渐近无标度地理网络的广义模型
A generalised model for asymptotically-scale-free geographical networks
摘要：
我们考虑了一个广义d维渐近无标度地理网络模型。当考虑到这类网络随时间的增长时，其核心是附着规则，即一个新节点附着到一个（或多个）先前存在的节点的概率。为了更真实，还考虑了网络的每个节点$i$的适应度参数$η_i \in [0,1]$，以反映节点吸引新节点的能力。我们的d维模型考虑了节点之间的地理距离，$η$具有不同的概率分布，这合理地修改了增长动力学。假设优先连接规则为$Π_i\propto k_i η_i r_{ij}^{-α_A} $，其中$k_i$是$i$第预先存在的站点的连接性，而$α_A$表征了欧几里得距离r对于网络增长的重要性。对于特殊的参数值，该模型分别恢复Bianconi-Barabási和Barabási-Albert的。目前的广义模型是渐近无标度在所有情况下，其度分布是非常好的拟合与q指数分布，优化非加性熵$S_q$，由$p(k) \propto e_q^{-k/κ} \equiv 1/[1+(q-1)k/κ]^{1/(q-1)}$，与$(q,κ)$唯一地依赖于比率$α_A/d$和健身分布。因此，这个模型构成了一个实现的渐近无标度的地理网络内的非广泛的统计力学，其中$k$起着能源的作用和$κ$起着温度的作用。一般的标度律也被发现为q作为模型参数的函数。
英文摘要：
We consider a generalised d-dimensional model for asymptotically-scale-free geographical networks. Central to many networks of this kind, when considering their growth in time, is the attachment rule, i.e. the probability that a new node is attached to one (or more) preexistent nodes. In order to be more realistic, a fitness parameter $η_i \in [0,1]$ for each node $i$ of the network is also taken into account to reflect the ability of the nodes to attract new ones. Our d-dimensional model takes into account the geographical distances between nodes, with different probability distribution for $η$ which sensibly modifies the growth dynamics. The preferential attachment rule is assumed to be $Π_i\propto k_i η_i r_{ij}^{-α_A} $ where $k_i$ is the connectivity of the $i$th pre-existing site and $α_A$ characterizes the importance of the euclidean distance r for the network growth. For special values of the parameters, this model recovers respectively the Bianconi-Barabási and the Barabási-Albert ones. The present generalised model is asymptotically scale-free in all cases, and its degree distribution is very well fitted with q-exponential distributions, which optimise the nonadditive entropy $S_q$, given by $p(k) \propto e_q^{-k/κ} \equiv 1/[1+(q-1)k/κ]^{1/(q-1)}$, with $(q,κ)$ depending uniquely only on the ratio $α_A/d$ and the fitness distribution. Hence this model constitutes a realization of asymptotically-scale-free geographical networks within nonextensive statistical mechanics, where $k$ plays the role of energy and $κ$ plays the role of temperature. General scaling laws are also found for q as a function of the parameters of the model.
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