In the analysis of large-scale network data, a fundamental operation is the comparison of observed phenomena to the predictions provided by null models: when we find an interesting structure in a family of real networks, it is important to ask whether this structure is also likely to arise in random networks with similar characteristics to the real ones. A long-standing challenge in network analysis has been the relative scarcity of reasonable null models for networks; arguably the most common such model has been the configuration model, which starts with a graph $G$ and produces a random graph with the same node degrees as $G$. This leads to a very weak form of null model, since fixing the node degrees does not preserve many of the crucial properties of the network, including the structure of its subgraphs. Guided by this challenge, we propose a new family of network null models that operate on the $k$-core decomposition. For a graph $G$, the $k$-core is its maximal subgraph of minimum degree $k$; and the core number of a node $v$ in $G$ is the largest $k$ such that $v$ belongs to the $k$-core of $G$. We provide the first efficient sampling algorithm to solve the following basic combinatorial problem: given a graph $G$, produce a random graph sampled nearly uniformly from among all graphs with the same sequence of core numbers as $G$. This opens the opportunity to compare observed networks $G$ with random graphs that exhibit the same core numbers, a comparison that preserves aspects of the structure of $G$ that are not captured by more local measures like the degree sequence. We illustrate the power of this core-based null model on some fundamental tasks in network analysis, including the enumeration of networks motifs.

中文翻译：

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带有规定的$ K $-核心序列的随机图：用于网络分析的新Null模型

在分析大型网络数据时，一项基本操作是将观察到的现象与空模型提供的预测进行比较：当我们在一个真实网络家族中找到一个有趣的结构时，重要的是要问这种结构是否也可能会出现在具有与真实特征相似的随机网络中。网络分析中的一个长期挑战是，对于网络而言，合理的空模型相对缺乏。可以说，最常见的此类模型是配置模型，该模型以图形$ G $开头，并生成一个具有与$ G $相同的节点度的随机图形。这导致空模型的形式非常薄弱，因为固定节点度并不能保留网络的许多关键属性，包括其子图的结构。在这一挑战的指引下，我们提出了一个新的网络空值模型系列，它们可以在$ k $ -core分解下运行。对于图$ G $，$ k $核心是其最小度为$ k $的最大子图；$ G $中节点$ v $的核心数是最大的$ k $，因此$ v $属于$ G $的$ k $核心。我们提供了第一个有效的采样算法来解决以下基本的组合问题：给定图$ G $，从具有与$ G $相同的核心编号序列的所有图中生成几乎均匀采样的随机图。这为将观察到的网络$ G $与显示相同核心数的随机图进行比较提供了机会，该比较保留了$ G $结构的某些方面，而这些方面并未被更多的本地度量（如度数序列）捕获。我们说明了这种基于核心的空模型在网络分析中某些基本任务上的强大功能，