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Router-Level Topology Modeling
Ellen Witte Zegura was recently asked to comment on router-level topology modeling. Comments to are welcome.

On occasion, people ask me whether they should use GT-ITM[1] to generate topologies for Internet simulations. Usually they mention the Faloutsos work observing power laws in measurements of Internet topology[2], and perhaps they point to some of the more recent topology generators such as Inet[3] and BRITE[4]. Here is an answer.

Let me start by explaining that the GT-ITM software package implements a collection of topology generation methods, including standard random graphs, Waxman's variant on random graphs, and the transit-stub method. The transit-stub method uses the other methods to build up a topology whose high-level structure arguably reflects the high-level structure of the Internet, and it is probably the most widely used method in GT-ITM.

In principle, any of the graphs generated by GT-ITM could be interpreted as modeling either the router-level Internet topology (where vertices are routers and edges are one-hop connectivity) or the autonomous-system level Internet topology (where vertices are autonomous systems and edges represent peering agreements). In practice, however, transit-stub graphs should be interpreted as router-level models, since they explicitly group vertices into domains, and reflect that grouping in the connectivity between vertices.

The graphs generated by the transit-stub method (and indeed by all the methods currently implemented in GT-ITM) will NOT generally have structure described by power laws. I say "generally" because the transit-stub method in particular has many parameters; the structure of the graphs that are generated can vary widely (even for the same size and number of edges) depending on the parameters. It is therefore difficult to make statements that apply to all instantiations of the method.

Does that mean you shouldn't use the transit-stub method? If you need an AS-level representation, I would not recommend it. However, if you need a router-level representation, I think it remains a reasonable choice. Let me explain why. First, the power law observations have primarily focused on the autonomous-system level representation of the Internet, because that data is available via BGP route tables. Data on the router-level topology of the Internet can only be obtained indirectly (e.g., by inference from traceroute measurements), since network administrators don't like to reveal details of internal topology. I believe it remains an open question whether the router-level Internet topology has power-law behavior. It wouldn't surprise me if it does, but it also wouldn't surprise me if router-level topology has more complex structural behavior, due to differences in intradomain and interdomain connectivity.

Second, many simulations will involve topologies with a few 100s of nodes, due to limitations on simulation speed. My intuition says that these topologies are too small for discussions of power laws (i.e., linear on a log-log plot) to make much sense. And if you are simulating an even smaller topology (few 10s of nodes) a multi-domain topology probably doesn't make much sense.

Third, the other options for router-level models are fairly limited. The newer topology generation methods that I know about (BRITE, Inet) target AS-level representation, not router-level representation. They do a fairly good job generating large graphs (1000s of vertices) with power-law behavior similar to that observed in the snapshots of the Internet AS-level topology. But they aren't intended as router-level models.

I don't claim the transit-stub method is the last word in router-level topology modeling. Indeed, I'm sure it's not, and I know of some specific things that would improve it. For example, transit domains should have explicitly designated border router vertices, to which all external edges are connected, rather than using all vertices as endpoints of external edges. However, I do think it is currently a reasonable choice for moderate size router-level topologies.

Ellen Zegura
January 2001

  2. Faloutsos, Faloutsos and Faloutsos, On power-law relationships of the Internet topology, Proceedings of Sigcomm 1999.
  3. Jin, Chen and Jamin, Inet: Internet topology generator, Technical report CSE-TR-433-00, Dept of EECS, U. of Michigan.
  4. Medina, Matta and Byers, On the origin of power-laws in Internet topologies, ACM Computer Communication Review, April 2000.
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