Degree (graph theory)

Core Image Degree (CID) in graph theory is a concept that measures the centrality of a node within a graph's core-periphery structure. The core-periphery model in network analysis distinguishes between a densely connected core and a sparsely connected periphery. CID specifically quantifies the importance or influence of a node based on its connections within the core of the network, offering insights into the structural properties of complex networks.

Definition

The Core Image Degree of a node is defined by its connections to the core nodes of a network. A core node is typically characterized by its high degree of connectivity or centrality within the network. The CID value of a node is calculated based on the number and strength of its links to these core nodes. This measure can be applied to both directed and undirected graphs, and it can incorporate weighted connections, where the strength of the links is taken into account.

Calculation

To calculate the CID of a node, one must first identify the core of the network. This can be done using various methods, such as k-core decomposition, which peels away layers of the network to reveal a densely connected core. Once the core is identified, the CID for a node i can be calculated as the sum of its connections to nodes within the core, often normalized by the maximum possible degree within the network to facilitate comparison across different networks or nodes.

Applications

CID finds applications in various fields, including social network analysis, biology (especially in studying protein interaction networks), information science (for analyzing citation networks), and computer science (in the study of the Internet's structure). By identifying nodes with high CID values, researchers can pinpoint influential individuals in social networks, key proteins in biological networks, or major hubs in transportation and communication networks.

Challenges and Limitations

One challenge in applying CID is determining the boundary between the core and periphery, which can vary depending on the method used to identify the core. Additionally, CID, like other centrality measures, may not fully capture the multifaceted roles that nodes can play in a network's structure and dynamics.

See Also

• Centrality
• Graph Theory
• Network Analysis
• K-core

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