There are forwarding tables and routing tables. Routing tables map destination to all possible paths. Then forwarding tables can be constructed by picking the best path to each destination.
$ route
Kernel IP routing table
Destination Gateway Genmask Flags Metric Ref Use Iface
172.16.197.0 * 255.255.255.0 U 0 0 0 vmnet8
172.16.76.0 * 255.255.255.0 U 0 0 0 vmnet1
128.31.32.0 * 255.255.252.0 U 2 0 0 wlan0
link-local * 255.255.0.0 U 1000 0 0 wlan0
default guest-wireless. 0.0.0.0 UG 0 0 0 wlan0
I also learned about AS's -- Autonomous Systems for BGP routing. Every ISP has an AS. Multiple IP address prefixes can be mapped to an AS. For example, MIT's AS is 3, and 18.0.0.0/8 and 128.30.0.0/15 map to it (the 0's mean it could be anything).
Then we learned about Tier 1 and Tier 2 networks. There are only 10ish tier 1 networks and they make a complete graph. The tier 2 networks have to pay tier 1 for transit links (tier 2 using a tier 1 as a router). There are free peering links that tier 2 networks can set up among themselves for free. There are also customer links connected to tier 2 for people buying that tier 2's service.
Every router may have many IP addresses, because it has different links/edges, and each IP address represent that router's end of the link/edge. So an IP address does not represent a machine, but represent endpoints of net links.
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