{"title": "Economic Properties of Social Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 633, "page_last": 640, "abstract": null, "full_text": " Economic Properties of Social Networks\n\n\n\n Sham M. Kakade Michael Kearns Luis E. Ortiz\n\n Robin Pemantle Siddharth Suri\n\n University of Pennsylvania\n Philadelphia, PA 19104\n\n\n Abstract\n\n We examine the marriage of recent probabilistic generative models\n for social networks with classical frameworks from mathematical eco-\n nomics. We are particularly interested in how the statistical structure of\n such networks influences global economic quantities such as price vari-\n ation. Our findings are a mixture of formal analysis, simulation, and\n experiments on an international trade data set from the United Nations.\n\n\n1 Introduction\n\nThere is a long history of research in economics on mathematical models for exchange mar-\nkets, and the existence and properties of their equilibria. The work of Arrow and Debreu\n[1954], who established equilibrium existence in a very general commodities exchange\nmodel, was certainly one of the high points of this continuing line of inquiry. The origins\nof the field go back at least to Fisher [1891].\n\nWhile there has been relatively recent interest in network models for interaction in eco-\nnomics (see Jackson [2003] for a good review), it was only quite recently that a network or\ngraph-theoretic model that generalizes the classical Arrow-Debreu and Fisher models was\nintroduced (Kakade et al. [2004]). In this model, the edges in a network over individual\nconsumers (for example) represent those pairs of consumers that can engage in direct trade.\nAs such, the model captures the many real-world settings that can give rise to limitations on\nthe trading partners of individuals (regulatory restrictions, social connections, embargoes,\nand so on). In addition, variations in the price of a good can arise due to the topology of\nthe network: certain individuals may be relatively favored or cursed by their position in the\ngraph.\n\nIn a parallel development over the last decade or so, there has been an explosion of interest\nin what is broadly called social network theory -- the study of apparently \"universal\"\nproperties of natural networks (such as small diameter, local clustering of edges, and heavy-\ntailed distribution of degree), and statistical generative models that explain such properties.\nWhen viewed as economic networks, the assumptions of individual rationality in these\nworks are usually either non-existent, or quite weak, compared to the Arrow-Debreu or\nFisher models.\n\nIn this paper we examine classical economic exchange models in the modern light of social\nnetwork theory. We are particularly interested in the interaction between the statistical\nstructure of the underlying network and the variation in prices at equilibrium. We quantify\nthe intuition that increased levels of connectivity in the network result in the equalization of\n\n\f\nprices, and establish that certain generative models (such as the the preferential attachment\nmodel of network formation (Barabasi and Albert [1999]) are capable of explaining the\nheavy-tailed distribution of wealth first observed by Pareto. Closely related work to ours is\nthat of Kranton and Minehart [2001], which also considers networks of buyers and sellers,\nthough they focus more on the economics of network formation.\n\nMany of our results are based on a powerful new local approximation method for global\nequilibrium prices: we show that in the preferential attachment model, prices computed\nfrom only local regions of a network yield strikingly good estimates of the global prices.\nWe exploit this method theoretically and computationally. Our study concludes with an\napplication of our model to United Nations international trade data.\n\n\n2 Market Economies on Networks\n\nWe first describe the standard Fisher model, which consists of a set of consumers and a set\nof goods. We assume that there are gj units of good j in the market, and that each good j is\nbe sold at some price pj. Each consumer i has a cash endowment ei, to be used to purchase\ngoods in a manner that maximizes the consumers' utility. In this paper we make the well-\nstudied assumption that the utility function of each consumer is linear in the amount of\ngoods consumed (see Gale [1960]), and leave the more general case to future research. Let\nuij 0 denote the utility derived by i on obtaining a single unit of good j. If i consumes\nxij amount of good j, then the utility i derives is u\n j ij xij .\n\nA set of prices {pj} and consumption plans {xij} constitutes an equilibrium if the follow-\ning two conditions hold:\n\n1. The market clears, i.e. supply equals demand. More formally, for each j, x\n i ij = gj .\n\n2. For each consumer i, their consumption plan {xij}j is optimal. By this we mean that\nthe consumption plan maximizes the linear utility function of i, subject to the constraint\nthat the total cost of the goods purchased by i is not more than the endowment ei.\n\nIt turns out that such an equilibrium always exists if each good j has a consumer which\nderives nonzero utility for good j -- that is, uij > 0 for some i (see Gale [1960]). Further-\nmore, the equilibrium prices are unique.\n\nWe now consider the graphical Fisher model, so named because of the introduction of a\ngraph-theoretic or network structure to exchange. In the basic Fisher model, we implicitly\nassume that all goods are available in a centralized exchange, and all consumers have equal\naccess to these goods. In the graphical Fisher model, we desire to capture the fact that each\ngood may have multiple vendors or sellers, and that individual buyers may have access\nonly to some, but not all, of these sellers. There are innumerable settings where such asym-\nmetries arise. Examples include the fact that consumers generally purchase their groceries\nfrom local markets, that social connections play a major role in business transactions, and\nthat securities regulations prevent certain pairs of parties from engaging in stock trades.\n\nWithout loss of generality, we assume that each seller j sells only one of the available\ngoods. (Each good may have multiple competing sellers.) Let G be a bipartite graph,\nwhere buyers and sellers are represented as vertices, and all edges are between a buyer-\nseller pair. The semantics of the graph are as follows: if there is an edge from buyer i to\nseller j, then buyer i is permitted to purchase from seller j. Note that if buyer i is connected\nto two sellers of the same good, he will always choose to purchase from the cheaper source,\nsince his utility is identical for both sellers (they sell the same good).\n\nThe graphical Fisher model is a special case of a more general and recently introduced\nframework (Kakade et al. [2004]). One of the most interesting features of this model is the\nfact that at equilibrium, significant price variations can appear solely due to structural prop-\nerties of the underlying network. We now describe some generative models of economies.\n\n\f\n3 Generative Models for Social Networks\n\nFor simplicity, in the sequel we will consider economies in which the numbers of buyers\nand sellers are equal. We will also restrict attention to the case in which all sellers sell the\nsame good1.\n\nThe simplest generative model for the bipartite graph G might be the random graph, in\nwhich each edge between a buyer i and a seller j is included independently with probability\np. This is simply the bipartite version of the classical Erdos-Renyi model (Bollobas [2001]).\n\nMany researchers have sought more realistic models of social network formation, in order\nto explain observed phenomena such as heavy-tailed degree distributions. We now describe\na slight variant of the preferential attachment model (see Mitzenmacher [2003]) for the case\nof a bipartite graph. We start with a graph in which one buyer is connected to one seller. At\neach time step, we add one buyer and one seller as follows. With probability , the buyer\nis connected to a seller in the existing graph uniformly at random; and with probability\n1 - , the buyer is connected to a seller chosen in proportion to the degree of the seller\n(preferential attachment). Simultaneously, a seller is attached in a symmetric manner: with\nprobability the seller is connected to a buyer chosen uniformly at random, and with\nprobability 1 - the seller is connected under preferential attachment. The parameter in\nthis model thus allows us to move between a pure preferential attachment model ( = 0),\nand a model closer to classical random graph theory ( = 1), in which new parties are\nconnected to random extant parties2.\n\nNote that the above model always produces trees, since the degree of a new party is always\n1 upon its introduction to the graph. We thus will also consider a variant of this model in\nwhich at each time step, a new seller is still attached to exactly one extant buyer, while\neach new buyer is connected to > 1 extant sellers. The procedure for edge selection is as\noutlined above, with the modification that the new edges of the buyer are added without\nreplacement -- meaning that we resample so that each buyer gets attached to exactly \ndistinct sellers. In a forthcoming long version, we provide results on the statistics of these\nnetworks.\n\nThe main purpose of the introduction of is to have a model capable of generating highly\ncyclical (non-tree) networks, while having just a single parameter that can \"tune\" the asym-\nmetry between the (number of) opportunities for buyers and sellers. There are also eco-\nnomic motivations: it is natural to imagine that new sellers of the good arise only upon\nobtaining their first customer, but that new buyers arrive already aware of several alterna-\ntive sellers.\n\nIn the sequel, we shall refer to the generative model just described as the bipartite (, )-\nmodel. We will use n to denote the number of buyers and the number of sellers, so the\nnetwork has 2n vertices. Figure 1 and its caption provide an example of a network gener-\nated by this model, along with a discussion of its equilibrium properties.\n\n4 Economics of the Network: Theory\n\nWe now summarize our theoretical findings. The proofs will be provided in a forthcoming\nlong version. We first present a rather intuitive \"frontier\" theorem, which implies a scheme\nin which we can find upper and lower bounds on the equilibrium prices using only local\ncomputations. To state the theorem we require some definitions. First, note that any subset\nV of buyers and sellers defines a natural induced economy, where the induced graph G\n\n 1From a mathematical and computational standpoint, this restriction is rather weak: when con-\nsidered in the graphical setting, it already contains the setting of multiple goods with binary utility\nvalues, since additional goods can be encoded in the network structure.\n 2We note that = 1 still does not exactly produce the Erdos-Renyi model due to the incremental\nnature of the network generation: early buyers and sellers are still more likely to have higher degree.\n\n\f\n B13\n B18\n\n\n\n\n\n B7\n S10: 1.00\n\n\n\n\n S9: 0.75\n S11: 1.00 S12: 1.00\n S4: 1.00\n\n\n\n\n S6: 0.67\n\n\n\n B15\n S3: 1.00\n S19: 0.75 B11\n B3 S1: 1.50\n\n\n S15: 0.67\n B1 B2\n S2: 1.00\n\n\n\n\n B0\n S16: 0.67\n\n\n\n S18: 0.75\n S13: 1.00 S5: 1.50\n S0: 1.50\n\n B6\n B17\n S7: 1.50\n\n\n B10\n\n\n S8: 1.00\n\n\n B9 B8 S14: 0.75\n B19\n\n\n\n\n\n B4\n B12 B14\n\n B16\n\n B5\n\n\n\n\n\n S17: 1.00\nFigure 1: Sample network generated by the bipartite ( = 0, = 2)-model. Buyers and sellers\nare labeled by `B' or `S' respectively, followed by an index indicating the time step at which they\nwere introduced to the network. The solid edges in the figure show the exchange subgraph -- those\npairs of buyers and sellers who actually exchange currency and goods at equilibrium. The dotted\nedges are edges of the network that are unused at equilibrium because they represent inferior prices\nfor the buyers, while the dashed edges are edges of the network that have competitive prices, but are\nunused at equilibrium due to the specific consumption plan required for market clearance. Each seller\nis labeled with the price they charge at equilibrium. The example exhibits non-trivial price variation\n(from 2.00 down to 0.33 per unit good). Note that while there appears to be a correlation between\nseller degree and price, it is far from a deterministic relation, a topic we shall examine later.\n\n\n\nconsists of all edges between buyers and sellers in V that are also in G. We say that G\nhas a buyer (respectively, seller) frontier if on every (simple) path in G from a node in V\nto a node outside of V , the last node in V on this path is a buyer (respectively, seller).\n\nTheorem 1 (Frontier Bound) If V has a subgraph G with a seller (respectively, buyer)\nfrontier, then the equilibrium price of any good j in the induced economy on V is a lower\nbound (respectively, upper bound) on the equilibrium price of j in G.\n\nTheorem 1 implies a simple price upper bound: the price commanded by any seller j is\nbounded by its degree d. Although the same upper bound can be seen from first principles,\nit is instructive to apply Theorem 1. Let G be the immediate neighborhood of j (which is j\nand its d buyers); then the equilibrium price in G is just d, since all d buyers are forced to\nbuy from seller j. This provides an upper bound since G has a buyer frontier. Since it can\nbe shown that the degree distribution obeys a power law in the bipartite (, )-model, we\nhave an upper bound on the cumulative price distribution. We use = (1 - )/(1 + ).\n\nTheorem 2 In the bipartite (, )-model, the proportion of sellers with price greater than\nw is O(w-1/). For example, if = 0 (pure preferential attachment) and = 1, the\nproportion falls off as 1/w2.\n\nWe do not yet have such a closed-form lower bound on the cumulative price distribution.\nHowever, as we shall see in Section 5, the price distributions seen in large simulation results\ndo indeed show power-law behavior. Interestingly, this occurs despite the fact that degree\nis a poor predictor of individual seller price.\n\n\f\n 0 3 3\n 10 10 10\n 0\n 10\n k=1\n =1\n -1\n 10\n 2 2\n -1 10 10\n 10 k=2\n\n -2 =2\n 10\n\n =3\n Average Error 1 1\n -2 10\n 10 k=3 =4 10\n -3\n 10\n\n Cumulative of Degree/Wealth Maximum to Minimum Wealth Maximum to Minimum Wealth\n\n k=4\n -4\n 10 -3 0\n 10 0\n -1 0 1 2 10 10\n 10 10 10 10 50 100 150 200 250 1 2\n 10 10 0 0.2 0.4 0.6 0.8 1\n Degree/Wealth N N alpha\n\n Figure 2: See text for descriptions.\n\n\nAnother quantity of interest is what we might call price variation -- the ratio of the price\nof the richest seller to the poorest seller. The following theorem addresses this.\n\nTheorem 3 In the bipartite (, )-model, if (2 + 1) < 1, then the ratio of the maximum\n 2 2\n -( +1)\nprice to the minimum price scales with number of buyers n as (n 1+ ). For the\nsimplest case in which = 0 and = 1, this lower bound is just (n).\n\nWe conclude our theoretical results with a remark on the price variation in the Erdos-Renyi\n(random graph) model. First, let us present a condition for there to be no price variation.\n\nTheorem 4 A necessary and sufficient condition for there to be no price variation, ie for\nall prices to be equal to 1, is that for all sets of vertices S, |N (S)| |S|, where N (S) is\nthe set of vertices connected by an edge to some vertex in S.\n\nThis can be viewed as an extremely weak version of standard expansion properties well-\nstudied in graph theory and theoretical computer science -- rather than demanding that\nneighbor sets be strictly larger, we simply ask that they not be smaller. One can further show\nthat for large n, the probability that a random graph (for any edge probability p > 0) obeys\nthis weak expansion property approaches 1. In other words, in the Erdos-Renyi model,\nthere is no variation in price -- a stark contrast to the preferential attachment results.\n\n5 Economics of the Network: Simulations\n\nWe now present a number of studies on simulated networks (generated according to the\nbipartite (, )-model). Equilibrium computations were done using the algorithm of Deva-\nnur et al. [2002] (or via the application of this algorithm to local subgraphs). We note that\nit was only the recent development of this algorithm and related ones that made possible\nthe simulations described here (involving hundreds of buyers and sellers in highly cyclical\ngraphs). However, even the speed of this algorithm limits our experiments to networks with\nn = 250 if we wish to run repeated trials to reduce variance. Many of our results suggest\nthat the local approximation schemes discussed below may be far more effective.\n\nPrice and Degree Distributions: The first (leftmost) panel of Figure 2 shows empirical\ncumulative price and degree distributions on a loglog scale, averaged over 25 networks\ndrawn according to the bipartite ( = 0.4, = 1)-model with n = 250. The cumulative\ndegree distribution is shown as a dotted line, where the y-axis represents the fraction of\nthe sellers with degree greater than or equal to d, and the degree d is plotted on the x-axis.\nSimilarly, the solid curve plots the fraction of sellers with price greater than some value w,\nwhere the price w is shown on the x-axis. The thin sold line has our theoretically predicted\nslope of -1 = -3.33, which shows that degree distribution is quite consistent with our\n \nexpectations, at least in the tails. Though a natural conjecture from the plots is that the\nprice of a seller is essentially determined by its degree, below we will see that the degree\n\n\f\nis a rather poor predictor of an individual seller price, while more complex (but still local)\nproperties are extremely accurate predictors.\n\nPerhaps the most interesting finding is that the tail of the price distribution looks linear, i.e.\nit also exhibits power law behavior. Our theory provided an upper bound, which is precisely\nthe cumulative degree distribution. We do not yet have a formal lower bound. This plot\n(and other experiments we have done) further confirm the robustness of the power law\nbehavior in the tail, for < 1 and = 1.\n\nAs discussed in the Introduction, Pareto's original observation was that the wealth (which\ncorresponds to seller price in our model) distribution in societies obey a power law, which\nhas been born out in many studies on western economies. Since Pareto's original observa-\ntion, there have been too many explanations of this phenomena to recount here. However,\nto our knowledge, all of these explanations are more dynamic in nature (eg a dynamical\nsystem of wealth exchange) and don't capture microscopic properties of individual ratio-\nnality. Here we have power law wealth distribution arising from the combination of certain\nnatural statistical properties of the network, and classical theories of economic equilibrium.\n\nBounds via Local Computations: Recall that Theorem 1 suggests a scheme by which we\ncan do only local computations to approximate the global equilibrium price for any seller.\nMore precisely, for some seller j, consider the subgraph which contains all nodes that are\nwithin distance k of j. In our bipartite setting, for k odd, this subgraph has a buyer frontier,\nand for k even, this subgraph has a seller frontier, since we start from a seller. Hence,\nthe equilibrium computation on the odd k (respectively, even k) subgraph will provide an\nupper (respectively, lower) bound.\n\nThis provides an heuristic in which one can examine the equilibrium properties of small\nregions of the graph, without having to do expensive global equilibrium computations.\nThe effectiveness of this heuristic will of course depend on how fast the upper and lower\nbounds tighten. In general, it is possible to create specific graphs in which these bounds\nare arbitrarily poor until k is large enough to encompass the entire graph. As we shall see,\nthe performance of this heuristic is dramatically better in the bipartite (, )-model.\n\nThe second panel in Figure 2 shows how rapidly the local equilibrium computations con-\nverge to the true global equilibrium prices as a function of k, and also how this conver-\ngence is influenced by n. In these experiments, graphs were generated by the bipartite\n( = 0, = 1) model. The value of n is given on the x-axis; the average errors (over\n5 trials for each value of k and n) in the local equilibrium computations are given on the\ny-axis; and there is a separate plot for each of 4 values for k. It appears that for each value\nof k, the quality of approximation obtained has either mild or no dependence on n.\n\nFurthermore, the regular spacing of the four plots on the logarithmic scaling of the y-axis\nestablishes the fact that the error of the local approximations is decaying exponentially\nwith increased k -- indeed, by examining only neighborhoods of 3 steps from a seller in an\neconomy of hundreds, we are already able to compute approximations to global equilibrium\nprices with errors in the second decimal place. Since the diameter for n = 250 was often\nabout 17, this local graph is considerably smaller than the global. However, for the crudest\napproximation k = 1, which corresponds exactly to using seller degree as a proxy for\nprice, we can see that this performs rather poorly. Computationally, we found that the time\nrequired to do all 250 local computations for k = 3 was about 60% less than the global\ncomputation, and would result in presumably greater savings at much larger values of n.\n\nParameter Dependencies: We now provide a brief examination of how price variation\ndepends on the parameters of the bipartite (, )-model. We first experimentally evaluate\nthe lower bounds provided in Theorem 3. The third panel of Figure 2 shows the maximum\nto minimum price as function of n (averaged over 25 trials) on a loglog scale. Each line is\nfor a fixed value of , and the values of range form 1 to 4 ( = 0).\n\n\f\n 2\nRecall from Theorem 3, our lower bound on the ratio is (n 1+ ) (using = 0). We\nconjecture that this is tight, and, if so, the slopes of lines (in the loglog plot) should\nbe 2 , which would be (1, 0.67, 0.5, 0.4). The estimated slopes are somewhat close:\n 1+\n(1.02, 0.71, 0.57, 0.53). The overall message is that for small values of , price variation\nincreases rapidly with the economy size n in preferential attachment.\n\nThe rightmost panel of Figure 2 is a scatter plot of vs. the maximum to minimum price\nin a graph (where n = 250) . Here, each point represents the maximum to minimum price\nratio in a specific network generated by our model. The circles are for economies generated\nwith = 1 and the x's are for economies generated with = 3. Here we see that in general,\nincreasing dramatically decreases price variation (note that the price ratio is plotted on a\nlog scale). This justifies the intuition that as is increased, more \"economic equality\" is\nintroduced in the form of less preferential bias in the formation of new edges. Furthermore,\nthe data for = 1 shows much larger variation, suggesting that a larger value of also has\nthe effect of equalizing buyer opportunities and therefore prices.\n\n6 An Experimental Illustration on International Trade Data\n\nWe conclude with a brief experiment exemplifying some of the ideas discussed\nso far. The statistics division of the United Nations makes available exten-\nsive data sets detailing the amounts of trade between major sovereign nations (see\nhttp://unstats.un.org/unsd/comtrade). We used a data set indicating, for each pair of na-\ntions, the total amount of trade in U.S. dollars between that pair in the year 2002.\n\nFor our purposes, we would like to extract a discrete network structure from this numerical\ndata. There are many reasonable ways this could be done; here we describe just one.\nFor each of the 70 largest nations (in terms of total trade), we include connections from\nthat nation to each of its top k trading partners, for some integer k > 1. We are thus\nincluding the more \"important\" edges for each nation. Note that each nation will have\ndegree at least k, but as we shall see, some nations will have much higher degree, since\nthey frequently occur as a top k partner of other nations. To further cast this extracted\nnetwork into the bipartite setting we have been considering, we ran many trials in which\neach nation is randomly assigned a role as either a buyer or seller (which are symmetric\nroles), and then computed the equilibrium prices of the resulting network economy. We\nhave thus deliberately created an experiment in which the only economic asymmetries are\nthose determined by the undirected network structure.\n\nThe leftmost panel of Figure 3 show results for 1000 trials under the choice k = 3. The\nupper plot shows the average equilibrium price for each nation, where the nations have been\nsorted by this average price. We can immediately see that there is dramatic price variation\ndue to the network structure; while many nations suffer equilibrium prices well under $1,\nthe most topologically favored nations command prices of $4.42 (U.S.), $4.01 (Germany),\n$3.67 (Italy), $3.16 (France), $2.27 (Japan), and $2.09 (Netherlands). The lower plot of the\nleftmost panel shows a scatterplot of a nation's degree (x-axis) and its average equilibrium\nprice (y-axis). We see that while there is generally a monotonic relationship, at smaller\ndegree values there can be significant price variation (on the order of $0.50).\n\nThe center panel of Figure 3 shows identical plots for the choice k = 10. As suggested\nby the theory and simulations, increasing the overall connectivity of each party radically\nreduces price variation, with the highest price being just $1.10 and the lowest just under $1.\nInterestingly, the identities of the nations commanding the highest prices (in order, U.S.,\nFrance, Switzerland, Germany, Italy, Spain, Netherlands) overlaps significantly with the\nk = 3 case, suggesting a certain robustness in the relative economic status predicted by\nthe model. The lower plot shows that the relationship between degree and price divides the\npopulation into \"have\" (degree above 10) and \"have not\" (degree below 10) components.\n\nThe preponderance of European nations among the top prices suggests our final experi-\n\n\f\n UN data network, top 3 links, full set of nations UN data network, top 10 links, full set of nations UN data network, top 3 links, EU collapsed nation set\n 5 1.4 8\n\n 1.2\n 4 6\n 1\n\n 3 0.8\n 4\n price price price\n 2 0.6\n\n 0.4 2\n 1\n 0.2\n\n 0 0 0\n 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 0 5 10 15 20 25 30 35 40\n price rank price rank price rank\n\n\n\n 5 1.15 8\n\n\n 4 1.1 6\n\n 3 1.05\n\n 4\n\n 2 1\n\n average price average price average price\n\n 2\n 1 0.95\n\n\n 0 0.9 0\n 0 5 10 15 20 25 0 5 10 15 20 25 30 35 0 2 4 6 8 10 12 14\n average degree average degree average degree\n\n Figure 3: See text for descriptions.\n\n\nment, in which we modified the k = 3 network by merging the 15 current members of the\nEuropean Union (E.U.) into a single economic nation. This merged vertex has much higher\ndegree than any of its original constituents and can be viewed as an (extremely) idealized\nexperiment in the economic power that might be wielded by a truly unified Europe.\n\nThe rightmost panel of Figure 3 provides the results, where we show the relative prices and\nthe degree-price scatterplot for the 35 largest nations. The top prices are now commanded\nby the E.U. ($7.18), U.S. ($4.50), Japan ($2.96), Turkey ($1.32), and Singapore ($1.22).\nThe scatterplot shows a clear example in which the highest degree (held by the U.S.) does\nnot command the highest price.\n\nAcknowledgments\n\nWe are grateful to Tejas Iyer and Vijay Vazirani for providing their software implementing\nthe Devanur et al. [2002] algorithm. Siddharth Suri acknowledges the support of NIH grant\nT32HG0046. Robin Pemantle acknowledges the support of NSF grant DMS-0103635.\n\nReferences\n\nKenneth J. Arrow and Gerard Debreu. Existence of an equilibrium for a competitive economy. Econo-\n metrica, 22(3):265290, July 1954.\n\nA. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286:509512, 1999.\n\nB. Bollobas. Random Graphs. Cambridge University Press, 2001.\n\nNikhil R. Devanur, Christos H. Papadimitriou, Amin Saberi, and Vijay V. Vazirani. Market equilib-\n rium via a primal-dual-type algorithm. In FOCS, 2002.\n\nIrving Fisher. PhD thesis, Yale University, 1891.\n\nD. Gale. Theory of Linear Economic Models. McGraw Hill, N.Y., 1960.\n\nMatthew Jackson. A survey of models of network formation: Stability and efficiency. In Group\n Formation in Economics: Networks, Clubs and Coalitions. Cambridge University Press, 2003.\n\nS. Kakade, M. Kearns, and L. Ortiz. Graphical economics. COLT, 2004.\n\nR. Kranton and D. Minehart. A theory of buyer-seller networks. American Economic Review, 2001.\n\nM. Mitzenmacher. A brief history of generative models for power law and lognormal distributions.\n Internet Mathematics, 1, 2003.\n\n\f\n", "award": [], "sourceid": 2599, "authors": [{"given_name": "Sham", "family_name": "Kakade", "institution": null}, {"given_name": "Michael", "family_name": "Kearns", "institution": null}, {"given_name": "Luis", "family_name": "Ortiz", "institution": null}, {"given_name": "Robin", "family_name": "Pemantle", "institution": null}, {"given_name": "Siddharth", "family_name": "Suri", "institution": null}]}