Uniform sampling from the set $\mathcal{G}(\mathbf{d})$ of graphs with a given degree-sequence $\mathbf{d} = (d_1, \dots, d_n) \in \mathbb N^n$ is a classical problem in the study of random graphs. We consider an analogue for temporal graphs in which the edges are labeled with integer timestamps. The input to this generation problem is a tuple $\mathbf{D} = (\mathbf{d}, T) \in \mathbb N^n \times \mathbb N_{>0}$ and the task is to output a uniform random sample from the set $\mathcal{G}(\mathbf{D})$ of temporal graphs with degree-sequence $\mathbf{d}$ and timestamps in the interval $[1, T]$. By allowing repeated edges with distinct timestamps, $\mathcal{G}(\mathbf{D})$ can be non-empty even if $\mathcal{G}(\mathbf{d})$ is, and as a consequence, existing algorithms are difficult to apply. We describe an algorithm for this generation problem which runs in expected time $O(M)$ if $\Delta^{2+\epsilon} = O(M)$ for some constant $\epsilon > 0$ and $T - \Delta = \Omega(T)$ where $M = \sum_i d_i$ and $\Delta = \max_i d_i$. Our algorithm applies the switching method of McKay and Wormald $[1]$ to temporal graphs: we first generate a random temporal multigraph and then remove self-loops and duplicated edges with switching operations which rewire the edges in a degree-preserving manner.

翻译：从具有给定度序列 $\mathbf{d} = (d_1, \dots, d_n) \in \mathbb N^n$ 的图集合 $\mathcal{G}(\mathbf{d})$ 中进行均匀采样是随机图研究中的一个经典问题。我们考虑时间图上的类比问题，其中边被标记为整数时间戳。该生成问题的输入是一个元组 $\mathbf{D} = (\mathbf{d}, T) \in \mathbb N^n \times \mathbb N_{>0}$，任务是从具有度序列 $\mathbf{d}$ 且时间戳位于区间 $[1, T]$ 内的时间图集合 $\mathcal{G}(\mathbf{D})$ 中输出均匀随机样本。通过允许具有不同时间戳的重复边，即使 $\mathcal{G}(\mathbf{d})$ 为空，$\mathcal{G}(\mathbf{D})$ 也可能非空，因此现有算法难以应用。我们描述了一种针对此生成问题的算法，该算法在期望时间 $O(M)$ 内运行，前提是对于某个常数 $\epsilon > 0$ 有 $\Delta^{2+\epsilon} = O(M)$，且 $T - \Delta = \Omega(T)$，其中 $M = \sum_i d_i$，$\Delta = \max_i d_i$。我们的算法将 McKay 和 Wormald [1] 的交换方法应用于时间图：首先生成随机时间多重图，然后通过交换操作移除自环和重复边，这些操作以保持度的方式重连边。