利用相对熵分析收缩随机网络的度分布向泊松分布的收敛性
Analysis of the convergence of the degree distribution of contracting random networks towards a Poisson distribution using the relative entropy
AI总结：
该研究针对三类节点删除场景的收缩随机网络，用相对熵证明配置模型网络的度分布收敛到泊松分布，对应结构收敛为ER图，还验证了不同初始度分布网络的收敛性。
摘要：
我们针对随机网络在收缩过程中的结构演化给出了分析结果，该收缩过程通过通用的节点删除场景实现，即随机删除、偏好删除和传播删除。聚焦于具有给定度分布$P_0(k)$且无关联的配置模型网络，我们通过严谨论证表明，在收缩过程中，这些网络的度分布会收敛到泊松分布。为此，我们采用相对熵$S_t = S[P_t(k) || \pi(k|\langle K \rangle_t)]$作为度分布$P_t(k)$（$t$时刻收缩网络的度分布）与对应泊松分布$\pi(k|\langle K \rangle_t)$（具有相同平均度$\langle K \rangle_t$）之间的距离度量。相对熵适合作为距离度量，因为对于任意度分布$P_t(k)$，其满足$S_t \ge 0$，仅当$P_t(k) = \pi(k|\langle K \rangle_t)$时等号成立。我们推导了网络收缩过程中相对熵的时间导数$dS_t/dt$的表达式，并证明在收缩过程中相对熵单调递减至零。因此我们得出结论，收缩的配置模型网络的度分布会收敛到泊松分布。由于收缩后的网络仍保持无关联，这意味着它们的结构会收敛到Erdős-Rényi（ER）图结构，证实了此前通过主方程直接积分和计算机模拟得到的结果[I. Tishby, O. Biham和E. Katzav，《物理评论E》第100卷，032314（2019）]。我们针对具有简并度分布（随机正则图）、指数度分布和幂律度分布（无标度网络）的配置模型网络，验证了该收敛性。
英文摘要：
We present analytical results for the structural evolution of random networks undergoing contraction processes via generic node deletion scenarios, namely, random deletion, preferential deletion and propagating deletion. Focusing on configuration model networks, which exhibit a given degree distribution $P_0(k)$ and no correlations, we show using a rigorous argument that upon contraction the degree distributions of these networks converge towards a Poisson distribution. To this end, we use the relative entropy $S_t=S[P_t(k) || π(k|\langle K \rangle_t)]$ of the degree distribution $P_t(k)$ of the contracting network at time $t$ with respect to the corresponding Poisson distribution $π(k|\langle K \rangle_t)$ with the same mean degree $\langle K \rangle_t$ as a distance measure between $P_t(k)$ and Poisson. The relative entropy is suitable as a distance measure since it satisfies $S_t \ge 0$ for any degree distribution $P_t(k)$, while equality is obtained only for $P_t(k) = π(k|\langle K \rangle_t)$. We derive an equation for the time derivative $dS_t/dt$ during network contraction and show that the relative entropy decreases monotonically to zero during the contraction process. We thus conclude that the degree distributions of contracting configuration model networks converge towards a Poisson distribution. Since the contracting networks remain uncorrelated, this means that their structures converge towards an Erd{\H o}s-Rényi (ER) graph structure, substantiating earlier results obtained using direct integration of the master equation and computer simulations [I. Tishby, O. Biham and E. Katzav, {\it Phys. Rev. E} {\bf 100}, 032314 (2019)]. We demonstrate the convergence for configuration model networks with degenerate degree distributions (random regular graphs), exponential degree distributions and power-law degree distributions (scale-free networks).
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