Influence maximization problem has received significant attention in recent years due to its application in various domains, such as product recommendation, public opinion dissemination, and disease propagation. This paper proposes a theoretical analysis framework for collective influence in hypergraphs, focusing on identifying a set of seeds that maximize influence in threshold models. First, we extend the message passing method from pairwise networks to hypergraphs to accurately describe the activation process in threshold models. Then, we introduce the concept of hypergraph collective influence (HCI) to measure the influence of nodes. Subsequently, we design an algorithm, HCI-TM, to select the influence maximization set, taking into account both node and hyperedge activation. Numerical simulations demonstrate that HCI-TM outperforms several competing algorithms in synthetic and real-world hypergraphs. Furthermore, we find that HCI can be used as a tool to predict the occurrence of cascading phenomena. Notably, we find that the HCI-TM algorithm works better for larger average hyperdegrees in Erdös–Rényi hypergraphs and smaller power-law exponents in scale-free hypergraphs.

The augmented cube $A{Q}_n$ is a variation of the hypercube $Q_n$ . This paper considers the fault-tolerant Panconnectivity of $A{Q}_n$ . Assume that $F\subseteq V(A{Q}_n)\cup E(A{Q}_n)$ and $n\ge 4$ . We prove that for any two fault-free vertices $u$ and $v$ with distance $d$ in $A{Q}_n$ , there exists a fault-free path $P_{uv}$ of each length from $max\{d+2,4\}$ to $2^n-f_v-1$ in $A{Q}_n-F$ if $\left|F\right|\le 2n-4$ , where $f_v$ is the number of faulty vertices in $A{Q}_n$ . Moreover, the bound is sharp.

The augmented cube $ AQ_n $ is an outstanding variation of the hypercube $ Q_n $. It possesses many of the favorable properties of $ Q_n $ as well as some embedded properties not found in $ Q_n $. This paper focuses on the fault-tolerant Hamiltonian connectivity of $ AQ_n $. Under the assumption that $ F\subset V(AQ_n)\cup E(AQ_n) $ with $ |F|\leq 2n-3 $, we proved that for any two different correct vertices $ u $ and $ v $ in $ AQ_n $, there exists a fault-free Hamiltonian path that joins vertices $ u $ and $ v $ with the exception of $ (u, v) $, which is a weak vertex-pair in $ AQ_n-F $($ n\geq4 $). It is worth noting that in this paper we also proved that if there is a weak vertex-pair in $ AQ_n-F $, there is at most one pair. This paper improved the current result that $ AQ_n $ is $ 2n-4 $ fault-tolerant Hamiltonian connected. Our result is optimal and sharp under the condition of no restriction to each vertex.

Benefiting from the improvement of positional encoding and the introduction of lexical knowledge, Transformer has achieved superior performance than the prevailing BiLSTM-based models in named entity recognition (NER) task. However, existing Transformer-based models for Chinese NER pay less attention to the information captured by the bottom layers of Transformer and the significance of representation subspace where each head of Transformer is projected. In this article, we propose M ulti- T ask L abel- W ise T ransformer (MTLWT). From a global perspective, we assign entity boundary prediction (EBP) and entity type prediction (ETP) tasks to the first two layers. In this way, we stimulate lower layers to participate more in constructing character representation. Besides, in each multi-head self-attention (MHSA) layer, we provide a specific focus for each individual head, making the head project into a significant subspace. Experiments on four datasets from different domains show that our proposed model achieves comparable performance with other state-of-the-art models. In particular, MTLWT outperforms the other frameworks without external knowledge on all the datasets.