A link stream is a set of triplets $(t, u, v)$ indicating that $u$ and $v$ interacted at time $t$. Link streams model numerous datasets and their proper study is crucial in many applications. In practice, raw link streams are often aggregated or transformed into time series or graphs where decisions are made. Yet, it remains unclear how the dynamical and structural information of a raw link stream carries into the transformed object. This work shows that it is possible to shed light into this question by studying link streams via algebraically linear graph and signal operators, for which we introduce a novel linear matrix framework for the analysis of link streams. We show that, due to their linearity, most methods in signal processing can be easily adopted by our framework to analyze the time/frequency information of link streams. However, the availability of linear graph methods to analyze relational/structural information is limited. We address this limitation by developing (i) a new basis for graphs that allow us to decompose them into structures at different resolution levels; and (ii) filters for graphs that allow us to change their structural information in a controlled manner. By plugging-in these developments and their time-domain counterpart into our framework, we are able to (i) obtain a new basis for link streams that allow us to represent them in a frequency-structure domain; and (ii) show that many interesting transformations to link streams, like the aggregation of interactions or their embedding into a euclidean space, can be seen as simple filters in our frequency-structure domain.
翻译:链接流是一组三重美元( t, u, v) 。 链接流是一组三重美元( t, u, v), 表明美元和美元在时间上互动。 链接流模型众多数据集及其适当研究在许多应用中至关重要。 在实践中, 原始链接流往往被汇总或转换成时间序列或图形, 从而做出决策。 然而, 原始链接流的动态和结构信息是如何进入转换对象的。 这项工作表明, 可以通过通过直径直线图和信号操作员来研究链接流, 从而澄清这一问题。 为此, 我们为分析链接流的流引入了一个全新的线性线性矩阵框架。 我们显示, 由于其线性矩阵的细直线性矩阵, 能够将它们分解成不同分辨率层次的结构; 以及 (二) 筛选图表, 使我们得以将精细的直线性矩阵用于分析链接流的流流, 使得我们能够将其结构流转化为新的链接; 将这些直径直径的直径直路图显示为我们的直径框架。