Distributional Signals for Node Classification in Graph Neural Networks
In graph neural networks (GNNs), both node features and labels are examples
of graph signals, a key notion in graph signal processing (GSP). While it is
common in GSP to impose signal smoothness constraints in learning and
estimation tasks, it is unclear how this can be done for discrete node labels.
We bridge this gap by introducing the concept of distributional graph signals.
In our framework, we work with the distributions of node labels instead of
their values and propose notions of smoothness and non-uniformity of such
distributional graph signals. We then propose a general regularization method
for GNNs that allows us to encode distributional smoothness and non-uniformity
of the model output in semi-supervised node classification tasks. Numerical
experiments demonstrate that our method can significantly improve the
performance of most base GNN models in different problem settings.
Feng Ji, See Hian Lee, Kai Zhao, Wee Peng Tay, Jielong Yang
# Distributional Signals for Node Classification in Graph Neural Networks

###### Abstract

In graph neural networks (GNNs), both node features and labels are examples of graph signals, a key notion in graph signal processing (GSP). While it is common in GSP to impose signal smoothness constraints in learning and estimation tasks, it is unclear how this can be done for discrete node labels. We bridge this gap by introducing the concept of distributional graph signals. In our framework, we work with the distributions of node labels instead of their values and propose notions of smoothness and non-uniformity of such distributional graph signals. We then propose a general regularization method for GNNs that allows us to encode distributional smoothness and non-uniformity of the model output in semi-supervised node classification tasks. Numerical experiments demonstrate that our method can significantly improve the performance of most base GNN models in different problem settings.

 Graph neural networks, graph signal processing, distributional signals, node classification, smoothness

## I Introduction

We consider the semi-supervised node classification problem [1] that determines class labels of nodes in graphs given sample observations and possibly node features. Numerous graph neural network (GNN) models have been proposed to tackle this problem. One of the first models is the graph convolutional network (GCN) [2]. Interpreted geometrically, a GCN aggregates information such as node features from the neighborhood of each node of the graph. Algebraically, this process is equivalent to applying a graph convolution filter to node feature vectors. Subsequently, many GNN models with different considerations are introduced. Popular models include the graph attention network (GAT) [3] that learns weights between pairs of nodes during aggregation, and the hyperbolic graph convolutional neural network (HGCN) [4] that considers embedding of nodes of a graph in a hyperbolic space instead of a Euclidean space. For inductive learning, GraphSAGE [5] is proposed to generate low-dimensional vector representations for nodes that are useful for graphs with rich node attribute information. While new models draw inspiration from GCN, GCN itself is built upon the foundation of graph signal processing (GSP).

GSP is a signal processing framework that handles graph-structured data [6, 7, 8]. A graph signal is a vector with each component corresponding to a node of a graph. Examples include node features and node labels. Moreover, convolutions used in models such as GCN are special cases of convolution filters in GSP [6]. All these show the close connections between GSP theory and GNNs.

In GSP, signal smoothness (over the graph) is widely used to regularize inference tasks. Intuitively, a signal is smooth if its values are similar at each pair of nodes connected by an edge. One popular way to formally define signal smoothness is to use the Laplacian quadratic form. There are numerous GSP tools that leverage a smooth prior of the graph signals. For example, Laplacian (Tikhonov) regularization is proposed for noise removal in [6] and signal interpolation [9]. In [10], it is used in graph signal in-painting and anomaly detection. In [11], the same technique is used for graph topology inference.

However, for GNNs, it is remarked in [12, Section 4.1.2] that "graph Laplacian regularization can hardly provide extra information that existing GNNs cannot capture". Therefore, a regularization scheme based on feature propagation is proposed. It is demonstrated to be effective by comparing with other methods such as [13] and [14] based on adversarial learning and [15] that co-trains GNN models with an additional agreement model, which gives the probability that two nodes have the same label. We partially agree with the above assertion regarding graph Laplacian regularization, while remaining reservative about its full correctness. In this paper, we propose a method that is inspired by Laplacian regularization. As our main contribution, we introduce the notion of distributional graph signals, instead of considering graph signals. Analogous to the graph signal smoothness defined using graph Laplacian, we define the smoothness of distributional graph signals. Together with another property known as non-uniformity, we devise a regularization scheme for GNNs in node classification tasks. This approach is easy to implement and can be used as a plug-in regularization term together with any given base GNN model. Its effectiveness is demonstrated with numerical results.

The rest of the paper is organized as follows. We formally introduce the notion of distributional graph signals in Section II. In Section II-B, we motivate by analyzing step graph signals naturally occurring in node classification problems. We study two key properties, namely smoothness and non-uniformity, of distributional graph signals in Section III-A and Section III-B. Based on the discussions, we describe the proposed regularization scheme in Section III-C. In Section III-D, we analyze the model and discuss its relations with related works. We present experimental results in Section IV and conclude in Section V.

## II Distributional graph signals

In this section, we motivate and introduce distributional graph signals based on GSP theory.