In a recent breakthrough STOC 2015 paper, a continuous diffusion process was considered on hypergraphs (which has been refined in a recent JACM 2018 paper) to define a Laplacian operator, whose spectral properties satisfy the celebrated Cheeger's inequality. However, one peculiar aspect of this diffusion process is that each hyperedge directs flow only from vertices with the maximum density to those with the minimum density, while ignoring vertices having strict in-between densities.

In this work, we consider a generalized diffusion process, in which vertices in a hyperedge can act as mediators to receive flow from vertices with maximum density and deliver flow to those with minimum density. We show that the resulting Laplacian operator still has a second eigenvalue satisfying the Cheeger's inequality.

Our generalized diffusion model shows that there is a family of operators whose spectral properties are related to hypergraph conductance, and provides a powerful tool to enhance the development of spectral hypergraph theory. Moreover, since every vertex can participate in the new diffusion model at every instant, this can potentially have wider practical applications.

在最近的突破性STOC 2015论文中，在超图上考虑了连续扩散过程（在最近的JACM 2018论文中进行了改进）以定义一个Laplacian算子，其谱特性满足著名的Cheeger不等式。但是，该扩散过程的一个特殊方面是，每个超边仅将流从具有最大密度的顶点引导到具有最小密度的顶点，而忽略了具有严格中间密度的顶点。

在这项工作中，我们考虑了广义扩散过程，其中超边中的顶点可以充当介质，以从最大密度的顶点接收流量并将流体传递给最小密度的顶点。我们证明了所得的拉普拉斯算子仍然具有满足Cheeger不等式的第二特征值。

我们的广义扩散模型表明，有一族算子的频谱特性与超图电导相关，并且为增强频谱超图理论的发展提供了强大的工具。而且，由于每个顶点都可以随时参与新的扩散模型，因此这可能具有更广泛的实际应用。