A PPROACH - 2 The central algorithm: ReviseGraph(G) function 1: G i - PowerPoint PPT Presentation

Introduction Multiscale Network Modeling Results M ULTIGRID APPROACH FOR MODELING NETWORKS F IELDS I NSTITUTE A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro University of Illinois at Chicago University of Texas at Austin Clemson University 2014 A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results O UTLINE 1 I NTRODUCTION 2 M ULTISCALE N ETWORK M ODELING 3 R ESULTS Examples Statistics Summary: The multiscale method (MUSKETEER) generates synthetic networks that match the properties of real networks. A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results M OTIVATION - THE MISSING DATA PROBLEM Networks are the central part of many complex systems, 1 e.g. infrastructure, social, neural systems We need to evaluate ideas/methods/algorithms on them, & 2 understand their structure Limitations of empirical data: 3 Difficult or Impossible to get 1 Insufficient: want to show robustness on 10 2 to 10 6 networks 2 A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results M ETHODS FOR N ETWORK M ODELING Network model: Erd˝ os-Rényi, Kronecker Graph, 1 ERGM, Watts-Strogatz, Liu-Chung expected degrees, Barabási-Albert, etc. Mechanistic model 2 Randomize empirical data 3 An application-specific topology generator: 4 BRITE, INET, Tiers, GT-IGM, PLOD, GridG, GeNGe, etc. New (5.): Multiscale network generation (MUSKETEER) Ref: “Multiscale Network Generation”. Free and Open source. arxiv.org/abs/1207.4266 A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results M ETHODS FOR N ETWORK M ODELING Network model: Erd˝ os-Rényi, Kronecker Graph, 1 ERGM, Watts-Strogatz, Liu-Chung expected degrees, Barabási-Albert, etc. Mechanistic model 2 Randomize empirical data 3 An application-specific topology generator: 4 BRITE, INET, Tiers, GT-IGM, PLOD, GridG, GeNGe, etc. New (5.): Multiscale network generation (MUSKETEER) Ref: “Multiscale Network Generation”. Free and Open source. arxiv.org/abs/1207.4266 A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results M ULTISCALE A LGORITHMS What is a multiscale/multigrid algorithm ? Iteratively coarsen i.e. reduce the number of variables in a 1 problem: L 1 → ··· → L k → ··· → L ′ 1 → L ′ → L 0 0 L i + 1 = P T L i P e.g. Solve in level k and then refine it back to level 0 2 Strengths: O ( m ) or O ( m log m ) performance for P or NP-hard problems Pitfalls: Enforcing constraints & Precision Very successful in large linear/nonlinear equation solvers Ref: Knepley/UC - PETSc A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results R EAL N ETWORKS Real Networks: Organized 1 hierarchically Refs: Ravasz & Barabasi Levels are dissimilar 2 Refs: Doyle et al. Connections are 3 usually local: low expansion, clustering, loops A Road Network Ref: Barabasi, Spielman A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results T HE MULTISCALE APPROACH The multiscale network modeling approach: Generates a hierarchy 1 of coarsened networks Edits at any level of 2 coarsening Synthethic nodes are 3 resampled Synthetic edges 4 preserve locality Version 1.2 (Dec): Fast editing algorithm A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results A PPROACH - 2 The central algorithm: ReviseGraph(G) function 1: G i + 1 ← Coarsen ( G i ) 2: ˜ G i + 1 ← ReviseGraph ( G i + 1 ) 3: G ′ i ← Interpolate (˜ G i + 1 ) 4: ˜ G i ← EditEdgesAndNodes ( G ′ i ) 5: ˜ G i ← UserDefinedAdjustment (˜ G i ) 6: Return ˜ G i Editing does not specifically attempt to enforce properties like degree distribution or clustering Preservation of local and global graph properties emerges as an approximate invariant of the editing process A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Multiscale Network Modeling Results A PPROACH - 2 The central algorithm: ReviseGraph(G) function 1: G i + 1 ← Coarsen ( G i ) 2: ˜ G i + 1 ← ReviseGraph ( G i + 1 ) 3: G ′ i ← Interpolate (˜ G i + 1 ) 4: ˜ G i ← EditEdgesAndNodes ( G ′ i ) 5: ˜ G i ← UserDefinedAdjustment (˜ G i ) 6: Return ˜ G i Editing does not specifically attempt to enforce properties like degree distribution or clustering Preservation of local and global graph properties emerges as an approximate invariant of the editing process A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results N ETWORKS Let’s make some networks ... A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results P RESERVATION OF H IDDEN P ROPERTIES A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results E XAMPLE : C OAUTHORSHIP Collaboration network (Newman): GCC 379 nodes growth rate: nodes [0, 0.3]; edges:[0, 0.1] A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results E XAMPLE : P OWER G RID Western Interconnection - a power grid with 4941 nodes edit rate: nodes [0, 0.1]; edges:[0, 0.1] A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results E VALUATION OF R ANDOM N ETWORKS A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results Q UALITY OF R ANDOM N ETWORKS - 1 Experimental simulation Level 0 edits: 8% nodes, 8% edges Level 1 edits: 7% nodes, 7% edges Generally, the choice of edit rates is based on the problem Colorado Springs HIV (left) and replica (right) Ref: Potterat et al. A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results Q UALITY OF R ANDOM N ETWORKS - 1 F IGURE : Colorado Springs Network Median of replicas num nodes 0 . 99 num edges 1 . 02 num comps 1 . 00 clustering 0 . 87 avg 1 . 02 degree total deg*deg 0 . 94 assortativity avg 0 . 97 eccentricity avg 1 . 01 distance harmonic avg 1 . 07 distance avg between. 1 . 03 centrality modularity 1 . 00 powerlaw exp 1 . 02 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 Relative to real network Diversity: 30% of nodes and 60% of edges are new or removed A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results Q UALITY OF R ANDOM N ETWORKS - 2 F IGURE : Colorado Springs Network 1 . 0 empirical generated networks 0 . 8 P [ Degree > k ] 0 . 6 0 . 4 0 . 2 0 . 0 0 5 10 15 20 25 k A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results Q UALITY OF R ANDOM N ETWORKS - 3 F IGURE : Western Interconnection (Watts & Strogatz) Median of replicas num nodes 1 . 02 num edges 1 . 01 num comps 1 . 00 clustering 1 . 07 avg degree 0 . 99 total deg*deg assortativity 0 . 96 avg distance 1 . 07 avg between. 1 . 06 centrality modularity 1 . 00 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 Relative to real network edge edit rate: [0, 0, 0, 0.2]; node edit rate:[0, 0, 0, 0, 0.2] A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks

Introduction Examples Multiscale Network Modeling Statistics Results Q UALITY OF R ANDOM N ETWORKS - 4 Erd˝ os-Rényi template Barabási-Albert template Median of Median of replicas replicas num nodes 1 . 00 num nodes 1 . 00 num edges num edges 0 . 98 0 . 99 num comps 1 . 00 num comps 1 . 00 clustering 1 . 02 clustering 0 . 97 avg avg degree 0 . 98 degree 1 . 00 total deg*deg 0 . 90 assortativity total deg*deg 1 . 11 assortativity avg 1 . 06 eccentricity avg eccentricity 1 . 05 avg 1 . 01 distance avg 1 . 00 harmonic avg distance 1 . 01 distance harmonic avg avg between. 1 . 00 distance 1 . 02 centrality avg between. 1 . 01 modularity centrality 1 . 06 modularity powerlaw exp 0 . 97 0 . 99 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 Relative to real network Relative to real network A. “Sasha” Gutfraind Lauren A. Meyers Ilya Safro Multigrid approach for modeling networks