Graph Convolution Networks for Cell Segmentation

Sachin Bahade, Michael Edwards and Xianghua Xie

Department of Computer Science, Swansea University, U.K.

Keywords:

Graph Convolution Network, Signal Processing, Cell Segmentation, Medical Imaging.

Abstract:

Graph signal processing is an emerging ﬁeld in deep learning, aiming to solve various non-Euclidean domain

problems. Pathologist have difﬁculty detecting diseases at an early stage due to the limitations of clinical meth-

ods and image analysis. For more accurate diagnosis of disease and early detection, automated segmentation

can play a vital role. However, efﬁciency and accuracy of the system depends on how the model learned. We

have found that traditional machine-learning methods, such as clustering and thresholding, are unsuited for

precise cell segmentation. Furthermore, the recent development of deep-learning techniques has demonstrated

promising results, especially for medical images. In this paper, we proposed two graph-based convolution

methods for cell segmentation to improve analysis of immunostained slides. Our proposed methods use ad-

vanced deep-learning, spectral-, and spatial-based graph signal processing approaches to learn features. We

have compared our results with state-of-the-art fully convolutional networks(FCN) method and found a sig-

niﬁcant of improvement of 2.2% in the spectral-based approach and 3.94% in the spatial-based approach in

pixel based accuracy.

1 INTRODUCTION

Humans are surrounded with data and it is present

everywhere. Such data have been processed. The

processing of data itself could be improved in order

to enhance the way social networks process data, or

to beneﬁt the medical and ﬁnancial sector, for exam-

ple. There are various techniques evolved in terms of

processing these data. In deep-learning, the convolu-

tional neural network has shown remarkable success

in various computer-vision problems such as segmen-

taion and classiﬁcation (Badrinarayanan et al., 2017;

Krizhevsky et al., 2012). The reason behind these ac-

complishments is that data lies in the Euclidean do-

main, have locality and order information which fa-

cilitates the convolution operation learning represen-

tative features. These Euclidean data have the limi-

tation of generality when processed by conventional

convolutional neural networks, for example convo-

lution operation on a social-network graph. Data

which do not belong to the Euclidean space is known

as ’non-Euclidean’ or ’irregular’ domain data. One

of the solutions is that such non-Euclidean data can

be represented on the graph. The study of graph-

signal processing and spectral-graph theory works

with irregular-domain data and these studies have

helped to design tools for various operations like con-

volution and ﬁltering on graph (Bronstein et al., 2017;

Sandryhaila and Moura, 2013). (Kipf and Welling,

2016) used graph-signal processing tools to formulate

convolution on graph as a multiplication of the sig-

nal with a ﬁlter in the Fourier domain (Shuman et al.,

2013). Another approach of graph-signal processing

is the spatial based graph convolution neural network,

where neighbouring information is gathered around a

centre node to process operations.

An image is usually considered as an array of pixel

values arranged in a 2D grid form. The data arranged

in this fashion can also be considered as a signal ly-

ing on a 2D grid graph, where pixel values lie on each

node as features. The connectivity of pixel neigh-

bourhoods is represented by an adjacency matrix de-

scribing the connectivity between pixels. This pixel

connectivity of an image in a 2d grid graph helps to

solve segmentation problems. The most used meth-

ods for these tasks are fully convolutional network

(FCN) (Long et al., 2015) and U-Net (Ronneberger

et al., 2015). These architectures are primarily based

upon convolutional neural network(CNN), where a

FCN requires more training data and the U-Net is de-

signed for biomedical images with few samples and

enough time with no dense layer. Our contribution

in this paper is to apply a graph convolution opera-

tor on the non-Euclidean data and propose a method

for segmentation task on biomedical cell image data.

We have experimented these data with different tra-

620

Bahade, S., Edwards, M. and Xie, X.

Graph Convolution Networks for Cell Segmentation.

DOI: 10.5220/0010324306200627

In Proceedings of the 10th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2021), pages 620-627

ISBN: 978-989-758-486-2

ditional methods like clustering, threshold and deep

learning methods such as FCN, spectral and spatial-

based graph convolution. In observation, we have

found that graph convolution networks improve seg-

mentation results. This paper is divided into the fol-

lowing sections. Section II provides a review of cell

segmentation and related deep-learning methods. In

Section III, our proposed graph-CNN architecture and

method is discussed. In section IV, we show the

implementation of the threshold-based Otsu method,

the k-means clustering method, and the deep-learning

and graph-based methods, to solve the same prob-

lem. In the results section, our method is qualitatively

and quantitatively compared with the state-of-the-art

FCN. Section VI concludes the paper.

2 RELATED WORK

In the medical ﬁeld, lymphoma cancer cells found

important elements that is a cluster of differentia-

tion 4 (CD4) and cluster of differentiation 8 (CD8)

where CD8 helps to kill infected virus cells and CD4

works as a signal activator. In a clinical setting,

the CD4/CD8 count ratio is used to judge the condi-

tion of the immune system. Automated segmentation

of such immunostaining images helps to carry out a

more accurate diagnosis of disease or even early de-

tection (Ronneberger et al., 2015). A review of var-

ious automated methods for cell detection and seg-

mentation is provided by (Thomas and John, 2017).

There are several approaches used for segmentation

including traditional and deep learning. Mainly we

are focusing on graph based approach convolution op-

eration to understand node based features.

2.1 Traditional Methods for

Segmentation

One approach for image segmentation is developed

using principal component analysis (PCA) and k-

means (Katkar et al., 2016). In order to identify tuber-

culosis bacteria, (Rulaningtyas et al., 2015), devel-

oped a new algorithm to help clinicians. They identify

the problem of local minima in k-means and solved

it by windowing technique. The whole image is di-

vided into small patches and segment each patch is

segmented using k-means clustering.

There is a feature classiﬁcation limitation of a

region-, edge-, and pixel-based segmentation with

color images. By transferring the RGB image into

CIELAB (L*a*b) color space, (Yadav et al., 2016)

analyze the features of each pixel of an image, classi-

fying the colors using k-means and adopting support

vector machine (SVM) classiﬁers to detect tumours

by comparing the clustered image with labelled data.

The two-colour components, a and b, from L*a*b

colour space can also be used as a feature for k-means

clustering in the segmentation of white blood cells

from microscopic images (Salem, 2014). Another

approach to obtain segmentation of white blood cells

for acute leukaemia images is to use a linear-contrast

segmentation technique using hue, saturation and in-

tensity (HSI) colour space (Abdul Nasir et al., 2011).

Recently there has been a growth in the use of

deep learning techniques for medical-image analysis,

especially segmentation tasks for the early detection

of disease. (Wang et al., 2016) used deep learning

techniques with the threshold-based Otsu method to

identify tissue: dividing image into patches and ap-

plying supervised classiﬁcation to detect a tumour.

Some of the promising works on deep learning for im-

age segmentation tasks gain most researchers’ atten-

tions (Ronneberger et al., 2015; Badrinarayanan et al.,

2017; Arora and Patil, 2017; Abdul Nasir et al., 2011;

Long et al., 2015).

2.2 Deep Learning based Segmentation

One of the state-of-the-art deep learning methods

used in this study is FCN-based segmentation (Long

et al., 2015). It is an encoder-decoder architecture

made by ﬁne-tuning a portion of the VGG-16 network

and it does not have a dense layer. It is replaced with

1 ×1 convolution to adapt the classiﬁer for dense pre-

diction, to classify each pixel within an image and as-

sign a label to them. To get the output size equal to

the original size, upsampling is used to expand the

features. There might be information loss through the

convolution process, so skip connection helps to re-

cover lost information (Long et al., 2015). (Arora

and Patil, 2017) adapt a model from (Laina et al.,

2016) and used transfer learning to solve the problem

of depth prediction of a scene from a single monoc-

ular image and pixel-wise semantic labelling. The

popular SegNet architecture (Badrinarayanan et al.,

2017) eliminates the need for learning to upsample

compared to FCN. U-net architecture is build based

on FCN architecture (Ronneberger et al., 2015). The

only difference is that the pooling operation of up-

sample part is replaced by up-convolution which in-

creases the resolution and concatenates skip connec-

tion instead of adding, helping to improve segmenta-

tion.

Graph Convolution Networks for Cell Segmentation

621

2.3 Graph-based Convolutional Neural

Networks

The popularity of graph-based convolutional neural

networks has been rapidly growing in recent years due

to the generic nature of irregular data. Graph con-

volution updates the node based on the neighbouring

node information and processed for the convolution

operation. Graph convolution networks is categorised

into two forms: spectral-, and spatial-based. Spectral-

based convolution uses ﬁlters from the perspective of

graph-signal processing while spatial-based convolu-

tion deﬁnes graph convolutions by information prop-

agation.

The Graph is a set of vertices and edges where

nodes are connected. Graph convolution is the con-

volution operation in the frequency domain. The ma-

trix representation of the graph is convolved with the

features matrix. The result multiply with the weights

on each nodes in the i

layer and passed through

the hidden layer non-linear function. In conventional

CNN, the pooling layer is used to reduce the resolu-

tion of input feature map but in the case of a graph,

there is no reduction of size due to the multiplication

of the ﬁlter with spectral signal (Edwards and Xie,

2016). To pool local feature output from the convo-

lution layer, it is required to perform graph coarsen-

ing which reduces the number of vertices, and handle

the edges between these vertices based on the similar

properties. In graph convolution, there is no reduction

of vertices, only changes in the output ﬁlter channel.

But for the precise classiﬁcation, pooling generalizes

features in the spatial domain. Agglomerative pool-

ing is a bottom-up approach to reduce vertices and

project the features on a new graph. There are various

methods to do graph coarsening such as graclus etc.

One of the common methods for selecting vertices is

to select a subset of the set of vertices or generate

new nodes. Algebraic Multigrid (AMG) is a graph

coarsening method which project a signal to a coarser

graph by greedy selection of vertices. This method is

used as a pooling operation on graph (Edwards and

Xie, 2016).

Spatial-based graph convolution follows the simi-

lar approach of convolutional operation of a conven-

tional CNN on an image. Their operation is based

on a node’s spatial relations. Images are represented

as a special form of 2D graph with each pixel repre-

senting a node and is directly connected to its nearby

pixels. Similar to conventional convolution operation,

spatial graph convolution perform convolution opera-

tion by considering its neighbours representations of

node and central node.

Figure 1: GCNN architecture: blue color box represents

output result of the graph, processed by operations GCN/G-

pooling. It also mentioned the size of the graph: number of

nodes and output channels.

Figure 2: Spatial GCNN architecture: blue color box rep-

resents output result of the graph, processed by operations

spatial GCNN. It also mentioned the size of the graph: num-

ber of nodes and output channels.

3 METHODS

3.1 Proposed Network Architecture

Motivated by the convolution neural network U-net

architecture, we propose a similar architecture us-

ing spectral based graph convolution and graph pool-

ing. The network architecture is illustrated in Fig-

ure 1. The encoder part consists of its three blocks,

each consisting of three graph convolutions and graph

pooling layers. For graph-pooling operation, we uti-

lize AMG coarsening to obtain the restriction and

projection matrices. The decoder part starts with

the graph up-pooling operation in sequence with the

last pooling operation of an encoder. It consists of

three layers of graph convolution, each initiated with a

graph up-pooling operation. In the case of up-pooling

operation, we used the projection matrix of previous

coarsened graph to reconstruct original size graph di-

mension.

In the spatial approach based graph convolution

architecture, we have used mixture model CNN

(MoNet) framework for the graph convolution oper-

ation where each convolution operation is followed

by activation. Here we have avoided the pooling

operation as the spatial approach used aggregative

methods of neighbour node used to learn efﬁciently

large graphs.

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622

3.2 Proposed Method of Utilizing

Spectral based Graph-CNN

To perform convolution on the graph, spectral theory

is used to deﬁne the analogue to convolution and for

the downsampling and upsampling operation, graph

coarsening as a pooling layer is deﬁned. While per-

forming the graph pooling, we partition the graph

into coarsened graph and use projection and restric-

tion for graph pooling and graph up pooling opera-

tion (Liu et al., 2014) (Masci et al., 2015). Training is

fed forward through the network to obtain output and

loss propagates backward to update the weights. The

graph holds spatial information about the connectivity

of nodes and allows graph processing tools, convolu-

tion and pooling, to operate on signals (Shuman et al.,

2013).

To perform the graph convolution, an image is

considered as a 2D grid graph, having a set of nodes

(V ), set of weighted edges (E) and adjacency matrix

(A). The graph possesses the property that each node

is connected with its neighbouring nodes which form

the basis of locality for the convolution operation.

The graph structure of the image represents an

irregular data and graph Laplacian is the core op-

erator for graph convolution layer. One form of

the Laplacian operation is represented as L = D − A

where D is the degree matrix and A is the adjacency

matrix. Normalized Laplacian matrix is L = I

−

−1/2

where I

is the identity matrix which

considers self node features. The Laplacian matrix

is decomposed into orthonormal vector U = u

i=1...N

where u

is eigenvector associating with eigenvalues

i=1...N

. Apply graph Laplacian and then eigende-

composition of graph Laplacian matrix which gives

the Fourier modes and graph frequencies (Defferrard

et al., 2016). In graph signal processing, a graph sig-

nal

is a feature vector that lies on the node of the

graph. Applying the graph Fourier transform (F

) us-

ing matrix U on signal s gives,

(s) = ˆs = U

s (1)

Then the inverse graph Fourier transform is applied

which gives original signal s (Defferrard et al., 2016),

−1

( ˆs) = U ˆs = UU

s = s, (2)

Now the convolution of signal s with a ﬁlter g in

Fourier domain is deﬁned as,

s ∗

g = F

−1

s  F

g), (3)

and can be represented in,

s ∗

g = ˆg(L)s, (4)

For the graph pooling operation, we are using the

AMG method to coarsen a graph and projecting sig-

nals on a new coarsened graph via a greedy selection

of vertices (Chevalier and Safro, 2009). The two-level

coarsening is shown in the Figure 5. Every AMG

coarsened graph provides the restriction matrix R and

the projection matrix P for the interpolation of the in-

put signal s. Downsampling operation is performed

by multiplication of signal and restriction matrix

= s

∗ R

and reverse pooling by multiplication with projection

matrix,

= s

∗ P

Where s

is the output of the downsampling operation

and s

is the output of up-sampling. R

and P

are the

restriction and projection matrices, respectively.

3.3 Proposed Method of Utilizing

Spatial based Graph-CNN

Similar to the conventional CNN on the two dimen-

sional grid image, spatial based graph convolution de-

ﬁnes the spatial relation of the node and its neighbour-

ing nodes on the graph. Each node is represented as

a vertex of the graph, and the value is the signal on

that vertex. In the spatial graph convolution network,

centre nodes are updated by averaging the neighbours

nodes analogous to the conventional CNN.

In our case, we are using a spatial based mixture

model CNN for graph (MoNet) where the pixel neigh-

borhood relationship can be represented by a pseudo-

coordinate. In this approach, x represents a vertex on

a graph, and yεN(x) are the vertices in the neighbour-

hood of x. Assign a d dimensional vector of pseudo-

coordinate u(x, y). The pseudo-coordinate calculates

the degree of nodes by the equation

u(x,y) = (1/

deg(x),1/

deg(y))

(5)

In this coordinate space, parametric learnable gaus-

sian kernel function is deﬁned as below:

(u) = exp(−1/2(u − u

)

(Σ

)

−

1(u − u

)) (6)

where Σ

and u

are learnable d × d and d × 1 covari-

ance matrix and mean vector of Gaussian kernel.

With the help of these kernel function, a patch op-

erator is used to perform the function of convolution

operation. This operator applies the Gaussian ker-

nel on each node pseudo-coordinate with all neigh-

bourhood and summoned up the results (Monti et al.,

2017).

(x) f =

∑

yεN(x)

(u(x,y)) f (y), j = 1,...J, (7)

The patch operator can be deﬁned by the above

equation, Where J represents the dimensionality of

Graph Convolution Networks for Cell Segmentation

623

extracted patch. The generalised graph convolution

operation is written as

( f ∗ g)(x) =

∑

j=1

(x) f , (8)

where g

is the learnable weight matrix.

4 EXPERIMENTATION

4.1 Generation of Hodgkin Lymphoma

(Ground Truth) Segmentation

The microscopic immunostaining images of Hodgkin

lymphoma show some patterns of stain based on the

colour of those stains. The slide is stained with

two immunostaining patterns CD4 and CD8. How-

ever, the data need to be cleaned to overcome a large

amount of additional noisy information. To create a

ground truth label for segmentation, manual labelling

of each cell is created by selecting the contour points

for each cell class and all the pixels are ﬁlled inside

the contour with the unique class ID. In this way, the

ground truth label is the same size image where every

pixel is assigned a class label. It can also be described

as dense labelling.

Further, the labelled data we have created is used

for supervised machine learning, to learn meaningful

features from the data and used in deep learning ex-

perimentation.

4.2 Segmentation using Clustering

Method

Clustering plays an important role in segmentation.

There are many methods that have been proposed for

segmentation. Among them, k-means becomes very

popular. Due to good results of the k-means clustering

method (Katkar and Baraskar, 2015), we have chosen

this method for our data and the result is shown in Fig-

ure 3. k, the number of clusters chosen is three. K-

means identiﬁes the nearest pixels assign to random

centroid and shifts centroid to a new position based

on the average of pixels in the cluster. The output im-

age shows the segmentation of CD4, CD8 and back-

ground.

4.3 Segmentation using Deep Learning

We used fully convolutions neural network as one of

the state-of-the-art methods for segmentation. We

prepared the data and created patches of size 224 ×

Figure 3: K-Means clustering of Hodgkin lymphoma im-

munostaining image. Top: Original image, Bottom: Seg-

mentation via k-means. red colors represent a CD4, yellow

shows CD8 and rest are background FOXP3 protein.

224 from 768 ×1366 size image. With the deep learn-

ing supervised approach, we need data images as well

as respective class label masks for each patch.

In this architecture, the encoder contains several

layers. Each layer is a combination of convolution

followed by pooling operation. At each convolu-

tion, 3 × 3 kernel convolve with the input image and

produces output feature maps. The decoder is used

to reconstruct the original image by upsampling and

skip connection. In the experimentation, initially

224 × 224 size patches fed into the network. The in-

tersection oven union (IOU) accuracy was observed

with varying accuracy when it was tested on the un-

seen image. The resulting output obtained from this

ﬁne-tuning is shown in Figure 4. Pixel based accu-

racy was found to be accuracy 0.8709 %, trained with

Adam optimizer and learning rate of 0.001. We have

used a total of 414 patch samples for training and val-

idation.

4.4 Segmentation using Spectral

Graph-CNN

Due to the resource limitation required for the spec-

tral based graph convolution, the original image was

divided into patches of size 32 × 32 × 3. To process

this patch using the graph, ﬁrst we need to construct

the graph which holds the signal that is 32 × 32 × 3

size. Let N be the number of nodes holding the 3

RGB pixel values at each node say d. The mathe-

ICPRAM 2021 - 10th International Conference on Pattern Recognition Applications and Methods

624

Figure 4: Left: Original Image, Right: FCN Fine-Tuning

method.

matical form of graph convolution operation is men-

tioned in section 3.2. However, simpliﬁed expression

of graph convolution operation explained here. This

expression is nicely derived from spectral base graph

convolution (Kipf and Welling, 2016). X is a feature

matrix of dimension N ×F

. N is the number of nodes

and F

is the number of features on each node. The

2D grid graph is created to hold this signal, with the

dimension N ×N. This N ×N binary matrix stores the

connectivity of the node, and is called as adjacency

matrix represented by A. Like conventional CNN, the

hidden layer in graph-CNN is represented by

= f (H

i−1

,A) (9)

as H

is i

hidden layer. For optimization and

training, weight is assign to edges between connect-

ing nodes as weight matrix W

and also consider the

self node features that are added into the adjacency

matrix. So, the new adjacency matrix is,

A = A + I

where I is an identity matrix. Graph convolution is

modiﬁed as, ﬁrst compute the node feature represen-

tation of each node by aggregating feature represen-

tation of its neighbours node and then transform it by

multiplying by the weight matrix. To avoid the gradi-

ent exploding, normalize the feature representation by

adding degree matrix D

−1

. Whole graph convolution

is represented as,

f (H

,A) = σ(D

0.5

) (10)

where σ is a nonlinear activation function. The

Laplacian matrix is decomposed into orthonormal

vector U . This eigendecomposition of graph Lapla-

cian gives the Fourier mode and graph frequencies.

So, the generalized equation of graph convolution is,

f (H

,A) = σ(U ∗ s) (11)

where, s is a signal on a graph.

4.5 Segmentation using Spatial

Graph-CNN

In the experimentation of spatial graph CNN we are

using a MoNet operator as described in the method

Figure 5: Graph representation: Top left: Original graph

of the 2D grid, Top right: ﬁrst coarsening level with AMG

pooling, Bottom: second coarsening level with AMG.

section. We have set the neighbourhood of each ver-

tex is 4 and based on the Euclidean distance their

respective four adjacent nodes has been collected in

sorted order and build an adjacency matrix of k near-

est neighbour graph. To perform a Gaussian ker-

nel operation, coordinate distance between source and

target node is used as a pseudo-coordinates. In the op-

eration, apply Gaussian kernel w

(u(x,y)) over each

pseudo coordinate on the nodes and their neighbours

yεN(x) the result is multiplied with the signal on

neighbour f (y) and summed all the neighbours re-

sults. We have used Gaussian kernel size 25 and

summed up the result of each Gaussian output patch.

All operations are deﬁned as the patch operator. This

patch operator multiplied with the learnable weight

matrix to perform convolution operation.

( f ∗ g)(x) =

∑

j=1

(x) f , (12)

Here g is the weighted matrix and D

(x) is patch op-

erator.

The architecture used in the model contains 4

graph convolution blocks with output feature size

32, 64, 32 and 3, giving the segmented output same

as input dimension. Each layer is followed with

LeakyRelu activation function except the last layer.

In the architecture diagram 2, the blue box shows the

output graph with the number of nodes and feature

size and edges describes the convolution operation.

For the training, ten thousand random patches from

each image size 1366 × 768 graph signal, has col-

lected for 21 such images patches. The data is trained

with 5 fold cross validation with RMSProp optimizer

and learning rate of 1e-5. We have set the the training

epoch 10 with batch size one that helps to enhance the

functionality of patch operator and learn the feature

Graph Convolution Networks for Cell Segmentation

625

better as compare to spectral based graph convolution

as shown in confusion matrix Figure 6.

Figure 6: Confusion Matrices of segmentation methods.

Top: FCN, Middle: Spectral Graph-CNN, Bottom: Spatial

Graph-CNN.

Table 1: Quantitative comparison of results.

Method Pixel Accuracy (%)

FCN 87.09

(Ours)Spectral Graph-CNN) 89.29

(Ours)Spatial Graph-CNN) 91.03

5 RESULT

For training and testing we have used a total of 23

images of size 1366 × 768 cell segmentation dataset.

For FCN approach training and validation data sam-

ple is divided into ratio of 70:30 for the total 414

patch samples. We report the average accuracy of

87.09% computed over a total of 18 patch samples of

size 224 ×224. Regarding the graph-based approach:

due to limitation of resources we used samples of

size 32 × 32 and total number of sample for training

and validation is 30800 with 70:30 ratio. The pixel

accuracy is taken over 1008 unseen image samples

for both spectral- and spatial-based approach with an

improvement of 2.2% and 3.94% compared with the

FCN approach. Their class based comparative analy-

sis of quantitative measure can be shown in the confu-

sion matrix of Figure 6 where both spectral and spatial

based methods shows signiﬁcant improvement.

The comparative quantitative and qualitative

result of the proposed method is shown in Table 1

and Figure 7. It is observed that the size feature of

the cell is better represented by graph-based approach

with an improved result.

(a) Ground Truth

(b) FCN result

(d) Spatial G-CNN result

Figure 7: Qualitative Comparison of Results. 7a: Ground

Truth of original image of different samples, 7b: Result ob-

tained by FCN method, 7c: Result obtained by G-CNN

method, 7d:Result obtained by Spatial G-CNN method.

Red color represent the CD4 stain cells and Green color cor-

responds to CD8.

ICPRAM 2021 - 10th International Conference on Pattern Recognition Applications and Methods

626

6 CONCLUSIONS

This study proposes a novel method of performing

segmentation on cell images using spectral and spa-

tial graph-CNN. It also allows patch-wise distribu-

tion of the original image for better feature learning.

Convolutions are performed in the spectral domain of

the graph Laplacian for learning of spatially localized

features. Spatial based graph convolution handles dif-

ferent graphs to learn locally, each node. Results are

provided on both conventional CNN and graph-based

CNN which shows graph-based CNN has the ability

to learn localized feature maps across multiple layers

of a network.

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