Delta
R_{\rm ch}$, where $A\\!=\\!N+Z$ is the mass number, and $N$ and $Z$ are the
neutron and proton number, respectively. In reality, however, the charge
symmetry is broken by the Coulomb interaction that pushes protons out relative
to neutrons, leading to a weaker correlation between $\Delta R_{\rm np}$ and
$\Delta R_{\rm ch}$. It turns out that $\Delta R_{\rm ch}$ is strongly
correlated with $|N-Z|$ $\times$ $L$ even when $|N-Z|$ is small. On the other
hand, $\Delta R_{\rm np}$ depends on both $|N-Z|\times L$ and the symmetry
energy with the $L$ dependence dominating at large $|N-Z|$ [6]. Such
experiments provide a clean and largely model independent complement to the
parity violating asymmetry experiments. In the present study, the mirror
charge radii formalism is applied to the
${}^{54}\mathrm{Ni}$-${}^{54}\mathrm{Fe}$ pair. The rms charge radius of 54Ni
was determined for the first time and then combined with the known radius of
stable 54Fe [28]. Although this pair has a smaller $|N-Z|$ = 2 relative to our
previous measurement on the ${}^{36}\mathrm{Ca}$-${}^{36}\mathrm{S}$ mirror
pair [29], the precise determination of the charge radius of 54Ni provides a
meaningful constraint on $L$, with input from modern nuclear models.

## Experiment —

This experiment took place at the National Superconducting Cyclotron
Laboratory at Michigan State University. A 58Ni primary beam was impinged upon
a beryllium target and the produced 54Ni($I^{\pi}=0^{+}$, $T_{1/2}=114$ ms)
beam was filtered out using the A1900 fragment separator. The isolated 54Ni
beam was then thermalized in a gas cell [30], extracted at an energy of 30 keV
and transported to the BECOLA facility [31, 32]. A typical rate of Ni+ ions at
the entrance of the BECOLA was typically 400/s. At BECOLA the Ni beam was
captured, cooled and bunched in a radio frequency quadrupole (RFQ) ion trap
[33]. The ion beam was extracted from the RFQ at an approximate energy of
29850 eV. Then the beam was neutralized in-flight in a charge-exchange cell
(CEC) [34]. The typical neutralization efficiency was 50%, and the metastable
$3d^{9}4s$ ${}^{3}\mathrm{D}_{3}$ state was populated, which was estimated by
a simulation to be 15% [35] of the total population. A small scanning
potential (typically 50 V) was applied to the CEC to change the velocity of
the incident ion beam and thus of the atom beam. This in turn Doppler-shifted
the laser frequency in the rest frame of the atoms, and effectively scanned
the laser frequency to measure the hyperfine spectrum. Ions in the metastable
state were excited with 352-nm laser light to the $3d^{9}4p$
${}^{3}\mathrm{P}_{2}$ state, and fluorescence light was recorded as a
function of the scanning voltage with a mirror-based fluorescence detection
system[31, 36]. A background suppression factor of 2$\times$105 was achieved
by performing time-resolved fluorescence measurements with the bunched beam
[32, 37, 38].

Figure 1: Resonance spectra for 54Ni (left) and 60Ni (right) relative to the
rest-frame transition frequency of 60Ni. The solid line is the fit to the
data.

A Penning Ionization Gauge (PIG) ion source [35] was used to generate beams of
stable 58,60Ni isotopes, and spectroscopy was performed every 4-6 hours
throughout the data taking time for 54Ni. The resonance frequencies of 58,60Ni
were used as the reference for the extraction of the 54Ni isotope shift as
well as to determine the kinetic beam energy with $10^{-5}$ relative accuracy
[39]. When changing between the isotopes, the laser frequency was adjusted to
perform spectroscopy at the same beam energy. The applied laser frequencies
were referenced against molecular iodine transition lines [40].

## Experimental Results —

The observed resonance line of 54Ni is shown in Fig. 1 (left). A Voigt
function with an exponential low-energy tail to describe the asymmetry caused
by inelastic collisions with the sodium vapor [34] was used to fit the 54Ni
spectrum, and the fit result is shown as a solid line. The asymmetry parameter
and the Lorentz width of the Voigt function were fixed to those obtained from
the reference measurements on 58Ni and 60Ni. A typical spectrum of 60Ni is
shown in Fig. 1 (right) as an example of a stable isotope measurement.

The isotope shifts defined as
$\delta\nu^{A,A^{\prime}}=\nu^{A}-\nu^{A^{\prime}}$ were extracted and
summarized in Table 1. The uncertainty is dominated by the statistical
uncertainty of the 54Ni resonance centroid (7.5 MHz). A discussion of the
systematic uncertainty contributions is detailed in [41]. From the obtained
isotope shifts, the differential mean square (ms) charge radius was extracted
as $\delta\langle
r^{2}\rangle^{A,A^{\prime}}=(\delta\nu^{A,A^{\prime}}-\mu^{A,A^{\prime}}K_{\alpha})/F+\mu^{A,A^{\prime}}\alpha$
[42] with the offset parameter $\alpha$, the field-shift factor $F$, the
offset-dependent mass-shift factor $K_{\alpha}$, and
$\mu^{A,A^{\prime}}=(m_{A}-m_{A^{\prime}})/\\{(m_{A}+m_{e})(m_{A^{\prime}}+m_{e})\\}$,
where $m_{A}$ and $m_{A^{\prime}}$ are the nuclear masses, and $m_{e}$ is the
electron mass. The $F$ and $K_{\alpha}$ were separately determined [41] by the
King-fit analysis [43] using re-measured isotope-shifts of the stable
isotopes, and are listed in Tab. 1 for 58Ni and 60Ni as reference isotopes.
Here, the offset parameter $\alpha$ was chosen to remove the correlation
between the field- and mass-shift parameters in the linear regression. The
obtained differential ms and the rms charge radii are also listed in Tab. 1.

Table 1: Isotope shift, atomic parameters, differential ms and rms charge radii of 54Ni for $A^{\prime}=58$ and $A^{\prime}=60$ as the reference isotope are summarized. | $A^{\prime}=58$ | $A^{\prime}=60$
---|---|---
$\delta\nu^{54,A^{\prime}}$ / $\mathrm{MHz}$ |  -1410.4 (8.2) |  -1919.7 (7.9)
$\alpha\,/\,\mathrm{u}\,\mathrm{fm}^{2}$ |  417 |  388
$K_{\alpha}\,/\,\mathrm{GHz/u}$ |  929.8 (2.2) |  954.0 (3.5)
$F\,/\,\mathrm{MHz/fm^{2}}$ |  -767 (70) |  -804 (66)
$\delta\big{\langle}r^{2}\big{\rangle}^{54,A^{\prime}}$/ fm2 |  -0.235 (29) |  -0.522 (20)
$R(^{54}$Ni) / fm |  3.738 (4) |  3.737 (3)

The differential ms charge radii were used together with the known rms charge
radii for reference isotopes to determine the rms charge radius of 54Ni as
$R(^{54}{\rm Ni})=\\{(R(^{A^{\prime}}{\rm Ni}))^{2}+\delta\langle
r^{2}\rangle^{54,A^{\prime}}\\}^{1/2}$. The rms charge radii of 58Ni, 60Ni and
54Fe were evaluated by combining tabulated values [28] for the Barrett radii
$R_{k\alpha}$ from muonic spectroscopy and for the ratio of the radial moments
$V_{2}$ from electron scattering, which yields the model-independent rms
charge radii $R_{\rm ch}$ = $R_{k\alpha}/V_{2}$ as 3.7698 (16) fm, 3.8059 (17)
fm and 3.6880 (17) fm, respectively. With the rms charge radii of
${}^{54}\mathrm{Fe}$ the difference in mirror charge radii was determined to
be $\Delta R_{\mathrm{ch}}=R({{}^{54}{\rm Ni}})-R({{}^{54}{\rm
Fe}})=0.049\,(4)\,\mathrm{fm}.$

## Theoretical radii —

Predictions were made for the difference in charge radii of 54Ni and 54Fe
using the 48 Skryme energy-density functionals (EDF) [6] and the covariant
density-functional (CODF) theory where a correlation between $\Delta R_{\rm
ch}$ and $L$ was also observed [27].

For the $A=36$ mirror pair [29], it was found that the Skyrme results are
sensitive to the isoscalar (IS) or the isoscalar plus isovector (IS$+$IV)
forms of the spin-orbit potential. However, the present $A=54$ pair turns out
to be insensitive to the forms. The IS results is about 0.003 fm larger in
$\Delta R_{\rm ch}$, which is negligible, and therefore we adapted the
standard IS$+$IV form in this paper.

The Skyrme [6] and CODF [44] calculations include the relativistic spin-obit
(RSO) correction to the charge radius [45], and were performed for spherical
nuclei. It is known that the quadrupole correlations increase the rms radii
when the saturation condition of isoscalar nuclear matter is taken into
account [46]. In the present work, the quadrupole deformation effects were
taken into account as a correction, which is discussed in the following.

The Bohr Hamiltonian starts with an expansion of the nuclear surface in terms
of of its multipole degrees of freedom

$R(\theta,\phi)=R_{0}\left[1+\displaystyle\sum_{\lambda,\mu}\alpha_{\lambda,\mu}Y_{\lambda,\mu}(\theta,\phi)\right],$
(1)

where $R_{0}$ is the radius of the nucleus when it has the spherical
equilibrium shape, and $Y_{\lambda,\mu}$ is the spherical harmonic. The
integrals of Eq. (1) involve $\beta^{2}=\sum_{\lambda\geq
2}\sum_{\mu}\mid\alpha_{\lambda,\mu}\mid^{2}$. To order $\beta^{2}$, the
volume integral of Eq. (1) is
$I_{0}=\\{R_{0}^{3}(4\pi+3\alpha_{0}\,\sqrt{4\pi}+3\beta^{2})\\}/3$. Proton
($q=p$), neutron ($q=n$) and matter ($q=m$) distributions are distinguished by
using $R_{0q}$, $\alpha_{0q}$ and $\beta_{q}$. For the matter density, if we
impose the condition of saturation (that the average interior density remains
constant), then the volume must be conserved, $I_{0}=4\pi R_{0m}^{3}/3$. This
condition can be imposed by having

$\alpha_{0m}=-\frac{\beta_{m}^{2}}{\sqrt{4\pi}}.$ (2)

To order $\beta^{2}$, the $r^{2}$ integral is
$I_{2}=\\{R_{0}^{5}(4\pi+5\alpha_{0}\,\sqrt{4\pi}+10\beta^{2})\\}/5$. With the
condition of volume conservation from Eq. (2), the matter ms radius is

$\left<r^{2}\right>_{m}=\frac{I_{2}}{I_{0}}=\left<r^{2}\right>_{0m}\left[1+\frac{5}{4\pi}\beta_{m}^{2}\right],$
(3)

where $\left<r^{2}\right>_{0m}=3R_{0m}^{2}/5$ is the ms radius with no
deformation. If $\beta_{p}$ = $\beta_{n}$ = $\beta_{m}$, then we can use Eq.
(3) for protons. But if $\beta_{p}\neq\beta_{n}$, one must make some
assumptions about the $\alpha_{0}$ term. If we take
$\alpha_{0p}=\alpha_{0n}=\alpha_{0m}$ for the volume correction, then

$\displaystyle\left<r^{2}\right>_{p}$
$\displaystyle=\left<r^{2}\right>_{0p}\left[1+\frac{2\alpha_{0p}}{\sqrt{4\pi}}+\frac{7}{4\pi}\beta_{p}^{2}\right]$
$\displaystyle=\left<r^{2}\right>_{0p}\left[1-\frac{2}{4\pi}\beta_{m}^{2}+\frac{7}{4\pi}\beta_{p}^{2}\right].$
(4)

For $\lambda$ = 2, the $\beta_{p}$ are related to the $B(E2,\uparrow)_{p}$ for
0+ to 2+ (in units of $e^{2}$) by
$\beta_{p}=4\pi\sqrt{B(E2,\uparrow)_{p}}/(5a_{q}\left<r^{2}\right>_{0p})$,
where $a_{q}=Z$ for protons. For $\beta_{n}$ and $\beta_{m}$ we have
equivalent expressions involving neutrons with $a_{q}=N$ and matter with
$a_{q}=A$. The calculated $B(E2,\uparrow)_{p}$ can be compared to
electromagnetic experimental results, whereas $B(E2,\uparrow)_{n}$ and
$B(E2,\uparrow)_{m}$ are much less well known from hadronic scattering
experiments. The $B(E2,\uparrow)_{q}$ was calculated from the full-basis
configuration interaction calculations in the $fp$ shell-model space with the
GFPX1A Hamiltonian [47]. The basic quantities calculated are the model-space
matrix elements denoted by $A_{q}$. The full matrix element
$M_{q}=\sqrt{B(E2,\uparrow)_{q}}$ was obtained with “effective charges”
$\delta_{cv}$ that arise from 2$\hbar\omega$ admixtures of core nucleons ($c$)
induced by the valence $(fp)$ nucleons ($v$) as
$M_{p}=A_{p}(1+\delta_{pp})+A_{n}\delta_{pn}$ and
$M_{n}=A_{n}(1+\delta_{nn})+A_{p}\delta_{np}$. Here $M_{m}=M_{p}+M_{n}$. We
take the approximation that $\delta_{pp}=\delta_{nn}$ and
$\delta_{pn}=\delta_{np}$. The isoscalar effective charge was evaluated using
the data in [48] as $\delta_{0}=\delta_{pp}+\delta_{pn}\approx 1.00$.
Comparing the $fp$ model-space calculations to data for mirror transitions in
$A=51$ [49] found $\delta_{1}\approx-0.60$. Therefore, $\delta_{pn}=0.80$ and
$\delta_{pp}=0.20$ were used. For the $A_{q}$ the radial matrix elements are
calculated with harmonic-oscillator radial wavefunctions with
$\hbar\omega=45A^{-1/3}-25A^{-2/3}$ [50].

For 54Fe we obtain $B(E2,\uparrow)_{p}=619\,$ e2 fm4 compared to the
experimental value of 640(13) e2 fm4 [51]. The value for 54Ni is
$B(E2,\uparrow)_{p}=467\,$ e2 fm4 compared to an experimental value of $<$ 800
e2 fm4 [51]. There is additional $E2$ strength up to about 4 MeV in excitation
energy in the calculations and in experiment [51]. Use of the collective model
for the radius change can be justified by treating the lowest 2+ state as a
member of the ground-state band. The results for ($\beta_{p}$, $\beta_{n}$,
$\beta_{m}$) are (0.185, 0.149, 0.166) for 54Fe and (0.149, 0.185, 0.166) for
54Ni. The deformation increase in the rms radius is 0.026 fm for 54Fe and
0.014 fm for 54Ni.

## Discussion —

The resulting quadrupole correction factor for $\Delta R_{\rm ch}$ is $-0.012$
fm, which is added to the Skyrme and CODF calculations performed in the
spherical basis. If we were to include the higher 2+ states in 54Ni the
quadrupole correction factor would be $-0.010$ fm. The results are shown in
Fig. 2 by the colored points. The color indicates the neutron skin of 208Pb:
0.12 fm (red), 0.16 fm (orange), 0.20 fm (green), and 0.24 fm (blue) for
Skyrme calculations. Also results of the CODF calculations are shown in
crosses. The quadrupole correlations are explicitly contained in the
CHFB$+$5DCH calculations using the D1S Hamiltonian given in [52, 53]. However,
their $B(E2,\uparrow)_{p}$ values of 1310 and 1575 e2 fm4 for 54Fe and 54Ni,
respectively, are very different from the $fp$ model-space calculations and
experiment.

Figure 2: $\Delta R_{\rm ch}$ as a function of $L$ at $\rho_{0}$. The
experimental result is shown as a horizontal gray band. The solid circles are
results of Skyrme EDF and the crosses are for the CODF calculations. The upper
figure shows comparison with the GW170817 and the PREX-2.

The Skyrme and CODF calculations show consistent agreement in the correlation
between $\Delta R_{\rm ch}$ and $L$. In comparison to these calculations, the
experimental one-sigma error band shown in Fig. 2 in gray implies a value of
$L$ in the range of 20-70 MeV. In the top panel of Fig. 2 we compare the
present result with the range for $L$ of 11-65 MeV deduced from GW170817 [54],
to which our result is consistent, suggesting a relatively soft neutron matter
EOS. The present result is also compared against the recent PREX-2 result of
$\Delta R_{\rm np}$ = 0.283 (71) fm [10] that implies $L=106\,(37)$ MeV [55].
Our result is barely consistent within 1$\sigma$ error bands with the PREX-2,
which indicates rather stiff EOS. It is noted that our previous results on the
mirror pair 36Ca-36S indicates the range of $L$ = 5-70 MeV [29], which is
consistent with the present results. This implies that the theoretical model
dependence is well under control. However, the $A$ = 36 result does not
include the quadrupole correlation. It is expected to be small, and once the
experimental $B(E2)$ for the $A$ = 36 pair become available, the range from
the $A$ = 36 will be updated.

Figure 3: “Data-to-data” relation between $\Delta R_{\rm ch}$ and $\Delta
R_{\rm pn}$(48Ca). The same marks and color coding are used as Fig. 2.

Finally the correlation between $\Delta R_{\rm ch}$ and $\Delta R_{\rm
np}(^{48}{\rm Ca})$ is shown in Fig. 3. Our $\Delta R_{\rm ch}$ restricts the
$\Delta R_{\rm np}(^{48}{\rm Ca})$ to the interval of
$\\!0.15\\!-\\!0.19\,{\rm fm}$. The connection to 48Ca is timely given that
the Calcium Radius EXperiment (CREX) has been completed [56], where
experimental error of about $\pm\,0.02\,{\rm fm}$ is expected, which is
comparable to the error obtained here. It is of particular interest whether
CREX will confirm the soft EOS or reveal a larger $\Delta R_{\rm np}$ as the
PREX-2.

## Summary —

The $\Delta R_{\rm ch}$ between mirror nuclei 54Ni-54Fe was evaluated, and
compared with the Skyrme EDFs and the CODF theories. The $\Delta R_{\rm ch}$
and $L$ correlation implies a range of $L$ = 20-70 MeV, and is consistent with
the $L$ from GW170817 and our previous result in the 36Ca-36S pair, suggesting
a soft neutron matter EOS. Our result is barely consistent within 1$\sigma$
error bands with the PREX-2 that indicates a stiff EOS. The present $\Delta
R_{\rm ch}$ also predicts the $\Delta R_{\rm np}$(48Ca) as
$0.15\\!-\\!0.19\,{\rm fm}$. More data on the mirror charge radii in different
mass regions is required to properly assess the model dependence and to set
tighter limits on the $L$.

## Acknowledgements

This work is support in part by the National Science Foundation grant No.
PHY-15-65546 and by the U.S. Department of Energy Office of Science, Office of
Nuclear Physics under Award DE-FG02-92ER40750, and by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID
279384907 - SFB 1245. We thank Nathalie Pillet for providing the CHFB$+$5DCH
calculation results for 54Ni and 54Fe.

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