Topological representations are rapidly becoming a popular way to capture and
encode higher-order interactions in complex systems. They have found
applications in disciplines as different as cancer genomics, brain function,
and computational social science, in representing both descriptive features of
data and inference models. While intense research has focused on the
connectivity and homological features of topological representations,
surprisingly scarce attention has been given to the investigation of the
community structures of simplicial complexes. To this end, we adopt recent
advances in symbolic embeddings to compute and visualize the community
structures of simplicial complexes. We first investigate the stability
properties of embedding obtained for synthetic simplicial complexes to the
presence of higher order interactions. We then focus on complexes arising from
social and brain functional data and show how higher order interactions can be
leveraged to improve clustering detection and assess the effect of higher order
interaction on individual nodes. We conclude delineating limitations and
directions for extension of this work.

本文介绍了如何使用符号嵌入来计算和可视化单纯复合体的社区结构，研究了综合性数据和来自大脑的拓扑表示的社区结构，并展示了如何利用高阶交互作用来改善聚类检测并评估高阶交互作用对个体节点的影响。

简单形式复合体社区检测的 Simplex2Vec 嵌入

Simplex2Vec embeddings for community detection in simplicial complexes

Convolutional neural networks (CNNs) have enabled the state-of-the-art
performance in many computer vision tasks. However, little effort has been
devoted to establishing convolution in non-linear space. Existing works mainly
leverage on the activation layers, which can only provide point-wise
non-linearity. To solve this problem, a new operation, kervolution (kernel
convolution), is introduced to approximate complex behaviors of human
perception systems leveraging on the kernel trick. It generalizes convolution,
enhances the model capacity, and captures higher order interactions of
features, via patch-wise kernel functions, but without introducing additional
parameters. Extensive experiments show that kervolutional neural networks (KNN)
achieve higher accuracy and faster convergence than baseline CNN.

本文介绍了一种新的 kervolution 运算，该方法可以在非线性空间内运行，扩展了卷积神经网络的模型容量，捕捉更高阶的特征交互，并通过基于补丁的核函数实现，而不引入额外参数。大量实验表明， kervolutional neural networks (KNN) 比基线卷积神经网络具有更高的准确性和更快的收敛速度。