Advancing Connectivity Analysis in Cubic Fuzzy Graphs for Disaster Management: A Study on Tsunami Threat Assessment

By Soham NandiReviewed by Susha Cheriyedath, M.Sc.Feb 7 2024

In an article published in the journal Plos One, researchers explored the application of cubic fuzzy graphs (CFG) in determining danger zones for tsunami threats triggered by earthquakes. They specifically explored the connectivity aspects and introduced the terms "partial cubic α−strong" and "partial cubic δ−weak edges" for these graphs, estimating connectivity index bounds and expressions for various CFGs.

Additionally, they defined the average connectivity index and discussed how removing vertices or edges may impact connectivity. The study introduced concepts like partial cubic connectivity enhancing and reducing nodes, applying these notions to identify tsunami-affected areas after earthquakes and comparing results with existing methods.

Background

Graph theory, a fundamental domain across multiple disciplines, encounters challenges in modeling uncertain and unclear networks. Fuzzy graphs offer a means to address this uncertainty. While traditional fuzzy graphs have been extensively studied, the advent of CFGs introduced a more versatile approach by combining interval-valued fuzzy (IVF) sets with fuzzy sets, allowing for enhanced handling of uncertainty.

Existing research has explored connectivity in crisp and fuzzy graphs, but CFGs remain relatively unexplored in this context. The paper introduced the novel concepts of partial cubic α−strong and partial cubic δ−weak edges to address limitations in the strict definitions of cubic α−strong and cubic δ−weak edges. The researchers delved into the critical aspect of connectivity within CFGs, motivated by their advantageous representation capabilities compared to IVF and general fuzzy graphs.

Moreover, they established the connectivity index for CFGs and determined its bounds for various CFG families. The study also explored the impact of edge removal on the connectivity index and introduced the average connectivity index, partial cubic connectivity enhancing node, and partial cubic connectivity reducing node for CFGs. These contributions filled gaps in existing literature, offering a more comprehensive understanding of complex systems modeled by CFGs and providing valuable insights for real-world applications, such as earthquake analysis and decision-making under uncertainty.

Bounds for connectivity index of CFG

The authors discussed bounds for the connectivity index (σ) of different families of CFGs. The provided theorems established relationships between σ and various CFG characteristics. The first theorem pertained to complete CFGs, presenting an upper bound for σ. The second theorem provided a bound for CFGs with a specific vertex set and mentioned complete CFGs.

The third theorem connected the α−strong property to σ, while the corollary linked σ to β-strong and partially δ-weak edges. The fourth theorem focused on CFGs structured as trees, providing an expression for σ. The fifth theorem established conditions for partial saturated CFG cycles, determining their σ values. The researchers contributed insights into the connectivity index of CFGs across diverse structures, providing valuable information for graph-theoretical analyses.

Average connectivity index of a CFG

The researchers delved into the fundamental concepts surrounding the average connectivity index in CFGs. The average connectivity index, denoted as σ̄, served as a pivotal metric, offering insights into the overall connectivity of CFGs. The definitions presented were crucial in characterizing nodes within CFGs based on their influence on connectivity. The introduced notions of partial cubic connectivity reducing nodes (PCCRNs), partial cubic connectivity enhancing nodes (PCCENs), and neutral nodes contributed to a nuanced understanding of how individual nodes impacted the network's connectivity dynamics.

Furthermore, the authors introduced classifications for CFGs based on the presence of PCCENs, PCCRNs, or neutral nodes, providing a comprehensive framework for analyzing connectivity alterations. The proposition established clear conditions for identifying PCCENs, PCCRNs, and neutral nodes within CFGs. These definitions and propositions collectively laid the groundwork for a more intricate exploration of CFGs, offering valuable insights for network analysis and interpretation. The average connectivity index, along with these classifications, emerged as a powerful tool in unraveling the connectivity intricacies inherent in CFGs, thereby enriching the overall understanding of complex network structures.

Application to determine danger zone of Tsunami threats

The authors explored the practical application of CFGs in determining danger zones for tsunami threats triggered by earthquakes. Natural disasters, especially earthquakes, pose significant challenges due to their unpredictability and potential for widespread destruction. The authors introduced a CFG-based tsunami threat model, where vertices represented areas with varying levels of past, present, and future tsunami threat values. The edges in this model signify the possibility of danger zones arising from a tsunami threat.

By employing the strength of connectivity between different areas, the study categorized danger zones into cubic α−strong, cubic β−strong, cubic δ−weak, partial cubic α−strong, and partial cubic δ−weak zones. The paper presented an algorithm for identifying affected areas and demonstrated its application using a set of areas surrounding an ocean. The connectivity analysis revealed distinct zones with varying levels of tsunami threat, providing a foundation for tailored disaster preparedness and mitigation efforts.

This approach, utilizing CFGs with lower and upper IVF-membership values and fuzzy-membership values, proved advantageous in capturing the nuanced nature of relationships between areas and enhancing decision-making in complex scenarios. The findings underscored the significance of CFGs in offering a more detailed representation of uncertainty in real-world problems, ultimately contributing to improved disaster management strategies.

Comparative analysis

The introduction of partial cubic α-strong and δ-weak edges represented a notable extension of the existing cubic α-strong and cubic δ-weak edges, particularly within the context of modeling earthquake-induced tsunami threats. In a comparative analysis, it became evident that the concepts of partial cubic α-strong and δ-weak edges offered specific advantages over their cubic counterparts. While cubic edges provided insights into certain edge conditions, they fell short in addressing scenarios where the IVF connectivity and fuzzy connectivity exhibited nuanced relationships with the corresponding membership values.

In contrast, the proposed partial cubic edges offered a more comprehensive understanding of edge conditions, especially when certain connectivity characteristics did not strictly align with membership values. This nuanced approach proved beneficial for precise zone delineation and a thorough examination of tsunami threat conditions across different temporal perspectives, marking a significant enhancement in the modeling framework.

Conclusion

In conclusion, CFGs proved invaluable in modeling complex systems with uncertain information. Introducing partial cubic α−strong and δ−weak edges enhanced the understanding of connectivity within CFGs, offering nuanced insights into complex networks. The proposed average connectivity index and the concepts of partial connectivity reducing/enhancing nodes contribute to effective decision-making.

Applying strong and weak edges in CFGs for assessing tsunami threats showcased practical utility. Future exploration involves integrating graph theory with Farmatean fuzzy set models, promising advancements in problem-solving and decision-making. Additionally, extending the concepts to cubic Intuitionistic fuzzy graphs opens avenues for further research.