图的进化：致密化和收缩直径
Graph Evolution: Densification and Shrinking Diameters
摘要：
真实的图形如何随时间演变？什么是社会、技术和信息网络的"正常"增长模式？许多研究已经发现了静态图中的模式，在大型网络的单个快照或非常少量的快照中识别属性;这些包括入度分布和出度分布的重尾，社区，小世界现象等。然而，由于缺乏关于长期网络演变的信息，很难将这些发现转化为关于长期趋势的陈述。在这里，我们研究了广泛的实图，我们观察到一些令人惊讶的现象。首先，这些图中的大多数随着时间的推移而变密，边的数量在节点数量中超线性增长。其次，节点之间的平均距离通常会随着时间的推移而缩小，而传统的观点认为，这种距离参数应该作为节点数量的函数（如O（log n）或O（log（log n）））缓慢增加。现有的图形生成模型没有表现出这些类型的行为，即使在定性水平。我们提供了一个新的图形生成器，基于“森林火灾”的传播过程，具有简单，直观的理由，需要很少的参数（如节点的“易燃性”），并产生显示在以前的工作和本研究中观察到的所有属性的图形。我们还注意到，“森林火灾”模型在稀疏图和密集图之间呈现出急剧的转变。节点之间的距离逐渐减小的图在这个过渡点周围生成。
英文摘要：
How do real graphs evolve over time? What are ``normal'' growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include heavy tails for in- and out-degree distributions, communities, small-world phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time. Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time, with the number of edges growing super-linearly in the number of nodes. Second, the average distance between nodes often shrinks over time, in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log(log n)). Existing graph generation models do not exhibit these types of behavior, even at a qualitative level. We provide a new graph generator, based on a ``forest fire'' spreading process, that has a simple, intuitive justification, requires very few parameters (like the ``flammability'' of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study. We also notice that the ``forest fire'' model exhibits a sharp transition between sparse graphs and graphs that are densifying. Graphs with decreasing distance between the nodes are generated around this transition point.
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