1. An Optimization-Driven Approach for Modeling AS-level Internet Connectivity Presented by: Hyunseok Chang hschang@eecs.umich.edu Joint work with Sugih Jamin (UM) and Walter Willinger (AT&T)

2. IPAM 2003 AS-level Internet graph Autonomous System (AS) Peering relationship Provider-customer type Peer-to-peer type Autonomous System (AS) Point of Presence (PoP)

3. IPAM 2003 Inferred Internet AS graph Highly variable AS vertex degree distribution.

4. IPAM 2003 Related research Generating Internet-like random graphs Focusing on the quality of a generated graph, not on the generation process itself. e.g., Inet generator. Modeling the Internet AS graph A graph generation process reflects actual Internet growth history. e.g., Barabasi-Albert model, Fabrikant - HOT model. Our study focuses on the modeling aspect.

5. IPAM 2003 Our research focus The related works have been very generic, based only on topological properties (e.g., node degree). Our starting point: Fabrikant-HOT ( H euristically O ptimized T rade-off) model for Internet growth.  Attempt to explain how inter-AS peering relationships are established in an optimization-driven fashion.

6. IPAM 2003 AS degree distributions At first, we focus on PC subgraph single-homed.

7. IPAM 2003 Fabrikant-HOT model Each new node solves the local optimization problem to find a target node to connect to. Each new node i connects to an existing node j that minimizes the weighted sum of two objectives: min (  d ij + h j ) d ij (last mile cost) = Euclidean distance from i to j h j (transmission delay cost) = average hop distance from j to all other nodes

8. IPAM 2003 Multi-PoP ISPs Fabrikant-HOT model assumes each node has a single PoP, whereas ISPs maintain multiple PoPs. In reality, d ij and h j may not be independent. AS#Name# of PoPs 2914Verio121 7018AT&T108 1221Telstra61 3356Level352 1239SprintLink43

9. IPAM 2003 Modified Fabrikant-HOT model Each node maintains multiple PoPs. The number of PoPs of an existing node increases over time.

10. IPAM 2003 Original Fabrikant-HOT model For each new node i, Find node j that minimizes Connect node i to node j. For each existing node u, increment | loc-set ( u )| with prob. K  rank ( u ) - .  | loc-set ( i )|  1 Modified

11. IPAM 2003 Modified Fabrikant-HOT model creates a hot spot! (a) Fabrikant-HOT model (b) Modified Fabrikant-HOT model

12. IPAM 2003 Our proposed models Univariate HOT model. Criteria: (i) AS geography. Bivariate HOT model. Criteria: (i) AS geography, (ii) AS business model. Various extensions.

13. IPAM 2003 Our proposed models Univariate HOT model Criteria: (i) AS geography. Bivariate HOT model Criteria: (i) AS geography, (ii) AS business model. Various extensions.

14. IPAM 2003 Univariate HOT model A single-objective optimization: minimize last-mile connection cost. A newly arriving node i connects to an existing node that has the closest PoP to i. An existing node u gradually increases the number of PoPs as later arriving nodes are attached to u.

15. IPAM 2003 Modified Fabrikant-HOT model For each new node i, | loc-set ( i )|  1 Find node j that minimizes Connect node i to node j. For each existing node u, increment | loc-set ( u )| with prob. K  rank ( u ) - . Univariate HOT model

16. IPAM 2003 Node size & degree distribution  =0.1 Exponential-type distribution for the number of locations per node.

17. IPAM 2003 Node size & degree distribution  =1.0 Highly-variable distribution for the number of locations per node.

18. IPAM 2003 AS size vs. AS degree Univariate model predicts that AS degree variability would be comparable to AS size (i.e., # of PoPs) variability. However, in the current Internet: Maximum AS degree ~ 10 3 Maximum # of PoPs per AS ~ 10 2 Q: Are there other criteria?

19. IPAM 2003 Proposed HOT models Univariate HOT model. Criteria: (i) AS geography. Bivariate HOT model. Criteria: (i) AS geography, (ii) AS business model. Various extensions.

20. IPAM 2003 Bivariate HOT model What if new AS i has multiple candidate providers in close geographic proximity? e.g., global criteria set  = {reliability, cost, customer service} and customer i ’s local criteria set  ( i ) = {reliability, cost}, ISP X : {99%, $100/Mbps, fair} ISP Y : {98%, $150/Mbps, good} ISP Z : {97%, $50/Mbps, bad} With respect to  ( i ), ISP X and Z are Pareto optimal. X >  ( i ) Y Y  ( i ) Z X  ( i ) Z

21. IPAM 2003 Bivariate HOT model Given a new node i, Initialize nbr-set ( i ) as containing all the nodes which have a PoP in close proximity to i. Remove any node in nbr-set ( i ), which is not Pareto-optimal in terms of  ( i ) ; an existing node u has quality vector Q( u )=(x 1,…,x N ). Connect node i to one randomly selected node from nbr-set ( i ).

22. IPAM 2003 Node size & degree distribution Bivariate model matches the Internet well!

23. IPAM 2003 Proposed HOT models Univariate HOT model. Criteria: (i) AS geography. Bivariate HOT model. Criteria: (i) AS geography, (ii) AS business model. Various extensions.

24. IPAM 2003 Extension #1: multiple providers per AS Observation: # of providers for an AS increases over time (similar to # of PoPs). In our original model: Every time a new node i is added to a graph, each existing node u gets a chance to: i) increment | loc-set ( u )| with prob. K  rank ( u ) - , ii) increment | prov-set ( u )|with prob. R  rank ( u ) -  ( R << K ). In our extended model:

25. IPAM 2003 Extension #2: peer-to-peer neighbors Observation: decision on providers is unilateral, but decision on peers is bilateral. For existing nodes u and v to become peering partners, we expect: i) u  nbr-set ( v ) (or, v  nbr-set ( u )), ii) u   ( u ) v and v   ( v ) u  ( u ),  ( v ) : peering criteria set for u and v.

26. IPAM 2003 Extension #3: AS evolution Node death & change-of-provider events. Role transition (e.g., provider peer). Evolving qualities.

27. IPAM 2003 Summary Our modeling approach: Explores the possibility of studying the Internet AS evolution in an optimization-based framework. Introduces domain-specific concepts in the modeling framework. Challenges: Model validation Application(s) Any kind of input is welcome!!