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We propose a novel learning framework based on neural mean-field dynamics for inference and estimation problems of diffusion on networks. Our new framework is derived from the Mori-Zwanzig formalism to obtain an exact evolution of the node infection probabilities, which renders a delay differential equation with memory integral approximated by learnable time convolution operators, resulting in a highly structured and interpretable RNN. Directly using cascade data, our framework can jointly learn the structure of the diffusion network and the evolution of infection probabilities, which are cornerstone to important downstream applications such as influence maximization. Connections between parameter learning and optimal control are also established. Empirical study shows that our approach is versatile and robust to variations of the underlying diffusion network models, and significantly outperform existing approaches in accuracy and efficiency on both synthetic and real-world data. 

Abstract Let be a multigraph with maximum degree and chromatic index . If is bipartite then . Otherwise, by a theorem of Goldberg, , where denotes the odd girth of . Stiebitz, Scheide, Toft, and Favrholdt in their book conjectured that if then contains as a subgraph a ring graph with the same chromatic index. Vizing's characterization of graphs with chromatic index attaining the Shannon's bound showed the above conjecture holds for . Stiebitz et al verified the conjecture for graphs with and . McDonald proved the conjecture when is divisible by . In this paper, we show that the chromatic index condition alone is not sufficient to give the conclusion in the conjecture. On the positive side, we show that the conjecture holds for every with , and the maximum degree condition is best possible. This positive result leans on the positive resolution of the Goldberg‐Seymour conjecture.