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Scaling Properties of the Internet GraphAditya Akella, CMU
With Shuchi Chawla, Arvind Kannan and Srinivasan Seshan
PODC 2003
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Internet Evolution
Grows with time…
AS-level graph
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Internet Evolution Say, network
doubles in size
Key: Where to add
capacity?
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Internet Evolution
Moore’s-law like scaling sufficient?
If so, good scaling!
Uniformly scale all capacities?
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Internet Evolution Scale some links faster?
Moore’s-law like scaling insufficient?
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Internet Evolution
Congested hot-spots
If so, poor scaling!!
Scale some links faster?
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Key Questions How does the worst congestion grow?
O(n)? O(n2)? How much of this is due to…
Topology? Power-law structure Other distributions
Routing algorithm? BGP-Policy routing
Traffic demand matrix? Uniform vs. non-uniform
What can be done? Redesign the network? Change routing?
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Outline
Analysis Overview – key result
Results from simulation
Discussion of results, network design
Conclusion
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Analysis in One Minute Simple evolutionary model
Preferential Connectivity Known to yield power-law graphs #nodes v with dv ≥ d is proportional to d-
Unit traffic between all node-pairs Routed along the shortest path Prefer paths through higher-degree nodes
How does maximum congestion depend on n, the number of vertices? Congestion on an edge == number of shortest path routes using
the edge Consider congestion on the edge between two highest degree
nodes
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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!
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Outline
Analysis Overview
Results from simulation
Discussion of results, network design
Conclusion
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Methodology: Outline Topology
Power-law #nodes v with dv ≥ d is proportional to d-
Real AS-level topologies Inet-3.0 generated synthetic
Exponential #nodes v with dv ≥ d is proportional to e-d
Inet-3.0 generated Density same as power-law graphs of same size
Tree-like Grown from the preferential connectivity model
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Methodology: Outline Routing algorithm
Shortest-path Prefer paths through high degree nodes
BGP routing Policy-based
Peers only provide transit to traffic to/from customers Customers don’t provide transit for providers and peers
Real graphs: past work on classifying edges Synthetic graphs: heuristically classify edges before
imposing policy routing Accurate maximum congestion
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Methodology: Outline
Traffic matrixUniform demands: Any-2-any
Between all pairsNon-uniform: Clout model
Between “stubs” Traffic depends on “popularity”
Popularity of node u depends on degree (du) and avg degree of neighbors (Au)
Traffic (uv) is proportional to popularity(u)
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Methodology: Outline
Given Topology X Routing X Traffic matrix
We seek Max edge congestion as a function of n
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Shortest-Path Routing (Any-2-any)
Exponential >> Power law graphs > Power-law trees
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Policy Routing (Any-2-Any)
Poor scaling just like shortest path
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Policy Routing vs. Shortest PathAny-2-Any
Synthetic Graphs
Real Graphs
Policy routing is never worse!
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The Clout Model
Shortest-path routing Scaling is even worse
than uniform
Policy routing Same true for policy Policy routing better than shortest path!
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Outline
Analysis overview
Results from simulation
Discussion of results, network design
Conclusion
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Discussion
Scaling according to Moore’s law insufficientCongested hot-spots in the “core”Policy routing has minimal impact
May have to change the networkRouting: diffuse demand in a centralized mannerStructure: add additional edges to the graph
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Adding Parallel Links
Intuition: Congestion higher on edges with higher average degree
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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
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Parallel Links (Shortest path, Any2Any)
Even min yields (n) scaling!Desirable extent of AS-AS peering
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Related Work
“Power law graphs have good congestion properties” [Mihail03]Allow routing with O(nlog2n) congestion Incorrectly extend to shortest path routingAlso find policy routing to be worse
Over smaller real graphs
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Conclusion
Congestion scales poorly in Internet-like graphs
Policy-routing does not worsen the congestion
Alleviation possible via simple, straight-forward mechanisms