Every Property of Hyperfinite Graphs is Testable Ilan Newman and Christian Sohler

1. Every Property of Hyperfinite Graphs isTestableIlan Newman and Christian Sohler

2. Property Testing Property Testing [Rubinfeld, Sudan, 96; Goldreich, Goldwasser, Ron, 98] • Oracle accesstohugeobject O • Decidewhether O has a property P orisfar-awayfrom P • Goal: Performthis relaxed decisiontask in sublinear orevenconstant time • Algorithmmayberandomized Definition • An objectise-farfrom P, ifitdiffers in morethan an e-fractionofitsdescriptionfromanyobjectwithproperty P

3. The Bounded-Degree Graph Model Boundeddegreegraph model [Goldreich, Ron, 02] • Graph G=(V,E) withmaximumdegree d (bounded-degreegraph) • Oracle access: (a) randomvertex (b) i-thedgeincidenttovertex v Definition (Graph Property; e-far) • A graphpropertyis a familyofgraphs P=  P(i), whereeach P(i) is a setofgraphs on i verticesthatisclosedunderisomorphism. • An n-vertexgraph G=(V,E) ise-farfrom a property P, ifitdiffersfromanybounded-degreegraph in P bymorethanednedges.

4. Testable Graph Properties Definition (testableproperty) • A graphproperty P iscallednon-uniformlytestable in thebounded-degreegraph model, ifthereis a function q=q(e,d) such thatforany n thereis an algorithm A(e,d,n) that on input an n-vertexgraph G withmaximumdegreeatmost d(a) makesatmost q queriesto G,(b) accepts G withprobabilityat least 2/3, if G hasproperty P, and(c) rejects G withprobabilityat least 2/3, if G ise-farfrom P.

5. Hyperfinite Graphs Definition (hyperfiniteness) • A graph G is(e,k)-hyperfinite, ifonecanremoveen edgesfrom G such thatonlyconnectedcomponentsofatmost k verticesremain. A family F ofgraphsisr-hyperfinite, ifforanye>0 everygraph GF is (e,r(e))-hyperfinite. Example • Planargraphsarer-hyperfinite forsomesuitablefunctionrbytheplanarseparatortheorem

6. Property Testing in Hyperfinite Graphs / of Hyperfinite Properties Previouswork • Every hereditarypropertyistestable in hyperfinite graphs[Czumaj, Shapira,Sohler ,09] • Every monotone hyperfinite propertyistestable[Benjamini, Schramm, Shapira, 08] • Every hereditary hyperfinite propertyistestable [Hassidim, Kelner, Nguyen, Onak, 09] • Every propertyistestable in graphsofboundedgrowth [Elek 07]

7. Ourresults Thiswork [Newman, S., 11] • Every propertyistestable in hyperfinite graphs. • Every hyperfinite graphpropertyistestable. • Hyperfinite graphisomorphismistestable. • Manygraphparametersareestimable. • The distributionof k-disksoftwo hyperfinite bounded-degreegraphsisclose, iffthegraphsareclose. Definition (k-disk) • A k-diskof a vertex v isthesubgraphrootedat v thatisinducedby all verticesatdistanceatmost k from v.

8. Somefurtherrelatedwork Motifs • A motifis a subgraphthatoccursmorefrequentlythan in a random graph. • Can beusedtocharacterizegraphsthatcomefromcertaindomains • „The motifssharedby ecological food webs were distinct from the motifs shared by the genetic networks of Escherichia coli and Saccharomycescerevisiae or from those found in the World Wide Web.” R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, U. Alon. Network Motifs: Simple Building Blocks of Complex Networks. Science, Vol. 298. no. 5594, pp. 824 - 827, 2002.

9. Constant sizegraphrepresentations Definition • For an inputgraph G, let G‘ be an (e,k)-representativeof G, if(a) G‘ hasconnectedcomponentsofsizeatmost k(b) G‘ canbeobtainedfrom G byremovingatmostednedges Observation • If k isindependentof n, then G‘ has a constantsizerepresentation: • List all D(k) graphs on k vertices H(1), H(2), …, H(D(k)) • Take a D(k)-dimensional vectorwhose i-thentryisthenumberofoccurencesofisomorphiccopiesof H(i) in G‘

10. A Property Tester Idea: • Reducepropertytesting in hyperfinite graphstographswhosecomponentshaveconstantsize The Tester: • Let P= P(i) be an arbitrarygraphproperty • Let P‘(n) = {H‘ : HP(n) and H‘ is an (e/2,k)-representativeofH} • Assumeweknowan (e/2,k)-representative G‘ of G • Accept, iff G‘P‘(n)

11. LocalPartitioningoracles LocalPlanarPartitioning Oracle [Hassidim, Kelner, Nguyen, Onak, 09] • Provides „localaccess“ to a partitionthat • hasconnectedcomponentsofsizeatmost k(e) and • hasbeenobtainedbyremoving, say, atmostedn/4 edgesfrom G • „Localaccess“ meansthatonlysome k-disksaresampledfrom G

12. LocalPartitioningoracles GlobalPartitioning(k,d) [Hassidim, Kelner, Nguyen, Onak, 09] • p = (p ,…, p ) = random permutation of the vertices • P =  • while G is not empty do • Let v be the first vertex in G according to p • if there exists a (k,d)-isolated neighborhood of v in G then • S = this neighborhood • else S = {v} • P = P  {S} • remove vertices in S from the graph 1 n (k,d)-isolated neighborhood of v: Connected set S of size at most k with vS and at most d|S| edges between S and V

13. Howtoobtaintherepresentativeof a graph? Algorithmforcomputingtherepresentative: • Estimatethefrequencyofconnectedcomponentsby uniform samplingusingtheplanarpartitioningoracle • (takesizeofcomponentintoaccount) Analysis: • Chernoffbounds

14. Second result Result: • The distributionof k-disksoftwo hyperfinite bounded-degreegraphsisclose, iffthecorrespondinggraphsareclose. Sketch ofproof: • „<=“: easy • „=>“: Weknowthateverypropertyof hyperfinite graphsistestable • So, forfixed n-vertexgraphconsiderthepropertyofbeing H • Let ALG beourpropertytestingalgorithmforthisproperty • On input H, ALG must acceptw.p. at least 2/3 • Weclaimthat, if a graph H‘ hassimilardistributionof k-disksas H, thenthetesterdoes not rejectw.p. > 1/2 • => H‘ ise-closeto H

15. Proofofclaim Claim • If H‘ has a similardistribution on k-disksas H, thenthetester on input H‘ does not rejectw.p.>1/2 Sketch ofProof: • The resultofthelocalpartitioningalgorithmdepends on thesampled k-disks (andinternalrandomness) • Ifthedistributionof k-disksissimilar, thenthebehaviourofthealgorithmsshouldbesimilarunlessthe k-disksintersect • However, as n goestoinfinity, thisprobabilitytendsto 0

16. Summary Testablegraphproperties (bounded-degree model) • Every hyperfinite property • Properties definedbythelocalstructure (forexample, trianglefreeness) • Certaincombinationsoftheabove (e.g. union) • Other properties, likeconnectivity… Open Questions • Characterize ‚remaining‘ testableproperties • Property testing in expandergraphs?

17. Thank you!