Scaling Properties of the Internet GraphAditya Akella

With Shuchi Chawla, Arvind Kannan and Srinivasan Seshan

PODC 2003

Internet Evolution

AS interconnects: varied capacities

AS-level graph

Internet Evolution Say, network

doubles in size

Internet Evolution

Moore’s-law like scaling sufficient?

If so, good scaling!

Double all capacities?

Internet Evolution Plain doubling

not enough?

Moore’s-law like scaling insufficient?

Internet Evolution

Congested hot-spots

If so, poor scaling!!

Plain doubling not enough?

Key Questions

How does the worst congestion grow? O(n)? O(n2)?

How much of this is due to… Power-law structure?

Other distributions Routing algorithm?

BGP-Policy routing Traffic demand matrix?

What can be done? Redesign the network? Change routing?

Outline

Analysis Overview

Results from simulation

Discussion of results, network design

Conclusion

Outline

Analysis OverviewOutline key observations

Results from simulation

Discussion of results, network design

Conclusion

Analysis

To understand scaling properties of power-law graphs Sanity check the (more realistic) simulation results

Simple evolutionary model Preferential Connectivity

Known to yield power-law graphs Unit traffic between all node-pairs

Routed along the shortest path

How does maximum congestion depend on n, the number of vertices? Congestion on an edge == number of shortest path routes using the

edge

Analysis mainly for intuition; simulation results have the final say.

Key Observations (I) e* -- edge between the top two degree nodes s1 and s2.

Observation 1: A significant fraction of single-source shortest path trees (n) trees) in the graph contain e*.

S1

S2

e*

S1

S2

e*

e* occurs in both trees

Key Observations (II)

Observation 2: In at least a constant fraction of the (n) shortest path trees, s1 and s2 retain at least a constant fraction of their degrees.

S1

S2

e*

4/4

4/5S1

S2

e*

5/5

3/4

S1 ,S2 retain most of their degrees

Key Observations (III)

Observation 3: The degrees of s1 and s2 are (n1/).

And

In each tree that e* belongs to, congestion on

e* min{degtree(s1), degtree(s2)}.

S1

S2

e*

So…

Congestion(e*) 3

Key Result

Theorem: The expected maximum edge congestion is (n1+1/) (shortest path routing, any-2-any).

(n1.8) or worse for the Internet. Bad Scaling!

Outline

Analysis Overview

Results from simulation

Discussion of results, network design

Conclusion

Outline

Analysis Overview

Results from simulationMethodologyA few plots

Discussion of results, network design

Conclusion

Methodology: Outline

TopologyPower-law

Real AS-level topologies Inet-3.0 generated synthetic

Exponential Inet-3.0 generated; density same as similar-

sized Inet power-law graphs

Tree-like Grown from the preferential connectivity model

Methodology: Outline

Routing algorithmShortest-pathBGP routing

Policy-based, valley-free Synthetic graphs: heuristically classify edges

before imposing policy routing

Methodology: Outline

Traffic matrixUniform demands: Any-2-any

Between all pairs

Non-uniform: Clout model Between “leaves” or “stubs” Popularity: average degree of the neighbors Stub identification

Methodology: Outline

Topology X Routing X Traffic matrix

We seek Max edge congestion as a function of n

Shortest-Path Routing (Any-2-any)

Exponential >> Power law graphs > Power-law trees

Policy Routing (Any-2-Any)

Poor scaling just like shortest path, but…

Policy Routing vs. Shortest PathAny-2-Any

Synthetic Graphs

Real Graphs

Policy routing is never worse!

The Clout Model

Scaling is even worse

Same true for policy… But policy routing is better again!

Outline

Analysis overview

Results from simulation

Discussion of results, network design

Conclusion

Discussion

Scaling according to Moore’s law insufficientCongested hot-spots in the “core”

May have to alter routing or the macroscopic structureRouting: Diffuse demand in a centralized

mannerStructure: Add additional edges to the graph

Adding Parallel Links

Intuition: Congestion higher on edges with higher avg degree

Adding Parallel Links

#parallel links is dependant on degrees of nodes at the ends of the edge

Candidate functionsMinimum, Maximum, Sum and Product of

degrees Shortest path routing, any-2-any New edge congestion = edge

congestion/#parallel links

Parallel Links

Even min yields (n) scaling!Desirable extent of AS-AS peering

Conclusion

Congestion scales poorly in Internet-like graphs

Policy-routing does not worsen the congestion

Alleviation possible via simple, straight-forward mechanisms