The eigentime identity for random walks on the weighted networks is the expected time for a walker going from a node to another node. Eigentime identity can be studied by the sum of reciprocals of all nonzero Laplacian eigenvalues on the weighted networks. In this paper, we study the weighted [Formula: see text]-flower networks with the weight factor [Formula: see text]. We divide the set of the nonzero Laplacian eigenvalues into three subsets according to the obtained characteristic polynomial. Then we obtain the analytic expression of the eigentime identity [Formula: see text] of the weighted [Formula: see text]-flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take [Formula: see text], [Formula: see text] as example, and show that the leading term of the eigentime identity on the weighted [Formula: see text]-flower networks obey superlinearly, linearly with the network size.

中文翻译：

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加权 (m,n)-花网络的特征时间恒等式

加权网络上随机游走的特征时间恒等式是步行者从一个节点到另一个节点的预期时间。特征时间恒等可以通过加权网络上所有非零拉普拉斯特征值的倒数和来研究。在本文中，我们研究了带有权重因子的加权[公式：见文]-花网络[公式：见文]。我们根据得到的特征多项式将非零拉普拉斯特征值集合划分为三个子集。然后我们利用拉普拉斯特征多项式和马尔可夫谱的递归结构得到加权[公式：见文本]-花网络的特征时间恒等式[公式：见文本]的解析表达式。我们以【公式：见正文】、【公式：见正文】为例，