Graph neural network (GNN) models have been widely used for learning graph-structured data. Due to the permutation-invariant requirement of graph learning tasks, a basic element in graph neural networks is the invariant and equivariant linear layers. Previous work (Maron et al., 2019b) provided a maximal collection of invariant and equivariant linear layers and a simple deep neural network model, called k-IGN, for graph data defined on k-tuples of nodes. It is shown that the expressive power of k-IGN is at least as good as the k-Weisfeiler-Leman (WL) algorithm in graph isomorphism tests. However, the dimension of the invariant layer and equivariant layer is the k-th and 2k-th bell numbers, respectively. Such high complexity makes it computationally infeasible for k-IGNs with k >= 3. In this paper, we show that a much smaller dimension for the linear layers is sufficient to achieve the same expressive power. We provide two sets of orthogonal bases for the linear layers, each with only 3(2^k-1)-k basis elements. Based on these linear layers, we develop neural network models GNN-a and GNN-b and show that for the graph data defined on k-tuples of data, GNN-a and GNN-b achieve the expressive power of the k-WL algorithm and the (k+1)-WL algorithm in graph isomorphism tests, respectively. In molecular prediction tasks on benchmark datasets, we demonstrate that low-order neural network models consisting of the proposed linear layers achieve better performance than other neural network models. In particular, order-2 GNN-b and order-3 GNN-a both have 3-WL expressive power, but use a much smaller basis and hence much less computation time than known neural network models.

Blockchain technology has gained growing popularity in recent years. While the technology works well for most applications, the vulnerability of the blockchain-based system is not well understood. The core part of a blockchain-based system is the consensus algorithm. It can be regarded as a protocol used by all network users to make decisions on the growth of the chain. Malicious users can exploit the vulnerabilities of the consensus algorithm to launch block withholding attacks to earn more profit by increasing their winning possibilities, which are significant threats to the security of blockchain-based applications. In this paper, we describe some scenarios of block withholding attacks and suggest effective mitigation methods accordingly. We propose two hybrid consensus algorithms against the block withholding attacks and evaluate the effectiveness of these algorithms, both analytically and through simulation. Simulation results from the SimBlock simulator verified the effectiveness of the proposed algorithms.

In this paper, we consider the power line outage identification problem as a graph signal classification problem, where the signal at each vertex is given as a time series. We propose graph convolutional networks (GCNs) for the task of classifying signals supported on graphs. An important element of the GCN design is filter design. We consider filtering signals in either the vertex (spatial) domain, or the frequency (spectral) domain. Two basic architectures are proposed. In the spatial GCN architecture, the GCN uses a graph shift operator as the basic building block to incorporate the underlying graph structure into the convolution layer. The spatial filter directly utilizes the graph connectivity information. It defines the filter to be a polynomial in the graph shift operator to obtain the convolved features that aggregate neighborhood information of each node. In the spectral GCN architecture, a frequency filter is used instead. A graph Fourier transform operator first transforms the raw graph signal from the vertex domain to the frequency domain, and then a filter is defined using the graph's spectral parameters. The spectral GCN then uses the output from the graph Fourier transform to compute the convolved features. There are additional challenges to classify the time-evolving graph signal as the signal value at each vertex changes over time. The GCNs are designed to recognize different spatiotemporal patterns from high-dimensional data defined on a graph. The application of the proposed methods to power line outage identification shows that these GCN architectures can successfully classify abnormal signal patterns and identify the outage location.