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This paper defines and evaluates Path Exploration Damping (PED) - a router-level mechanism for reducing the volume of propagation of likely transient update messages within a BGP network and decreasing average time to restore reachability compared to current BGP Update damping practices. PED selectively delays and suppresses the propagation of BGP updates that either lengthen an existing AS Path or vary an existing AS Path without shortening its length. We show how PED impacts on convergence time compared to currently deployed mechanisms like Route Flap Damping (RFD), Minimum Route Advertisement Interval (MRAI) and Withdrawal Rate Limiting (WRATE). We replay Internet BGP update traffic captured at two Autonomous Systems to observe that a PED-enabled BGP speaker can reduce the total number of BGP announcements by up to 32% and reduce Path Exploration by 77% compared to conventional use of MRAI. We also describe how PED can be incrementally deployed in the Internet, as it interacts well with prevailing MRAI deployment, and enables restoration of reachability more quickly than MRAI.