High-order Graph Neural Networks (HO-GNNs) have been developed to infer
consistent latent spaces in the heterophilic regime, where the label
distribution is not correlated with the graph structure. However, most of the
existing HO-GNNs are hop-based, i.e., they rely on the powers of the transition
matrix. As a result, these architectures are not fully reactive to the
classification loss and the achieved structural filters have static supports.
In other words, neither the filters' supports nor their coefficients can be
learned with these networks. They are confined, instead, to learn combinations
of filters. To address the above concerns, we propose Diffusion-jump GNNs a
method relying on asymptotic diffusion distances that operates on jumps. A
diffusion-pump generates pairwise distances whose projections determine both
the support and coefficients of each structural filter. These filters are
called jumps because they explore a wide range of scales in order to find bonds
between scattered nodes with the same label. Actually, the full process is
controlled by the classification loss. Both the jumps and the diffusion
distances react to classification errors (i.e. they are learnable).
Homophiliation, i.e., the process of learning piecewise smooth latent spaces in
the heterophilic regime, is formulated as a Dirichlet problem: the known labels
determine the border nodes and the diffusion-pump ensures a minimal deviation
of the semi-supervised grouping from a canonical unsupervised grouping. This
triggers the update of both the diffusion distances and, consequently, the
jumps in order to minimize the classification error. The Dirichlet formulation
has several advantages. It leads to the definition of structural heterophily, a
novel measure beyond edge heterophily. It also allows us to investigate links
with (learnable) diffusion distances, absorbing random walks and stochastic
diffusion.

提出使用渐进扩散差距的方法进行跳跃的扩散 - hops exploration，从而产生更具有动态支持的结构过滤器，建议用 Dirichlet 问题作为一种方法来在异类亲缘而不相关的标签分布的情况下学习逐段平滑的积淀空间，并使用 (可学习的) 渐进扩散距离、吸收随机漫步和 (具有可学习性) 随机扩散进行链接探索。

通过可学习度量过滤器的扩散跳跃 GNNs 实现同质化

Diffusion-Jump GNNs: Homophiliation via Learnable Metric Filters

Node classification on graphs can be formulated as the Dirichlet problem on
graphs where the signal is given at the labeled nodes, and the harmonic
extension is done on the unlabeled nodes. This paper considers a time-dependent
version of the Dirichlet problem on graphs and shows how to improve its
solution by learning the proper initialization vector on the unlabeled nodes.
Further, we show that the improved solution is at par with state-of-the-art
methods used for node classification. Finally, we conclude this paper by
discussing the importance of parameter t, pros, and future directions.

本文研究了基于图的节点分类问题，提出了可以通过机器学习方法改进解决该问题的时间相关狄利克雷问题，其在标记和未标记节点上的正确初始化向量选择是得到最优解的关键。同时，该方法在该领域处于领先水平。

基于时间相关的谐波扩展学习标签初始化

Learning Label Initialization for Time-Dependent Harmonic Extension

This paper links the field of potential theory -- i.e. the Dirichlet and
Neumann problems for the heat and Laplace equation -- to that of the Feynman
path integral, by postulating that the potential is equal to plus/minus the
Laplacian of the indicator of the domain D. The Laplacian of the indicator is a
generalized function: it is the d-dimensional analogue of the Dirac
delta'-function. This function has -- according to the author's best knowledge
-- not formally been defined before.
We show, first, that the path integral's perturbation series (or Born series)
matches the classical single and double boundary layer series of potential
theory, thereby connecting two hitherto unrelated fields. Second, we show that
the perturbation series is valid for all domains D that allow Green's theorem
(i.e. with a finite number of corners, edges and cusps), thereby expanding the
classical applicability of boundary layers. Third, we show that the minus
(plus) in the potential holds for the Dirichlet (Neumann) boundary condition;
showing for the first time a particularly close connection between these two
classical problems. Fourth, we demonstrate that the perturbation series of the
path integral converges in a monotone/alternating fashion, depending on the
convexity/concavity of the domain.
We also discuss the third boundary problem (which poses Robin boundary
conditions) and discuss an extension to moving domains.

本文将势理论和费曼路径积分领域联系起来，通过提出势等于域 D 的指示函数的拉普拉斯算子的正 / 负值，证明了路径积分的微扰级数与势理论的经典单层和双层边界级数匹配，并扩展了边界层的经典适用性。