成簇无标度网络中的自回避游动与连接常数
Self-avoiding walks and connective constants in clustered scale-free networks
摘要：
近年来，复杂网络上的各种类型的行走已经被用于模拟几种系统中的搜索和导航，特别强调随机行走。这提供了关于网络性质的有价值的信息，但自回避行走（SAW）可能比无限制随机行走更适合研究复杂系统的长距离特性。在这里，我们研究SAW在集群无标度网络，其特征在于程度分布的形式$P(k) \sim k^{-γ}$大$k$。在这些网络中，通过插入三节点循环（三角形）来引入聚类。SAW的长距离行为为我们提供了这种网络的渐近特性的信息。通过直接枚举得到了自避免行走的次数$a_n$，这使我们能够确定这些网络的{\em连接常数} $μ$作为比率$a_n / a_{n-1}$的大-$n$极限。给出了一种分析方法来解释步行计数法的结果，两种方法给出的结果相互一致。一般来说，SAW的平均数量$a_n$是更大的集群网络比非集群的具有相同的度分布。大系统尺寸$N$的连接常数的渐近极限取决于度分布的指数$γ$：对于$γ> 3$，$μ$收敛到一个有限值$N \to \infty$;对于$γ= 3$，尺寸相关的$μ_N$发散为$\ln N$，对于$γ< 3$，我们有$μ_N \sim N^{(3 - γ) / 2}$。
英文摘要：
Various types of walks on complex networks have been used in recent years to model search and navigation in several kinds of systems, with particular emphasis on random walks. This gives valuable information on network properties, but self-avoiding walks (SAWs) may be more suitable than unrestricted random walks to study long-distance characteristics of complex systems. Here we study SAWs in clustered scale-free networks, characterized by a degree distribution of the form $P(k) \sim k^{-γ}$ for large $k$. Clustering is introduced in these networks by inserting three-node loops (triangles). The long-distance behavior of SAWs gives us information on asymptotic characteristics of such networks. The number of self-avoiding walks, $a_n$, has been obtained by direct enumeration, allowing us to determine the {\em connective constant} $μ$ of these networks as the large-$n$ limit of the ratio $a_n / a_{n-1}$. An analytical approach is presented to account for the results derived from walk enumeration, and both methods give results agreeing with each other. In general, the average number of SAWs $a_n$ is larger for clustered networks than for unclustered ones with the same degree distribution. The asymptotic limit of the connective constant for large system size $N$ depends on the exponent $γ$ of the degree distribution: For $γ> 3$, $μ$ converges to a finite value as $N \to \infty$; for $γ= 3$, the size-dependent $μ_N$ diverges as $\ln N$, and for $γ< 3$ we have $μ_N \sim N^{(3 - γ) / 2}$.
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