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The Internet can be considered as the most important infrastructure for modern society and businesses. A loss of Internet connectivity has strong negative financial impacts for businesses and economies. Therefore, assessing Internet connectivity, in particular beyond their own premises and area of direct control, is of growing importance in the face of potential failures, accidents, and malicious attacks. This paper presents CORIA, a software framework for an easy analysis of connectivity risks based on large network graphs. It provides researchers, risk analysts, network managers and security consultants with a tool to assess an organization's connectivity and paths options through the Internet backbone, including a user-friendly and insightful visual representation of results. CORIA is flexibly extensible in terms of novel data sets, graph metrics, and risk scores that enable further use cases. The performance of CORIA is evaluated by several experiments on the Internet graph and further randomly generated networks.