稀疏图的第二大特征对统计量
Second largest Eigenpair Statistics for Sparse Graphs
摘要：
我们开发了一种形式主义来计算第二大特征对的统计加权稀疏图$N\gg 1$节点，有限平均连通性和有界的最大程度，在顶部特征对统计是已知的情况下。在对原始矩阵模型进行适当的收缩之后，可以将问题转换为具有虚拟温度的球体上的二次型的优化。我们使用的腔和副本的方法来找到解决方案的自洽方程的辅助概率密度函数，这可以解决一个改进的人口动力学算法强制本征向量正交的飞行。分析结果与大（加权）邻接矩阵的数值对角化完全一致，集中在随机规则和Erdens-Rényi图的情况下。我们进一步分析了无偏随机游动的稀疏马尔可夫转移矩阵的情况，其第二大特征对描述了具有最大松弛时间的非平衡模式。我们还表明，人口规模$N_P$的人口动力学算法实际上并没有捕获的热力学极限$N\to\infty$通常假设：人口动力学算法的准确性具有强烈的非单调行为作为$N_P$的函数，从而意味着必须选择一个最佳的大小$N_P^\star=N_P^\star(N)$，以最好地再现结果从数值对角化的图形的有限大小$N$。
英文摘要：
We develop a formalism to compute the statistics of the second largest eigenpair of weighted sparse graphs with $N\gg 1$ nodes, finite mean connectivity and bounded maximal degree, in cases where the top eigenpair statistics is known. The problem can be cast in terms of optimisation of a quadratic form on the sphere with a fictitious temperature, after a suitable deflation of the original matrix model. We use the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by an improved population dynamics algorithm enforcing eigenvector orthogonality on-the-fly. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, focussing on the cases of random regular and Erdős-Rényi graphs. We further analyse the case of sparse Markov transition matrices for unbiased random walks, whose second largest eigenpair describes the non-equilibrium mode with the largest relaxation time. We also show that the population dynamics algorithm with population size $N_P$ does not actually capture the thermodynamic limit $N\to\infty$ as commonly assumed: the accuracy of the population dynamics algorithm has a strongly non-monotonic behaviour as a function of $N_P$, thus implying that an optimal size $N_P^\star=N_P^\star(N)$ must be chosen to best reproduce the results from numerical diagonalisation of graphs of finite size $N$.
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