Conversational Emotion Recognition (CER) is a crucial task in Natural Language Processing (NLP) with wide applications. Prior works in CER generally focus on modeling emotion influences solely with utterance-level features, with little attention paid on phrase-level semantic connection between utterances. Phrases carry sentiments when they are referred to emotional events under certain topics, providing a global semantic connection between utterances throughout the entire conversation. In this work, we propose a two-stage Summarization and Aggregation Graph Inference Network (SumAggGIN), which seamlessly integrates inference for topic-related emotional phrases and local dependency reasoning over neighbouring utterances in a global-to-local fashion. Topic-related emotional phrases, which constitutes the global topic-related emotional connections, are recognized by our proposed heterogeneous Summarization Graph. Local dependencies, which captures short-term emotional effects between neighbouring utterances, are further injected via an Aggregation Graph to distinguish the subtle differences between utterances containing emotional phrases. The two steps of graph inference are tightly-coupled for a comprehensively understanding of emotional fluctuation. Experimental results on three CER benchmark datasets verify the effectiveness of our proposed model, which outperforms the state-of-the-art approaches.