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Detecting the core of a network by the centralities of the nodes

July 2024
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Many networks exhibit the core/periphery structure. Core/periphery structure is a type of meso-scale structure that consists of densely connected core nodes and sparsely connected peripheral nodes. Core nodes tend to be well-connected, both among themselves and to peripheral nodes, which tend not to be well-connected to other nodes. In this brief report, we propose a new method to detect the core of a network by the centrality of each node. It is discovered that such nodes with non-negative centralities often consist in the core of the networks. The simulation is carried out on different real networks. The results are checked by the objective function. The checked results may show the effectiveness of the simulation results by the centralities of the nodes on the real networks. Furthermore, we discuss the characters of networks with the single core/periphery structure and point out the scope of the application of our method at the end of this paper.

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Chin. Phys. B 33, 088903 (2024)

Detecting the core of a network by the centralities of the nodes

Peijie Ma(马佩杰), Xuezao Ren(任学藻)†, Junfang Zhu(朱军芳)‡, and Yanqun Jiang(蒋艳群)

School of Mathematics and Science, Southwest University of Science and Technology, Mianyang 621010, China

(Received 23 January 2024; revised manuscript received 15 April 2024; accepted manuscript online 17 May 2024)

Many networks exhibit the core/periphery structure. Core/periphery structure is a type of meso-scale structure that

consists of densely connected core nodes and sparsely connected peripheral nodes. Core nodes tend to be well-connected,

both among themselves and to peripheral nodes, which tend not to be well-connected to other nodes. In this brief report, we

propose a new method to detect the core of a network by the centrality of each node. It is discovered that such nodes with

non-negative centralities often consist in the core of the networks. The simulation is carried out on different real networks.

The results are checked by the objective function. The checked results may show the effectiveness of the simulation results

by the centralities of the nodes on the real networks. Furthermore, we discuss the characters of networks with the single

core/periphery structure and point out the scope of the application of our method at the end of this paper.

Keywords: complex network, core/periphery structure, the objective function

PACS: 89.75.Hc DOI: 10.1088/1674-1056/ad4cd4

1. Introduction

Many networks exhibit the core/periphery structure,

which is important for scientists to understand the properties

and the dynamics of the networks.[1]The constitution of the

structure often refers to a simple two-class partition.[2–4]The

deﬁnition of core/periphery has been formalized in Ref. [2].

The notion of the core in the network is often used to describe

the connectivity between the nodes with large degree, and it

has been applied to proﬁle meso-scale properties in networks

by examining the density of connections.[5–10]The core in the

network inﬂuences the functionality of itself. It is demon-

strated in the transmission of rumors or information in social

networks,[11,12]in the organization of the human connection

in neurodevelopment,[13,14]in the transportation networks of

airline ﬂights,[15]and in the characterizing data patterns. [16]

The core of the network is often regarded to be comprised

of the densely inter-connected high-degree nodes, which im-

pact adaptability, ﬂexibility, and controllability.[17,18]Lots of

proﬁling methods have been proposed based on optimizing a

suitable ﬁtness function using the coreness value to deﬁne the

density of links inside the core,[2]referring to a quality index

with respect to the size of the expected core and the fuzziness

of the boundary,[19]or applying Markov chains to describe

random walks to index the coreness of individual nodes.[20]

These methods rely on subjective ﬁne-tuning due to the pres-

ence of one or more free parameters, which could be obtained

arbitrarily or by techniques. Other examples include maxi-

mizing the closeness centrality within the core by an ensemble

of random networks to deﬁne a coefﬁcient that characterizes

the core.[3]Recently, some scientists have proposed to use

an inﬂuence propagation process to detect the pairs of core-

periphery nodes.[21]Generally, these methods always tend to

be relatively complex in nature, and therefore scalability is-

sues are likely to be encountered when applied to the huge

networks.

Interestingly, the presence of a network with

core/periphery structure could be divided into two parts. How-

ever, there is, at present, no general method to deﬁne the core

nodes. Due to the importance of the core/periphery structure

in the networks, we present a method to proﬁle such structure

in this brief report. First, we deﬁne the centrality of each node.

It could be considered that the core of a network consists of

the nodes with non-negative centralities. Second, the results

can be checked by the objective function. It is discovered that

the nodes with non-negative centralities may be included to

the core by detecting the core/periphery structure. Third, we

discuss the character of the network with single core/periphery

structure on the synthetic network. Furthermore, we describe

the scope of the application of our method.

2. Method

Consider an unweighted and undirected network. A com-

plex network can be described as a graph set G(V,E). The

node set is V, and the connection set is E. In this paper, it

is considered that these connections are undirected in the net-

work. Each element in the adjacency matrix Aof the network

is 0 or 1, when node iis connected with node j, the element

ai j =1, otherwise ai j =0. The degree of node iis ki=∑jai j .

Then, we can deﬁne the centrality C(i)of the node iin the net-

work. It is considered that the core node in the star network

may have the highest centrality, while the periphery nodes

have the lowest centrality. The centrality of node iis deﬁned

†Corresponding author. E-mail: rxz63@aliyun.com

‡Corresponding author. E-mail: zjfbird@mail.ustc.edu.cn

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Detecting the core of a network by the centralities of the nodes

Abstract

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