Poverty status identification is the first obstacle to eradicating poverty. Village-level poverty identification is very challenging due to the arduous field investigation and insufficient information. The development of the Web infrastructure and its modeling tools provides fresh approaches to identifying poor villages. Upon those techniques, we build a village graph for village poverty status identification. By modeling the village connections as a graph through the geographic distance, we show the correlation between village poverty status and its graph topological position and identify two key factors (Centrality, Homophily Decaying effect) for identifying villages. We further propose the first graph-based method to identify poor villages. It includes a global Centrality2Vec module to embed village centrality into the dense vector and a local graph distance convolution module that captures the decaying effect. In this paper, we make the first attempt to interpret and identify village-level poverty from a graph perspective.