This paper studies the distribution of a family of rankings, which includes Google's PageRank, on a directed configuration model. In particular, it is shown that the distribution of the rank of a randomly chosen node in the graph converges in distribution to a finite random variable scriptR* that can be written as a linear combination of i.i.d. copies of the attracting endogenous solution to a stochastic fixed‐point equation of the form R=scriptD∑i=1NscriptCiscriptRi+Q, where (Q,N,{scriptCi}) is a real‐valued vector with N∈{0,1,2,…}, P(|Q|>0)>0, and the {scriptRi} are i.i.d. copies of scriptR, independent of (Q,N,{scriptCi}). Moreover, we provide precise asymptotics for the limit scriptR*, which when the in‐degree distribution in the directed configuration model has a power law imply a power law distribution for scriptR* with the same exponent. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 237–274, 2017

Given two distributions F and G on the nonnegative integers we propose an algorithm to construct in-and out-degree sequences from samples of i.i.d. observations from F and G, respectively, that with high probability will be graphical, that is, from which a simple directed graph can be drawn. We then analyze a directed version of the configuration model and show that, provided that F and G have finite variance, the probability of obtaining a simple graph is bounded away from zero as the number of nodes grows. We show that conditional on the resulting graph being simple, the in-and out-degree distributions are (approximately) F and G for large size graphs. Moreover, when the degree distributions have only finite mean we show that the elimination of self-loops and multiple edges does not significantly change the degree distributions in the resulting simple graph.1. Introduction. In order to study complex systems such as the World Wide Web (WWW) 1 or the Twitter network we propose a model for generating a simple directed random graph with prescribed degree distributions. The ability to match degree distributions to real graphs is perhaps the first characteristic one would desire from a model, and although several models that accomplish this for undirected graphs have been proposed in the recent literature [8,10,11,20], not much has been done for the directed case. In the WWW example that motivates this work, vertices represent webpages and the edges represent the links between them; for the Twitter graph vertices represent people and an edge from one vertex to another means that the first person is "following" the second. Empirical studies (e.g., [9,15]) suggest that both the in-degree and out-degree, number of links pointing to a page and the number of outbound links of a page, respectively, follow a power-law distribution, a characteristic often referred to as the scale-free property.

Given two distributions F and G on the nonnegative integers we propose an algorithm to construct in-and out-degree sequences from samples of i.i.d. observations from F and G, respectively, that with high probability will be graphical, that is, from which a simple directed graph can be drawn. We then analyze a directed version of the configuration model and show that, provided that F and G have finite variance, the probability of obtaining a simple graph is bounded away from zero as the number of nodes grows. We show that conditional on the resulting graph being simple, the in-and out-degree distributions are (approximately) F and G for large size graphs. Moreover, when the degree distributions have only finite mean we show that the elimination of self-loops and multiple edges does not significantly change the degree distributions in the resulting simple graph.1. Introduction. In order to study complex systems such as the World Wide Web (WWW) 1 or the Twitter network we propose a model for generating a simple directed random graph with prescribed degree distributions. The ability to match degree distributions to real graphs is perhaps the first characteristic one would desire from a model, and although several models that accomplish this for undirected graphs have been proposed in the recent literature [8,10,11,20], not much has been done for the directed case. In the WWW example that motivates this work, vertices represent webpages and the edges represent the links between them; for the Twitter graph vertices represent people and an edge from one vertex to another means that the first person is "following" the second. Empirical studies (e.g., [9,15]) suggest that both the in-degree and out-degree, number of links pointing to a page and the number of outbound links of a page, respectively, follow a power-law distribution, a characteristic often referred to as the scale-free property.

Dynamic pricing is designed to increase the revenues or profits of firms by adjusting prices in response to changes in the marginal value of capacity. We examine the impact of dynamic pricing on social welfare and consumers' surplus. We present a dynamic pricing formulation designed to maximize welfare and show that the welfare-maximizing dynamic pricing policy has the same structural properties as the revenue-maximizing policy. For systems with scarce capacity, we show that the revenuemaximizing dynamic pricing policy and the market-clearing price are both asymptotically optimal for welfare. We also find in most cases that revenue-maximizing dynamic pricing improves consumers' surplus compared to the revenue-maximizing static price. Our findings can potentially transform the public image of dynamic pricing and provide new managerial insights as well as policy implications: (1) in large-scale systems with scarce capacity, a central planner would essentially implement the same pricing policy as a firm with monopoly power; (2) the revenue-maximizing dynamic pricing policy can benefit customers when the demand elasticity is in a small bounded interval as is the case for several important demand functions.