Geometric deep learning has demonstrated a great potential in non-Euclidean
data analysis. The incorporation of geometric insights into learning
architecture is vital to its success. Here we propose a curvature-enhanced
graph convolutional network (CGCN) for biomolecular interaction prediction, for
the first time. Our CGCN employs Ollivier-Ricci curvature (ORC) to characterize
network local structures and to enhance the learning capability of GCNs. More
specifically, ORCs are evaluated based on the local topology from node
neighborhoods, and further used as weights for the feature aggregation in
message-passing procedure. Our CGCN model is extensively validated on fourteen
real-world bimolecular interaction networks and a series of simulated data. It
has been found that our CGCN can achieve the state-of-the-art results. It
outperforms all existing models, as far as we know, in thirteen out of the
fourteen real-world datasets and ranks as the second in the rest one. The
results from the simulated data show that our CGCN model is superior to the
traditional GCN models regardless of the positive-to-negativecurvature ratios,
network densities, and network sizes (when larger than 500).

本文提出了一种曲率增强的图卷积神经网络，使用 Ollivier-Ricci 曲率为节点邻域的特征聚合赋权，用于生物分子相互作用预测，结果显示该模型表现优异。

基于曲率增强的图卷积网络用于生物分子相互作用预测

Curvature-enhanced Graph Convolutional Network for Biomolecular  Interaction Prediction

We have performed an empirical comparison of two distinct notions of discrete
Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and
Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci
curvature were developed based on different properties of the classical smooth
notion, and thus, the two notions shed light on different aspects of network
structure and behavior. Nevertheless, our extensive computational analysis in a
wide range of both model and real-world networks shows that the two
discretizations of Ricci curvature are highly correlated in many networks.
Moreover, we show that if one considers the augmented Forman-Ricci curvature
which also accounts for the two-dimensional simplicial complexes arising in
graphs, the observed correlation between the two discretizations is even
higher, especially, in real networks. Besides the potential theoretical
implications of these observations, the close relationship between the two
discretizations has practical implications whereby Forman-Ricci curvature can
be employed in place of Ollivier-Ricci curvature for faster computation in
larger real-world networks whenever coarse analysis suffices.

本研究对图或网络的两种离散 Ricci 曲率形式，即 Forman-Ricci 曲率和 Ollivier-Ricci 曲率进行了实证比较分析。在更广泛的模型和实际网络中进行了广泛的计算分析，表明了两种离散曲率在许多网络中高度相关，具有相似的结构和行为特性。此外，通过引入转化 Forman-Ricci 曲率，研究显示，两种曲率之间的相关性更高，特别是在真实网络中，可以利用 Forman-Ricci 曲率来替代 Ollivier-Ricci 曲率进行更快速的计算和分析。

复杂网络中两种离散的 Ricci 曲率的比较分析

Comparative analysis of two discretizations of Ricci curvature for  complex networks

We study the Ollivier-Ricci curvature of graphs as a function of the chosen
idleness. We show that this idleness function is concave and piecewise linear
with at most $3$ linear parts, with at most $2$ linear parts in the case of a
regular graph. We then apply our result to show that the idleness function of
the Cartesian product of two regular graphs is completely determined by the
idleness functions of the factors.

研究了作为所选空置时间函数的图的 Ollivier-Ricci 曲率，发现这个空置时间函数是凹的并且分段线性，最多有 $3$ 个线性部分，在正则图的情况下最多有 $2$ 个线性部分。然后应用我们的结果，证明了两个正则图笛卡尔积的空置时间函数完全由因子的空置时间函数确定。