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.

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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.

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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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