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American Mathematical Society
Colloquium PublicationsVolume 60
Large Networks and Graph Limits
László Lovász
Large Networks and Graph Limits
American Mathematical Society
Colloquium PublicationsVolume 60
Large Networks and Graph Limits
László Lovász
American Mathematical SocietyProvidence, Rhode Island
http://dx.doi.org/10.1090/coll/060
Editorial Board
Lawrence C. EvansYuri Manin
Peter Sarnak (Chair)
2010 Mathematics Subject Classification. Primary 58J35, 58D17, 58B25, 19L64, 81R60,19K56, 22E67, 32L25, 46L80, 17B69.
For additional information and updates on this book, visitwww.ams.org/bookpages/coll-60
ISBN-13: 978-0-8218-9085-1
Copying and reprinting. Individual readers of this publication, and nonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research. Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publicationis permitted only under license from the American Mathematical Society. Requests for suchpermission should be addressed to the Acquisitions Department, American Mathematical Society,201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made bye-mail to [email protected].
c© 2012 by the author. All rights reserved.Printed in the United States of America.
©∞ The paper used in this book is acid-free and falls within the guidelinesestablished to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org/
10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12
To Kati
as all my books
Contents
Preface xi
Part 1. Large graphs: an informal introduction 1
Chapter 1. Very large networks 31.1. Huge networks everywhere 31.2. What to ask about them? 41.3. How to obtain information about them? 51.4. How to model them? 81.5. How to approximate them? 111.6. How to run algorithms on them? 181.7. Bounded degree graphs 22
Chapter 2. Large graphs in mathematics and physics 252.1. Extremal graph theory 252.2. Statistical physics 32
Part 2. The algebra of graph homomorphisms 35
Chapter 3. Notation and terminology 373.1. Basic notation 373.2. Graph theory 383.3. Operations on graphs 39
Chapter 4. Graph parameters and connection matrices 414.1. Graph parameters and graph properties 414.2. Connection matrices 424.3. Finite connection rank 45
Chapter 5. Graph homomorphisms 555.1. Existence of homomorphisms 555.2. Homomorphism numbers 565.3. What hom functions can express 625.4. Homomorphism and isomorphism 685.5. Independence of homomorphism functions 725.6. Characterizing homomorphism numbers 755.7. The structure of the homomorphism set 79
Chapter 6. Graph algebras and homomorphism functions 836.1. Algebras of quantum graphs 836.2. Reflection positivity 88
vii
viii CONTENTS
6.3. Contractors and connectors 946.4. Algebras for homomorphism functions 1016.5. Computing parameters with finite connection rank 1066.6. The polynomial method 108
Part 3. Limits of dense graph sequences 113
Chapter 7. Kernels and graphons 1157.1. Kernels, graphons and stepfunctions 1157.2. Generalizing homomorphisms 1167.3. Weak isomorphism I 1217.4. Sums and products 1227.5. Kernel operators 124
Chapter 8. The cut distance 1278.1. The cut distance of graphs 1278.2. Cut norm and cut distance of kernels 1318.3. Weak and L1-topologies 138
Chapter 9. Szemeredi partitions 1419.1. Regularity Lemma for graphs 1419.2. Regularity Lemma for kernels 1449.3. Compactness of the graphon space 1499.4. Fractional and integral overlays 1519.5. Uniqueness of regularity partitions 154
Chapter 10. Sampling 15710.1. W -random graphs 15710.2. Sample concentration 15810.3. Estimating the distance by sampling 16010.4. The distance of a sample from the original 16410.5. Counting Lemma 16710.6. Inverse Counting Lemma 16910.7. Weak isomorphism II 170
Chapter 11. Convergence of dense graph sequences 17311.1. Sampling, homomorphism densities and cut distance 17311.2. Random graphs as limit objects 17411.3. The limit graphon 18011.4. Proving convergence 18511.5. Many disguises of graph limits 19311.6. Convergence of spectra 19411.7. Convergence in norm 19611.8. First applications 197
Chapter 12. Convergence from the right 20112.1. Homomorphisms to the right and multicuts 20112.2. The overlay functional 20512.3. Right-convergent graphon sequences 20712.4. Right-convergent graph sequences 211
CONTENTS ix
Chapter 13. On the structure of graphons 21713.1. The general form of a graphon 21713.2. Weak isomorphism III 22013.3. Pure kernels 22213.4. The topology of a graphon 22513.5. Symmetries of graphons 234
Chapter 14. The space of graphons 23914.1. Norms defined by graphs 23914.2. Other norms on the kernel space 24214.3. Closures of graph properties 24714.4. Graphon varieties 25014.5. Random graphons 25614.6. Exponential random graph models 259
Chapter 15. Algorithms for large graphs and graphons 26315.1. Parameter estimation 26315.2. Distinguishing graph properties 26615.3. Property testing 26815.4. Computable structures 276
Chapter 16. Extremal theory of dense graphs 28116.1. Nonnegativity of quantum graphs and reflection positivity 28116.2. Variational calculus of graphons 28316.3. Densities of complete graphs 28516.4. The classical theory of extremal graphs 29316.5. Local vs. global optima 29416.6. Deciding inequalities between subgraph densities 29916.7. Which graphs are extremal? 307
Chapter 17. Multigraphs and decorated graphs 31717.1. Compact decorated graphs 31817.2. Multigraphs with unbounded edge multiplicities 325
Part 4. Limits of bounded degree graphs 327
Chapter 18. Graphings 32918.1. Borel graphs 32918.2. Measure preserving graphs 33218.3. Random rooted graphs 33818.4. Subgraph densities in graphings 34418.5. Local equivalence 34618.6. Graphings and groups 349
Chapter 19. Convergence of bounded degree graphs 35119.1. Local convergence and limit 35119.2. Local-global convergence 360
Chapter 20. Right convergence of bounded degree graphs 36720.1. Random homomorphisms to the right 36720.2. Convergence from the right 375
x CONTENTS
Chapter 21. On the structure of graphings 38321.1. Hyperfiniteness 38321.2. Homogeneous decomposition 393
Chapter 22. Algorithms for bounded degree graphs 39722.1. Estimable parameters 39722.2. Testable properties 40222.3. Computable structures 405
Part 5. Extensions: a brief survey 413
Chapter 23. Other combinatorial structures 41523.1. Sparse (but not very sparse) graphs 41523.2. Edge-coloring models 41623.3. Hypergraphs 42123.4. Categories 42523.5. And more... 429
Appendix A. Appendix 433A.1. Mobius functions 433A.2. The Tutte polynomial 434A.3. Some background in probability and measure theory 436A.4. Moments and the moment problem 441A.5. Ultraproduct and ultralimit 444A.6. Vapnik–Chervonenkis dimension 445A.7. Nonnegative polynomials 446A.8. Categories 447
Bibliography 451
Author Index 465
Subject Index 469
Notation Index 473
Preface
Within a couple of months in 2003, in the Theory Group of Microsoft Researchin Redmond, Washington, three questions were asked by three colleagues. MichaelFreedman, who was working on some very interesting ideas to design a quantumcomputer based on methods of algebraic topology, wanted to know which graphparameters (functions on finite graphs) can be represented as partition functionsof models from statistical physics. Jennifer Chayes, who was studying internetmodels, asked whether there was a notion of “limit distribution” for sequencesof graphs (rather than for sequences of numbers). Vera T. Sos, a visitor fromBudapest interested in the phenomenon of quasirandomness and its connections tothe Regularity Lemma, suggested to generalize results about quasirandom graphsto multitype quasirandom graphs. It turned out that these questions were veryclosely related, and the ideas which we developed for the answers have motivatedmuch of my research for the next years.
Jennifer’s question recalled some old results of mine characterizing graphsthrough homomorphism numbers, and another paper with Paul Erdos and JoelSpencer in which we studied normalized versions of homomorphism numbers andtheir limits. Using homomorphism numbers, Mike Freedman, Lex Schrijver and Ifound the answer to Mike’s question in a few months. The method of solution, theuse of graph algebras, provided a tool to answer Vera’s. With Christian Borgs, Jen-nifer Chayes, Lex Schrijver, Vera Sos, Balazs Szegedy, and Kati Vesztergombi, westarted to work out an algebraic theory of graph homomorphisms and an analytictheory of convergence of graph sequences and their limits. This book will try togive an account of where we stand.
Finding unexpected connections between the three questions above was stim-ulating and interesting, but soon we discovered that these methods and results areconnected to many other studies in many branches of mathematics. A couple ofyears earlier Itai Benjamini and Oded Schramm had defined convergence of graphsequences with bounded degree, and constructed limit objects for them (our maininterest was, at least initially, the convergence theory of dense graphs). Similarideas were raised even earlier by David Aldous. The limit theories of dense andbounded-degree graphs have lead to many analogous questions and results, andeach of them is better understood thanks to the other.
Statistical physics deals with very large graphs and their local and global prop-erties, and it turned out to be extremely fruitful to have two statistical physicists(Jennifer and Christian) on the (informal) team along with graph theorists. Thisput the burden to understand the other person’s goals and approaches on all of us,but at the end it was the key to many of the results.
xi
xii PREFACE
Another important connection that was soon discovered was the theory of prop-erty testing in computer science, initiated by Goldreich, Goldwasser and Ron sev-eral years earlier. This can be viewed as statistics done on graphs rather than onnumbers, and probability and statistics became a major tool for us.
One of the most important application areas of these results is extremal graphtheory. A fundamental tool in the extremal theory of dense graphs is Szemeredi’sRegularity Lemma, and this lemma turned out to be crucial for us as well. Graphlimit theory, we hope, repaid some of this debt, by providing the shortest andmost general formulation of the Regularity Lemma (“compactness of the graphonspace”). Perhaps the most exciting consequence of the new theory is that it allowsthe precise formulation of, and often the exact answer to, some very general ques-tions concerning algorithms on large graphs and extremal graph theory. Indepen-dently and about the same time as we did, Razborov developed the closely relatedtheory of flag algebras, which has lead to the solution of several long-standing openproblems in extremal graph theory.
Speaking about limits means, of course, analysis, and for some of us graph the-orists, it meant hard work learning the necessary analytical tools (mostly measuretheory and functional analysis, but even a bit of differential equations). Involvinganalysis has advantages even for some of the results that can be stated and provedpurely graph-theoretically: many definitions and proofs are shorter, more trans-parent in the analytic language. Of course, combinatorial difficulties don’t justdisappear: sometimes they are replaced by analytic difficulties. Several of theseare of a technical nature: Are the sets we consider Lebesgue/Borel measurable? Ina definition involving an infimum, is it attained? Often this is not really relevantfor the development of the theory. Quite often, on the other hand, measurabilitycarries combinatorial meaning, which makes this relationship truly exciting.
There were some interesting connections with algebra too. Balazs Szegedysolved a problem that arose as a dual to the characterization of homomorphismfunctions, and through his proof he established, among others, a deep connectionwith the representation theory of algebras. This connection was later further de-veloped by Schrijver and others. Another one of these generalizations has lead toa combinatorial theory of categories, which, apart from some sporadic results, hasnot been studied before. The limit theory of bounded degree graphs also found verystrong connections to algebra: finitely generated infinite groups yield, through theirCayley graphs, infinite bounded degree graphs, and representing these as limits offinite graphs has been studied in group theory (under the name of sofic groups)earlier.
These connections with very different parts of mathematics made it quite diffi-cult to write this book in a readable form. One way out could have been to focus ongraph theory, not to talk about issues whose motivation comes from outside graphtheory, and sketch or omit proofs that rely on substantial mathematical tools fromother parts. I felt that such an approach would hide what I found the most excitingfeature of this theory, namely its rich connections with other parts of mathematics(classical and non-classical). So I decided to explain as many of these connectionsas I could fit in the book; the reader will probably skip several parts if he/she doesnot like them or does not have the appropriate background, but perhaps the flavorof these parts can be remembered.
PREFACE xiii
The book has five main parts. First, an informal introduction to the math-ematical challenges provided by large networks. We ask the “general questions”mentioned above, and try to give an informal answer, using relatively elementarymathematics, and motivating the need for those more advanced methods that aredeveloped in the rest of the book.
The second part contains an algebraic treatment of homomorphism functionsand other graph parameters. The two main algebraic constructions (connectionmatrices and graph algebras) will play an important role later as well, but theyalso shed some light on the seemingly completely heterogeneous set of “graph pa-rameters”.
In the third part, which is the longest and perhaps most complete within itsown scope, the theory of convergent sequences of dense graphs is developed, andapplications to extremal graph theory and graph algorithms are given.
The fourth part contains an analogous theory of convergent sequences of graphswith bounded degree. This theory is more difficult and less well developed thanthe dense case, but it has even more important applications, not only becausemost networks arising in real life applications have low density, but also becauseof connections with the theory of finitely generated groups. Research on this topichas been perhaps the most active during the last months of my work, so the topicwas a “moving target”, and it was here where I had the hardest time drawing theline where to stop with understanding and explaining new results.
The fifth part deals with extensions. One could try to develop a limit theoryfor almost any kind of finite structures. Making a somewhat arbitrary selection,we only discuss extensions to edge-coloring models and categories, and say a fewwords about hypergraphs, to much less depth than graphs are discussed in partsIII and IV.
I included an Appendix about several diverse topics that are standard mathe-matics, but due to the broad nature of the connections of this material in mathe-matics, few readers would be familiar with all of them.
One of the factors that contributed to the (perhaps too large) size of this bookwas that I tried to work out many examples of graph parameters, graph sequences,limit objects, etc. Some of these may be trivial for some of the readers, others maybe tough, depending on one’s background. Since this is the first monograph on thesubject, I felt that such examples would help the reader to digest this quite diversematerial.
In addition, I included quite a few exercises. It is a good trick to squeeze alot of material into a book through this, but (honestly) I did try to find exercisesabout which I expected that, say, a graduate student of mathematics could solvethem with not too much effort.
Acknowledgements. I am very grateful to my coauthors of those papers thatform the basis of this book: Christian Borgs, Jennifer Chayes, Michael Freedman,Lex Schrijver, Vera Sos, Balazs Szegedy, and Kati Vesztergombi, for sharing theirideas, knowledge, and enthusiasm during our joint work, and for their advice and ex-tremely useful criticism in connection with this book. The creative atmosphere andcollaborative spirit at Microsoft Research made the successful start of this researchproject possible. It was a pleasure to do the last finishing touches on the book inRedmond again. The author acknowledges the support of ERC Grant No. 227701,
xiv PREFACE
OTKA grant No. CNK 77780, the hospitality of the Institute for Advanced Studyin Princeton, and in particular of Avi Wigderson, while writing most of this book.
My wife Kati Vesztergombi has not only contributed to the content, but hasprovided invaluable professional, technical and personal help all the time.
Many other colleagues have very unselfishly offered their expertise and adviceduring various phases of our research and while writing this book. I am particularlygrateful to Miklos Abert, Noga Alon, Endre Csoka, Gabor Elek, Guus Regts, SvanteJanson, David Kunszenti-Kovacs, Gabor Lippner, Russell Lyons, Jarik Nesetril,Yuval Peres, Oleg Pikhurko, the late Oded Schramm, Miki Simonovits, Vera Sos,Kevin Walker, and Dominic Welsh. Without their interest, encouragement andhelp, I would not have been able to finish my work.
Bibliography
M. Abert and T. Hubai: Benjamini-Schramm convergence and the distribution of chromatic rootsfor sparse graphs, http://arxiv.org/abs/1201.3861
J. Adamek, H. Herrlich and G.E. Strecker: Abstract and Concrete Categories: The Joy of Cats,Reprints in Theory and Applications of Categories 17 (2006), 1–507.
S. Adams: Trees and amenable equivalence relations, Ergodic Theory Dynam. Systems 10 (1990),1–14.
R. Ahlswede and G.O.H. Katona: Graphs with maximal number of adjacent pairs of edges, ActaMath. Hung. 32 (1978), 97–120.
R. Albert and A.-L. Barabasi: Statistical mechanics of complex networks, Rev. Modern Phys. 74(2002), 47–97.
D.J. Aldous: Representations for partially exchangeable arrays of random variables, J. Multivar.Anal. 11 (1981), 581–598.
D.J. Aldous: Tree-valued Markov chains and Poisson-Galton-Watson distributions, in: Microsur-veys in Discrete Probability (D. Aldous and J. Propp, editors), DIMACS Ser. Discrete Math.Theoret. Comput. Sci. 41, Amer. Math. Soc., Providence, RI. (1998), 1–20.
D. Aldous and R. Lyons: Processes on Unimodular Random Networks, Electron. J. Probab. 12,Paper 54 (2007), 1454–1508.
D.J. Aldous and M. Steele: The Objective Method: Probabilistic Combinatorial Optimizationand Local Weak Convergence, in: Discrete and Combinatorial Probability (H. Kesten, ed.),Springer (2003) 1–72.
N. Alon (unpublished)
N. Alon, R.A. Duke, H. Lefmann, V. Rodl and R. Yuster: The algorithmic aspects of the regularitylemma, J. Algorithms 16 (1994), 80–109.
N. Alon, E. Fischer, M. Krivelevich and M. Szegedy: Efficient testing of large graphs, Combina-torica 20 (2000) 451–476.
N. Alon, W. Fernandez de la Vega, R. Kannan and M. Karpinski: Random sampling and approx-imation of MAX-CSPs, J. Comput. System Sci. 67 (2003) 212–243.
N. Alon, E. Fischer, I. Newman and A. Shapira: A Combinatorial Characterization of the TestableGraph Properties: It’s All About Regularity, Proc. 38th ACM Symp. on Theory of Comput.(2006), 251–260.
N. Alon and A. Naor: Approximating the Cut-Norm via Grothendieck’s Inequality SIAM J. Com-put. 35 (2006), 787–803.
N. Alon, P.D. Seymour and R. Thomas: A separator theorem for non-planar graphs, J. Amer.Math. Soc. 3 (1990), 801–808.
N. Alon and A. Shapira: A Characterization of the (natural) Graph Properties Testable withOne-Sided Error, SIAM J. Comput. 37 (2008), 1703–1727.
N. Alon, A. Shapira and U. Stav: Can a Graph Have Distinct Regular Partitions? SIAM J. Discr.Math. 23 (2009), 278–287.
N. Alon and J. Spencer: The Probabilistic Method, Wiley–Interscience, 2000.
N. Alon and U. Stav: What is the furthest graph from a hereditary property? Random Struc.Alg. 33 (2008), 87–104.
451
452 BIBLIOGRAPHY
O. Angel and B. Szegedy (unpublished)
D. Aristoff and C. Radin: Emergent structures in large networks,http://arxiv.org/abs/1110.1912
S. Arora, D. Karger and M. Karpinski: Polynomial time approximation schemes for dense instancesof NP-hard problems, Proc. 27th ACM Symp. on Theory of Comput. (1995), 284–293.
T. Austin: On exchangeable random variables and the statistics of large graphs and hypergraphs,Probability Surveys 5 (2008), 80–145.
T. Austin and T. Tao: On the testability and repair of hereditary hypergraph properties, RandomStruc. Alg. 36 (2010), 373–463.
E. Babson and D. Kozlov: Topological obstructions to graph colorings, Electr. Res. Announc.AMS 9 (2003), 61–68.
E. Babson and D.N. Kozlov: Complexes of graph homomorphisms, Isr. J. Math. 152 (2006),285–312.
E. Babson and D.N. Kozlov: Proof of the Lovasz conjecture, Annals of Math. 165 (2007), 965–
1007.
A. Bandyopadhyay and D. Gamarnik: Counting without sampling. Asymptotics of the log-partition function for certain statistical physics models, Random Struc. Alg. 33 (2008), 452–479.
A.-L. Barabasi: Linked: The New Science of Networks, Perseus, Cambridge, MA (2002).
A.-L. Barabasi and R. Albert: Emergence of scaling in random networks, Science 286 (1999)509–512.
M. Bayati, D. Gamarnik and P. Tetali: Combinatorial Approach to the Interpolation Methodand Scaling Limits in Sparse Random Graphs, Proc. 42th ACM Symp. on Theory of Comput.(2010), 105–114.
J.L. Bell and A.B. Slomson: Models and Ultraproducts: An Introduction, Dover Publ. (2006).
I. Benjamini, L. Lovasz: Global Information from Local Observation, Proc. 43rd Ann. Symp. onFound. of Comp. Sci. (2002), 701–710.
I. Benjamini, G. Kozma, L. Lovasz, D. Romik and G. Tardos: Waiting for a bat to fly by (inpolynomial time), Combin. Prob. Comput. 15 (2006), 673–683.
I. Benjamini and O. Schramm: Recurrence of Distributional Limits of Finite Planar Graphs,Electronic J. Probab. 6 (2001), paper no. 23, 1–13.
I. Benjamini, O. Schramm and A. Shapira: Every Minor-Closed Property of Sparse Graphs isTestable, Advances in Math. 223 (2010), 2200–2218.
C. Berg, J.P.R. Christensen and P. Ressel: Positive definite functions on Abelian semigroups,
Math. Ann. 223 (1976), 253–272.
C. Berg and P.H. Maserick: Exponentially bounded positive definite functions, Ill. J. Math. 28(1984), 162–179.
A. Bertoni, P. Campadelli and R. Posenato: An upper bound for the maximum cut mean value,Graph-theoretic concepts in computer science, Lecture Notes in Comp. Sci., 1335, Springer,Berlin (1997), 78–84.
P. Billingsley: Convergence of Probability Measures, Wiley, New York (1999).
A. Bjorner and L. Lovasz: Pseudomodular lattices and continuous matroids, Acta Sci. Math.Szeged 51 (1987), 295-308.
G.R. Blakley and P.A. Roy: A Holder type inequality for symmetric matrices with nonnegativeentries, Proc. Amer. Math. Soc. 16 (1965) 1244–1245.
M. Boguna and R. Pastor-Satorras: Class of correlated random networks with hidden variables,Physical Review E 68 (2003), 036112.
B. Bollobas: Relations between sets of complete subgraphs, in: Combinatorics, Proc. 5th BritishComb. Conf. (ed. C.St.J.A. Nash-Williams, J. Sheehan), Utilitas Math. (1976), 79–84.
BIBLIOGRAPHY 453
B. Bollobas: A probabilistic proof of an asymptotic formula for the number of labelled regulargraphs, Europ. J. Combin. 1 (1980), 311–316.
B. Bollobas: Random Graphs, Second Edition, Cambridge University Press, 2001.
B. Bollobas, C. Borgs, J. Chayes and O. Riordan: Percolation on dense graph sequences, Ann.Prob. 38 (2010), 150–183.
B. Bollobas, S. Janson and O. Riordan: The phase transition in inhomogeneous random graphs,Random Struc. Alg. 31 (2007) 3–122.
B. Bollobas, S. Janson and O. Riordan: The cut metric, random graphs, and branching processes,J. Stat. Phys. 140 (2010) 289–335.
B. Bollobas, S. Janson and O. Riordan: Monotone graph limits and quasimonotone graphs, In-ternet Mathematics 8 (2012), 187–231.
B. Bollobas and V. Nikiforov: An abstract Szemeredi regularity lemma, in: Building bridges,Bolyai Soc. Math. Stud. 19, Springer, Berlin (2008), 219–240.
B. Bollobas and O. Riordan: A Tutte Polynomial for Coloured Graphs, Combin. Prob. Comput.
8 (1999), 45–93.
B. Bollobas and O. Riordan: Metrics for sparse graphs, in: Surveys in combinatorics 2009,Cambridge Univ. Press, Cambridge (2009) 211–287.
B. Bollobas and O. Riordan: Random graphs and branching processes, in: Handbook of large-scalerandom networks, Bolyai Soc. Math. Stud. 18, Springer, Berlin (2009), 15–115.
C. Borgs: Absence of Zeros for the Chromatic Polynomial on Bounded Degree Graphs, Combin.Prob. Comput. 15 (2006), 63–74.
C. Borgs, J. Chayes, J. Kahn and L. Lovasz: Left and right convergence of graphs with boundeddegree, http://arxiv.org/abs/1002.0115
C. Borgs, J. Chayes and L. Lovasz: Moments of Two-Variable Functions and the Uniqueness ofGraph Limits, Geom. Func. Anal. 19 (2010), 1597–1619.
C. Borgs, J. Chayes, L. Lovasz, V.T. Sos and K. Vesztergombi: Counting graph homomorphisms,in: Topics in Discrete Mathematics (ed. M. Klazar, J. Kratochvil, M. Loebl, J. Matousek,R. Thomas, P. Valtr), Springer (2006), 315–371.
C. Borgs, J.T. Chayes, L. Lovasz, V.T. Sos and K. Vesztergombi: Convergent Graph SequencesI: Subgraph frequencies, metric properties, and testing, Advances in Math. 219 (2008), 1801–1851.
C. Borgs, J.T. Chayes, L. Lovasz, V.T. Sos and K. Vesztergombi: Convergent Graph SequencesII: Multiway Cuts and Statistical Physics, Annals of Math. 176 (2012), 151–219.
C. Borgs, J.T. Chayes, L. Lovasz, V.T. Sos and K. Vesztergombi: Limits of randomly grown graphsequences, Europ. J. Combin. 32 (2011), 985–999.
C. Borgs, J.T. Chayes, L. Lovasz, V.T. Sos, B. Szegedy and K. Vesztergombi: Graph Limits andParameter Testing, STOC38 (2006), 261–270.
L. Bowen: Couplings of uniform spanning forests, Proc. Amer. Math. Soc. 132 (2004), 2151–2158.
G. Brightwell and P. Winkler: Graph homomorphisms and long range action, in Graphs, Mor-phisms and Statistical Physics, DIMACS Ser. Disc. Math. Theor. CS, American MathematicalSociety (2004), 29–48.
W.G. Brown, P. Erdos and M. Simonovits: Extremal problems for directed graphs, J. Combin.Theory B 15 (1973), 77–93.
W.G. Brown, P. Erdos and M. Simonovits: On multigraph extremal problems, Problemes combi-natoires et theorie des graphes, Colloq. Internat. CNRS 260 (1978), 63–66.
O.A. Camarena, E. Csoka, T. Hubai, G. Lippner and L. Lovasz: Positive graphs,http://arxiv.org/abs/1205.6510
S. Chatterjee and P. Diaconis: Estimating and understanding exponential random graph models,http://arxiv.org/abs/1102.2650
S. Chatterjee and S.R.S Varadhan: The large deviation principle for the Erdos–Renyi randomgraph, Europ. J. Combin. 32 (2011), 1000–1017.
454 BIBLIOGRAPHY
F.R.K. Chung: From Quasirandom graphs to Graph Limits and Graphletshttp://arxiv.org/abs/1203.2269
F.R.K. Chung, R.L. Graham and R.M. Wilson: Quasi-random graphs, Combinatorica 9 (1989),345–362.
F.R.K Chung and R.L. Graham: Quasi-Random Hypergraphs, Proc. Nat. Acad Sci. 86 (1989),pp. 8175–8177.
F.R.K Chung and R.L. Graham: Quasi-random subsets of Zn, J. Combin. Theory A 61 (1992),64–86.
D.L. Cohn: Measure Theory, Birkheuser, Boston, 1980.
D. Conlon and J. Fox: Bounds for graph regularity and removal lemmas,http://arxiv.org/abs/1107.4829
D. Conlon, J. Fox and B Sudakov: An approximate version of Sidorenko’s conjecture, Geom.Func. Anal. 20 (2010), 1354–1366.
J. Cooper: Quasirandom Permutations, J. Combin. Theory A 106 (2004), 123–143.
J. Cooper: A permutation regularity lemma, Electr. J. Combin. 13 R# 22 (2006).
E. Csoka: Local algorithms with global randomization on sparse graphs (manuscript)
E. Csoka: An undecidability result on limits of sparse graphs,http://arxiv.org/abs/1108.4995
E. Csoka: Maximum flow is approximable by deterministic constant-time algorithm in sparsenetworks, http://arxiv.org/abs/1005.0513
N.G. de Bruijn: Algebraic theory of Penrose’s non-periodic tilings of the plane I-II, Indag. Math.84 (1981) 39–52, 53–66.
P. Diaconis and D. Freedman: On the statistics of vision: the Julesz Conjecture, J. Math. Psy-chology 2 (1981) 112–138.
P. Diaconis and D. Freedman: The Markov Moment Problem and de Finetti’s Theorem: Part I,Math. Zeitschrift 247 (2004), 183–199.
P. Diaconis, S. Holmes and S. Janson: Threshold graph limits and random threshold graphs,Internet Math. 5 (2008), 267–320.
P. Diaconis, S. Holmes and S. Janson: Interval graph limits, http://arxiv.org/abs/1102.2841
P. Diaconis and S. Janson: Graph limits and exchangeable random graphs, Rendiconti di Matem-atica 28 (2008), 33–61.
R.L. Dobrushin: Estimates of semi-invariants for the Ising model at low temperatures, in: Topicsin Statistical and Theoretical Physics (AMS Translations, Ser. 2) bf 177 (1996), pp. 59–81.
W. Dorfler and W. Imrich: Uber das starke Produkt von endlichen Graphen, Osterreich. Akad.Wiss. Math.-Natur. Kl. S.-B. II 178 (1970), 247–262.
J. Draisma, D.C. Gijswijt, L. Lovasz, G. Regts and A. Schrijver: Characterizing partition functionsof the vertex model, J. of Algebra 350 (2012), 197–206.
G. Elek: On limits of finite graphs, Combinatorica 27 (2007), 503–507.
G. Elek: The combinatorial cost, Enseign. Math. bf53 (2007), 225–235.
G. Elek: L2-spectral invariants and convergent sequences of finite graphs, J. Functional Analysis254 (2008), 2667–2689.
G. Elek: Parameter testing with bounded degree graphs of subexponential growth, Random Struc.Alg. 37 (2010), 248–270.
G. Elek: On the limit of large girth sequences, Combinatorica 30 (2010), 553–563.
G. Elek: Finite graphs and amenability, http://arxiv.org/abs/1204.0449
G. Elek: Samplings and observables. Convergence and limits of metric measure spaces ,http://arxiv.org/abs/1205.6936
G. Elek and G. Lippner: Borel oracles. An analytical approach to constant-time algorithms, Proc.
Amer. Math. Soc. 138 (2010), 2939–2947.
BIBLIOGRAPHY 455
G. Elek and G. Lippner: An analogue of the Szemeredi Regularity Lemma for bounded degreegraphs, http://arxiv.org/abs/0809.2879
G. Elek and B. Szegedy: A measure-theory approach to the theory of dense hypergraphs, Advancesin Math. 231 (2012), 1731–1772.
P. Erdos: On sequences of integers no one of which divides the product of two others and on somerelated problems, Mitt. Forsch.-Inst. Math. Mech. Univ. Tomsk 2 (1938), 74–82;
P. Erdos: On some problems in graph theory, combinatorial analysis and combinatorial numbertheory, in: Graph Theory and Combinatorics, Academic Press, London (1984), 1–17.
P. Erdos, L. Lovasz and J. Spencer: Strong independence of graphcopy functions, in: GraphTheory and Related Topics, Academic Press (1979), 165-172.
P. Erdos and A. Renyi: On random graphs I, Publ. Math. Debrecen 6 (1959), 290–297.
P. Erdos and A. Renyi: On the evolution of random graphs, MTA Mat. Kut. Int. Kozl. 5 (1960),17–61.
P. Erdos and M. Simonovits: A limit theorem in graph theory, Studia Sci. Math. Hungar. 1
(1966), 51–57.
P. Erdos and A.H. Stone: On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946),1087–1091.
W. Feller, An Introduction to Probability Theory and its Applications, Second edition, Wiley, NewYork (1971).
E. Fischer: The art of uninformed decisions: A primer to property testing, The ComputationalComplexity Column of the Bulletin of the European Association for Theoretical ComputerScience 75 (2001), 97-126.
E. Fischer and I. Newman: Testing versus Estimation of Graph Properties, Proc. 37th ACM Symp.on Theory of Comput. (2005), 138–146.
D.C. Fisher: Lower bounds on the number of triangles in a graph, J. Graph Theory 13 (1989),505–512.
J. Fox: A new proof of the graph removal lemma, Annals of Math. 174 (2011), 561–579.
J. Fox and J. Pach (unpublished)
F. Franek and V. Rodl: Ramsey Problem on Multiplicities of Complete Subgraphs in NearlyQuasirandom Graphs, Graphs and Combin. 8 (1992), 299–308.
P. Frankl and V. Rodl: The uniformity lemma for hypergraphs, Graphs and Combin. 8 (1992),309–312.
P. Frankl and V. Rodl: Extremal problems on set systems, Random Struc. Alg. 20 (2002),131–164.
M. Freedman, L. Lovasz and A. Schrijver: Reflection positivity, rank connectivity, and homomor-phisms of graphs, J. Amer. Math. Soc. 20 (2007), 37–51.
A. Frieze and R. Kannan: Quick approximation to matrices and applications, Combinatorica 19(1999), 175–220.
D. Gaboriau: Invariants l2 de relations d’equivalence et de groupes, Publ. Math. Inst. Hautes.
Etudes Sci. 95 (2002), 93–150.
O. Gabber and Z. Galil: Explicit Constructions of Linear-Sized Superconcentrators, J. Comput.Syst. Sci. 22 (1981), 407–420.
D. Gamarnik: Right-convergence of sparse random graphs, http://arxiv.org/abs/1202.3123
D. Garijo, A. Goodall and J. Nesetril: Graph homomorphisms, the Tutte polynomial and “q-statePotts uniqueness”, Elect. Notes in Discr. Math. 34 (2009), 231–236.
D. Garijo, A. Goodall and J. Nesetril: Contractors for flows, Electronic Notes in Discr. Math. 38(2011), 389–394.
H.-O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter, Berlin, 1988.
S. Gerke and A. Steger: The sparse regularity lemma and its applications, Surveys in Combina-torics (2005), 227–258.
456 BIBLIOGRAPHY
E.N. Gilbert: Random graphs, Ann. Math. Stat. 30 (1959), 1141–1144.
B. Godlin, T. Kotek and J.A. Makowsky: Evaluations of Graph Polynomials, 34th Int. Workshopon Graph-Theoretic Concepts in Comp. Sci., Lecture Notes in Comp. Sci. 5344, Springer,Berlin–Heidelberg (2008), 183–194.
C. Godsil and G. Royle: Algebraic Graph Theory, Grad. Texts in Math. 207, Springer, New York(2001).
O. Goldreich (ed): Property Testing: Current Research and Suerveys, LNCS 6390, Springer, 2010.
O. Goldreich, S. Goldwasser and D. Ron: Property testing and its connection to learning andapproximation, J. ACM 45 (1998), 653–750.
O. Goldreich and D. Ron: A sublinear bipartiteness tester for bounded degree graphs, Combina-torica 19 (1999), 335–373.
O. Goldreich and D. Ron: Property testing in bounded degree graphs, Algorithmica 32 (2008),302–343.
O. Goldreich and L. Trevisan: Three theorems regarding testing graph properties, Random Struc.
Alg. , 23 (2003), 23–57.
A.W. Goodman: On sets of aquaintences and strangers at any party, Amer. Math. Monthly 66(1959) 778–783.
W.T. Gowers: Lower bounds of tower type for Szemeredi’s Uniformity Lemma, Geom. Func. Anal.7 (1997), 322–337.
W.T. Gowers: A new proof of Szemeredi’s theorem, Geom. Func. Anal. 11 (2001), 465–588.
W.T. Gowers: Quasirandomness, counting and regularity for 3-uniform hypergraphs, Combin.Prob. Comput. 15 (2006), 143–184.
W.T. Gowers: Hypergraph regularity and the multidimensional Szemeredi theorem, Annals ofMath. 166 (2007), 897–946.
B. Green and T. Tao: The primes contain arbitrarily long arithmetic progressions. Ann. of Math.167 (2008), 481–547.
B. Green, T. Tao and T. Ziegler: An inverse theorem for the Gowers Us+1[N ]-norm. Electron.Res. Announc. Math. Sci. 18 (2011), 69–90.
G.R. Grimmett and D.R. Stirzaker: Probability and Random Processes, Oxford University Press(1982).
M. Gromov: Metric structures for Riemannian and non-Riemannian spaces, Birkhauser (1999).
A. Grzesik: On the maximum number of C5’s in a triangle-free graph, J. Combin. Theory B 102(2012), 1061–1066.
P. Halmos: Measure theory, D. van Nostrand and Co., 1950.
V. Harangi: On the density of triangles and squares in regular finite and unimodular randomgraphs, http://www.renyi.hu/~harangi/papers/d3d4.pdf
P. de la Harpe and V.F.R. Jones: Graph Invariants Related to Statistical Mechanical Models:Examples and Problems, J. Combin. Theory B 57 (1993), 207–227.
A. Hassidim, J.A. Kelner, H.N. Nguyen and K. Onak: Local Graph Partitions for Approximationand Testing, Proc. 50th Ann. IEEE Symp. on Found. Comp. Science (2009), 22–31.
H. Hatami: Graph norms and Sidorenko’s conjecture, Israel J. Math. 175 (2010), 125–150.
H. Hatami, J. Hladky, D. Kral, S. Norine and A. Razborov: On the number of pentagons intriangle-free graphs, http://arxiv.org/abs/1102.1634
H. Hatami, J. Hladky, D. Kral, S. Norine and A. Razborov: Non-three-colorable common graphsexist, http://arxiv.org/abs/1105.0307
H. Hatami, L. Lovasz and B. Szegedy: Limits of local-global convergent graph sequenceshttp://arxiv.org/abs/1205.4356
H. Hatami and S. Norine: Undecidability of linear inequalities in graph homomorphism densities,J. Amer. Math. Soc. 24 (2011), 547–565.
BIBLIOGRAPHY 457
F. Hausdorff: Summationsmethoden und Momentfolgen I–II, Math. Z. 9 (1921), 74–109 and280–299.
D. Haussler and E. Welzl: ε-nets and simplex range queries, Discr. Comput. Geom. 2 (1987),127–151.
J. Haviland and A. Thomason: Pseudo-random hypergraphs. Graph theory and combinatorics(Cambridge, 1988). Discrete Math. 75 (1989), 255–278.
J. Haviland and A. Thomason: On testing the ”pseudo-randomness” of a hypergraph. DiscreteMath. 103 (1992), 321–327.
P. Hell and J. Nesetril: On the complexity of H-colouring, J. Combin. Theory B 48 (1990), 92–110.
P. Hell and J. Nesetril: Graphs and Homomorphisms, Oxford University Press, 2004.
J. Hladky, D. Kral and S. Norine: Counting flags in triangle-free digraphs, Electronic Notes inDiscr. Math. 34 (2009), 621–625.
J. Hladky: Bipartite subgraphs in a random cubic graph, Bachelor Thesis, Charles University,2006.
S. Hoory and N. Linial: A counterexample to a conjecture of Lovasz on the χ-coloring complex,J. Combin. Theory B 95 (2005), 346–349.
D. Hoover: Relations on Probability Spaces and Arrays of Random Variables, Institute for Ad-vanced Study, Princeton, NJ, 1979.
C. Hoppen, Y. Kohayakawa, C.G. Moreira, B. Rath and R. Menezes Sampaio: Limits of permu-tation sequences, http://arxiv.org/abs/1103.5844
C. Hoppen, Y. Kohayakawa, C.G. Moreira and R. Menezes Sampaio: Limits of permutationsequences through permutation regularity, http://arxiv.org/abs/1106.1663
Y.E. Ioannidis and R. Ramakrishnan: Containment of conjunctive queries: Beyond relations assets, ACM Transactions on Database Systems, 20 (1995), 288–324.
J. Isbell: Some inequalities in hom sets, J. Pure Appl. Algebra 76 (1991), 87–110.
Y. Ishigami: A Simple Regularization of Hypergraphs, http://arxiv.org/abs/math/0612838
C. Jagger, P. Stovıcek and A. Thomason: Multiplicities of subgraphs, Combinatorica 16 (1996),123–141.
S. Janson: Connectedness in graph limits, http://arxiv.org/abs/math/0802.3795v2
S. Janson: Graphons, cut norm and distance, couplings and rearrangements,http://arxiv.org/pdf/1009.2376.pdf
S. Janson: Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529–563.
S. Janson: Limits of interval orders and semiorders, J. Combinatorics 3 (2012), 163–183.
S. Janson: Graph limits and hereditary properties, http://arxiv.org/abs/1102.3571
S. Janson, T. �Luczak and A. Ruczynski: Random Graphs, Wiley, 2000.
V. Kaimanovich: Amenability, hyperfiniteness, and isoperimetric inequalities, C. R. Acad. Sci.Paris Sr. I Math. 325 (1997), 999–1004.
O. Kallenberg: Probabilistic Symmetries and Invariance Principles, Springer, New York, 2005.
Z. Kallus, P. Haga, P. Matray, G. Vattay and S. Laki: Complex geography of the internet network,Acta Phys. Pol. B 52, 1057–1069.
G.O.H. Katona: Continuous versions of some extremal hypergraph problems, in: Combinatorics,(Keszthely, Hungary, 1976), Coll. Math. Soc. J. Bolyai 18 (J. Bolyai Math. Soc., Budapest,1978), 653–678.
G.O.H. Katona: Continuous versions of some extremal hypergraph problems II, Acta Math. Hung.35 (1980), 67–77.
G.O.H. Katona: Probabilistic inequalities from extremal graph results (a survey), Ann. DiscreteMath. 28 (1985), 159–170.
A. Kechris and B.D. Miller: Topics in orbit equivalence theory, Lecture Notes in Mathematics1852. Springer-Verlag, Berlin, 2004.
458 BIBLIOGRAPHY
A. Kechris, S. Solecki and S. Todorcevic: Borel chromatic numbers, Advances in Math. 141(1999) 1–44.
H.G. Kellerer: Duality Theorems for Marginal Problems, Z. Wahrscheinlichkeitstheorie verw.Gebiete 67 (1984), 399–432.
K. Kimoto: Laplacians and spectral zeta functions of totally ordered categories, J. RamanujanMath. Soc. 18 (2003), 53–76.
K. Kimoto: Vandermonde-type determinants and Laplacians of categories, preprint No. 2003-24,Graduate School of Mathematics, Kyushu University (2003).
J. Kock: Frobenius Algebras and 2D Topological Quantum Field Theories, London Math. Soc.Student Texts, Cambridge University Press (2003).
Y. Kohayakawa: Szemeredi’s regularity lemma for sparse graphs, in: Sel. Papers Conf. Found. ofComp. Math., Springer (1997), 216–230.
Y. Kohayakawa, B. Nagle, V. Rodl and M. Schacht: Weak hypergraph regularity and linearhypergraphs, J. Combin. Theory B 100 (2010), 151–160.
Y. Kohayakawa and V. Rodl: Szemeredi’s regularity lemma and quasi-randomness, in: RecentAdvances in Algorithms and Combinatorics, CMS Books Math./Ouvrages Math. SMC 11,Springer, New York (2003), 289–351.
Y. Kohayakawa, V. Rodl and J. Skokan: Hypergraphs, quasi-randomness, and conditions for
regularity, J. Combin. Theory A 97 (2002), 307–352.
I. Kolossvary and B. Rath: Multigraph limits and exchangeability, Acta Math. Hung. 130 (2011),1–34.
J. Komlos, J. Pach and G. Woeginger: Almost Tight Bounds for epsilon-Nets, Discr. Comput.Geom. 7 (1992), 163–173.
J. Komlos and M. Simonovits: Szemeredi’s Regularity Lemma and its applications in graph the-ory, in: Combinatorics, Paul P. Erdos is Eighty (D. Miklos et. al, eds.), Bolyai SocietyMathematical Studies 2 (1996), pp. 295–352.
S. Kopparty and B. Rossman: The Homomorphism Domination Exponent, Europ. J. Combin.32 (2011), 1097–1114.
S. Kopparty: Local Structure: Subgraph Counts II, Rutgers University Lecture Notes,http://www.math.rutgers.edu/~sk1233/courses/graphtheory-F11/hom-inequalities.pdf
D. Kozlov: Combinatorial Algebraic Topology, Springer, Berlin, 2008.
B. Kra: The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point ofview, Bull. of the AMS 43 (2005), 3–23.
D. Kral and O. Pikhurko: Quasirandom permutations are characterized by 4-point densities,http://arxiv.org/abs/1205.3074
D. Kunszenti-Kovacs (unpublished)
M. Laczkovich: Closed sets without measurable matchings, Proc. of the Amer. Math. Soc. 103(1988), 894–896.
M. Laczkovich: Equidecomposability and discrepancy: a solution to Tarski’s circle squaring prob-lem, J. Reine und Angew. Math. 404 (1990), 77–117.
M. Laczkovich: Continuous max-flow min-cut theorems, Report 19. Summer Symp. in Real Anal-ysis, Real Analysis Exchange 21 (1995–96), 39.
J.B. Lasserre: A sum of squares approximation of nonnegative polynomials, SIAM Review 49(2007), 651–669.
J.L.X. Li and B. Szegedy: On the logarithimic calculus and Sidorenko’s conjecture,http://arxiv.org/abs/math/1107.1153v1
B. Lindstrom: Determinants on semilattices, Proc. Amer. Math. Soc. 20 (1969), 207–208.
R. Lipton and R.E. Tarjan: A separator theorem for planar graphs, SIAM Journal on AppliedMathematics 36 (1979), 177–189.
BIBLIOGRAPHY 459
P.A. Loeb: An introduction to non-standard analysis and hyperfinite probability theory, Prob-abilistic Analysis and Related Topics 2 (A.T. Bharucha-Reid, editor), Academic Press, NewYork (1979), 105–142.
D. London: Inequalities in quadratic forms, Duke Mathematical Journal, 33 (1966), 511–522.
L. Lovasz: Operations with structures, Acta Math. Hung. 18 (1967), 321–328.
L. Lovasz: On the cancellation law among finite relational structures, Periodica Math. Hung. 1(1971), 145–156.
L. Lovasz: Direct product in locally finite categories, Acta Sci. Math. Szeged 23 (1972), 319–322.
L. Lovasz: Kneser’s conjecture, chromatic number, and homotopy, J. Combin. Theory A 25(1978), 319-324.
L. Lovasz: Combinatorial Problems and Exercises, Akademiai Kiado - North Holland, Budapest,1979; reprinted by AMS Chelsea Publishing (2007).
L. Lovasz: Connection matrices, in: Combinatorics, Complexity and Chance, A Tribute to Do-minic Welsh Oxford Univ. Press (2007), 179–190.
L. Lovasz: The rank of connection matrices and the dimension of graph algebras, Europ. J. Com-bin. 27 (2006), 962–970.
L. Lovasz: Very large graphs, in: Current Developments in Mathematics 2008 (eds. D. Jerison,B. Mazur, T. Mrowka, W. Schmid, R. Stanley, and S. T. Yau), International Press, Somerville,MA (2009), 67–128.
L. Lovasz: Subgraph densities in signed graphons and the local Sidorenko conjecture, Electr.J. Combin. 18 (2011), P127 (21pp).
L. Lovasz: Notes on the book: Large networks, graph homomorphisms and graph limits,http://www.cs.elte.hu/~lovasz/book/homnotes.pdf
L. Lovasz and A. Schrijver: Graph parameters and semigroup functions, Europ. J. Combin. 29(2008), 987–1002.
L. Lovasz and A. Schrijver: Semidefinite functions on categories, Electr. J. Combin. 16 (2009),no. 2, Special volume in honor of Anders Bjorner, Research Paper 14, 16 pp.
L. Lovasz and A. Schrijver: Dual graph homomorphism functions, J. Combin. Theory A , 117(2010), 216–222.
L. Lovasz and M. Simonovits: On the number of complete subgraphs of a graph, in: Combina-torics, Proc. 5th British Comb. Conf. (ed. C.St.J.A.Nash-Williams, J.Sheehan), Utilitas Math.(1976), 439–441.
L. Lovasz and M. Simonovits: On the number of complete subgraphs of a graph II, in: Studies inPure Math., To the memory of P. Turan (ed. P. Erdos), Akademiai Kiado (1983), 459-495.
L. Lovasz and V.T. Sos: Generalized quasirandom graphs, J. Combin. Theory B 98 (2008),146–163.
L. Lovasz and L. Szakacs (unpublished)
L. Lovasz and B. Szegedy: Limits of dense graph sequences, J. Combin. Theory B 96 (2006),933–957.
L. Lovasz and B. Szegedy: Szemeredi’s Lemma for the analyst, Geom. Func. Anal. 17 (2007),252–270.
L. Lovasz and B. Szegedy: Contractors and connectors in graph algebras, J. Graph Theory 60(2009), 11–31.
L. Lovasz and B. Szegedy: Testing properties of graphs and functions, Israel J. Math. 178 (2010),113–156.
L. Lovasz and B. Szegedy: Regularity partitions and the topology of graphons, in: An IrregularMind, Szemeredi is 70, J. Bolyai Math. Soc. and Springer-Verlag (2010), 415–446.
L. Lovasz and B. Szegedy: Finitely forcible graphons, J. Combin. Theory B 101 (2011), 269–301.
L. Lovasz and B. Szegedy: Random Graphons and a Weak Positivstellensatz for Graphs, J. GraphTheory 70 (2012) 214–225.
460 BIBLIOGRAPHY
L. Lovasz and B. Szegedy: Limits of compact decorated graphs http://arxiv.org/abs/1010.5155
L. Lovasz and B. Szegedy: The graph theoretic moment problemhttp://arxiv.org/abs/1010.5159
L. Lovasz and K. Vesztergombi: Nondeterministic property testing,http://arxiv.org/abs/1202.5337
N. Lusin: Lecons par les Ensembles Analytiques et leurs Applications, Gauthier–Villars, Paris(1930).
R. Lyons: Asymptotic enumeration of spanning trees, Combin. Prob. Comput. 14 (2005) 491–522.
R. Lyons and F. Nazarov: Perfect Matchings as IID Factors on Non-Amenable Groups, Europ.J. Combin. 32 (2011), 1115–1125.
J.A. Makowsky: Connection Matrices for MSOL-Definable Structural Invariants, in: Logic andIts Applications, ICLA 2009 (eds. R. Ramanujam, S. Sarukkai) (2009), 51–64.
W. Mantel: Problem 28, solution by H. Gouwentak, W. Mantel, J. Teixeira de Mattes, F. Schuhand W.A. Wythoff, Wiskundige Opgaven 10 (1907), 60–61.
G.A. Margulis: Explicit construction of concentrators, Probl. Pered. Inform. 9 (1973), 71–80;English translation: Probl. Inform. Transmission, Plenum, New York (1975).
Yu.V. Matiyasevich: The Diophantineness of enumerable sets. Dokl. Akad. Nauk SSSR, 191:279–282, 1970.
J. Matousek: Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combina-torics and Geometry, Springer (2003).
B.D. McKay: Maximum Bipartite Subgraphs of Regular Graphs with Large Girth, Proc. 13th SEConf. on Combin. Graph Theory and Computing, Boca Raton, Florida (1982), 436.
R. McKenzie: Cardinal multiplication of structures with a reflexive relation, Fund. Math. 70(1971), 59–101.
J.W. Moon and L. Moser: On a problem of Turan, Mat. Kut. Int. Kozl. 7 (1962), 283–286.
H.P. Mulholland and C.A.B. Smith: An inequality arising in genetical theory, Amer. Math.Monthly 66 (1959), 673–683.
V. Muller: The edge reconstruction hypothesis is true for graphs with more than n · log2 n edges,J. Combin. Theory B 22 (1977), 281–283.
B. Nagle, V. Rodl and M. Schacht: The counting lemma for regular k-uniform hypergraphs,Random Struc. Alg. 28 (2006), 113–179.
J. Nesetril and P. Ossona de Mendez: How many F’s are there in G? Europ. J. Combin. 32(2011), 1126–1141.
Ilan Newman and Christian Sohler : Every Property of Hyperfinite Graphs is Testable, Proc. 43th
ACM Symp. on Theory of Comput. (2011) 675–684.
H. N. Nguyen and K. Onak, Constant-time approximation algorithms via local improvements,Proc. 49th Ann. IEEE Symp. on Found. Comp. Science (2008), 327–336.
V. Nikiforov: The number of cliques in graphs of given order and size, Trans. Amer. Math. Soc.363 (2011), 1599–1618.
G. Palla, L. Lovasz and T. Vicsek, Multifractal network generator, Proc. NAS 107 (2010), 7640–7645.
F.V. Petrov, A. Vershik: Uncountable graphs and invariant measures on the set of universalcountable graphs, Random Struc. Alg. 37 (2010), 389–406.
O. Pikhurko: An analytic approach to stability, Discrete Math, 310 (2010) 2951–2964.
B. Pittel: On a random graph evolving by degrees, Advances in Math. 223 (2010), 619–671.
A. Pultr: Isomorphism types of objects in categories determined by numbers of morphisms, ActaSci. Math. Szeged 35 (1973), 155–160.
C. Radin and M. Yin: Phase transitions in exponential random graphs,http://arxiv.org/abs/1108.0649v2
BIBLIOGRAPHY 461
B. Rath and L. Szakacs: Multigraph limit of the dense configuration model and the preferentialattachment graph, Acta Math. Hung. 136 (2012), 196–221.
A.A. Razborov: Flag Algebras, J. Symbolic Logic, 72 (2007), 1239–1282.
A.A. Razborov: On the minimal density of triangles in graphs, Combin. Prob. Comput. 17 (2008),603–618.
A.A. Razborov: On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discr. Math.24 (2010), 946–963.
G. Regts: The rank of edge connection matrices and the dimension of algebras of invariant tensors,Europ. J. Combin. 33 (2012), 1167–1173.
C. Reiher: Minimizing the number of cliques in graphs of given order and edge density (manu-script).
N. Robertson and P. Seymour: Graph Minors. XX. Wagner’s conjecture, J. Combin. Theory B92, 325–357.
V. A. Rohlin: On the fundamental ideas of measure theory, Translations Amer. Math. Soc., Series
1, 10 (1962), 1–54. Russian original: Mat. Sb. 25 (1949), 107–150.
K. Roth: Sur quelques ensembles d’entiers, C. R. Acad. Sci. Paris 234 (1952), 388–390.
V. Rodl and M. Schacht: Regular partitions of hypergraphs: Regularity Lemmas, Combin. Prob.Comput. 16 (2007), 833–885.
V. Rodl and M. Schacht: Regular partitions of hypergraphs: Counting Lemmas, Combin. Prob.Comput. 16 (2007), 887–901.
V. Rodl and J. Skokan: Regularity lemma for k-uniform hypergraphs, Random Struc. Alg. 25(2004), 1–42.
R. Rubinfeld and M. Sudan: Robust characterization of polynomials with applications to programtesting, SIAM J. Comput. 25 (1996), 252–271.
I.Z. Ruzsa and E. Szemeredi: Triple systems with no six points carrying three triangles, in: Com-binatorics, Proc. Fifth Hungarian Colloq., Keszthely, Bolyai Society–North Holland (1976),939–945.
R.H. Schelp and A. Thomason: Remark on the number of complete and empty subgraphs, Combin.Prob. Comput. 7 (1998), 217–219.
O. Schramm: Hyperfinite graph limits, Elect. Res. Announce. Math. Sci. 15 (2008), 17–23.
A. Schrijver: Tensor subalgebras and first fundamental theorems in invariant theory, J. of Algebra319 (2008) 1305–1319.
A. Schrijver: Graph invariants in the edge model, in: Building Bridges Between Mathematics andComputer Science (eds. M. Grotschel, G.O.H. Katona), Springer (2008), 487–498.
A. Schrijver: Graph invariants in the spin model, J. Combin. Theory B 99 (2009) 502–511.
A. Schrijver: Characterizing partition functions of the vertex model by rank growth (manuscript)
A. Scott and A. Sokal: On Dependency Graphs and the Lattice Gas, Combin. Prob. Comput. 15(2006), 253–279.
A. Scott: Szemeredi’s regularity lemma for matrices and sparse graphs, Combin. Prob. Comput.20 (2011), 455–466.
A.F. Sidorenko: Classes of hypergraphs and probabilistic inequalities, Dokl. Akad. Nauk SSSR254 (1980), 540–543.
A.F. Sidorenko: Extremal estimates of probability measures and their combinatorial nature (Rus-sian), lzv. Acad. Nauk SSSR 46 (1982), 535–568.
A.F. Sidorenko: Inequalities for functionals generated by bipartite graphs (Russian) Diskret. Mat.3 (1991), 50–65; translation in Discrete Math. Appl. 2 (1992), 489–504.
A.F. Sidorenko: A correlation inequality for bipartite graphs, Graphs and Combin. 9 (1993),201–204.
A.F. Sidorenko: Randomness friendly graphs, Random Struc. Alg. 8 (1996), 229–241.
B. Simon: The Statistical Mechanics of Lattice Gasses, Princeton University Press (1993).
462 BIBLIOGRAPHY
M. Simonovits: A method for solving extremal problems in graph theory, stability problems, in:Theory of Graphs, Proc. Colloq. Tihany 1966, Academic Press (1968), 279–319.
M. Simonovits: Extremal graph problems, degenerate extremal problems, and supersaturatedgraphs, in: Progress in Graph Theory, NY Academy Press (1984), 419–437.
M. Simonovits and V.T. Sos: Szemeredi’s partition and quasirandomness, Random Struc. Alg. 2(1991), 1–10.
M. Simonovits and V.T. Sos: Hereditarily extended properties, quasi-random graphs and notnecessarily induced subgraphs. Combinatorica 17 (1997), 577–596.
M. Simonovits and V.T. Sos: Hereditary extended properties, quasi-random graphs and inducedsubgraphs, Combin. Prob. Comput. 12 (2003), 319–344.
Ya.G. Sinai: Introduction to ergodic theory, Princeton Univ. Press (1976).
B. Szegedy: Edge coloring models and reflection positivity, J. Amer. Math. Soc. 20 (2007),969–988.
B. Szegedy: Edge coloring models as singular vertex coloring models, in: Fete of Combinatorics
(eds. G.O.H. Katona, A. Schrijver, T. Szonyi), Springer (2010), 327–336.
B. Szegedy: Gowers norms, regularization and limits of functions on abelian groups,http://arxiv.org/abs/1010.6211
B. Szegedy (unpublished).
E. Szemeredi: On sets of integers containing no k elements in arithmetic progression”, ActaArithmetica 27 (1975) 199–245.
E. Szemeredi: Regular partitions of graphs, Colloque Inter. CNRS (J.-C. Bermond, J.-C. Fournier,M. Las Vergnas and D. Sotteau, eds.) (1978) 399–401.
T. Tao: A variant of the hypergraph removal lemma, J. Combin. Theory A 113 (2006), 1257–1280.
T.C. Tao: Szemeredi’s regularity lemma revisited, Contrib. Discrete Math. 1 (2006), 8–28.
T.C. Tao: The dichotomy between structure and randomness, arithmetic progressions, and theprimes, in: Proc. Intern. Congress of Math. I (Eur. Math. Soc., Zurich, 2006) 581–608.
A. Thomason: Pseudorandom graphs, in: Random graphs ’85 North-Holland Math. Stud. 144,North-Holland, Amsterdam, 1987, 307–331.
A. Thomason: A disproof of a conjecture of Erdos in Ramsey theory, J. London Math. Soc. 39
(1898), 246–255.
P. Turan: Egy grafelmeleti szelsoertekfeladatrol. Mat. Fiz. Lapok bf 48 (1941), 436–453.
J.H. van Lint and R.M. Wilson: A Course in Combinatorics, Cambridge University Press (1992).
V. Vapnik and A. Chervonenkis: On the uniform convergence of relative frequencies of events totheir probabilities, Theor. Prob. Appl. 16 (1971), 264–280.
A.M. Vershik: Classification of measurable functions of several arguments, and invariantly dis-tributed random matrices, Funkts. Anal. Prilozh. 36 (2002), 12–28; English translation: Funct.Anal. Appl. 36 (2002), 93–105.
A.M. Vershik: Random metric spaces and universality, Uspekhi Mat. Nauk 59 (2004), 65–104;English translation: Russian Math. Surveys 59 (2004), 259–295.
D. Welsh: Complexity: Knots, Colourings and Counting, London Mathematical Society LectureNotes 186. Cambridge Univ. Press, Cambridge, 1993.
D. Welsh and C. Merino: The Potts model and the Tutte polynomial, Journal of MathematicalPhysics, 41 (2000), 1127–1152.
H. Whitney: The coloring of graphs, Ann. of Math. 33 (1932), 688–718.
H.S. Wilf: Hadamard determinants, Mobius functions, and the chromatic number of a graph,
Bull. Amer. Math. Soc. 74 (1968), 960–964.
D. Williams: Probability with Martingales, Cambridge Univ. Press, 1991.
E. Witten: Topological quantum field theory, Comm. Math. Phys. 117 (1988), 353–386.
E. Zeidler: Nonlinear functional analysis and its applications. Part I, Springer Verlag, New York,1985.
BIBLIOGRAPHY 463
A.A. Zykov: On Some Properties of Linear Complexes, Mat. Sbornik 24 (1949), 163–188; Amer.Math. Soc. Transl. 79 (1952), 1–33.
Author Index
Abert, Miklos, xiv
Adamek, Jiri, 447
Adams, Scot, 332
Ahlswede, Rudolph, 28
Albert, Reka, 10
Aldous, David, 23, 180, 184, 338, 341, 344,346, 357, 358
Alon, Noga, xiv, 8, 128, 143, 151, 154, 160,230, 272, 274, 276, 279, 294, 358, 385
Angel, Omer, 393
Aristoff, David, 261
Arora, Sanjeev, 19, 265
Artin, Emil, 446
Austin, Timothy, 279
Babson, Eric, 80
Bandyopadhyay, Antar, 377, 378
Barabasi, Albert-Laszlo, 10
Bayati, Mohsen, 399
Bell, John, 444
Benjamini, Itai, xi, 6, 23, 338, 351, 354,356, 360, 385, 402, 403
Berg, Christian, 443
Bertoni, Alberto, 376
Billingsley, Patrick, 185, 438
Bjorner, Anders, 430
Blakley, George, 28
Boguna, Marian, 157
Bollobas, Bela, 8, 26, 27, 123, 135, 141,157, 158, 221, 247, 286, 287, 399, 415
Borgs, Christian, xiii, 67, 73, 151, 153, 158,160, 164, 169, 174, 185, 187, 195, 201,210, 218, 221, 264, 265, 317, 375, 382
Bowen, Lewis, 357
Brightwell, Graham, 79
Brown, William, 307
Campadelli, Paola, 376
Chatterjee, Sourav, 260
Chayes, Jennifer, xi, xiii, 16, 67, 73, 151,153, 158, 160, 164, 169, 174, 185, 187,195, 201, 210, 218, 221, 264, 265, 317,375, 382
Chervonenkis, Alexey, 446
Christensen, Jens, 443
Chung, Fan, 9, 197, 308, 421, 422Conlon, David, 142, 143, 296Cooper, Joshua, 429Csoka, Endre, xiv, 405, 409, 410
de la Harpe, Pierre, 34, 63Diaconis, Persi, 8, 14, 157, 178, 180, 184,
193, 218, 221, 253, 257, 260, 441
Dobrushin, Roland, 67, 367, 370, 372, 378Dorfler, Willibald, 72Draisma, Jan, 418
Duke, Richard, 279
Elek, Gabor, xiv, 23, 180, 184, 199, 338,341, 357, 360, 383, 384, 391, 393, 398,399, 411, 421, 424, 430
Erdos, Paul, xi, 8, 10, 13, 16, 28, 29, 73, 74,157, 260, 293, 297, 307, 430
Feller, William, 441
Fischer, Eldar, 20, 143, 271, 272, 276Fisher, David, 27Fox, Jacob, 142, 143, 198, 296, 384
Franek, Frantisek, 297, 298Frankl, Peter, 424Freedman, David, 8, 14, 157, 441Freedman, Michael, xi, xiii, 47, 64, 75
Frieze, Alan, 127, 142, 145, 146, 230, 279,428
Frucht, Robert, 433
Gabber, Ofer, 384Gaboriau, Damien, 336Galil, Zvi, 384
Gamarnik, David, 377, 378, 382, 399Garijo, Delia, 97Gerke, Stefanie, 395, 415
Gijswijt, Dion, 418Gilbert, Edgar, 8Godlin, Benny, 46, 51Goldreich, Oded, xii, 5, 19, 20, 263, 265,
403Goldwasser, Shafi, xii, 5, 19, 265Goodall, Andrew, 97
465
466 AUTHOR INDEX
Goodman, Al, 26
Gowers, Timothy, 15, 142, 421, 422, 424,430
Graham, Ronald, 9, 197, 308, 421, 422Green, Benjamin, 430
Grimmet, Geoffrey, 191
Gromov, Mikhail, 430
Grothendieck, Alexander, 128Grzesik, Andrzej, 307
Haga, Peter, 22
Halmos, Paul, 445Harangi, Viktor, 359
Hassidim, Avinathan, 389
Hatami, Hamed, 30, 239–241, 287, 295–297,299, 303, 307, 348, 360, 361, 391
Hell, Pavol, 55, 425, 429Herrlich, Horst, 447
Hilbert, David, 446
Hladky, Jan, 297, 389
Holmes, Susan, 193, 218, 253Hoory, Shlomo, 80
Hoover, Douglas, 180, 184
Hoppen, Carlos, 429
Imrich, Wilfried, 72
Ioannidis, Yannis, 299
Isbell, John, 425Ishigami, Yoshiyasu, 424
Jagger, Chris, 297, 298Janson, Svante, xiv, 8, 122, 123, 131, 157,
158, 178, 180, 184, 193, 218, 221, 247,248, 253, 257, 429
Jones, Vaughan, 34, 63
Julesz, Bela, 14
Kahn, Jeff, 67, 73, 375, 382
Kaimanovich, Vadim, 391
Kallenberg, Olav, 184, 221Kallus, Zsofia, 22
Kannan, Ravindran, 127, 142, 145, 146,160, 230, 279, 428
Karger, David, 19, 265
Karpinski, Marek, 20, 160, 265Katalin, Vesztergombi, 280
Katona, Gyula O.H., 26–30, 287, 288
Kechris, Alexander, 330, 336, 349, 383
Kellerer, Hans, 438Kelner, Jonathan, 389
Kimoto, Kazufumi, 425
Kock, Joachim, 84
Kohayakawa, Yoshiharu, 395, 415, 422, 429Kolmogorov, Andrey, 374
Kolossvary, Istvan, 318
Komlos, Janos, 446
Kopparty, Swastik, 59, 299Kotek, Tomer, 46, 51
Kozlov, Dmitri, 80
Kozma, Gadi, 6
Kral, Daniel, 297
Kra, Bryna, 430
Krivelevich, Michael, 143, 272
Kruskal, Joseph, 26–28, 30, 287, 288
Kunszenti-Kovacs, David, xiv, 239
Laczkovich, Miklos, 138, 337, 338
Laki, Sandor, 22
Lasserre, Jean, 303
Lefmann, Hanno, 279
Li, Xiang, 296
Lindstrom, Bernt, 433
Linial, Nati, 80
Lippner, Gabor, xiv, 338, 393, 411
Lipton, Richard, 384
Loeb, Peter, 445
London, David, 28
�Los, Jerzy, 445
Lovasz, Laszlo, 6, 16, 27, 30, 47, 64, 67–70,72–78, 80, 98, 101, 125, 139, 141, 143,149, 151, 153, 157, 158, 160, 164, 166,167, 169, 174, 178, 180, 185, 187, 195,201, 210, 218, 221, 226, 232, 245, 253,257, 264, 265, 268, 280, 294, 296, 303,308, 309, 311, 317, 318, 324, 348, 360,361, 375, 382, 391, 418, 425–428, 430,446
�Luczak, Tomasz, 8
Lusin, Nikolai, 330
Lyons, Russell, xiv, 23, 67, 338, 341, 344,346, 357, 358, 399, 415
Matray, Peter, 22
Makowski, Janos, 46, 51
Mantel, W., 25, 26
Margulis, Grigory, 384
Maserick, Peter, 443
Matiyasevich, Yuri, 299, 447
Matousek, Jiri, 80
McKay, Brandan, 389
McKenzie, Ralph, 72
Menezes Sampaio, Rudini, 429
Merino, Criel, 64
Miller, Benjamin, 330, 336, 349, 383
Moon, John, 293
Moreira, Carlos Gustavo, 429
Moser, Leo, 293
Mulholland, Hugh P., 28
Muller, Vladimir, 69
Nagle, Brendan, 424
Naor, Assaf, 128
Nesetril, Jarik, xiv, 55, 59, 97, 425, 429
Newman, Ilan, 271, 276, 404
Nguyen, Huy, 24, 389, 405, 407
Nikiforov, Vlado, 27, 141, 289
Norine, Serguey, 30, 287, 297, 299, 303, 307
AUTHOR INDEX 467
Onak, Krzysztof, 24, 389, 405, 407
Ossona de Mendez, Patrice, 59
Pach, Janos, 384, 446
Paley, Raymond, 9Palla, Gergely, 158Pastor-Satorras, Romualdo, 157
Penrose, Roger, 355Peres, Yuval, xiv
Petrov, Fedor, 178Pikhurko, Oleg, xiv, 135, 139, 153, 155
Posenato, Roberto, 376Pultr, Ales, 425
Radin, Charles, 261Ramakrishnan, Raghu, 299Rath, Balazs, 318, 429
Razborov, Alexander, xii, 25, 27, 284, 288,289, 297, 307, 425, 427
Regts, Guus, xiv, 418, 421
Reiher, Christian, 27, 289Renyi, Alfred, 8, 10, 157, 260
Ressel, Paul, 443Riordan, Oliver, 123, 135, 157, 158, 221,
247, 415
Robertson, Neil, 49Rodl, Vojtech, 279, 297, 298, 421, 422, 424
Romik, Dan, 6Ron, Dana, xii, 5, 19, 265, 403
Rossman, Benjamin, 59Rota, Gian-Carlo, 433
Roth, Klaus, 430Roy, Prabir, 28Rubinfeld, Ronitt, 19
Ruczinski, Andrzej, 8Ruzsa, Imre, 198
Sauer, Norbert, 446Schacht, Matthias, 421, 424
Schelp, Richard, 286, 292Schramm, Oded, xi, xiv, 23, 338, 351, 354,
356, 360, 383, 385, 389, 402, 403
Schrijver, Alexander, xi–xiii, 53, 64, 75, 76,108, 174, 185, 418, 419, 421, 425–427
Schutzenberger, Marcel-Paul, 433
Scott, Alexander, 67, 395, 415Seymour, Paul, 49, 385
Shapira, Asaf, 154, 274, 276, 385, 402, 403Shelah, Saharon, 446Sidorenko, Alexander, 17, 29, 295, 297
Simonovits, Miklos, xiv, 9, 27, 29, 293, 295,307
Sinai, Yakov, 121
Skokan, Jozef, 421, 422, 424Slomson, Alan, 444
Smith, Cedric, 28Sohler, Christian, 404
Sokal, Alan, 67Solecki, Slawomir, 330
Sos, Vera T., xi, xiii, xiv, 9, 16, 151, 153,160, 164, 169, 174, 185, 187, 195, 201,210, 221, 264, 265, 309, 317
Spencer, Joel, xi, 8, 16, 73, 74
Stav, Uri, 154, 294
Steger, Angelika, 395, 415
Stirzaker, David, 191
Stone, Arthur, 29, 293
Stovıcek, Pavel, 297, 298
Strecker, George, 447
Sudakov, Benjamin, 296
Sudan, Madhu, 19
Szakacs, Laszlo, 318, 326
Szegedy, Balazs, xi, xiii, 16, 30, 77, 78, 98,108, 125, 139, 141, 143, 149, 157, 166,167, 178, 180, 184, 199, 226, 232, 234,245, 253, 257, 268, 294–296, 303, 308,311, 318, 324, 348, 360, 361, 391, 393,418, 421, 424, 431, 446
Szegedy, Mario, 143, 272
Szemeredi, Endre, xii, 13, 15, 21, 141, 142,167, 180, 198, 199, 276, 279, 428, 430
Tao, Terence, 13, 141, 279, 421, 424, 430
Tardos, Gabor, 6
Tarjan, Robert, 384
Tetali, Prasad, 399
Thomas, Robin, 385
Thomason, Andrew, 9, 286, 292, 297, 298
Todorcevic, Stevo, 330
Trevisan, Lucca, 263
Turan, Pal, 13, 25–27, 430
Tutte, William, 48, 64, 434, 453, 462
van Lint, Jakobus, 434
Vapnik, Vladimir, 446
Varadhan, Srinivasa, 260
Vattay, Gabor, 22
Vershik, Anatoly, 430
Vershik, Antoly, 178
Vesztergombi, Katalin, xi, xiii, 16, 151, 153,160, 164, 169, 174, 185, 187, 195, 201,202, 210, 221, 264, 265, 317
Vicsek, Tamas, 158
von Neumann, John, 430
Walker, Kevin, xiv
Welsh, Dominic, xiv, 47, 64, 435
Whitney, Hassler, 74
Wilf, Herbert, 433
Williams, David, 441
Wilson, Richard, 9, 197, 308, 434
Winkler, Peter, 79
Witten, Edward, 86
Woeginger, Gerhard, 446
Yin, Mei, 261
Yuster, Raphael, 279
468 AUTHOR INDEX
Zeidler, Eberhard, 312Ziegler, Tamar, 431
Subject Index
adjacency matrix, 39
automorphism, 234average ε-net, 228Azuma’s Inequality, 163, 265, 440, 441
ball distance, 339
Bernoulli lift, 348Bernoulli shift, 342blowup of graph, 40
bond, 38Borel coloring, 330Borel graph, 329
Cauchy sequence, 174chromatic invariant, 435chromatic polynomial, 47, 435circle graph, 416
cluster of homomorphisms, 80component map, 341concatenation of graphs, 85
conjugate in concatenation algebra, 85connection matrix, 43
dual, 76flat, 43
connection rank, 44, 107connector, 94constituent of quantum graph, 83
continuous geometry, 430contractor, 94convergent graph sequence, 173
Counting Lemma, 167Inverse, 169
cutmaximum, 64
cut distance, 12, 128, 132cut metric, 128cut norm, 127, 131
distance from a norm, 135distinguishable by sampling, 266dough folding map, 342
edge-coloring, 417edge-coloring function, 417edge-coloring model, 417
edge-connection matrix, 416
edit distance, 11, 128, 352
epi-mono decomposition, 448
epimorphism, 448
equipartition, 37
equitable partition, 37
equivalence graphon, 255
expansion
proper, 94
flag algebra, 289, 427
formula
first order, 50
monadic second order, 50
node-monadic second order, 50
free energy, 265
Frobenius algebra, 84
Frobenius identity, 84
grandmother graph, 341
graph
H-colored
isomorphic, 71
ε-homogeneous, 141
ε-regular, 142
k-broken, 416
bi-labeled, 85
decorated, 120
directed, 429
edge-weighted, 38
eulerian, 63, 89
expander, 384
flat, 39
fully labeled, 39
gaudy, 51
H-colored, 71
hyperfinite, 383
labeled, 39
looped-simple, 38
measure preserving, 332
multilabeled, 39
norming, 239
partially labeled, 39, 85
planar, 383
469
470 SUBJECT INDEX
random
evolving, 8
involution invariant, 340
multitype, 8
randomly weighted, 77
seminorming, 239
series-parallel, 88
signed, 39
simple, 38
simply labeled, 39
W-random, 157
weakly norming, 239
weighted, 38
Graph of Graphs, 339
graph parameter, 41
additive, 41
contractible, 95
estimable, 263
finite rank, 44
gaudy, 52
isolate indifferent, 257
isolate-indifferent, 41
maxing, 41
minor-monotone, 42
multiplicative, 41
normalized, 41
reflection positive, 44
flatly, 44
simple, 41
graph property, 41
distinguishable by sampling, 402
hereditary, 42, 247
minor-closed, 42
monotone, 41
random-free, 247
robust, 273
testable, 272, 402
graph sequence
bounded growth, 384
hyperfinite, 383
locally convergent, 16, 353
quasirandom, 9, 187, 197
multitype, 10
subexponential growth, 384
graphing, 23, 332
bilocally isomorphic, 347
cyclic, 330
cylic, 334
hyperfinite, 385
local isomorphism, 346
locally equivalent, 346
random rooted component, 341
stationary distribution, 333
graphon, 16, 115, 217
degree function, 116
finitely forcible, 308
infinitesimally finitely forcible, 314
triangle-free, 247
graphon property, 247
testable, 268
graphon variety, 250
r-graph, 421
Grothendieck norm, 128
ground state energy, 65
microcanonical, 204
Holder property, 240
Hadwiger number, 54
half-graph, 13
Hamilton cycle, 47
Hausdorff Moment Problem, 442
homomorphism, 55
homomorphism density, 6, 58
homomorphism entropy, 202, 204, 375
homomorphism frequency, 59
hypergraph
r-uniform, 421
complete, 422
complete r-partite, 422hypergraph sequence
quasirandom, 422
idempotent
degree, 91
resolve, 89
support, 102
idempotent basis, 89
internet, 3
intersection graph, 38
interval graph, 218
isomorphism up to a nullset, 121
kernel, 115, 217
connected, 122
direct sum, 122
pullback, 217
pure, 222
regular, 250
tensor product, 123
twin-free, 219
unlabeled, 132
weakly isomorphic, 121kernel variety, 250
simple, 250
K-graphon, 322
left-convergent, 173
left-generator, 448
limit
dense graph sequence, 180
weak, 175
line-graph, 38, 331
local distance, 331
Mobius function, 433
Mobius inverse, 42
Mobius matrix, 433
SUBJECT INDEX 471
Markov chain, 416, 439
reversible, 439stationary distribution, 439
step distribution, 439matching
perfect, 84
measureergodic, 259
unimodular, 340measure preserving, 336
Minkowski dimension, 231moment matrix, 443
monomorphism, 448Monotone Reordering Theorem, 253, 442morphism
coproduct, 448left-isomorphic, 447
product, 447pullback, 448
pushout, 448multicut, 64
restricted, 204multigraph, 38multiplicative over coproducts, 426
near-blowup of graph, 40
neighborhood distance, 222neighborhood sampling, 22
networks, 3node
labeled, 28node cover number, 46node evaluation function, 417
norminvariant, 135
object
terminal, 447zero, 447
oblivious testing, 272orientation
eulerian, 49, 97, 119overlay
fractional, 129
Polya urn, 191
Paley graph, 9, 237parameter estimation, 18
partially ordered set, 429partition
fractional, 211, 212legitimate, 225
partition function, 265
Penrose tiling, 355perfect matching, 46, 417
preferential attachment graphgrowing, 191
prefix attachment graph, 188probability space, 436
atom, 436
Borel, 436countably generated, 436
separating, 436standard, 437
product of graphs, 40
gluing, 42property testing, 5, 19
quantum graph, 83
k-labeled, 84loopless, 84
simple, 84quantum morphism, 427
quasirandomness, 422quotient
graph
fractional, 212quotient graph, 40, 141, 144
simple, 40quotient set, 211
fractional, 212restricted, 211
Rado graph, 178random graph
ultimate, 256random graph model, 175
consistent, 175countable, 177
local, 175local countable, 177
random graphon model, 256random variable
exchangeable, 184
random walk, 439rank of kernel, 254
Reconstruction Conjecture, 69(α, β, k)-regularization, 424
Regularity Lemma, xii, 14, 142, 145, 199Strong, 143, 148
Weak, 142Removal Lemma, 197, 198, 273right-generator, 448
root, 339
S-flow, 63sample concentration, 158, 159
sampling distance, 12, 351nondeterministic, 360
Sampling LemmaFirst, 160Second, 164, 165
Schatten norm, 135semidefinite programming, 304
sequencewell distributed, 185
sequence of distributionsconsistent, 353
472 SUBJECT INDEX
involution invariant, 354similarity distance, 226square-sum, 282stability number, 46stable set polynomial, 65stationary walk, 439stepfunction, 115, 442
stepping operator, 144subgraph sampling, 5support graph, 336
template graph, 8, 141tensor network, 418tensoring method, 240test parameter, 263test property, 266, 268threshold graph
random, 193threshold graphon, 187, 253topology
local, 331weak, 226
treespanning, 48
tree-decomposition, 107
tree-width, 107Turan graph, 25Tutte polynomial, 48twin nodes, 39, 101twin points, 219twin reduction, 40
ultrafilter, 444principal, 444
ultralimit, 445ultrametric, 331ultraproduct, 444uniform attachment graph, 188unlabeling, 86
Vapnik–Chervonenkis dimension,VC-dimension, 446
variation distance, 12Voronoi cell, 228
weak convergence, 139
zeta matrix, 433
Notation Index
�,�A indicator function, 37
A Borel sets in the graph of graphs, 339
A � 0 positive semidefinite, 37
a× b product in category, 447
a⊕ b coproduct in category, 448
A ·B dot product of matrices, 37
α ∨ β pullback, 448
α ∧ β morphisms in pushout, 448
αϕ nodeweight of homomorphism, 56
AG adjacency matrix, 39
α(G) stability number, 41
A(G,x) characteristic polynomial, 67
αH nodeweight-sum, 58
αi(H) nodeweight, 38
A+ sigma-algebra of weighted rootedgraphs, 343
Aut(W ) automorphism group, 234
a1W1 ⊕ a2W2 ⊕ . . . direct sum of kernels,122
BG,r(v) ball of radius r about v, 38
B(H, r) neighborhood of root, 339
βi,j(H) edgeweight, 38
Bm m-bond, 38
Bm•• labeled bond, 39
Br r-balls, 339
C complex numbers, 37
Ca cyclic graphing, 330
cep(G;u, v) cluster expansion polynomial,
434
chr(G, q) chromatic polynomial, 435
chr0(G, k) number of colorations, 435
Cn cycle, 38
col(G,h) edge-coloring function, 417
Conn(G) connected subgraphs, 38
cri(G) chromatic invariant, 435
Csp(G) connected spanning subgraphs, 38
C(U,H), C(U,W ) overlay functional, 205
cut(G,B) maximum multicut, 64
d1(G,G′) edit distance, 128, 352
d�(G,G′) labeled cut distance, 128, 129
δ�(G,G′), δ�(G,G′) unlabeled cutdistance, 129
d�(G,G′, X) overlay cut distance, 129
δ�(U,W ) cut distance, 132d�(U,W ) cut norm distance, 131
deg,degA, degGA degree, 331
degc(ϕ, v) edge-colored degree, 417
deg(H) degree of root, 339dG(X,Y ) edge density between sets, 141
dHaus(A,B) Hausdorff distance, 208d•(H1,H2) ball distance, 339
dimVC(H) Vapnik–Chervonenkisdimension, 446
δ(r,k) (G1, G2), δnd (G1, G2)
nondeterministic sampling distances,360
δN (U,W ), δ1(U,W ), δ2(U,W ) distancesderived from norm, 135
dob(W ) Dobrushin value, 370
δr(F,F ′), δ(F, F ′) sampling distance, 351
δsamp(G,G′), δsamp(W,W ′) samplingdistance, 12, 158
dsim(s, t) similarity distance, 277
d◦(u, v) local distance, 331
dvar(α, β) variation distance, 12dW (x) normalized degree, 116
E+, E− signed edge-sets, 39
e(G) = |E(G)| number of edges, 38
η, ηG edge measure, 333eG(X,Y ) set of edges between X and Y , 38
ent(G,H) homomorphism entropy, 202
ent∗(G,H) homomorphism entropytypical, 204
ent∗(G,W ) sparse homomorphism entropy,375
Eul eulerian property, 63−→eul(F ) number of eulerian orientations, 49
f↑, f↓, f⇓ Mobius inverses, 42
F1F2 gluing product, 42F †, F ‡, F labeled quantum graphs from
simple graph, 283
F ◦G concatenation of graphs, 85
473
474 NOTATION INDEX
F signed graph from simple graph, 57
F(K),Fn(K) compact decorated graphs,318
Fsimpk simple graphs on [k], 38
Fk,m bi-labeled graphs, 85, 86
F•k ,F•
S k-labeled and S-labeled graphs, 39
Fmultk multigraphs on [k], 38
F stabk k-labeled graphs with stable labeled
set, 95
flo(G, q) nowhere-zero q-flows, 436
F ∗ conjugate in concatenation algebra, 85
G1�G2 Cartesian sum, 40
G1 ×G2 categorical product, 40
G1 �G2 strong product, 40
G1 ∗G2 gluing k-broken graphs, 416
G→ edge-rooted graphs, 340
G•,G• rooted graphs, 339
G•F extensions of F , 339
G(H) randomized weighted graph, 157
G(k,G) random induced subgraph, 157
G×k categorical power, 40
G(m) blow-up of G, 40
G(n, p),G(n,m),G(n;H) random graphs, 8
G+ Bernoulli lift, 348
G+ weighted rooted graphs, 343
Gpan growing preferential attachment graph,
191
Gpfxn prefix attachment graph, 188
G/P quotient graph, 40, 83, 141, 211
GP averaged graph, 141
G/ρ fractional quotient graph, 212
G[S] induced subgraph, 38
Gsimp simple version of G, 38
G(n,W ),G(S,W ) W -random graphs, 157
Guan uniform attachment graph, 188
[[G]] unlabeling, 39
[[G]]S removing labels not in S, 39
Gx connected component containing x, 338
h(α) head of morphism, 447
H(a,B)) weighted graph with nodeweightvector a and edgeweight matrix B, 38
H graph of graphs, 339
H+ graph of weighted graphs, 343
hom(F,G) homomorphism number, 6, 56,77, 319
Hom(F,G) set of homomorphisms, 56
Hom(F,G) graph of homomorphisms, 79
Hom(F,G) complex of homomorphisms, 80
homϕ(F,G) weight of homomorphism, 56,58
hom∗(G,H) typical homomorphismnumber, 204
hom(F,X), inj(F,X) homomorphism
polynomials, 108
H(n,W ),H(S,W ) W -random weightedgraphs, 157
I(G) set of stable sets, 65ind(F,G) number of embeddings, 56inj(F,G) injective homomorphism number,
56, 57I(W ) entropy functional, 260I(W ) induced subgraphs of graphon, 247
[k] = {1, 2, . . . , k}, 12K••
1 2-multilabeled graph on one node, 39K(a, b) morphisms, 447K•
a,b, K••a,b partially labeled complete
bipartite graphs, 39Kin
a ,Kouta morphisms into and out of a, 447
χ(G) chromatic number, 41Kn complete graph, 38K◦
n looped complete graph, 38K•
n, K••n partially labeled complete graphs,
39Kr
n complete hypergraph, 422
L(C) intersection graph, 38L(G) line-graph, 38, 331λi(G), λ′
i(G), λi(W ), λ′i(W ) eigenvalues,
195limω ultralimit, 445ln, log, log∗ natural, binary and iterated
logarithm, 37Lrn complete r-partite hypergraph, 422
L(W ) space of functions txy, 311
Maxcut(G),maxcut(G) maximum cut, 64M(f, k), Msimp(f, k), Mmult(f, k),
Mflat(f, k),M(f,N) connectionmatrices, 43
M ′(f, k) edge-connection matrix, 416MG Mobius inverse of ZG, 83MA
hom,MAinj,M
Asurj,M
Aaut matrices of
homomorphism numbers, 73
Mk(f) moments of a function, 442μ(x, y), μP Mobius function, 433, 434
N,N∗ nonnegative integers and positiveintegers, 37
N(f, k) dual connection matrix, 76
ν(G) matching number, 41NG(v) = N(v) neighbors of v, 38(n)k = n(n− 1) . . . (n− k + 1), 37Nk(f) annihilator of quantum graphs, 84∇(v) edges incident with v, 38
Ob(K) objects of category, 447ω(G) size of maximum clique, 41On edgeless graph, 38
Pa path graphing, 330πG,H , πG,W , πy distribution on
homomorphisms, 368Π(α) partitions of [0, 1], 205Π(n),Π(n,α) partitions of [n], 204Π∗(n,α) fractional partitions, 212
NOTATION INDEX 475
pm(G) number of perfect matchings, 41
Pn path, 38
P •n , P
••n partially labeled paths, 39∏
ω ultraproduct, 444
P closure of graph property, 247
P(T ) probability measures on Borel sets,438
Qq(G),Qa(G) quotient sets, 211
Q∗q(G),Q∗
a(G) fractional quotient sets, 212
Qk k-labeled quantum graphs, 84
Qk/f , Qk,k/f factor algebras, 84
Qstabk k-labeled quantum graphs with stable
labeled set, 95
qr(H) quasirandomness, 422
R,R+ reals and nonnegative reals, 37
Rcε graphons far from R, 270
r(f, k) connection rank function, 44
ρG,r neighborhood sample distribution, 22
root(H) root of graph, 339
rW (a, b) similarity distance, 226
rW (x, y) neighborhood distance, 222
S[0,1], S[0,1] measure preserving maps, 437
σG,k subgraph sample distribution, 5
Sn star, 38
σ+ distribution of weighted graphs, 343
Spec(W ) spectrum, 124σ∗(A) probability measure scaled by
degrees, 339stab(G) number of stable sets, 46
stab(G, x) stable set polynomial, 65
surj(F,G) surjective homomorphismnumber, 56
t(α) tail of morphism, 447
t(F,G), tinj(F,G), tind(F,G)homomorphism densities, 58, 319
t(F,W ), tind(F,W ) homomorphismdensities in graphons, 116, 117
t(F,w) decorated homomorphism density,120, 323
τ(G) node cover number, 46
τ(H) node-cover of hypergraph, 446Tk tensors with k slots, 419
T (n, r) Turan graph, 25
tree(G) number of spanning trees, 435
t∗(F,G), t∗inj(F,G), t∗ind(F,G)
homomorphism frequencies, 59
tut(G;x, y) Tutte polynomial, 434
TW kernel operator, 124
tx(F,W ) labeled homomorphism density,117
UW,U ◦W,U ⊗W products of kernels, 123
U [X] submatrix of kernel, 160
v(G) = |V (G)| number of nodes, 38
W,W0,W1 spaces of kernels and graphons,115
W, W0, W1 unlabeled kernels, 132Wϕ(x, y) variable transformation, 121WH graphon from graph, 115W(K) K-graphons, 322Wn,W ◦n,W⊗n powers of kernels, 123
W/P quotient graph, 144WP (x, y) stepping operator, 144
x ≥ 0 (for U) nonnegativity of quantumgraphs, 281
x ≡ y (mod f) congruence of quantumgraphs, 84
Z,Zq integers and integers modulo q, 37ZG sum of quotients, 83
Published Titles in This Series
60 Laszlo Lovasz, Large Networks and Graph Limits, 2012
58 Freydoon Shahidi, Eisenstein Series and Automorphic L-Functions, 2010
57 John Friedlander and Henryk Iwaniec, Opera de Cribro, 2010
56 Richard Elman, Nikita Karpenko, and Alexander Merkurjev, The Algebraic andGeometric Theory of Quadratic Forms, 2008
55 Alain Connes and Matilde Marcolli, Noncommutative Geometry, Quantum Fields andMotives, 2008
54 Barry Simon, Orthogonal Polynomials on the Unit Circle: Part 1: Classical Theory; Part2: Spectral Theory, 2005
53 Henryk Iwaniec and Emmanuel Kowalski, Analytic Number Theory, 2004
52 Dusa McDuff and Dietmar Salamon, J-holomorphic Curves and Symplectic Topology,Second Edition, 2012
51 Alexander Beilinson and Vladimir Drinfeld, Chiral Algebras, 2004
50 E. B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, 2002
49 Vladimir V. Chepyzhov and Mark I. Vishik, Attractors for Equations ofMathematical Physics, 2002
48 Yoav Benyamini and Joram Lindenstrauss, Geometric Nonlinear Functional Analysis,2000
47 Yuri I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, 1999
46 J. Bourgain, Global Solutions of Nonlinear Schrodinger Equations, 1999
45 Nicholas M. Katz and Peter Sarnak, Random Matrices, Frobenius Eigenvalues, andMonodromy, 1999
44 Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol,The Book of Involutions, 1998
43 Luis A. Caffarelli and Xavier Cabre, Fully Nonlinear Elliptic Equations, 1995
42 Victor W. Guillemin and Shlomo Sternberg, Variations on a Theme by Kepler, 1991
40 R. H. Bing, The Geometric Topology of 3-Manifolds, 1983
39 Nathan Jacobson, Structure and Representations of Jordan Algebras, 1968
38 O. Ore, Theory of Graphs, 1962
37 N. Jacobson, Structure of Rings, 1956
36 Walter Helbig Gottschalk and Gustav Arnold Hedlund, Topological Dynamics, 1955
34 J. L. Walsh, The Location of Critical Points of Analytic and Harmonic Functions, 1950
33 J. F. Ritt, Differential Algebra, 1950
32 R. L. Wilder, Topology of Manifolds, 1949
31 E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, 1996
30 Tibor Rado, Length and Area, 1948
29 A. Weil, Foundations of Algebraic Geometry, 1946
28 G. T. Whyburn, Analytic Topology, 1942
27 S. Lefschetz, Algebraic Topology, 1942
26 N. Levinson, Gap and Density Theorems, 1940
25 Garrett Birkhoff, Lattice Theory, 1940
24 A. A. Albert, Structure of Algebras, 1939
23 G. Szego, Orthogonal Polynomials, 1939
22 Charles N. Moore, Summable Series and Convergence Factors, 1938
21 Joseph Miller Thomas, Differential Systems, 1937
20 J. L. Walsh, Interpolation and Approximation by Rational Functions in the ComplexDomain, 1935
19 N. Wiener and R. C. Paley, Fourier Transforms in the Complex Domain, 1934
18 M. Morse, The Calculus of Variations in the Large, 1934
17 J. H. M. Wedderburn, Lectures on Matrices, 1934
PUBLISHED TITLES IN THIS SERIES
16 Gilbert Ames Bliss, Algebraic Functions, 1933
15 M. H. Stone, Linear Transformations in Hilbert Space and Their Applications toAnalysis, 1932
14 Joseph Fels Ritt, Differential Equations from the Algebraic Standpoint, 1932
13 R. L. Moore, Foundations of Point Set Theory, 1932
12 Solomon Lefschetz, Topology, 1930
11 Dunham Jackson, The Theory of Approximation, 1930
10 Arthur B. Coble, Algebraic Geometry and Theta Functions, 1929
9 George D. Birkhoff, Dynamical Systems, 1927
8 L. P. Eisenhart, Non-Riemannian Geometry, 1927
7 Eric T. Bell, Algebraic Arithmetic, 1927
6 Griffith Conrad Evans, The Logarithmic Potential: Discontinuous Dirichlet andNeumann Problems, 1927
5 Griffith Conrad Evans and Oswald Veblen, The Cambridge Colloquium, 1918
4 Leonard Eugene Dickson and William Fogg Osgood, The Madison Colloquium, 1914
3 Gilbert Ames Bliss and Edward Kasner, The Princeton Colloquium, 1913
2 Max Mason, Eliakim Hastings Moore, and Ernest Julius Wilczynski, The NewHaven Colloquium, 1910
1 Frederick Shenstone Woods, Henry Seely White, and Edward Burr Van Vleck,The Boston Colloquium, 1905
Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathemati-cal theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as “property testing” in computer science and regularity par-tition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connec-tions with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization).
This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits.
This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future.
—Persi Diaconis, Stanford University
This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding math-ematician who is also a great expositor.
—Noga Alon, Tel Aviv University, Israel
Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovász’s book exemplifies this phe-nomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory.
—Terence Tao, University of California, Los Angeles, CA
László Lovász has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovász’s position as the main architect of this rapidly developing theory. The book is a must for combinato-rialists, network theorists, and theoretical computer scientists alike.
—Bela Bollobas, Cambridge University, UK
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