几类复杂网络非线性暂态动力学的多样性
On the Diversity of Non-Linear Transient Dynamics in Several Types of Complex Networks
摘要：
动态系统以多样化的演化为特征，不仅更灵活，而且对攻击、故障和变化的条件更具弹性。本文讨论了量化的多样性的非线性瞬态动力学在无向和无权重的复杂网络中获得的自避免随机游动的结果。从特定节点$i$开始的行走的多样性根据由沿着每个初始步骤的节点访问概率的熵组成的签名来量化。复杂网络的六个理论模型被认为是：Erdens-Rényi，Barabási-Albert，Watts-Strogatz，一个地理模型，以及两个最近推出的针织网络形成的路径。在网络类别和单个节点的水平上探索随机游走多样性。由于在连续步骤的步行的多样性往往是相关的，主成分分析系统地应用，以确定更相关的线性组合的多样性熵，并获得最佳的降维。几个有趣的结果的报道，包括这样的事实，即瞬时多样性往往会增加与所有考虑的网络模型的平均程度，瓦特和Strogatz和地理模型往往会产生多样性熵的步骤的数量逐渐增加，与其他四个考虑模型验证的急剧增加形成鲜明对比。主成分分析方法确定的diluent的主线性组合，允许一个有趣的表征个别节点，以及分区的网络到子图的类似的多样性。
英文摘要：
Dynamic systems characterized by diversified evolutions are not only more flexible, but also more resilient to attacks, failures and changing conditions. This article addresses the quantification of the diversity of non-linear transient dynamics obtained in undirected and unweighted complex networks as a consequence of self-avoiding random walks. The diversity of walks starting at a specific node $i$ is quantified in terms of a signature composed by the entropies of the node visit probabilities along each of the initial steps. Six theoretical models of complex networks are considered: Erdős-Rényi, Barabási-Albert, Watts-Strogatz, a geographical model, as well as two recently introduced knitted networks formed by paths. The random walk diversity is explored at the level of network categories and of individual nodes. Because the diversity at successive steps of the walks tends to be correlated, principal component analysis is systematically applied in order to identify the more relevant linear combinations of the diversity entropies and to obtain optimal dimensionality reduction. Several interesting results are reported, including the facts that the transient diversity tends to increase with the average degree for all considered network models and that the Watts and Strogatz and geographical models tend to yield diversity entropies which increase more gradually with the number of steps, contrasting sharply with the steep increases verified for the other four considered models. The principal linear combination of the diversities identified by the principal component analysis method is shown to allow an interesting characterization of individual nodes as well as partitioning of networks into subgraphs of similar diversity.
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