Optimizing SNARK networks via double metric dimension

Muhammad Ahmad; Muhammad Faheem; Sanaa A. Bajri; Zohaib Zahid; Muhammad Javaid; Hamiden Abd El-Wahed Khalifa; Muhammad Ahmad; Muhammad Faheem; Sanaa A. Bajri; Zohaib Zahid; Muhammad Javaid; Hamiden Abd El-Wahed Khalifa

doi:10.3934/math.20241074

AIMS Mathematics

2024, Volume 9, Issue 8: 22091-22111. doi: 10.3934/math.20241074

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Research article

Optimizing SNARK networks via double metric dimension

1.
Department of Mathematics, University of Management and Technology, Lahore 54000, Pakistan
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia
4.
Department of Operations and Management Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt

Received: 30 April 2024 Revised: 23 June 2024 Accepted: 01 July 2024 Published: 15 July 2024
MSC : 05C12

Doubly resolving sets (DRSs) provide a promising approach for source detection. They consist of minimal subsets of nodes with the smallest cardinality, referred to as the double metric dimension (DMD), that can uniquely identify the location of any other node within the network. Utilizing DRSs can improve the accuracy and efficiency of the identification of the origin of a diffusion process. This ability is crucial for early intervention and control in scenarios such as epidemic outbreaks, misinformation spreading in social media, and fault detection in communication networks. In this study, we computed the DMD of flower snarks $ J_{m} $ and quasi-flower snarks $ G_{m} $ by describing their minimal doubly resolving sets (MDRSs). We deduce that the DMD for the flower snarks $ J_{m} $ is finite and depends on the network's order, and the DMD for the quasi-flower snarks $ G_{m} $ is finite and independent of the network's order. Furthermore, our findings offer valuable insights into the structural features of complex networks. This knowledge can offer direction for future studies in network theory and its practical implementations.
- optimization,
- complex networks,
- metric dimension,
- doubly resolving sets,
- flower snarks,
- quasi-flower snarks
Citation: Muhammad Ahmad, Muhammad Faheem, Sanaa A. Bajri, Zohaib Zahid, Muhammad Javaid, Hamiden Abd El-Wahed Khalifa. Optimizing SNARK networks via double metric dimension[J]. AIMS Mathematics, 2024, 9(8): 22091-22111. doi: 10.3934/math.20241074

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Abstract

Doubly resolving sets (DRSs) provide a promising approach for source detection. They consist of minimal subsets of nodes with the smallest cardinality, referred to as the double metric dimension (DMD), that can uniquely identify the location of any other node within the network. Utilizing DRSs can improve the accuracy and efficiency of the identification of the origin of a diffusion process. This ability is crucial for early intervention and control in scenarios such as epidemic outbreaks, misinformation spreading in social media, and fault detection in communication networks. In this study, we computed the DMD of flower snarks $ J_{m} $ and quasi-flower snarks $ G_{m} $ by describing their minimal doubly resolving sets (MDRSs). We deduce that the DMD for the flower snarks $ J_{m} $ is finite and depends on the network's order, and the DMD for the quasi-flower snarks $ G_{m} $ is finite and independent of the network's order. Furthermore, our findings offer valuable insights into the structural features of complex networks. This knowledge can offer direction for future studies in network theory and its practical implementations.

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