Erdens-Rényi网络上非回溯随机游动首中时的分布
The distribution of first hitting times of non-backtracking random walks on Erdős-Rényi networks
摘要：
本文给出了有限$N$节点的Erdens-Rényi网络上非回溯随机游动首中时分布的解析结果。步行者在网络上的相邻节点之间随机跳跃，而不会退回到前一个节点，直到他们遇到之前已经访问过的节点或被困在死端节点。此时，路径终止。生成路径的长度$d$称为首次命中时间。利用递归方程，我们得到了从随机起始点出发的非回溯随机游动的首中时间$P(d>\ell)$，$\ell=0,1,2,\dots$的尾部分布的解析结果。事实证明，分布$P(d>\ell)$由离散瑞利分布和指数分布的乘积给出。结果发现，非回溯随机游动的路径，其终止在第一次击中的时间，是更长的，平均而言，比相应的简单的RW。然而，它们比同一网络上的SAW短，SAW在最后一次击中时终止。我们得到的概率，$p_{\rm ret}$和$p_{\rm trap}$，路径将终止追溯，即步入一个已经访问过的节点，或通过捕获，即进入一个节点的程度$k=1$，其中没有出口链接，分别分析结果。结果表明，在稀释网络的主要终止方案是陷阱，而在密集的网络中，大多数路径终止回溯。我们得到表达式的条件尾分布的路径长度，$P(d>\ell|{\rm ret})$和$P(d>\ell|\rm{trap})$，为那些路径终止追溯或捕获，分别。我们还研究了一类广义的无回溯随机游动模型，它不仅避免了回溯到前一个节点，而且避免了回溯到最后一个访问过的$S$节点。
英文摘要：
We present analytical results for the distribution of first hitting times of non-backtracking random walks on finite Erdős-Rényi networks of $N$ nodes. The walkers hop randomly between adjacent nodes on the network, without stepping back to the previous node, until they hit a node which they have already visited before or get trapped in a dead-end node. At this point, the path is terminated. The length, $d$, of the resulting path, is called the first hitting time. Using recursion equations, we obtain analytical results for the tail distribution of first hitting times, $P(d>\ell)$, $\ell=0,1,2,\dots$, of non-backtracking random walks starting from a random initial node. It turns out that the distribution $P(d>\ell)$ is given by a product of a discrete Rayleigh distribution and an exponential distribution. It is found that the paths of non-backtracking random walks, up to their termination at the first hitting time, are longer, on average, than those of the corresponding simple RWs. However, they are shorter than those of SAWs on the same network, which terminate at the last hitting time. We obtain analytical results for the probabilities, $p_{\rm ret}$ and $p_{\rm trap}$, that a path will terminate by retracing, namely stepping into an already visited node, or by trapping, namely entering a node of degree $k=1$, which has no exit link, respectively. It is shown that in dilute networks the dominant termination scenario is trapping while in dense networks most paths terminate by retracing. We obtain expressions for the conditional tail distributions of path lengths, $P(d>\ell|{\rm ret})$ and $P(d>\ell|\rm{trap})$, for those paths which terminate by retracing or by trapping, respectively. We also study a class of generalized non-backtracking random walk models which not only avoid the backtracking step into the previous node but avoid stepping into the last $S$ visited nodes.
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