What Is The Internet? Partial Connectivity in the Internet Core

"A collection of interconnected networks" defines what the Internet is, but not what it is not. Events threaten Internet fragmentation: politics suggest countries or ISPs may secede or be de-peered, disputes between ISPs result in persistent unreachability between their customers, and architectural changes risk breaking the "one" Internet. Understanding such threats benefits from a testable definition of what the Internet is and is not, enabling discussion and quantification of partial connectivity. We provide a conceptual definition giving an idealized asymptote of connectivity. It implies peninsulas of persistent, partial connectivity, and islands when one or more computers are partitioned from the main Internet. We provide algorithms to measure, operationally, the number, size, and duration of peninsulas and islands. We apply these algorithms in rigorous measurement from two complementary measurement systems, one observing 5M networks from a few locations, and the other a few destinations from 10k locations. Results show that peninsulas (partial connectivity) are about as common as Internet outages, quantifying this long-observed problem. Root causes show that most peninsula events (45%) are routing transients, but most peninsula-time (90%) is from a few long-lived events (7%). Our analysis helps interpret DNSmon, a system monitoring the DNS root, separating measurement error and persistent problems from underlying differences and operationally important transients. Finally, our definition confirms the international nature of the Internet: no single country can unilaterally claim to be "the Internet", but countries can choose to leave.