Faculty of Engineering, HTWK Leipzig University of Applied Sciences, Leipzig, Germany.
J Math Biol. 2023 Jun 7;87(1):3. doi: 10.1007/s00285-023-01937-1.
The paper deals with two interrelated topics: (1) identifying transient amplifiers in an iterative process, and (2) analyzing the process by its spectral dynamics, which is the change in the graph spectra by edge manipulation. Transient amplifiers are networks representing population structures which shift the balance between natural selection and random drift. Thus, amplifiers are highly relevant for understanding the relationships between spatial structures and evolutionary dynamics. We study an iterative procedure to identify transient amplifiers for death-Birth updating. The algorithm starts with a regular input graph and iteratively removes edges until desired structures are achieved. Thus, a sequence of candidate graphs is obtained. The edge removals are guided by quantities derived from the sequence of candidate graphs. Moreover, we are interested in the Laplacian spectra of the candidate graphs and analyze the iterative process by its spectral dynamics. The results show that although transient amplifiers for death-Birth updating are generally rare, a substantial number of them can be obtained by the proposed procedure. The graphs identified share structural properties and have some similarity to dumbbell and barbell graphs. We analyze amplification properties of these graphs and also two more families of bell-like graphs and show that further transient amplifiers for death-Birth updating can be found. Finally, it is demonstrated that the spectral dynamics possesses characteristic features useful for deducing links between structural and spectral properties. These feature can also be taken for distinguishing transient amplifiers among evolutionary graphs in general.
(1)在迭代过程中识别瞬态放大器,以及(2)通过其频谱动态分析该过程,即通过边缘操作改变图谱谱。瞬态放大器是表示种群结构的网络,这些结构改变了自然选择和随机漂移之间的平衡。因此,放大器对于理解空间结构和进化动态之间的关系非常重要。我们研究了一种用于死亡-出生更新的识别瞬态放大器的迭代过程。该算法从一个规则的输入图开始,并通过迭代删除边缘,直到达到所需的结构。因此,获得了一系列候选图。边缘删除由候选图序列中的数量指导。此外,我们对候选图的拉普拉斯谱感兴趣,并通过其频谱动态分析迭代过程。结果表明,尽管死亡-出生更新的瞬态放大器通常很少,但可以通过所提出的方法获得大量的瞬态放大器。识别出的图谱具有共享的结构属性,并且与哑铃和杠铃图谱具有一些相似性。我们分析了这些图谱的放大特性以及另外两种钟形图谱族,并表明可以找到更多的死亡-出生更新的瞬态放大器。最后,证明了频谱动态具有有用的特征,可以用于推断结构和光谱性质之间的联系。这些特征也可以用于在一般的进化图谱中区分瞬态放大器。