Theoretical and empirical evidence suggests that joint graph embedding algorithms induce correlation across the networks in the embedding space. In the Omnibus joint graph embedding framework, previous results explicitly delineated the dual effects of the algorithm-induced and model-inherent correlations on the correlation across the embedded networks. Accounting for and mitigating the algorithm-induced correlation is key to subsequent inference, as sub-optimal Omnibus matrix constructions have been demonstrated to lead to loss in inference fidelity. This work presents the first efforts to automate the Omnibus construction in order to address two key questions in this joint embedding framework: the correlation-to-OMNI problem and the flat correlation problem. In the flat correlation problem, we seek to understand the minimum algorithm-induced flat correlation (i.e., the same across all graph pairs) produced by a generalized Omnibus embedding. Working in a subspace of the fully general Omnibus matrices, we prove both a lower bound for this flat correlation and that the classical Omnibus construction induces the maximal flat correlation. In the correlation-to-OMNI problem, we present an algorithm -- named corr2Omni -- that, from a given matrix of estimated pairwise graph correlations, estimates the matrix of generalized Omnibus weights that induces optimal correlation in the embedding space. Moreover, in both simulated and real data settings, we demonstrate the increased effectiveness of our corr2Omni algorithm versus the classical Omnibus construction.

翻译：理论和实证证据表明，联合图嵌入算法会在嵌入空间中诱导网络间的相关性。在Omnibus联合图嵌入框架中，先前的研究结果明确阐述了算法诱导相关性和模型固有相关性对嵌入网络间相关性的双重影响。由于次优的Omnibus矩阵构造已被证明会导致推断保真度的损失，因此考虑并减轻算法诱导相关性是后续推断的关键。本研究首次尝试自动化构建Omnibus矩阵，以解决该联合嵌入框架中的两个关键问题：相关性到Omnibus问题和平坦相关性问题。在平坦相关性问题中，我们试图理解广义Omnibus嵌入产生的最小算法诱导平坦相关性（即所有图对间相同）。在完全一般Omnibus矩阵的子空间中工作，我们证明了该平坦相关性的下界，并证明经典Omnibus构造会诱导最大平坦相关性。在相关性到Omnibus问题中，我们提出了一种名为corr2Omni的算法，该算法从给定的估计成对图相关性矩阵出发，估计能诱导嵌入空间最优相关性的广义Omnibus权重矩阵。此外，在模拟和真实数据场景中，我们证明了corr2Omni算法相较于经典Omnibus构造具有更高的有效性。