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Recent years have witnessed the great success of deep learning on various point cloud analysis tasks, e.g., classification and semantic segmentation. Since point cloud data is sparse and irregularly distributed, one key issue for point cloud data processing is extracting useful information from local regions. To achieve this, previous works mainly extract the points' features from local regions by learning the relation between each pair of adjacent points. However, these works ignore the relation between edges in local regions, which encodes the local shape information. Associating the neighbouring edges could potentially make the point-to-point relation more aware of the local structure and more robust. To explore the role of the relation between edges, this paper proposes a novel Adaptive Edge-to-Edge Interaction Learning module, which aims to enhance the point-to-point relation through modelling the edge-to-edge interaction in the local region adaptively. We further extend the module to a symmetric version to capture the local structure more thoroughly. Taking advantage of the proposed modules, we develop two networks for segmentation and shape classification tasks, respectively. Various experiments on several public point cloud datasets demonstrate the effectiveness of our method for point cloud analysis.