Granulation of a network is crucial for the structural analysis of a network. One of the efficient granulation methodology is provided by rough set theory (RST) whereas graph theory provides different metrics for network analysis. Symmetry and symmetry-breaking in a network provides information about the network dynamics. In this paper, we provide a novel method of studying the symmetries of a graph and finding all the possible fixing sets of a graph under RST. We study granulation in simple undirected graphs and zero-divisor graphs of finite commutative rings <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2430_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2430_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\prod _{i=1}^{k}\mathbb {Z}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∏</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> by defining a symmetry-based indiscernibility relation on its vertex set V. We study the partition structure of V, approximations of subsets of V and associate an indiscernibility partition lattice with V. We define the discernibility matrix and provide some of its properties. We show that the reducts obtained from the discernibility function are the fixing sets of graph. Furthermore, using the proposed method, we study a social network based on opinion conflict relationship associated with multi-criteria decision problem.

Symmetry-based granulation in networks associated with commutative rings: application in social network dynamics

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发表日期：

2025年3月9日

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Imran Javaid,
Abeer Fatima,
Muhammad Akram

摘要

Granulation of a network is crucial for the structural analysis of a network. One of the efficient granulation methodology is provided by rough set theory (RST) whereas graph theory provides different metrics for network analysis. Symmetry and symmetry-breaking in a network provides information about the network dynamics. In this paper, we provide a novel method of studying the symmetries of a graph and finding all the possible fixing sets of a graph under RST. We study granulation in simple undirected graphs and zero-divisor graphs of finite commutative rings $\mathbb {Z}_{n}$ $Z_{n}$ and $\prod _{i=1}^{k}\mathbb {Z}_{2}$ $\prod_{i = 1}^{k} Z_{2}$ by defining a symmetry-based indiscernibility relation on its vertex set V. We study the partition structure of V, approximations of subsets of V and associate an indiscernibility partition lattice with V. We define the discernibility matrix and provide some of its properties. We show that the reducts obtained from the discernibility function are the fixing sets of graph. Furthermore, using the proposed method, we study a social network based on opinion conflict relationship associated with multi-criteria decision problem.