Title: Discrete Heat Kernels on Simplicial Complexes and Its Application to Functional Brain Networks Show Chinese title

Title: 离散热核在单纯复形上的应用及其在功能脑网络中的应用

Abstract: Networks constitute fundamental organizational structures across biological systems, although conventional graph-theoretic analyses capture exclusively pairwise interactions, thereby omitting the intricate higher-order relationships that characterize network complexity. This work proposes a unified framework for heat kernel smoothing on simplicial complexes, extending classical signal processing methodologies from vertices and edges to cycles and higher-dimensional structures. Through Hodge Laplacian, a discrete heat kernel on a finite simplicial complex $\mathcal{K}$ is constructed to smooth signals on $k$-simplices via the boundary operator $\partial_k$. Computationally efficient sparse algorithms for constructing boundary operators are developed to implement linear diffusion processes on $k$-simplices. The methodology generalizes heat kernel smoothing to $k$-simplices, utilizing boundary structure to localize topological features while maintaining homological invariance. Simulation studies demonstrate qualitative signal enhancement across vertex and edge domains following diffusion processes. Application to parcellated human brain functional connectivity networks reveals that simplex-space smoothing attenuates spurious connections while amplifying coherent anatomical architectures, establishing practical significance for computational neuroscience applications. Show Chinese abstract

Abstract: 网络在生物系统中构成了基本的组织结构，尽管传统的图论分析仅捕捉成对相互作用，从而忽略了表征网络复杂性的复杂高阶关系。 本工作提出了一种统一的框架，用于单纯复形上的热核平滑，将经典的信号处理方法从顶点和边扩展到循环和高维结构。 通过霍奇拉普拉斯算子，在有限单纯复形$\mathcal{K}$上构建了一个离散的热核，通过边界算子$\partial_k$在$k$-单形上平滑信号。 开发了计算高效的稀疏算法来构建边界算子，以在$k$-单形上实现线性扩散过程。 该方法将热核平滑推广到$k$-单形，利用边界结构来定位拓扑特征，同时保持同调不变性。 模拟研究显示，在扩散过程之后，顶点和边域中的信号质量得到定性增强。 应用于分块的人类大脑功能连接网络表明，单形空间平滑减少了虚假连接，同时增强了连贯的解剖结构，为计算神经科学应用建立了实际意义。