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Graph in the Information Age

n the past decade, graph theory has gone through a remarkable shift and a profound transformation. The change is in large part due to the humongous amount of informa- tion that we are confronted with. A main way Ito sort through massive data sets is to build and examine the network formed by interrelations. For example, Google’s successful Web search al- gorithms are based on the WWW graph, which contains all Web pages as vertices and hyper- links as edges. There are all sorts of information networks, such as biological networks built from biological and social networks formed by email, phone calls, instant messaging, etc., as well as various types of physical networks. Of particular interest to mathematicians is the col- laboration graph, which is based on the data from Mathematical Reviews. In the , every mathematician is a , and two mathe- maticians who wrote a joint paper are connected by an edge. Figure 1illustrates a portionofthe collaboration Figure 1. An of the graph consisting of about 5,000 vertices, repre- collaboration graph. senting mathematicians with Erd˝os number 2 (i.e., mathematicians who wrote a paper with a coauthor of Paul Erd˝os). been used in a wide range of areas. However, Graph theory has two hundred years of history never before have we confronted graphs of not studying the basic mathematical structures called only such tremendous sizes but also extraordinary graphs. A graph G consists of a collection V of richness and complexity, both at a theoretical and vertices and a collection E of edges that connect a practical level. Numerous challenging problems pairs of vertices. In the past, graph theory has have attracted the attention and imagination of Fan Chung is professor of at the University researchers from , , engi- of California, San Diego. Her email address is fan@ucsd. neering, , social science, and mathematics. edu. The new area of “” emerged, call- ing for a sound scientific foundation and rigorous ∗This article is based on the Noether Lecture given at the AMS-MAA-SIAM Annual Meeting, January 2009, Wash- analysis for which graph theory is ideally suited. In ington D. C. the other direction, examples of real-world graphs

726 Notices of the AMS Volume 57, Number 6 lead to central questions and new directions for graph in G(n, p) has the same expected research in graph theory. at every vertex, and therefore G(n, p) does not These real-world networks are massive and capture some of the main behaviors of real-world complex but illustrate amazing coherence. Empir- graphs. Nevertheless, the approaches and methods ically, most real-world graphs have the following in classical theory provide the properties: foundation for the study of random graphs with sparsity—The number of edges is within general degree distributions. • a constant multiple of the number of Many random graph models have been pro- vertices. posed in the study of information network graphs, small world phenomenon—Any two ver- but there are basically two different approaches. • tices are connected by a short . Two The “online” model mimics the growth or decay of vertices having a common neighbor are a dynamically changing network, and the “offline” more likely to be neighbors. model of random graphs consists of specified fam- powerlawdegree distribution—The degree ilies of graphs as the spaces together • of a vertex is the number of its neighbors. with some specified probability distribution. The number of vertices with degree j (or One online model is the so-called preferential β attachment scheme, which can be described as having j neighbors) is proportional to j− for some fixed constant β. “the rich get richer”. The preferential attachment scheme has been receiving much attention in the To deal with these information networks, many recent study of complex networks [11, 57], but its basic questions arise: What are basic structures of history can be traced back to Vilfredo Pareto in such large networks? How do they evolve? What 1896, among others. At each tick of the clock (so are the underlying principles that dictate their to speak), a new edge is added, with each of its behavior? How are subgraphs related to the large endpoints chosen with probability proportional to (and often incomplete) host graph? What are the their degrees. It can be proved [15, 31, 57] that the main graph invariants that capture the myriad preferential attachment scheme leads to a power properties of such large graphs? law . There are several other Toanswertheseproblems,wefirstdelveintothe online models, including the duplication model wealth of knowledge from the past, although it is (which seems to be more feasible for biological often not enough. In the past thirty years there has networks, see [35]), as well as many recent exten- been a great deal of progress in combinatorial and sions, such as adding more parameters concerning probabilistic methods, as well as spectralmethods. the “talent” or “fitness” of each node [50]. However, traditional probabilistic methods mostly There are two main offline graph models consider the same probability distribution for all for graphs with general degree distribution—the vertices or edges while real graphs are uneven configuration model and random graphs with ex- and clustered. The classical algebraic and ana- pected degree sequences. A random graph in lytic methods are efficient in dealing with highly the configuration model with degree sequences symmetric structures, whereas real-world graphs d1, d2, . . . , dn is defined by choosing a random are quite the opposite. Guided by examples of on Pi di “pseudo nodes”, where the real-world graphs, we are compelled to improvise, pseudo nodes are partitioned into parts of sizes extend, and create new theory and methods. Here di , for i 1, . . . , n. Each part is associated with a we will discuss the new developments in several vertex.By= using results of Molloy andReed[58, 59], topics in graph theory that are rapidly developing. it can be shown [2] that under some mild condi- The topics include a general random graph theory tions, a random power law graph with exponent for any given degree distribution, in β almost surely has no giant if β β0 general host graphs, PageRank for representing ≥ where β0 is a solution to the equation involving the quantitative correlations among vertices, and the Riemann zeta function ζ(β 2) 2ζ(β 1) 0. game aspects of graphs. The general random graph− model− G(−w) =with expected degree sequence w (w1, w2, . . . , wn) Random Graph Theory for General Degree follows the spirit of the Erd˝os-Rényi= model. The Distributions probability of having an edge between the ith and The primary subject in the study of random graph jth vertices is defined to be wi wj /Vol (G), where w theory is the classical random graph G(n, p), Vol (G) denotes Pi wi . Furthermore, in G( ) each introduced by Erd˝os and Rényi in 1959 [38, 39] edge is chosen independently of the others, and (also independently by Gilbert [44]). In G(n, p), therefore the analysis can be carried out. It was every pair of a of n vertices is chosen to proved in [28] that if the expected average degree be an edge with probability p. In a series of is strictly greater than 1 in a random graph papers, Erd˝os and Rényi gave an elegant and in G(w), then there is a (i.e., comprehensive analysis describing the evolution a connected component of volume a positive of G(n, p) as p increases. Note that a random fraction of that of the whole graph). Furthermore,

June/July 2010 Notices of the AMS 727 the giant component almost surely has volume the host graph and vice versa. It is of interest to δVol (G) O(√n log3.5 n), where δ is the unique understand the connections between a graph and nonzero root+ of the following equation [29]: its subgraphs. What invariants of the host graph n n canorcannotbetranslatedtoitssubgraphs?Under wi δ (1) X wi e− (1 δ) X wi . what conditions can we predict the behavior of all = − i 1 i 1 or any subgraphs? Can a sparse subgraph have = = Because of the robustness of the G(w) model, very different behavior from its host graph? Here many properties can be derived. For example, a we discuss some of the work in this direction. randomgraphin G(w) has averagedistance almost Many information networks or social networks surely equal to (1 o(1)) log n , and the diameter have very small diameters (in the range of log n), as + log w˜ dictated by the so-called small world phenomenon. is almost surely ( log n ), where w˜ w 2/ w log w˜ Pi i Pi i However, in a recent paper by Liben-Nowell and Θ =w provided some mild conditions on are satisfied Kleinberg [52], it was observed that the -like [27]. For the range 2 < β < 3, where the power law subgraphs derived from some chain-letter data exponents β for numerous real networks reside, seem to have relatively large diameter. In the the power law graph can be roughly described as study of the Erd˝os-Rényi graph model G(n, p), it an “octopus” with a dense subgraph having small was shown [60] that the diameter of a random diameter O(log log n) as the , while the overall is of order √n, in contrast with diameter is O(log n) and the average is the fact that the diameter of the host graph Kn O(log log n) (see [31]). is 1. Aldous [4] proved that in a For the spectra of power law graphs, there are G with a certain spectral bound σ , the diameter basically two competing approaches. One is to of a random spanning tree T of G, denoted by prove analogues of Wigner’s semicircle law (which diam(T ), has expected value satisfying is the case for G(n, p)), while the other predicts cσ √n c√n log n that the eigenvalues follow a power law distri- E(diam(T )) bution [40]. Although the semicircle law and the log n ≤ ≤ √σ power law have very different descriptions, both for some absolute constant c. In [32], it was assertions are essentially correct if the appropriate shown that for a general host graph G, with high matrices associated with a graph are considered probability the diameter of a random spanning [33, 34]. For β > 2.5, the largest eigenvalue of the tree of G is between c√n and c′√n log n, where c adjacency of a random power law graph and c′ depend on the spectral gap of G and the is almost surely (1 o(1))√m, where m is the ratio of the moments of the degree sequence. + maximum degree. Moreover, the k largest eigen- One way to treat random subgraphs of a given values have power law distribution with exponent graph G is as a (bond) percolation problem. For 2β 1 if the maximum degree is sufficiently large a positive value p 1, we consider G , which is − p and k is bounded above by a function depending formed by retaining≤ each edge independently with on β, m and w. When 2 < β < 2.5, the largest probability p and discarding the edge with proba- 3 β eigenvalue is heavily concentrated at cm − for bility 1 p. A fundamental problem of interest is some constant c depending on β and the aver- to determine− the critical probability p for which age degree. Furthermore, the eigenvalues of the Gp contains a giant connected component. In the (normalized) Laplacian satisfy the semicircle law applications of epidemics, we consider a general under the condition that the minimum expected host graph being a , consisting of degree is relatively large [34]. edges formed by pairs of people with possible The online model is obviously much harder contact. The question of determining the critical to analyze than the offline model. One possible probability then corresponds to the problem of approach is to couple the online model with the finding the epidemic threshold for the spreading offline model of random graphs with a similar of the disease. degree distribution. This means to find the appro- Percolation problems have long been studied priate conditions under which the online model [45, 49] in theoretical physics, especially with the can be sandwiched by two offline models within host graph being the Zk. Percolation some error bounds. In such cases, we can apply problems on lattices are known to be notoriously the techniques from the offline model to predict difficult even for low dimensions and have only the behavior of the online model (see [30]). been resolved very recently by bootstrap perco- lation [8, 9]. In the past, percolation problems Random Subgraphs in Given Host Graphs have been examined for a number of special host information networks that we observe graphs. Ajtai, Komlós, and Szemerédi considered are subgraphs of some host graphs that often the percolation on hypercubes [3]. Their work was have sizes prohibitively large or with incomplete further extended to Cayley graphs [16, 17, 18, 55] information. A natural question is to attempt to and regular graphs [42]. For expander graphs with deduce the properties of a random subgraph from degrees bounded by d, Alon, Benjamini, and Stacey

728 Notices of the AMS Volume 57, Number 6 [5] proved that the is greater for the Web graph, the concept and definitions than or equal to 1/(2d). In the other direction, work well for any graph. Indeed, PageRank has Bollobás, Borgs, Chayes, and Riordan [13] showed become a valuable tool for examining the correla- that for dense graphs (where the degrees are of tions of pairs of vertices (or pairs of subsets)in any order (n)), the giant component threshold is 1/ρ, given graph and hence leads to many applications whereΘρ is the largest eigenvalue of the adjacency in graph theory. matrix. The special case of having the complete The starting point of the PageRank is a typical graph Kn as the host graph concerns the Erd˝os- random walk on a graph G with edge weights wuv Rényi graph G(n, p), which is known to have the for edge u, v. The probability transition matrix P wuv critical probability at 1/n, as well as the “double is defined by: P(u, v) , where du wu,v . = du = Pv jump” near the threshold. For a preference vector s, and a jumping constant For general host graphs, the answer has been α > 0, the PageRank, denoted by pr(α, s) as a row elusive. One way to address such questions is vector, can be expressed as a series of random to search for appropriate conditions on the host walks as follows: graph so that percolations can be controlled. ∞ Recently it has been shown [26] that if a given (2) pr α X(1 α)ksP k. α,f = − host graph G satisfies some (mild) conditions k 0 = depending on its spectral gap and higher moments Equivalently, pr(α, s) satisfies the following recur- of its degree sequence, for any ǫ > 0, if p > rence : (1 ǫ)/d˜, then asymptotically almost surely the + (3) pr(α, s) αs (1 α)pr(α, s)P. percolated subgraph Gp has a giant component. In = + − the other direction, if p < (1 ǫ)/d˜, then almost In the original definition of Brin and Page [19], s is − surely the percolated subgraph Gp contains no taken to be the constant function with value 1/n at giant component. We note that the second order every vertex motivated by modeling the behavior ˜ ˜ 2 average degree d is d d /( dv ), where dv of a typical surfer who moves to a random page = Pv v Pv denotes the degree of v. with probability α and clicks a linked page with In general, subgraphs can have spectral gaps probability 1 α . − very different from those of the host graph. How- Because of the close connection of PageRank ever, if a graph G has all its nontrivial eigenvalues with random walks, there are very efficient and ro- of the (normalized) Laplacian lying in the range bust for computing and approximating within σ from the value 1, then it can be shown PageRank [6, 12, 47]. This leads to numerous ap- [25] that almost surely a random subgraph Gp has plications, including the basic problem of finding all its nontrivial eigenvalues in the same range (up a “good” in a graph. A quantitative measure to a lower-order term) if the degrees are not too for the “goodness” of a cut that separates a subset small. S of vertices is the Cheeger ratio: E(S, S)¯ PageRank and Local Partitioning h(S) | | , In graph theory there are many essential geo- = vol (S) metrical notions, such as (typically, the where E(S, S)¯ denotes the set of edges leaving S number of hops required to reach one vertex from and vol (S) Pv S dv . The Cheeger constant hG another), cuts (i.e., subsets of vertices/edges that of a graph is= the minimum∈ Cheeger ratio over all separate a part of the graph from the rest), flows subsets S with vol (S) vol (G)/2. The traditional (i.e., combinations of paths for routing between divide-and-conquer strategy≤ in algorithmic design given vertices), and so on. However, real-world relies on finding a cut with small Cheeger ratio. graphs exhibit the small world phenomenon, so Since the problem of finding any cut that achieves any pair of vertices are connected through a very the Cheeger constant of G is NP-hard [43], one of short path. Therefore the usual notion of graph the most widely used approximation algorithms distance is no longer very useful. Instead, we need was a spectral partitioning . By using a quantitative and precise formulation to differen- eigenvectors to line up the vertices, the spectral tiate among nodes that are “local” from “global” partitioning algorithm reduces the number of and “akin” from “dissimilar”. This is exactly what cuts under consideration from an exponential PageRank is meant to achieve. number of possibilities to a linear number of In 1998 Brin and Page [19] introduced the notion choices. Nevertheless, there is still a performance of PageRank for Google’s Web search algorithm. guarantee provided by the Cheeger inequality: Different from the usual methods in pattern match- h2 2 ing previously used in data retrieval, the novelty f hG 2hG λ , of PageRank relies entirely on the underlying Web ≥ ≥ 2 ≥ 2 graph to determine the “importance” of a Web where hf is the minimum Cheeger ratio among page. Although PageRank is originally designed subsets that are initial segments in the order

June/July 2010 Notices of the AMS 729 determined by the eigenvector f associated with route and uses strategies to maximize possible the spectral gap λ. payoff. In other words, we face a combination of For large graphs with billions of nodes, it is and graph theory for dealing with not feasible to compute eigenvectors. In addition, large networks both in quantitative analysis and it is of interest to have local cuts in the sense algorithm design. Many questions arise. Instead that for given seeds and the specified size for the of just proving the existence of Nash equilibrium, parts to be separated, it is desirable to find a cut we would like to design algorithms to effectively near the seeds separating a subset of the desired compute or approximate the Nash equilibrium. size. Furthermore, the cost/complexity of finding How rapidly can such algorithms converge? such a cut should be proportional to the specified There has been a great deal of progress in the size of the separated part but independent of computational complexity of Nash equilibrium the total size of the whole graph. Here, PageRank [22, 37]. comes into play. Earlier, Spielman and Teng [63] The analysis of selfish routing comes naturally introduced local partitioning algorithms by using in network management. How much does unco- random walks with the performance analysis us- ordinated routing affect the performance of the ing a mixing result of Lovász and Simonovitz [54] network, such as stability, congestion, and delay? (also see [56]). As it turns out, by using PageRank What are the trade-offs for some limited regula- instead of random walks, there is an improved partitioning algorithm [6] for which the perfor- tion? The so-called price of anarchy refers to the mance is supported by a local Cheeger inequality worst-case analysis to evaluate the loss of collec- for a subset S of vertices in a graph G: tive welfare from selfish routing. There has been extensive research done on selfish routing [62]. h2 2 g hS The reader is referred to several surveys [41, 51] hS λS , ≥ ≥ 8 log vol (S) ≥ 8 log vol (S) and some recent books on this topic [61].

where λS is the Dirichlet eigenvalue of the induced Many classical problems in graph theory can be subgraph on S, hS is the local Cheeger constant reexamined from the perspective of game theory. of S defined by hS minT S h(T ), and hg is the One popular topic on graphs is chromatic graph = ⊆ minimum Cheeger ratio over all PageRank g with theory. For a given graph G, what is the minimum the seed as a vertex in S and α appropriately number of colors needed to color the vertices of chosen depending only on the volume of S. This G so that adjacent vertices have different colors? approximation partition algorithm can be further In addition to theoretical interests, the graph improved using the fact that the set of seeds for coloring problem has numerous applications in which the PageRank leads to the Cheeger ratio the setting of conflict resolution. For example,each satisfying the above local Cheeger inequality is faculty member (as a vertex) wishes to schedule quite large (about half of the volume of S). We classes in a limited number of classrooms (as note that the local partitioning algorithm can also colors). Two faculty members who have classes be used as a subroutine for finding balanced cuts with overlapping time are connected by an edge, for the whole graph. and then the problem of classroom scheduling can Note that PageRank is expressed as a geometric be viewed as a problem. Instead sum of random walks in (2). Instead, we can con- of having a central agency to make assignments, sider an exponential sum of random walks, called we can imagine a game-theoretic scenario that the heat kernel pagerank, which in turn satisfies the faculty members coordinate among themselves to heat equation. The heat kernel pagerank leads to an improved local Cheeger inequality [23, 24] decide a nonconflicting assignment. Suppose there by removing the logarithmic factor in the lower is a payoff of 1 unit for each player (vertex) if its bound. Numerous problems in graph theory can color is different from all its neighbors. A proper possibly take advantage of PageRank and its vari- coloring is then a Nash equilibrium, since no player ations, and the full implications of these ideas has an incentive to change his/her strategy. remain to be explored. Kearns et al. [48] conducted an experimental study of several coloring games on specified net- Network Games works. Many examples were given to illustrate the In morning traffic, every commuter chooses difficulties in analyzing the dynamics of large net- his/her most convenient way to get to work works in which each node takes simple but selfish without paying attention to the consequences of steps. This calls for rigorous analysis, especially the decision to others. The Internet network can along the line of the combinatorial probabilistic be viewed as a similar macrocosm that functions methods and generalized Martingale approaches neither by the control of a central authority nor by that have been developed in the past ten years coordinated rules. The basic motivation for each [20]. Some work in this direction has been done on individual can only be deduced by greed and self- a multiple round model of graph coloring games ishness. Every player chooses the most convenient [20], but more work is needed.

730 Notices of the AMS Volume 57, Number 6 Summary [14] B. Bollobás, Y. Kohayakawa, and T. Łuczak, It is clear that we are at the beginning of a new The evolution of random subgraphs of the cube, Random Structures and Algorithms 3(1), 55–90, journey in graph theory, emerging as a central (1992). part of the information revolution. It is a long way [15] B. Bollabás and O. Riordan, Robustness and vul- from the “seven bridges of Königsberg”, a problem nerability of scale-free random graphs, Internet posed by in 1736. In contrast Math. 1 (2003) no. 1, 1–35. to its origin in , graph [16] C. Borgs, J. Chayes, R. van der Hofstad, G. Slade, theory today uses sophisticated combinatorial, and J. Spencer, Random subgraphs of finite graphs: I. The scaling window under the triangle condition, probabilistic, and spectral methods with deep Random Structures and Algorithms 27 (2005), 137– connections with a variety of areas in mathematics 184. and computer science. In this article, some vibrant [17] , Random subgraphs of finite graphs: II. The new directions in graph theory have been selected lace expansion and the triangle condition, Annals of 33 and described to illustrate the richness of the Probability (2005), 1886–1944. [18] , Random subgraphs of finite graphs: III. The mathematics involved, as well as the utilization phase transition for the n-cube, Combinatorica 26 through major threads of current technology. The (2006), 395–410. list of the sampled topics is by no means complete, [19] S. Brin and . Page, The anatomy of a large-scale hy- since these areas of graph theory are still rapidly pertextual Web search engine, Computer Networks developing. Abundant opportunities in research, and ISDN Systems 30 (1–7), (1998), 107–117. [20] K. Chaudhuri, F. Chung, and M. S. Jamall,A theoretical and applied, remain to be explored. network game, Proceedings of WINE 2008, Lecture Notes in Computer Science, Vol. 5385, Springer References Berlin/Heidelberg (2008), 522–530. [1] D. Achlioptas, R. M. D’Souza, and J. Spencer, Ex- [21] J. Cheeger, A lower bound for the smallest eigen- plosive percolation in random networks, Science value of the Laplacian, Problems in Analysis (R. C. 323, no. 5920, (2009), 1453–1455. Gunning, ed.), Princeton Univ. Press, Princeton, NJ, [2] W. Aiello, F. Chung, and L. Lu, A random graph 1970, pp. 195–199. model for massive graphs, Proceedings of the [22] X. Chen and X. Deng, Settling the complexity of Thirty-Second Annual ACM Symposium on Theory 2-player Nash-equilibrium, The 47th Annual IEEE of Computing, ACM Press, New York, 2000, pp. Symposium on Foundations of Computer Science, 171–180. FOCS 2006, pp. 261–272. [3] M. Ajtai, J. Komlós, and E. Szemerédi, Largest [23] F. Chung, The heat kernel as the pagerank of a random component of a k-cube, Combinatorica 2 graph, Proc. Nat. Acad. Sciences 105 (50) (2007), (1982), 1–7. 19735–19740. [4] D. Aldous, The random walk construction of uni- [24] , A local graph partitioning algorithm using form spanning trees, SIAM J. Discrete Math. 3 heat kernel pagerank, WAW 2009, Lecture Notes in (1990), 450–465. Computer Science, vol. 5427, Springer, 2009, pp. 62– [5] N. Alon, I. Benjamini, and A. Stacey, Percola- 75. tion on finite graphs and isoperimetric inequalities, [25] F. Chung and P. Horn, The spectral gap of a Annals of Probability 32, no. 3 (2004), 1727–1745. random subgraph of a graph, Special issue of [6] R. Andersen, F. Chung, and K. Lang, Local graph WAW2006, Internet Math 2 (2007), 225–244. partitioning using pagerank vectors, Proceedings of [26] F. Chung, P. Horn, and L. Lu, The giant component the 47th Annual IEEE Symposium on Foundation of in a random subgraph of a given graph, Proceedings Computer Science (FOCS’2006), pp. 475–486. of WAW2009, Lecture Notes in Computer Science, [7] , Detecting sharp drops in PageRank and vol. 5427, Springer, 2009, 38–49. a simplified local partitioning algorithm, Theory [27] F. Chung and L. Lu, The average distances in and Applications of Models of Computation, Pro- random graphs with given expected degrees, Pro- ceedings of TAMC 2007, LNCS 4484, Springer ceedings of National Academy of Sciences 99 (2002), Berlin/Heidelberg, (2007), pp. 1–12. 15879–15882. [8] J. Balogh, B. Bollobás, and R. Morris, Boot- [28] , Connected components in random graphs strap percolation in three dimensions, Annals of with given expected degree sequences, Annals of Probability, to appear. 6 (2002), 125–145. [9] , The sharp threshold for r-neighbour [29] , The volume of the giant component of a bootstrap percolation, preprint. random graph with given expected degrees, SIAM [10] Y. Bao, G. Feng, T.-Y. Liu, Z.-M. Ma, and Y. Wang, J. Discrete Math. 20 (2006), 395–411. Describe importance of in probabilistic [30] , Coupling online and offline analyses for ran- view, Internet Math., to appear. dom power law graphs, Internet Math. 1 (2004), [11] A.-L. Barabási and R. Albert, of scaling 409–461. in random networks, Science 286 (1999), 509–512. [31] , Complex Graphs and Networks, CBMS Lec- [12] P. Berkhin, Bookmark-coloring approach to per- ture Series, No. 107, AMS Publications, 2006, vii + sonalized pagerank computing, Internet Math., to 264 pp. appear. [32] F. Chung, P. Horn, and L. Lu, Diameter of random [13] B. Bollobás, C. Borgs, J. Chayes, and O. Riordan, spanning trees in a given graph, preprint. Percolation on sequences, preprint.

June/July 2010 Notices of the AMS 731 [33] F. Chung, L. Lu, and V. Vu, Eigenvalues of ran- [53] L. Lovász, Random walks on graphs, Combina- dom power law graphs, Annals of Combinatorics 7 torics, Paul Erd˝os Is Eighty, Vol. 2, Bolyai Society (2003), 21–33. Mathematical Studies, 2, Keszthely (Hungary), 1993, [34] , The spectra of random graphs with given 1–46. expected degrees, Proceedings of National Academy [54] L. Lovász and M. Simonovits, Random walks in a of Sciences 100, no. 11 (2003), 6313–6318. convex body and an improved volume algorithm, [35] F. Chung, L. Lu, G. Dewey, and D. J. Galas, Random Structures and Algorithms 4 (1993), 359– Duplication models for biological networks, J. 412. 10, no. 5 (2003), 677–687. [55] C. Malon and I. Pak, Percolation on finite Cayley [36] F. Chung and S.-T. Yau, Coverings, heat kernels and graphs, Lecture Notes in Computer Science 2483, spanning trees, Electronic Journal of Combinatorics Springer, Berlin, 2002, pp. 91–104. 6 (1999), #R12. [56] M. Mihail, and convergence of [37] C. Daskalakis, P. Goldberg and C. Papadim- Markov chains: A combinatorial treatment of ex- itriou, Computing a Nash equilibrium is PPAD- panders, Foundation of Computer Science (1989), complete, to appear in SIAM J. on Computing. 526–531. [38] P. Erd˝os and A. Rényi, On random graphs, I. Publ. [57] M. Mitzenmacher, A brief history of generative Math. Debrecen 6 (1959), 290–297. models for power law and lognormal distributions, [39] , On the evolution of random graphs, Magyar Internet Math. 1 (2004), 226–251. Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61. [58] M. Molloy and B. Reed, A critical point for ran- [40] M. Faloutsos, P. Faloutsos, and C. Faloutsos, dom graphs with a given degree sequence. Random On power-law relationships of the Internet topol- Structures and Algorithms 6 (1995), 161–179. ogy, Proceedings of the ACM SIGCOM Conference, [59] M. Molloy and B. Reed, The size of the giant ACM Press, New York, 1999, 251–262. component of a random graph with a given de- [41] R. Feldmann, M. Gairing, T. Lucking, B. Monien, gree sequence, Combin. Probab. Comput. 7 (1998), and M. Rode, Selfish routing in noncooperative net- 295–305. works: A survey, Lecture Notes in Computer Science, [60] A. Rényi and G. Szekeres, On the height of trees, J. vol. 2746, Springer, Berlin/Heidelberg, 2003, pp. Austral. Math. Soc. 7 (1967), 497–507. 21–45. [61] T. Roughgarden, Selfish Routing and the Price of [42] A. Frieze, M. Krivelevich, and R. Martin, The Anarchy, MIT Press, 2006. emergence of a giant component of pseudo-random [62] T. Roughgarden and E. Tardos, How bad is selfish graphs, Random Structures and Algorithms 24, ACM routing? Journal of the ACM 49 (2002), 236–259. New York (2004), 42–50. [63] D. Spielman and S.-H. Teng, Nearly-linear time algo- [43] M. R. Garey and D. S. Johnson, Computers rithms for graph partitioning, graph sparsification, and Intractability, A Guide to the Theory of and solving linear systems, Proceedings of the 36th NP-Completeness, W. H. Freeman and Co., San Annual ACM Symposium on Theory of Computing, Francisco, 1979, x + 338 pp. ACM Press, New York 2004, pp. 81–90. [44] E. N. Gilbert, Random graphs, Annals of Mathemat- ical 30 (1959), 1141–1144. [45] G. Grimmett, Percolation, Springer, New York, 1989. [46] J. Hopcroft and D. Sheldon, Manipulation- resistant reputations using hitting time, WAW 2007, Lecture Notes in Computer Science, vol. 4863, Springer, Berlin/Heidelberg, 2007, pp. 68–81. [47] G. Jeh and J. Widom, Scaling personalized Web search, Proceedings of the 12th Conference WWW, ACM Press, New York, 2003, pp. 271–279. [48] M. Kearns, S. Suri, and N. Montfort, An exper- imental study of the coloring problem on human subject networks, Science 322, no. 5788 (2006), 824–827. [49] H. Kesten, for mathematicians, volume 2 of Progress in Probability and Statistics, Birkhäuser, Boston, MA., 1982. [50] J. S. Kong, N. Sarshar, and V. P. Roychowdhury, Experience versus talent shapes the structure of the Web, PNAS 105, no. 37 (2008), 13724–13729. [51] S. Kontogiannis and P. Spirakis, Atomic self- ish routing in networks: A survey, Lecture Notes in Computer Science, vol. 3828, Springer, Berlin/Heidelberg, 2005, pp. 989–1002. [52] David Liben-Nowell and Jon Kleinberg, Tracing information flow on a global scale using Internet, PNAS 105, no. 12 (2008), 4633–4638.

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