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)

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AS-level Internet graph

Autonomous System (AS)Peering relationship

Provider-customer typePeer-to-peer type

Autonomous System (AS) Point of Presence (PoP)

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Inferred Internet AS graph

Highly variable AS vertex degree distribution.

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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.

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Our research focus The related works have been very generic, based

only on topological properties (e.g., node degree). Our starting point: Fabrikant-HOT (Heuristically Op

timized Trade-off) model for Internet growth.

Attempt to explain how inter-AS peering relationships are established in an optimization-driven fashion.

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AS degree distributions

At first, we focus on PC subgraph single-homed.

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Fabrikant-HOT model Each new node solves the local optimization p

roblem 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 (dij + hj) dij (last mile cost) = Euclidean distance from i to j hj (transmission delay cost) = average hop distanc

e from j to all other nodes

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Multi-PoP ISPs Fabrikant-HOT model assumes each node has

a single PoP, whereas ISPs maintain multiple PoPs.

In reality, dij and hj may not be independent.

AS# Name # of PoPs

2914 Verio 121

7018 AT&T 108

1221 Telstra 61

3356 Level3 52

1239 SprintLink 43

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Modified Fabrikant-HOT model Each node maintains multiple PoPs. The number of PoPs of an existing node increa

ses over time.

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Original Fabrikant-HOT model

ij jd h

For each new node i,

Find node j that minimizes Connect node i to node j.

( )l loc set j jild hmin

For each existing node u, increment |loc-set(u)| with prob. Krank(u)-.

|loc-set(i)| 1

Modified

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Modified Fabrikant-HOT model creates a hot spot!

(a) Fabrikant-HOT model (b) Modified Fabrikant-HOT model

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Our proposed models Univariate HOT model.

Criteria: (i) AS geography. Bivariate HOT model.

Criteria: (i) AS geography, (ii) AS business model. Various extensions.

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Our proposed models Univariate HOT model

Criteria: (i) AS geography. Bivariate HOT model

Criteria: (i) AS geography, (ii) AS business model. Various extensions.

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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 nu

mber of PoPs as later arriving nodes are attached to u.

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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. Krank(u)-.

( )l loc set j jild hmin

Univariate HOT model

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Node size & degree distribution =0.1

Exponential-type distribution for the number of locations per node.

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Node size & degree distribution =1.0

Highly-variable distribution for the number of locations per node.

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AS size vs. AS degree Univariate model predicts that AS degree varia

bility would be comparable to AS size (i.e., # of PoPs) variability.

However, in the current Internet: Maximum AS degree ~ 103

Maximum # of PoPs per AS ~ 102

Q: Are there other criteria?

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Proposed HOT models Univariate HOT model.

Criteria: (i) AS geography. Bivariate HOT model.

Criteria: (i) AS geography, (ii) AS business model. Various extensions.

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Bivariate HOT model What if new AS i has multiple candidate provid

ers in close geographic proximity? e.g., global criteria set = {reliability, cost, cu

stomer 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

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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 P

areto-optimal in terms of (i); an existing node u has quality vector Q(u)=(x1,…,xN).

Connect node i to one randomly selected node from nbr-set(i).

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Node size & degree distribution

Bivariate model matches the Internet well!

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Proposed HOT models Univariate HOT model.

Criteria: (i) AS geography. Bivariate HOT model.

Criteria: (i) AS geography, (ii) AS business model. Various extensions.

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Extension #1: multiple providers per AS Observation: # of providers for an AS increase

s 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. Krank(u)-, ii) increment |prov-set(u)|with prob. Rrank(u)- (R<<K).

In our extended model:

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Extension #2:peer-to-peer neighbors Observation: decision on providers is unilatera

l, but decision on peers is bilateral. For existing nodes u and v to become peering

partners, we expect: i) unbr-set(v) (or, vnbr-set(u)), ii) u (u) v and v (v) u (u), (v) : peering criteria set for u and v.

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Extension #3: AS evolution Node death & change-of-provider

events. Role transition (e.g., provider peer). Evolving qualities.

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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!!