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The mathematical challenge of large networks László Lovász Eötvös Loránd University, Budapest Joint work with Christian Borgs, Jennifer Chayes, Balázs Szegedy, Vera Sós and Katalin Vesztergombi May 20121
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Minimize x 3 -6x over x 0 minimum is not attained in rationals Minimize 4-cycle density in graphs with edge-density 1/2 minimum is not attained among graphs always >1/16, arbitrarily close for random graphs Real numbers are useful Graph limits are useful May 20122 Graph limits: Why are they needed?
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Two extremes: - dense (cn 2 edges) - bounded degree well developed good warm-up case How dense is the graph? May 20123 Inbetween??? Bollobas-Riordan Chung Conlon-Fox-Zhao less developed, more difficult most applications
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May 20124 How is the graph given? - Graph is HUGE. - Not known explicitly, not even the number of nodes. Idealize: define minimum amount of info.
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May 20125 Dense case: cn 2 edges. - We can sample a uniform random node a bounded number of times, and see edges between sampled nodes. Bounded degree ( d) - We can sample a uniform random node a bounded number of times, and explore its neighborhood to a bounded depth. How is the graph given? „Property testing”: Arora-Karger-Karpinski, Goldreich-Goldwasser-Ron, Rubinfeld-Sudan, Alon-Fischer-Krivelevich-Szegedy, Fischer, Frieze-Kannan, Alon-Shapira
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May 20126 Lecture plan Want to construct completion of the set of graphs. - What is the distance of two graphs? - Which graph sequences are convergent? - How to represent the limit object? - How does this completion space look like? - How to approximate graphs? (Regularity Lemmas and sampling)
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May 20127 Lecture plan Applications in (dense) extremal graph theory - Are extremal graph problems decidable? - Which graphs are extremal? - Local vs. global extrema - Is there always an extremal graph?
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May 20128 Lecture plan Applications in property testing - Deterministic and non-deterministic sampling (P=NP) - Which properties are testable by sampling?
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May 20129 Lecture plan The bounded degree case - Different limit objects: involution-invariant distributions, graphings - Local algorithms and distributed computing
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G 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 AGAG WGWG Pixel pictures May 201210
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A random graph with 100 nodes and with 2500 edges May 201211 Pixel pictures
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Rearranging rows and columns May 201212 Pixel pictures
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May 201213 Pixel pictures A randomly grown uniform attachment graph on 200 nodes At step n: - a new node is born; - any two nodes are joined with probability 1/n Ignore multiplicity of edges
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Approximation by small: Regularity Lemma Szemerédi 1975 May 201214
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Nodes can be so ordered essentially random Approximation by small: Regularity Lemma May 201215
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The nodes of any graph can be partitioned into a small number of essentially equal parts so that the bipartite graphs between 2 parts are essentially random (with different densities). with k 2 exceptions for subsets X,Y of parts V i,V j # of edges between X and Y is p ij |X||Y| ± ( n/k) 2 Given >0 difference at most 1 Approximation by small: Regularity Lemma May 201216
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May 201217 Original Regularity Lemma Szemerédi 1976 “Weak” Regularity Lemma Frieze-Kannan 1999 “Strong” Regularity Lemma Alon – Fisher - Krivelevich - M. Szegedy 2000 Approximation by small: Regularity Lemma
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May 201218 A randomly grown uniform attachment graph on 200 nodes Graph limits: Examples
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Knowing the limit W knowing many properties (approximately). Graph limits: Why are they useful? triangle density May 201219
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May 201220 Limit objects: the math distribution of k-samples is convergent for all k t(F,G): Probability that random map V(F) V(G) preserves edges (G 1,G 2,…) convergent: F t(F,G n ) is convergent
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May 201221 Limit objects: the math W 0 = { W: [0,1] 2 [0,1], symmetric, measurable } G n W : F: t(F,G n ) t(F,W) "graphon"
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Randomly grown prefix attachment graph At step n: - a new node is born; - connects to a random previous node and all its predecessors May 201222 Limit objects: an example
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A randomly grown prefix attachment graph with 200 nodes Is this graph sequence convergent at all? Yes, by computing subgraph densities! This tends to some shades of gray; is that the limit? No, by computing triangle densities! May 201223
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A randomly grown prefix attachment graph with 200 nodes (ordered by degrees) This also tends to some shades of gray; is that the limit? No… May 201224 Limit objects: an example
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The limit of randomly grown prefix attachment graphs May 201225 Limit objects: an example
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The distance of two graphs May 201226 (a) (b) cut distance (c) blow up nodes finite definition by fractional overlay
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May 201227 Examples: The distance of two graphs
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May 201228 p ij : density between S i and S j G P : complete graph on V(G) with edge weights p ij “Weak” Regularity Lemma
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May 201229 Frieze-Kannan “Weak” Regularity Lemma
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May 201230 Counting lemma: Inverse counting lemma: If for all graphs F with k nodes, then Counting Lemmas
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May 201231 Equivalence of convergence notions A graph sequence (G 1,G 2,...) is convergent iff it is Cauchy in .
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December 200832 The distance of two graphons
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May 201233 The semimetric space ( W 0, ) is compact. “Strong” Regularity Lemma
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May 201234 Limit objects: the math For every convergent graph sequence (G n ) there is a W W 0 such that G n W. W is essentially unique (up to measure-preserving transformation). Conversely, W (G n ) such that G n W.
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May 201235 Limit objects: the math For every convergent graph sequence (G n ) there is a W W 0 such that G n W. W is essentially unique (up to measure-preserving transformation). Conversely, W (G n ) such that G n W. Regularity Lemma + martingales L-Szegedy Exchangeable random variables Aldous; Diaconis-Janson Ultraproducts Elek-Szegedy
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May 201236 Limit objects: the math For every convergent graph sequence (G n ) there is a W W 0 such that G n W. W is essentially unique (up to measure-preserving transformation). Conversely, W (G n ) such that G n W. W-random graphs (sampling from a graphon)
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May 201237 Limit objects: the math For every convergent graph sequence (G n ) there is a W W 0 such that G n W. W is essentially unique (up to measure-preserving transformation). Conversely, W (G n ) such that G n W. Constructing canonical representation Borgs-Chayes-L Exchangeable random variables Kallenberg; Diaconis-Janson Inverse Counting Lemma, measure compactness Bollobas-Riordan
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