Stance detection on social media aims to identify if an individual is in support of or against a specific target. Most existing stance detection approaches primarily rely on modeling the contextual semantic information in sentences and neglect to explore the pragmatics dependency information of words, thus degrading performance. Although several single-task learning methods have been proposed to capture richer semantic representation information, they still suffer from semantic sparsity problems caused by short texts on social media. This article proposes a novel multigraph sparse interaction network (MG-SIN) by using multitask learning (MTL) to identify the stances and classify the sentiment polarities of tweets simultaneously. Our basic idea is to explore the pragmatics dependency relationship between tasks at the word level by constructing two types of heterogeneous graphs, including task-specific and task-related graphs (tr-graphs), to boost the learning of task-specific representations. A graph-aware module is proposed to adaptively facilitate information sharing between tasks via a novel sparse interaction mechanism among heterogeneous graphs. Through experiments on two real-world datasets, compared with the state-of-the-art baselines, the extensive results exhibit that MG-SIN achieves competitive improvements of up to 2.1% and 2.42% for the stance detection task, and 5.26% and 3.93% for the sentiment analysis task, respectively.