一种网络协同性测度的算法及其复杂性
Algorithm and Complexity for a Network Assortativity Measure
摘要：
我们发现，找到一个图实现与最小的Randić指数为给定的度序列是可解决的多项式时间，制定的问题作为一个最小重量完美的b-匹配问题。然而，通过这种归约找到的实现并不保证是连通的。近似的最小权重的b-匹配问题的连接约束下，被证明是NP难的。对于最小Randić指数问题的最优解不连通的情况，我们描述了一种启发式方法，使用保留度序列的成对边交换来连接图。在我们的计算实验中，启发式算法表现良好，我们的启发式算法后实现的Randić指数平均在无约束最优值的3%以内。虽然我们专注于最小化Randić指数，我们的结果扩展到最大化Randić指数以及。提供了Randić指数在哺乳动物中控制呼吸的神经元网络的同步和在诊断患有痴呆的个体中使皮质厚度网络标准化方面的应用。
英文摘要：
We show that finding a graph realization with the minimum Randić 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 Randić 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 Randić index of the realization after our heuristic is within 3% of the unconstrained optimal value on average. Although we focus on minimizing the Randić index, our results extend to maximizing the Randić index as well. Applications of the Randić index to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia are provided.
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