迭代独立模型
The Iterative Independent Model
摘要：
已经发现使用迭代生成算法的确定性复杂网络比传统的均匀随机图模型更接近真实世界网络中发现的属性。在本文中，我们介绍了一个新的，迭代独立模型（IIM），推广以前定义的模型。这些模型使用结构平衡理论的思想，通过克隆的概念产生边缘，其中“我朋友的朋友是我的朋友”，反克隆的概念产生边缘，其中“我敌人的敌人是我的朋友”。在本文中，我们极大地推广了这些概念，允许每个顶点添加在一个给定的时间步，以选择独立于其他顶点，如果它将被克隆或反克隆。虽然关注随机模型似乎很自然，我们随机决定是否克隆任何给定的顶点，但我们发现一般的确定性模型表现出某些结构特性，而不管概率如何。这使得应用程序可以探索细节，同时让理论模型解释在所有可能的场景中发生的结构现象。在整个文件中，我们证明了所有的IIM图有界远离零，这表明聚类特性也发现在社交网络的频谱间隙。此外，我们显示的直径，控制数和团数的界限进一步表明IIM图的聚类行为。最后，对于任何固定图$F$，所有IIM图最终将包含$F$的诱导副本。
英文摘要：
Deterministic complex networks that use iterative generation algorithms have been found to more closely mirror properties found in real world networks than the traditional uniform random graph models. In this paper we introduce a new, Iterative Independent Model (IIM), generalizing previously defined models. These models use ideas from Structural Balance Theory to generate edges through a notion of cloning where ``the friend of my friend is my friend'' and anticloning where ``the enemy of my enemy is my friend''. In this paper, we vastly generalize these notions by allowing each vertex added at a given time step to choose independently of the other vertices if it will be cloned or anticloned. While it may seem natural to focus on a randomized model, where we randomly determine whether or not to clone any given vertex, we found the general deterministic model exhibited certain structural properties regardless of the probabilities. This allows applications to then explore the particulars, while having the theoretical model explain the structural phenomenons that occur in all possible scenarios. Throughout the paper we demonstrate that all IIM graphs have spectral gap bounded away from zero, which indicates the clustering properties also found in social networks. Furthermore, we show bounds on the diameter, domination number, and clique number further indicating the well clustered behaviour of IIM graphs. Finally, for any fixed graph $F$ all IIM graphs will eventually contain an induced copy of $F$.
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