Structural sparsity in the real world Erik Demaine , Felix Reidl , - PowerPoint PPT Presentation

Structural sparsity in the real world Erik Demaine ∗ , Felix Reidl , Peter Rossmanith, Fernando Sánchez Villaamil, Blair D. Sullivan † and Somnath Sikdar Theoretical Computer Science ∗ MIT † NCSU @Bergen 2015

Contents The Program Structural Sparseness Models Algorithms Empirical Sparseness

The Program

Complex networks Structural graph theory Ubiquitous in real world Well-researched Empirical structure Deep structural theorems • Small-world • WQO by minor relation • Heavy-tailed degree seq. • Decomposition theorems • Clustering • Grid-theorem Algorithmic applications Great algorithmic properties • Disease spreading • (E)PTAS • Attack resilience • Subexponential algorithms • Fraud detection • Linear kernels • Drug discovery • Model-checking Can we bring these two fields together?

The idea 1 Bridge the gap by identifying a notion of sparseness that applies to complex networks. 2 Develop algorithmic tools for network related problems. 3 Show experimentally that the above is useful in practice.

The idea 1 Bridge the gap by identifying a notion of sparseness that applies to complex networks. • Need general and stable notion of sparseness. • How to prove that it holds for complex networks? 2 Develop algorithmic tools for network related problems. • Unclear what problems are interesting. 3 Show experimentally that the above is useful in practice. • Show that structural sparseness appears in the real world. • Show that algorithms can compete with known approaches.

Structural Sparseness

ω ∇ ∇ ∇ Nowhere dense r Locally bounded r Locally excluding Bounded expansion expansion a minor Excluding a topological minor r Locally bounded Excluding a minor treewidth Bounded genus Planar Bounded treewidth Bounded degree Outerplanar Bounded treedepth Forests Star forests Linear forests

Bounded expansion A graph class has bounded expansion if the density of its minors only depends on their depth. The following operations on a class of bounded expansion result again in a class of bounded expansion: • Taking shallow minors/immersions (in particular subgraphs) • Adding a universal vertex • Replacing each vertex by a small clique (lexicographic product)

Models

1 /6 1 /3 1 /5 Stochastic Perturbed bounded Kleinberg Block degree 1 ∏ (k) ∝ k 4 E[d] 3 Con � guration Barabasi-Albert Chung-Lu Heavy-tailed degree distribution

The positive side Name Definition f ( d ) Parameters d − γ Power law γ > 2 d − γ e − λd Power law w/ cutoff γ > 2 , λ > 0 e − λd Exponential λ > 0 d β − 1 e − λd β Stretched exponential λ, β > 0 exp( − ( d − µ ) 2 Gaussian ) µ, σ 2 σ 2 d − 1 exp( − (log d − µ ) 2 Log-normal ) µ, σ 2 σ 2 Theorem Let D be an asymptotic degree distribution with finite mean. Then random graphs generated by the Configuration Model or the Chung-Lu model with parameter D have bounded expansion with high probability.

The positive side Theorem The perturbed bounded degree model has bounded expansion with high probability. Perturbing forests of S √ n results in a somewhere dense class.

The negative side Theorem The Kleinberg Model is somewhere dense with high probability. Theorem The Barabási-Albert Model is somewhere dense with non-vanishing probability.

Bounded expansion Somewhere dense 1 /6 1 /3 1 /5 Stochastic Perturbed bounded Kleinberg Block degree 1 ∏ (k) ∝ k 4 E[d] 3 Con � guration Barabasi-Albert Chung-Lu Heavy-tailed degree distribution

Algorithms

Neighbourhood sizes Measure Definition Localized d ( v, u )) − 1 d ( v, u )) − 1 Closeness ( � ( � u ∈ V ( G ) u ∈ N r ( v ) d ( v, u ) − 1 d ( v, u ) − 1 Harmonic � � u ∈ V ( G ) u ∈ N r ( v ) |{ v | d ( v, v ) < ∞}| 2 | N r [ v ] | 2 Lin’s index � u ∈ V ( G ): d ( v,u ) < ∞ d ( v, u ) � u ∈ N r [ v ] d ( v, u ) Theorem Let G be a graph class of bounded expansion. There is an algorithm that for every r ∈ N and G ∈ G computes the size of the i -th neighbourhood of every vertex of G , for all i ≤ r , in linear time.

Closeness centrality ArashRa fi ey ArashRa fi ey d ( v, u )) − 1 RezaSaei RezaSaei ( � u ∈ N 1 ( v ) RémyBelmonte RémyBelmonte Pimvan'tHof Pimvan'tHof LeneM.Favrholdt LeneM.Favrholdt MortenMjelde MortenMjelde PetrA.Golovach PetrA.Golovach RodicaMihai RodicaMihai QinXin QinXin PinarHeggernes PinarHeggernes ErikJanvanLeeuwen ErikJanvanLeeuwen MichalPilipczuk MichalPilipczuk DieterKratsch DieterKratsch JohannesLangguth JohannesLangguth JeanR.S.Blair JeanR.S.Blair JesperNederlof JesperNederlof FredrikManne FredrikManne MarcinPilipczuk MarcinPilipczuk JiríFiala JiríFiala PålGrønåsDrange PålGrønåsDrange AssefawHadishGebremedhin AssefawHadishGebremedhin IoanT IoanT odinca odinca MarkusSortlandDregi MarkusSortlandDregi PetterKristiansen PetterKristiansen BengtAspvall BengtAspvall ManuBasavaraju ManuBasavaraju SadiaSharmin SadiaSharmin YngveVillanger YngveVillanger SigveHortemoSæther SigveHortemoSæther MartinVatshelle MartinVatshelle MagnúsM.Halldórsson MagnúsM.Halldórsson BartM.P BartM.P .Jansen .Jansen FedorV.Fomin FedorV.Fomin CharisPapadopoulos CharisPapadopoulos AndrzejProskurowski AndrzejProskurowski JanArneT JanArneT elle elle DimitriosM.Thilikos DimitriosM.Thilikos ArchontiaC.Giannopoulou ArchontiaC.Giannopoulou FedericoMancini FedericoMancini DanielLokshtanov DanielLokshtanov Binh-MinhBui-Xuan Binh-MinhBui-Xuan DanielMeister DanielMeister FredericDorn FredericDorn YuriRabinovich YuriRabinovich IsoldeAdler IsoldeAdler SaketSaurabh SaketSaurabh MikeFellows MikeFellows M.S.Ramanujan M.S.Ramanujan ChristianSloper ChristianSloper SergeGaspers SergeGaspers FahadPanolan FahadPanolan FranRosamond FranRosamond AlexeyA.Stepanov AlexeyA.Stepanov Network provided by Pål

Closeness centrality ArashRa fi ey ArashRa fi ey RezaSaei RezaSaei d ( v, u )) − 1 ( � RémyBelmonte RémyBelmonte u ∈ N 2 ( v ) Pimvan'tHof Pimvan'tHof LeneM.Favrholdt LeneM.Favrholdt MortenMjelde MortenMjelde PetrA.Golovach PetrA.Golovach RodicaMihai RodicaMihai QinXin QinXin PinarHeggernes PinarHeggernes ErikJanvanLeeuwen ErikJanvanLeeuwen MichalPilipczuk MichalPilipczuk DieterKratsch DieterKratsch JohannesLangguth JohannesLangguth JeanR.S.Blair JeanR.S.Blair JesperNederlof JesperNederlof FredrikManne FredrikManne MarcinPilipczuk MarcinPilipczuk JiríFiala JiríFiala PålGrønåsDrange PålGrønåsDrange AssefawHadishGebremedhin AssefawHadishGebremedhin IoanT IoanT odinca odinca MarkusSortlandDregi MarkusSortlandDregi PetterKristiansen PetterKristiansen BengtAspvall BengtAspvall ManuBasavaraju ManuBasavaraju SadiaSharmin SadiaSharmin YngveVillanger YngveVillanger SigveHortemoSæther SigveHortemoSæther MartinVatshelle MartinVatshelle MagnúsM.Halldórsson MagnúsM.Halldórsson BartM.P BartM.P .Jansen .Jansen FedorV.Fomin FedorV.Fomin CharisPapadopoulos CharisPapadopoulos AndrzejProskurowski AndrzejProskurowski DimitriosM.Thilikos DimitriosM.Thilikos JanArneT JanArneT elle elle FedericoMancini FedericoMancini ArchontiaC.Giannopoulou ArchontiaC.Giannopoulou DanielLokshtanov DanielLokshtanov Binh-MinhBui-Xuan Binh-MinhBui-Xuan DanielMeister DanielMeister FredericDorn FredericDorn IsoldeAdler IsoldeAdler YuriRabinovich YuriRabinovich SaketSaurabh SaketSaurabh MikeFellows MikeFellows M.S.Ramanujan M.S.Ramanujan ChristianSloper ChristianSloper SergeGaspers SergeGaspers FahadPanolan FahadPanolan FranRosamond FranRosamond AlexeyA.Stepanov AlexeyA.Stepanov Network provided by Pål

Closeness centrality ArashRa fi ey ArashRa fi ey RezaSaei RezaSaei d ( v, u )) − 1 ( � RémyBelmonte RémyBelmonte u ∈ N 3 ( v ) Pimvan'tHof Pimvan'tHof LeneM.Favrholdt LeneM.Favrholdt MortenMjelde MortenMjelde PetrA.Golovach PetrA.Golovach RodicaMihai RodicaMihai QinXin QinXin PinarHeggernes PinarHeggernes ErikJanvanLeeuwen ErikJanvanLeeuwen MichalPilipczuk MichalPilipczuk DieterKratsch DieterKratsch JohannesLangguth JohannesLangguth JeanR.S.Blair JeanR.S.Blair JesperNederlof JesperNederlof FredrikManne FredrikManne MarcinPilipczuk MarcinPilipczuk JiríFiala JiríFiala PålGrønåsDrange PålGrønåsDrange AssefawHadishGebremedhin AssefawHadishGebremedhin IoanT IoanT odinca odinca MarkusSortlandDregi MarkusSortlandDregi PetterKristiansen PetterKristiansen BengtAspvall BengtAspvall ManuBasavaraju ManuBasavaraju SadiaSharmin SadiaSharmin YngveVillanger YngveVillanger SigveHortemoSæther SigveHortemoSæther MartinVatshelle MartinVatshelle MagnúsM.Halldórsson MagnúsM.Halldórsson BartM.P BartM.P .Jansen .Jansen FedorV.Fomin FedorV.Fomin CharisPapadopoulos CharisPapadopoulos AndrzejProskurowski AndrzejProskurowski DimitriosM.Thilikos DimitriosM.Thilikos JanArneT JanArneT elle elle FedericoMancini FedericoMancini ArchontiaC.Giannopoulou ArchontiaC.Giannopoulou DanielLokshtanov DanielLokshtanov Binh-MinhBui-Xuan Binh-MinhBui-Xuan DanielMeister DanielMeister FredericDorn FredericDorn IsoldeAdler IsoldeAdler YuriRabinovich YuriRabinovich SaketSaurabh SaketSaurabh MikeFellows MikeFellows M.S.Ramanujan M.S.Ramanujan ChristianSloper ChristianSloper SergeGaspers SergeGaspers FahadPanolan FahadPanolan FranRosamond FranRosamond AlexeyA.Stepanov AlexeyA.Stepanov Network provided by Pål