Algorithm and Complexity for a Network Assortativity Measure

TL;DR

This paper investigates the computational complexity of minimizing the Randić index in graph realizations with given degree sequences, proposing polynomial-time solutions, NP-hardness results, and heuristics with practical applications in neuroscience and medical diagnosis.

Contribution

It formulates the minimum Randić index problem as a polynomial-time solvable b-matching problem and introduces a heuristic for connectivity, extending understanding of network optimization.

Findings

Minimum Randić index realization is polynomial-time solvable via b-matching.

Connectivity constraints make the problem NP-Hard.

Heuristic achieves solutions within 3% of optimal on average.

Abstract

We show that finding a graph realization with the minimum Randi\'c index for a given degree sequence is solvable in polynomial time by formulating the problem as a minimum weight perfect b-matching problem. However, the realization found via this reduction is not guaranteed to be connected. Approximating the minimum weight b-matching problem subject to a connectivity constraint is shown to be NP-Hard. For instances in which the optimal solution to the minimum Randi\'c index problem is not connected, we describe a heuristic to connect the graph using pairwise edge exchanges that preserves the degree sequence. In our computational experiments, the heuristic performs well and the Randi\'c index of the realization after our heuristic is within 3% of the unconstrained optimal value on average. Although we focus on minimizing the Randi\'c index, our results extend to maximizing the Randi\'c index as well. Applications of the Randi\'c index to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia are provided.