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Emotional support conversation aims at reducing the emotional distress of the help-seeker, which is a new and challenging task. It requires the system to explore the cause of help-seeker's emotional distress and understand their psychological intention to provide supportive responses. However, existing methods mainly focus on the sequential contextual information, ignoring the hierarchical relationships with the global cause and local psychological intention behind conversations, thus leads to a weak ability of emotional support. In this paper, we propose a Global-to-Local Hierarchical Graph Network to capture the multi-source information (global cause, local intentions and dialog history) and model hierarchical relationships between them, which consists of a multi-source encoder, a hierarchical graph reasoner, and a global-guide decoder. Furthermore, a novel training objective is designed to monitor semantic information of the global cause. Experimental results on the emotional support conversation dataset, ESConv, confirm that the proposed GLHG has achieved the state-of-the-art performance on the automatic and human evaluations. The code will be released in here \footnote{\small{~https://github.com/pengwei-iie/GLHG}}.