Closeness and Vertex Residual Closeness Metrics for Harary Graphs

Centrala begrepp

The paper analyzes the closeness and vertex residual closeness parameters of Harary Graphs, which are well-known constructs with n vertices that are k-connected with the least possible number of edges.

Sammanfattning

The paper examines the outcomes of the closeness and vertex residual closeness parameters in Harary Graphs. It provides a detailed analysis of these graph vulnerability metrics under different conditions based on the parity of k and n.
Key highlights:

Closeness is one of the most commonly used vulnerability metrics in network analysis. The paper explores various formulations of closeness, including the one proposed by Dangalchev, which provides ease of application to disconnected structures.
Vertex residual closeness is a newer and more sensitive parameter compared to other existing parameters, introduced by Dangalchev. It measures how the removal of a vertex from the graph impacts the graph's vulnerability.
The paper derives closed-form expressions for the closeness and vertex residual closeness values of Harary Graphs under different cases, considering the parity of k and n, as well as the diameter of the graph.
For the case of even k, the paper establishes a connection between the Harary Graph and the Consecutive Circulant Graph, allowing the closeness value of the latter to be expressed in terms of the former.
The analysis provides insights into the most sensitive nodes in Harary Graphs, which can be useful for identifying critical points in complex network structures.

Statistik

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Closeness and Residual Closeness of Harary Graphs

by Hande Tuncel... på arxiv.org 04-17-2024

https://arxiv.org/pdf/2308.11056.pdf

Djupare frågor

How can the closeness and vertex residual closeness metrics be extended to other classes of graphs beyond Harary Graphs?

The closeness and vertex residual closeness metrics can be extended to other classes of graphs by considering the structural properties and connectivity patterns specific to each type of graph. For instance, in the case of regular graphs, where each vertex has the same degree, the closeness metric can be adapted to account for the uniform connectivity. Similarly, for random graphs or scale-free networks, where the degree distribution follows a specific pattern, the closeness metric can be modified to capture the unique characteristics of these graphs.
To extend these metrics to other classes of graphs, one can explore variations in the calculation of distances between vertices, taking into consideration the specific graph topology and connectivity constraints. Additionally, the concept of vertex residual closeness can be applied by analyzing the impact of vertex removal on the overall network vulnerability in different types of graphs. By adapting the formulas and definitions of closeness and vertex residual closeness to suit the characteristics of various graph classes, a more comprehensive analysis of network vulnerability can be achieved.

What are the potential applications of these graph vulnerability measures in real-world network analysis and optimization problems?

The graph vulnerability measures, such as closeness and vertex residual closeness, have significant applications in real-world network analysis and optimization problems across various domains. Some potential applications include:

Infrastructure Networks: In transportation networks, power grids, and communication systems, understanding the vulnerability of critical nodes can help in designing robust infrastructure and developing efficient routing strategies to minimize disruptions.

Social Networks: Analyzing the closeness and vertex residual closeness in social networks can aid in identifying influential individuals or key connectors whose removal could impact information flow or network cohesion.

Cybersecurity: In cybersecurity, these metrics can be used to assess the resilience of computer networks against cyber attacks and identify vulnerable points that need strengthening to enhance network security.

Supply Chain Management: By applying graph vulnerability measures to supply chain networks, companies can optimize their logistics operations, identify potential risks, and develop contingency plans to mitigate disruptions.

Epidemiology: In studying the spread of diseases, analyzing network vulnerability can help in identifying critical locations for intervention strategies and understanding the impact of targeted interventions on disease transmission.

What other graph-theoretic parameters could be investigated to provide a more comprehensive understanding of the robustness and resilience of Harary Graphs?

In addition to closeness and vertex residual closeness, several other graph-theoretic parameters can be investigated to enhance the understanding of the robustness and resilience of Harary Graphs. Some parameters that could be explored include:

Betweenness Centrality: This metric identifies nodes that act as bridges between different parts of the network, playing a crucial role in maintaining connectivity.

Eccentricity: Examining the eccentricity of vertices in Harary Graphs can provide insights into the network's diameter and overall structure.

Clustering Coefficient: Analyzing the clustering coefficient can reveal the level of local connectivity and the presence of tightly-knit clusters within the graph.

Edge Connectivity: Studying the edge connectivity of Harary Graphs can help in understanding the minimum number of edges that need to be removed to disconnect the graph.

By investigating these and other graph-theoretic parameters, a more comprehensive analysis of the robustness and resilience of Harary Graphs can be achieved, leading to better-informed network design and optimization strategies.