随机正则图最短路长分布的均值和方差
The mean and variance of the distribution of shortest path lengths of random regular graphs
摘要：
随机网络的最短路径长度（DSPL）分布提供了有关其大尺度结构的有用信息。在随机正则图（RRG）的特殊情况下，它由度为$c \ge 3$的$N$节点组成，DSPL，表示为$P(L=\ell)$，遵循离散Gompertz分布。使用离散拉普拉斯变换，我们推导出一个封闭形式的表达的矩生成函数的DSPL的RRG。从矩生成函数，我们得到封闭形式的表达式的平均值和方差的DSPL。更具体地说，我们发现不同节点对之间的平均距离由$\langle L \rangle = \frac{\ln N}{\ln (c-1)} + \frac{1}{2} - \frac{ \ln c - \ln (c-2) +γ}{\ln (c-1)} + \mathcal{O} \left( \frac{\ln N}{N} \right)$给出，其中$γ$是Euler-Mascheroni常数。虽然领先的长期是已知的，这一结果包括一个新的校正项，这产生了非常好的协议，从直接数值评估$\langle L \rangle$通过尾和公式，并从计算机模拟获得的结果。然而，它并没有说明作为$c$或$N$的函数的$\langle L \rangle$的振荡行为。这些振荡在稀疏网络中可以忽略，但在密集网络中可以检测到。我们还推导出一个表达式的方差${\rm Var}(L)$的DSPL，它捕获的整体依赖的方差$c$，但不占振荡。振荡是由于壳体结构在随机节点周围的离散性质。它们反映了随着$N$的增加，新髋臼杯的填充情况。的平均值和方差的结果进行比较，在其他类型的随机网络中获得的相应结果。讨论了平均距离与直径的关系。
英文摘要：
The distribution of shortest path lengths (DSPL) of random networks provides useful information on their large scale structure. In the special case of random regular graphs (RRGs), which consist of $N$ nodes of degree $c \ge 3$, the DSPL, denoted by $P(L=\ell)$, follows a discrete Gompertz distribution. Using the discrete Laplace transform we derive a closed-form expression for the moment generating function of the DSPL of RRGs. From the moment generating function we obtain closed-form expressions for the mean and variance of the DSPL. More specifically, we find that the mean distance between pairs of distinct nodes is given by $\langle L \rangle = \frac{\ln N}{\ln (c-1)} + \frac{1}{2} - \frac{ \ln c - \ln (c-2) +γ}{\ln (c-1)} + \mathcal{O} \left( \frac{\ln N}{N} \right)$, where $γ$ is the Euler-Mascheroni constant. While the leading term is known, this result includes a novel correction term, which yields very good agreement with the results obtained from direct numerical evaluation of $\langle L \rangle$ via the tail-sum formula and with the results obtained from computer simulations. However, it does not account for an oscillatory behavior of $\langle L \rangle$ as a function of $c$ or $N$. These oscillations are negligible in sparse networks but detectable in dense networks. We also derive an expression for the variance ${\rm Var}(L)$ of the DSPL, which captures the overall dependence of the variance on $c$ but does not account for the oscillations. The oscillations are due to the discrete nature of the shell structure around a random node. They reflect the profile of the filling of new shells as $N$ is increased. The results for the mean and variance are compared to the corresponding results obtained in other types of random networks. The relation between the mean distance and the diameter is discussed.
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