We study the problem of determining the configuration of $n$ points, referred
to as mobile nodes, by utilizing pairwise distances to $m$ fixed points known
as anchor nodes. In the standard setting, we have information about the
distances between anchors (anchor-anchor) and between anchors and mobile nodes
(anchor-mobile), but the distances between mobile nodes (mobile-mobile) are not
known. For this setup, the Nystr\"om method is a viable technique for
estimating the positions of the mobile nodes. This study focuses on the setting
where the anchor-mobile block of the distance matrix contains only partial
distance information. First, we establish a relationship between the columns of
the anchor-mobile block in the distance matrix and the columns of the
corresponding block in the Gram matrix via a graph Laplacian. Exploiting this
connection, we introduce a novel sampling model that frames the position
estimation problem as low-rank recovery of an inner product matrix, given a
subset of its expansion coefficients in a special non-orthogonal basis. This
basis and its dual basis--the central elements of our model--are explicitly
derived. Our analysis is grounded in a specific centering of the points that is
unique to the Nystr\"om method. With this in mind, we extend previous work in
Euclidean distance geometry by providing a general dual basis approach for
points centered anywhere.

基于配对距离，利用锚点的距离信息确定移动节点的配置，通过基于图拉普拉斯函数的关联，引入一个新的采样模型来解决位置估计问题。

一种包含缺失距离的 Nyström 方法

A Nyström method with missing distances

Deep neural networks, while generalize well, are known to be sensitive to
small adversarial perturbations. This phenomenon poses severe security threat
and calls for in-depth investigation of the robustness of deep learning models.
With the emergence of neural networks for graph structured data, similar
investigations are urged to understand their robustness. It has been found that
adversarially perturbing the graph structure and/or node features may result in
a significant degradation of the model performance. In this work, we show from
a different angle that such fragility similarly occurs if the graph contains a
few bad-actor nodes, which compromise a trained graph neural network through
flipping the connections to any targeted victim. Worse, the bad actors found
for one graph model severely compromise other models as well. We call the bad
actors ``anchor nodes'' and propose an algorithm, named GUA, to identify them.
Thorough empirical investigations suggest an interesting finding that the
anchor nodes often belong to the same class; and they also corroborate the
intuitive trade-off between the number of anchor nodes and the attack success
rate. For the dataset Cora which contains 2708 nodes, as few as six anchor
nodes will result in an attack success rate higher than 80\% for GCN and other
three models.

该研究探讨了在图神经网络中存在诸多锚节点（即恶意节点），即使存在很少的锚节点，也会对模型产生严重的破坏性影响，并提出了一种名为 GUA 的算法来识别锚节点。

图形通用对抗攻击：个别恶意行为破坏图学习模型

Graph Universal Adversarial Attacks: A Few Bad Actors Ruin Graph  Learning Models

Learning node embeddings that capture a node's position within the broader
graph structure is crucial for many prediction tasks on graphs. However,
existing Graph Neural Network (GNN) architectures have limited power in
capturing the position/location of a given node with respect to all other nodes
of the graph. Here we propose Position-aware Graph Neural Networks (P-GNNs), a
new class of GNNs for computing position-aware node embeddings. P-GNN first
samples sets of anchor nodes, computes the distance of a given target node to
each anchor-set,and then learns a non-linear distance-weighted aggregation
scheme over the anchor-sets. This way P-GNNs can capture positions/locations of
nodes with respect to the anchor nodes. P-GNNs have several advantages: they
are inductive, scalable,and can incorporate node feature information. We apply
P-GNNs to multiple prediction tasks including link prediction and community
detection. We show that P-GNNs consistently outperform state of the art GNNs,
with up to 66% improvement in terms of the ROC AUC score.

本文提出 P-GNN 方法，通过采样锚节点集并计算给定目标节点相对于所有锚节点的距离，从而学习距离加权聚合方案以产生具有位置感知能力的节点嵌入。实验结果表明该方法能有效地提升节点在图结构中的位置特征提取能力，达到了较好的实验效果。