Attention-Based Graph Convolutional Network for Zero-Shot Learning with Pre-Training

Zero-shot learning (ZSL) is a powerful and promising learning paradigm for classifying instances that have not been seen in training. Although graph convolutional networks (GCNs) have recently shown great potential for the ZSL tasks, these models cannot adjust the constant connection weights between the nodes in knowledge graph and the neighbor nodes contribute equally to classify the central node. In this study, we apply an attention mechanism to adjust the connection weights adaptively to learn more important information for classifying unseen target nodes. First, we propose an attention graph convolutional network for zero-shot learning (AGCNZ) by integrating the attention mechanism and GCN directly. Then, in order to prevent the dilution of knowledge from distant nodes, we apply the dense graph propagation (DGP) model for the ZSL tasks and propose an attention dense graph propagation model for zero-shot learning (ADGPZ). Finally, we propose a modified loss function with a relaxation factor to further improve the performance of the learned classifier. Experimental results under different pre-training settings verified the effectiveness of the proposed attention-based models for ZSL.

A Case Study on Stochastic Games on Large Graphs in Mean Field and Sparse Regimes

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semiexplicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph’s normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, that is, when the degrees diverge in a suitable sense. Although equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.

Switch-Based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on $n$ vertices. As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least $n/2+7$ can be transformed into each other by switch operations of size at most 10, implying that the switch Markov chain using switches of size at most 10 is irreducible. As a proof of concept, we also show that this Markov chain is rapidly mixing on dense monotone graphs.