A Mathematical Framework for Structural Connectivity and Complexity Analysis of Processor Interconnection Networks

Sushil Kumar Dwivedi; Rakesh Kumar Katare

A Mathematical Framework for Structural Connectivity and Complexity Analysis of Processor Interconnection Networks

Author(s)	Mr. Sushil Kumar Dwivedi, Prof. Dr. Rakesh Kumar Katare
Country	India
Abstract	Processor interconnection networks provide the communication infrastructure for parallel computing, multiprocessor systems, many-core architectures, distributed computing, and high-performance computing. Their topology directly influences processor connectivity, communication distance, structural redundancy, fault tolerance, scalability, and implementation requirements. A systematic structural analysis is therefore essential for understanding interconnection-network characteristics before topology optimization or intelligent routing is applied. This paper presents a mathematical framework for the structural connectivity and complexity analysis of processor interconnection networks using graph-theoretic models. An interconnection network is represented as an undirected graph $G=(V,E)$, where vertices represent processors and edges represent communication links. The framework integrates multiple structural measures, including node degree, degree variance, link density, network diameter, average shortest-path length, clustering coefficient, betweenness and closeness centrality, path diversity, connectivity ratio, structural redundancy, and fault-tolerance retention. A normalized Structural Complexity Index (SCI) is formulated to jointly characterize communication distance and link requirements within a common comparative framework. The framework is demonstrated using five representative processor interconnection topologies: Hypercube, Torus, Fat-Tree, Dragonfly, and Perfect Difference Network (PDN). A 64-node configuration is used as the representative structural demonstration to examine differences in connectivity, communication distance, redundancy, and structural requirements across the selected topologies. The framework provides a unified structural baseline that can be extended to larger processor configurations and failure-oriented studies. Detailed stochastic failure experiments and probabilistic reliability estimation are beyond the scope of the present study.
Keywords	Processor interconnection networks, graph theory, network topology, structural connectivity, structural complexity, path diversity, fault tolerance, graph-based analysis, Structural Complexity Index.
Field	Computer > Network / Security
Published In	Volume 7, Issue 5, September-October 2026
Published On	2026-09-18

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A Mathematical Framework for Structural Connectivity and Complexity Analysis of Processor Interconnection Networks

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