Computer Science

Unveiling Hidden Topologies:Applications, Algorithms and Measurements

John Byers

Department of Computer Science, Boston University

Given an underlying graph G whose vertices are known wholly or partially in

advance edges are unknown in advance

Identify properties of the edge set via a sequence of probes (samples). Application-specific probing Statistical vs. topological properties Exact vs. approximate guarantees Adaptive vs. non-adaptive sampling

Hidden Topologies: What

Traditional science: study of “found” objects. Protein-protein interaction networks Metabolic networks Genome mapping

Emerging domain: study engineered artifacts, with scientific posture accorded to “found” objects. Internet topology: “map” is not known

Size, proprietary information, distributed Router-level topology vs. AS-level topology

Dynamic topologies: up-to-date maps are infeasible to maintain.

Examples: P2P, overlays, large testbeds

Hidden Topologies: Where

Compelling applications Existing approaches are “point

solutions” Cross-cutting theory is not yet well

developed Pitfalls/weaknesses not widely

disseminated Impact of models:

better models may make for better algorithms principles inform probing process

Hidden Topologies: Why

Traceroute exploration of many graphs yields heavy-tailed subnets [LBCX ’03, ACKM ’05]

Random subnets of scale-free graphs are not scale-free [Stumpf, Wiuf, May: PNAS 3/22/05] Parsimonious subgraph generation model:

Vertices selected uniformly at random. Edge (i, j) included iff both i and j selected.

Two examples of strong sampling bias.

Hidden Topologies: Foundations

Outline

Motivating hidden graphs

Case study 1: Hidden graphs in genome sequencing applications.

Case study 2: Internet mapping studies.

Case study 3: Locating constrained, annotated Internet subgraphs.

Discussion

Protein-protein interaction (PPI) networks; genomics. Nodes correspond to (known) chemicals. Edges correspond to observable chemical reactions. Probe: Combine an arbitrary subset S of chemicals. Binary probe QG (S):

0: non-existence of any edge within S 1: existence of at least one edge in S

Example: Genome sequencing of “contigs”: Model: at most one incident edge per node Goal: identify hidden matching efficiently.

Shotgun sequencing: parallelize probe process

Example: Interaction networks

[Grebinski, Kucherov ’97, ‘98]: Asymptotically optimal query bounds for hidden Hamiltonian cycles.

[Beigel, Alon, Apaydin, Fortnow, Kasif ’01]: Asymptotically optimal query bounds for hidden matchings.

[Alon, Beigel, Kasif, Rudich, Sudakov ’02]: Nearly-tight upper and lower bounds on hidden matchings for both deterministic and randomized algorithms.

[Angluin, Chen ’04]: Learning hidden graphs in O(m log n) queries.

[Alon, Asodi ’04]: Learning hidden subgraphs. [Angluin, Chen ’04]: Learning hidden hypergraphs.

(Numerous experimental biology papers).

Querying hidden graphs

Matching Query

(Slides courtesy of Simon Kasif)

Matching Query

Matching Query

Upper and Lower Bounds

Deterministic ProbabilisticNonadaptive

0.5(n choose 2)0.32(n choose 2)

1.44n logn0.5n logn

2 rounds (5/4)n3/2

0.5n logn0.72n logn0.5n logn

k rounds N1+1/(2k-2) polylogn0.5n logn

0.72n logn0.5n logn

1- or 2-Round Probabilistic Algorithm

1. Form O(n logn) tubes of size O(n) independently at random. Test each tube to see if it contains an edge.

1a.For each pair {u,v}, see if {u,v} is contained in a tube that tested negative in step 1. If so, {u,v} is a nonedge.

1b.For each pair {p,q}, see if {p,q} is contained in a tube that tested positive in step 1 but in which every other pair was determined to be a nonedge in step 1a. If so, {p,q} is an edge.

2. Test all pairs whose status is still unknown.

Probabilistic Algorithms (optimizing constants)

Procedure RPP (random projective plane) Assume n = p 2 + p + 1 Randomly permute all n vertices and

identify them with the points of the projective plane P of order p.

Perform one test for each line in P.Fix {x,y}. {x,y} belongs to a unique line in P.

The probability that that line contains no matched edge (except perhaps {x,y} itself) is e-1/2.

2-rounds, 0.74n logn tests, 0-sided error

Perform procedure RPP d logn times independently in parallel.

The probability that every line containing {x,y} contains an edge (besides {x,y}) is at most (d) ((1-e-

1/2)d logn. Choosing d 0.74, (d) 1/n.

The remaining edges (at most n/2 on average) are tested in round 2.

Highly structured hidden graph Clean abstraction Flexible probing process Amenable to randomization,

parallelization Practically useful

Modeling and algorithms success

Outline

Motivating hidden graphs

Case study 1: Hidden matchings in genome sequencing.

Case study 2: Internet mapping studies.

Case study 3: Locating constrained, annotated Internet subgraphs.

Discussion

Internet mapping efforts

Goal: Discover the Internet router-level topology• Vertices represent routers.• Edges connect routers that are one IP hop apart.

Measurement Primitive: traceroute Reports the IP path from A to B.

source destination

212.12.5.77

212.12.58.3 163.55.221.98

163.55.1.41

163.55.1.10

• k sources: Few active sources, strategically located.

• m destinations: Many passive destinations, globally dispersed.

• Union of many traceroute paths.

(k,m)-traceroute study

Most experimental traceroute studies[Pansiot et al ’98, Govindan et al ’00, Broido et al ’01-’05, etc.]

Destinations

Sources

A thought experiment

Idea: Simulate topology measurements on a random graph.

1. Generate a sparse Erdös-Rényi random graph, G=(V,E). Each edge present independently with probability pAssign weights: w(e) = 1 + , where in

2. Pick k unique source nodes, uniformly at random

3. Pick m unique destination nodes, uniformly at random

4. Simulate traceroute from k sources to m destinations, i.e. learn shortest paths between k sources and m destinations.

5. Let Ĝ be union of shortest paths.

Ask: How does Ĝ compare with G ?

||

1,

||

1

VV

Ĝ is a biased sample of G that looks heavy-tailedAre heavy tails a measurement artifact?

MeasuredGraph, Ĝ

Underlying Random Graph, G

Underlying Graph: N=100000, p=0.00015Measured Graph: k=3, m=1000

log(Degree)

log

(Pr[

X>

x])

Are nodes sampled unevenly?

• Conjecture: Shortest path routing favors higher degree nodes nodes sampled unevenly

• Validation:Examine true degrees of nodes in measured graph, Ĝ.

Expect true degrees of nodes in Ĝ to be higher than degrees of nodes in G, on average.

True Degrees of nodes in Ĝ

Degrees of all nodes in G

Measured Graph: k=5,m=1000

• Conclusion: Difference between true degrees of Ĝ and degrees of G is insignificant; dismiss conjecture.

Are edges sampled unevenly?

• Conjecture:Edges selected incident to a node in Ĝ not proportional to true degree.

• Validation:For each node in Ĝ, plot true degree vs. measured degree.

If unbiased, ratio of true to measured degree should be constant. Points clustered around y=cx line (c<1).

• Conclusion: Edges incident to a node are sampled disproportionately; supports conjecture.

Ob

serv

ed D

egre

e

True Degree

What does this suggest?

Summary:

Edges are sampled unevenly by (k,m)-traceroute methods.

Edges close to the source are sampled more often than edges further away.

Intuitive Picture:

Neighborhood near sources is well explored, but visibility of edges declines sharply with hop distance from sources.

Hop1lo

g(P

r[X

>x]

)

log(Degree)

Hop2

Hop3

Underlying Graph

Measured Graph

Hop4

Choose k sources and m destinations at random. Consider the subgraph G’ = (V’, E’) induced by routes from R

between all (source, dest) pairs. How do expected values of |V’| and |E’| scale as a function of k

and m for various graph models? One special case for k = 1 well understood.

Chuang-Sirbu multicast scaling law: |E’| ~ m 0.8

Analysis in [Phillips et al ’99, van Mieghem et al ’02] Formulations for general k are open.

Also of interest: quantification of marginal utility of adding k+1 ’st source or destination [BBBC ’01].

Non-Adaptive Scaling Laws

Statistical Test #1

2

)1(2)1(

)1(]Pr[

v

ek

Cut vertex set in half: N (near) and F (far), by distance from nearest source.Let v : (0.01) |V|

k : fraction of v that lies in N

Can bound likelihood k deviates from 1/2 using Chernoff bounds:

H0C1

Reject hypothesis with confidence 1- if:

2

)1()1(

v

e

C1: Are the highest-degree nodes near the source? If so, then consistent with bias.

The 1% highest degree nodes occur at random with distance to nearest source.

Statistical Test #2

Partition vertices across median distance: N (near) and F (far)

Compare degree distribution of nodes in N and F, using the Chi-Square Test:

l

iiii EEO

1

22 /)(

where O and E are observed and expected degree frequencies and l is histogram bin size.

Reject hypothesis with confidence 1- if:

H0C2

2]1,[

2 l

C2: Is the degree distribution of nodes near the source different from those further away? If so, consistent with bias.

Chi Square Test succeeds on degree distribution for nodes near the source and far from the source.

Testing C1

H0C1 The 1% highest degree nodes occur at random with distance

to source.

Pansiot-Grad: 93% of the highest degree nodes are in NMercator: 90% of the highest degree nodes are in NSkitter: 84% of the highest degree nodes are in N

Testing C2

H0C2

Pansiot-Grad Mercator Skitter

log

(Pr[

X>

x])

log(Degree)

Near

Far

All

Near

Far

All

Near

Far

All

Several possible explanations

1. Degree distribution is distance-independent, but sampling is biased.

2. Degree distribution is distance-dependent, and nodes further from the source really do have below-average degree.

3. Others?

In practice, it appears to be a combination of factors.

Other traceroute questions

Suppose you had the ability to conduct adaptive measurements (recently feasible, e.g. scriptroute). How to maximize edge coverage on a fixed

measurement budget? Traceroute @ home (SIGMETRICS ’05) DIMES (INFOCOM ’05)

“AS-level traceroute” [SIGCOMM ’03] Leverage to probe a hidden multigraph?

Unknown hidden graph Misconceptions about which caused us to

“bark up the wrong tree” Unclean abstraction Awkward, inflexible probing process

probes interdependent on underlying graph Amenable to parallelization Power of adaptation not yet known

Modeling and algorithms: mixed bag

Outline

Motivating hidden graphs

Case study 1: Hidden matchings in genome sequencing.

Case study 2: Internet mapping studies.

Case study 3: Locating constrained, annotated Internet subgraphs.

Discussion

Simulation “Blank slate” for crafting experiments Fine-grained control, specifying all details No external surprises, not especially realistic

Emulation All the benefits of simulation, plus: running real protocols on real systems

Internet experimentation Even more realistic Much harder to set up, control experiments

Experimental Methodologies

Our question: Can we bridge over some of the attractive

features of simulation and emulation into wide-area testbed or overlay experimentation?

Towards an answer: Which services would be useful? Outline design of a set of interesting services. Today’s talk:

specify parameters of an experiment on a blank slate locate one or more sub-topologies matching specification

Controlled Internet Experimentation

User specifies an envisioned target topology T edges and bounds on their attributes or more interesting: only path attributes (RTTs)

Then, given an overlay network G whose: vertices are known in advance and whose paths have measurable, multi-dimensional

attributes not known in advance

Conduct a set of adaptive probes to: locate a hidden instance (feasible embedding) of

T into G respecting constraints. more generally: sample from feasible embeddings

Annotated topologies: problem statement

Specifying Topologies

N nodes in testbed, k nodes in specification

k x k constraint matrix C = {ci,j}

Entry ci,j constrains the end-to-end path between embedding of virtual nodes i and j.

For example, place bounds on RTTs:ci,j = [li,j, hi,j] represents lower and upper bounds on target RTT.

Constraints can be multi-dimensional. Constraints can also be placed on nodes. More complex specifications possible...

Feasible Embeddings

Def’n: A feasible embedding is a mapping f such that for all i, j where f(i) = x and f(j) = y:

li,j ≤ d (x, y) ≤ hi,j

Do not need to know d (x, y) exactly, only thatli,j ≤ l’(x, y) ≤ d (x, y) ≤ h’ (x, y) ≤ hi,j

Key point: Testbed need not be exhaustively characterized, only sufficiently well to embed.

Hardness

Finding an embedding is as hard as subgraph isomorphism (NP-complete)

Counting or sampling from set of feasible embeddings is #P-hard.

Approximation algorithms are not much better.

Current Best-Practice

Brute force search [CBM ’03, HotNets-II]. No joke. Situation is not quite as dire as it sounds.

Several methods for pruning the search tree.

Adaptive measurement heuristics. Many (almost all?) user problem instances not

near boundary of solubility and insolubility. Prototype service on PlanetLab

Off-line searches up to thousands of nodes. On-line searches up to hundreds of nodes.

Current Best-Practice (cont.)

Good news: many of the hardness results are based on (unrealistic?) modeling assumptions

Bad news: better models for annotated topologies are notably absent

Why? Measurements that might assist in model-

building are just getting underway. Capturing the practical issues in a

parsimonious model is a formidable challenge.

Hidden graph is dynamic Abstraction is reasonably clean Combinatorial optimization issues may

pose thorny problems for analysis Model-based approaches could help

Modeling and algorithms: virgin territory

Numerous hidden graphs in science; more emerging as engineered artifacts.

Principled measurement/modeling/validation will be needed.

Forums for discussion and dissemination of ideas across disciplines will help.

Takeaway messages

[AA 04] N. Alon and V. Asodi, “Learning a hidden subgraph,” Proc. of 31st ICALP, 2004.

[AC 04] D. Angluin and J. Chen, “Learning a hidden graph using O(log n) queries per edge,” COLT 2004.

[ABK+ 02] N. Alon, R. Beigel, S. Kasif, S. Rudich and B. Sudakov, “Learning a hidden matching: Combinatorial identification of hidden matchings with applications to whole geonme sequencing”, SIAM Journal on Computing, 2004.

[ACKM 05] D. Achlioptas, A. Clauset, D. Kempe and C. Moore, “On the bias of traceroute sampling,” Proc. of ACM STOC 2005.

[BAA+ 01] R. Beigel, N. Alon, S. Apaydin, L. Fortnow and S. Kasif, “An optimal procedure for gap closing in whole genome shotgun sequencing,” Proc. of ACM RECOMB 2001.

[BBBC 01] P. Barford, A. Bestavros, J. Byers and M. Crovella, “On the marginal utility of network topology measurements,” Proc. of 1st ACM SIGCOMM Internet Measurement Workshop, 2001.

[BC 01 (Skitter)] A. Broido and K. Claffy, “Connectivity of IP graphs,” Proc. of SPIE ITCom, August 2001.

[CBM 03] J. Considine, J. Byers and K. Mayer-Patel, “A constraint satisfaction approach to testbed embedding services,” Proc. of ACM HotNets Workshop, 2003.

[DIMES] The DIMES project. www.netdimes.org. [DRFC 05] B. Donnet, P. Raoult, T. Friedman, and M. Crovella, “Efficient algorithms

for large-scale topology discovery,” to appear in Proc. of ACM SIGMETRICS, 2005. [FFF 99] M. Faloutsos, P. Faloutsos and C. Faloutsos, “On power-law relationships

of the Internet topology,” Proc. of ACM SIGCOMM ’99.

References cited (p. 1 of 2)

[GK 98] V. Grebinski and G. Kucherov, “Reconstructing a Hamiltonian cycle by querying the graph: Application to DNA physical mapping,” Discrete Applied Math. 88 (1998).

[GT 00 (Mercator)] R. Govindan and H. Tangmunarunkit, “Heuristics for Internet map discovery,” Proc. of IEEE INFOCOM 2000.

[LBCX 03] A. Lakhina, J. Byers, M. Crovella and P. Xie, “Sampling biases in IP topology measurements,” Proc. of IEEE INFOCOM 2003.

[MHH 01] P. van Mieghem, G. Hooghiemstra, R. van der Hofstad, “On the efficiency of multicast,” IEEE/ACM Transactions on Networking, May 2001.

[PG 98] J. Pansiot and D. Grad, “On routes and multicast trees in the Internet,” ACM Computer Communications Review, 28(1), 1998.

[PST 99] G. Philips, S. Shenker and H. Tangmunarunkit, “Scaling of multicast trees: comments on the Chuang-Sirbu scaling law,” Proc. Of ACM SIGCOMM ’99.

[SMW 02] N. Spring, R. Mahajan and D. Wetherall, “Measuring ISP topologies with Rocketfuel,”, Proc. of ACM SIGCOMM 2002.

[SWM 05] M. Stumpf, C. Wiuf and R. May, “Subnets of scale-free networks are not scale-free: Sampling properties of networks,” PNAS 102(12), March 22, 2005.

References cited