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Metric-Based Granular Computing in Networks

Hibba Arshad, Imran Javaid

TL;DR

The paper develops a metric-based granular computing framework for networks by modeling networks as information systems and using distance-based indiscernibility to form granules. It demonstrates an equivalence between rough-set reducts and resolving sets, enabling two algorithms to compute all minimal resolving sets, and introduces a discernibility-matrix based method for systematic identification of resolving subsets. The approach is illustrated on simple graphs and zero-divisor graphs arising from rings such as $\mathbb{Z}_n$ and $\prod_{i=1}^k \mathbb{Z}_2$, including both twin and twin-free cases, with applications to social and political networks. Taken together, these contributions provide concrete tools for identifying essential node subsets in networks and for analyzing network structure through metric granularity.

Abstract

Networks can be highly complex systems with numerous interconnected components and interactions. Granular computing offers a framework to manage this complexity by decomposing networks into smaller, more manageable components, or granules. In this article, we introduce metric-based granular computing technique to study networks. This technique can be applied to the analysis of networks where granules can represent subsets of nodes or edges and their interactions can be studied at different levels of granularity. We model the network as an information system and investigate its granular structures using metric representation. We establish that the concepts of reducts in rough set theory and resolving sets in networks are equivalent. Through this equivalence, we present a novel approach for computing all the minimal resolving sets of these networks.

Metric-Based Granular Computing in Networks

TL;DR

The paper develops a metric-based granular computing framework for networks by modeling networks as information systems and using distance-based indiscernibility to form granules. It demonstrates an equivalence between rough-set reducts and resolving sets, enabling two algorithms to compute all minimal resolving sets, and introduces a discernibility-matrix based method for systematic identification of resolving subsets. The approach is illustrated on simple graphs and zero-divisor graphs arising from rings such as and , including both twin and twin-free cases, with applications to social and political networks. Taken together, these contributions provide concrete tools for identifying essential node subsets in networks and for analyzing network structure through metric granularity.

Abstract

Networks can be highly complex systems with numerous interconnected components and interactions. Granular computing offers a framework to manage this complexity by decomposing networks into smaller, more manageable components, or granules. In this article, we introduce metric-based granular computing technique to study networks. This technique can be applied to the analysis of networks where granules can represent subsets of nodes or edges and their interactions can be studied at different levels of granularity. We model the network as an information system and investigate its granular structures using metric representation. We establish that the concepts of reducts in rough set theory and resolving sets in networks are equivalent. Through this equivalence, we present a novel approach for computing all the minimal resolving sets of these networks.

Paper Structure

This paper contains 11 sections, 33 theorems, 7 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Information System and Set Partitions
  4. Networks
  5. Political Network Modeled by Graphs Associated to Algebraic Structure
  6. Granular Computing in Networks
  7. Dependency Measures and Approximations
  8. The Discernibility Matrix and Resolving sets
  9. Applications in Social Networking
  10. Communication Network
  11. Conclusions

Key Result

Proposition 3.1

For $\mathcal{I}$, let $\mathbb{A}\subseteq V$ and $v_{i}, v_j \in V\setminus \mathbb{A}$ then the following statements are equivalent. (i) $v_{i}\equiv_{\mathbb{A}}v_j$ (ii) $d(v_{i}, u)=d(v_j, u)$$\forall u\in \mathbb{A}$ (iii) $\gamma(v_i|\mathbb{A})=\gamma(v_j|\mathbb{A})$

Figures (3)

  • Figure 1: Political Network
  • Figure 2: Customer Purchase Network
  • Figure 3: Employees Connected by Diverse skills

Theorems & Definitions (73)

  • Definition 3.1
  • Example 3.1
  • Remark 3.1
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.2
  • Proof
  • Proposition 3.3
  • Proof
  • Proposition 3.4
  • ...and 63 more