Heterogeneous Hypergraph Embedding for Graph Classification

Abstract

Recently, graph neural networks have been widely used for network embedding because of their prominent performance in pairwise relationship learning. In the real world, a more natural and common situation is the coexistence of pairwise relationships and complex non-pairwise relationships, which is, however, rarely studied. In light of this, we propose a graph neural network-based representation learning framework for heterogeneous hypergraphs, an extension of conventional graphs, which can well characterize multiple non-pairwise relations. Our framework first projects the heterogeneous hypergraph into a series of snapshots and then we take the Wavelet basis to perform localized hypergraph convolution. Since the Wavelet basis is usually much sparser than the Fourier basis, we develop an efficient polynomial approximation to the basis to replace the time-consuming Laplacian decomposition. Extensive evaluations have been conducted and the experimental results show the superiority of our method. In addition to the standard tasks of network embedding evaluation such as node classification, we also apply our method to the task of spammers detection and the superior performance of our framework shows that relationships beyond pairwise are also advantageous in the spammer detection.To make our experiment repeatable, source codes and related datasets are available at https://xiangguosun.mystrikingly.com.

A Hypergraph Approach for Estimating Growth Mechanisms of Complex Networks

“…In other words, we simply ignore 𝐵 1 part. In calculating (4), if the observed node set 𝐵 𝑡 ,𝑙 contains only existing nodes, one uses (3), whereas if it contains some newcomer nodes, one uses (8). Note that in (8), all calculations occur solely on existing nodes.…”

“…In calculating (4), if the observed node set 𝐵 𝑡 ,𝑙 contains only existing nodes, one uses (3), whereas if it contains some newcomer nodes, one uses (8). Note that in (8), all calculations occur solely on existing nodes. Therefore, we can remove the selection bias and obtain a stable estimate of 𝐴 𝑘 (𝑘 > 0) at the same time.…”

“…At each iteration of 𝑡 in the procedure, the set of nodes that acquire a new hyperedge (𝑣 exist 𝑖 at line 5) is sampled from the Hyper PA model with the HyperPA procedure, which is described at the bottom of Algorithm 1 as a subroutine. We use the conditional probability given in (8) of Hyper PA to separate the effects of the node birth process from the hyperedge acquisition process. In our experiments, the set of newcomer nodes (𝑣 new 𝑖 at line 4) is simply taken from the history of a dataset; this is not explicitly described in Input though.…”

“…The term 𝑃 𝐵 1 ∪𝐵 2 |𝐵 1 =𝐵 * (𝑡) corresponds to the desired conditional probability in (8). Note that we denote 𝐵 * ⊂ 𝑉 new and not 𝐵 * = 𝑉 new because we allow the temporal hypergraphs to add multiple hyperedges at each time-step 𝑡.…”

“…The size of a hyperedge indicates the number of co-authors of the corresponding paper, and the hyperdegree of a node corresponds to the number of papers the corresponding researcher has written in the past. Hypergraphs have been applied successfully in a wide variety of domains, including recommender systems [6], bioinformatics [7], classification [8], clustering [9], and document retrieval [10].…”