Generative Models for Networks and Applications to E-Commerce

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1 Generative Models for Networks and Applications to E-Commerce Patrick J. Wolfe (with David C. Parkes and R. Kang-Xing Jin) Division of Engineering and Applied Sciences Department of Statistics Harvard University Radcliffe Workshop Harvard, October 20, 2006 Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

2 Introduction Network Analysis in Electronic Markets Statistical analysis of networks plays a critical role in the context of economics and the social sciences. How can such analyses aid in the identification of preferences and behavior in network-based economies? Consider the case of ebay transactions: How to posit a network representation of this market? What can we learn from it? More generally, if we represent data via networks, how do we model and discover network structure (temporal, semantic, relational, etc.)? Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

3 Network Analysis in Electronic Markets Empirical study of Internet auctions is a relatively new field First studies of bidder communities on ebay seem to have appeared in 2006 (Reichardt & Bornholdt; DCP et al.) Reichardt & Bornholdt logged 10 days worth of data from ebay.de, and generated a network with bidders as nodes and edges linking bidders in a common auction Here we consider the Canon digital camera market on ebay Auctions are associated with nodes and weighted edges between nodes capture the number of bidders competing in a pair of auctions. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

4 Ebay Auction Data Data collected by searching closed listings on ebay.com Information includes the following elements for each auction: 1 Title of auction, name of seller, type of auction, reserve price, reputatation of seller. 2 Whether or not the item sold. The high bid in the auction, and the start and end time. 3 The name of each bidder, the time the bid was placed (to the bidder proxy) and the value of each bid. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

5 Identification of Preferences A central problem in the analysis of economic systems is to identify the preferences of the actors within the economy. We may then study both the allocative efficiency of market institutions, as well as propose new institutions in order to improve market efficiency. Specifically, on ebay we are interested in the following two questions: Can substitute goods be automatically identified from bidder behavior? Can complement goods be automatically identified from bidder behavior? Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

6 Substitutes and Complements Roughly, two items are substitutes if a typical bidder with value for one item also has value for the other item. Examples of substitutes goods include two models of 19 LCD monitors. Items are complements if a typical bidder has superadditive value for the pair of items. Examples of complements goods include a digital camera and a memory stick. This macro-level structural information on the preferences of participants in a marketplace can be useful for a number of reasons: Design of user interfaces, pre-bundling of complements, cross-selling, etc. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

7 Substitutes and Complements Here, the question of substitutes is in part motivated by the related problem of categorization: ebay has millions of widely varying items What is a scalable way to organize these items into categories? We construct a graph to represent the semantic information about goods that is revealed through bidder behavior: Auctions are nodes and an edge is drawn between auctions sharing a common bidder. Thus, an edge conveys information about preferences: The presence of bidding across auctions provides revealed preference, indicating (for example) that the items in the associated auctions are substitutes. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

8 Communities in Networks One formal definition of a community is (Reichardt & Bornholdt): Given a graph G with N nodes and M edges, a community of n nodes 2m and m edges is one satisfying n(n 1) > 2M N(N 1) > m nn n(n n), where m nn is the number of edges connecting the community to the network. Each of these terms represents an edge density the number of edges divided by the maximum number of edges. The first inequality requires that within-community density be greater than the average network density. The second inequality requires that the average network density be greater than the density of edges leaving the community. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

9 Communities in Networks Modularity is a global metric that has been widely used to compare the quality of different community structures and determine an optimal one (Newman, 2004): Let e ij be the fraction of all edges in the network that lie between community i and community j and d v be the degree of vertex v. Then the modularity Q of a network and community division is defined as Q = i e ii a 2 i, where a i = 1 2M v i d v, and a i gives the fraction of ends of edges in the network that are in the community i. The term a i 2 represents the expected fraction of network connections having both ends in community i if edges were to be randomly assigned, conditioned upon the same community division. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

10 Substitutes and Complements Figure: Canon communities Community detection applied to the Canon digital cameras market, containing all auctions containing Canon in the Digital Cameras category over a period from January 10, 2006 until January 25, Ten of these communities have exactly one camera model keyword (sd500, a620, etc.), and two communities have two camera keywords. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

11 Substitutes and Complements Figure: Community assignments Community detection applied to 10% of the ebay LCD market, with nodes shown as auctions and edges drawn between any two auctions that share a common bidder. Elsewhere, this method has been demonstrated to identify communities in artificial and real-world networks. Question: is it possible to introduce a formal statistical model? Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

12 From Community Detection to Generative Modeling Positing a good and representative model Imagine trying to generate networks from scratch according to a model: What characteristics might a typical network model possess? Particular types of structure, but also stochastic Transitivity, homophily by attributes, clustering, etc. What does it mean to generate from scratch in this way? Vertices, edges, weights, etc. chosen randomly according to an appropriately chosen probability distribution. If we analyze a pre-existing network, these randomly generated quantities would then become the unknown parameters of our model. If we choose a good and representative model, then generated networks will share the characteristics of those that we wish to study. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

13 What are Bayesian Methods? The connection to generative models A particular philosophy of statistical inference A mechanism for building generative models for data A formalism for learning from data via Bayes rule: evidence + assumptions = inference Given prior information about x and data y, Bayes rule provides a means of getting from prior to posterior: p(x, y) = p(x y)p(y) p(x y) = p(x y) p(y x) p(x) p(y x) p(x) p(y) ln p(x y) = ln p(y x) + ln p(x) + C Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

14 The Bottom Line Generative Models and Bayesian Inference Given a network inference task, how can we posit models that describe variation in network structure and admit inference? In the Bayesian paradigm, all inference stems from a description of the (posterior) probability distribution associated with a given model after having observed the data in question. While many realistic scenarios do not admit analytical solutions, we may construct algorithms so-called stochastic computation methods to perform inference via simulation in these cases. These simulation techniques present a valuable tool in certain situations, and have generated much recent interest in the statistical community. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

15 Open Questions Generative Models and E-Commerce Networks Model Elicitation: Hierarchical models of clustering and community detection. (Categories and sub-categories on ebay). Dynamic evolution of marketplaces over time (Current results obtained by sampling and hence averaging sequential effects). Model Fitting: Posterior sampling via stochastic computation. Importance sampling on graphs Trans-dimensional Markov chain Monte Carlo Computationally efficient approximations to exact inference. Wolfe et al. (Harvard University) Networks and E-Commerce October 20, / 15

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