Final answer:
Two routers might influence each other's routing tables through protocols like BGP or OSPF, where changes to routes advertised by one router lead to adjustments in the other router's routing table to reflect the most efficient paths.
Step-by-step explanation:
Routers influence each other's routing tables through various protocols designed for sharing routing information. An example of how two routers might influence each other's routing tables is by using a protocol like Border Gateway Protocol (BGP) or Open Shortest Path First (OSPF). When Router A advertises a new route to Router B, Router B will process this new information and update its routing table to include the new route, considering the best path based on the metric used by the protocol (like the number of hops in BGP or the cumulative cost in OSPF).
Alternatively, if Router A were to lose connectivity to a network it previously advertised, it would update its own table and communicate this change to Router B, prompting Router B to also remove this route from its table or find an alternative path. This dynamic interaction ensures that network traffic can be rerouted in case of failures or topology changes, optimizing the path data takes through the network.