Competitive economy as a ranking device over networks☆
Introduction
In a world with abundant information, ranking systems are of utmost importance. Well known examples include Google's PageRank, which helps Internet users to identify web pages that are more likely to interest them, and the citation count,4 which helps academic researchers to assess the quality of academic articles. In both cases, as well as in many other contexts, the ranking of items is based solely on the information embodied in the network such as links between web sites or citations between scientific articles.
Whereas the citation count merely counts the number of citations from other articles and does not discriminate by the source of the citation, more sophisticated ranking systems attempt to give more weight to votes from items which are of a higher rank (as ascribed by the system itself). The approach taken by PageRank, for example, is based on the idea of translating the link structure to a Markov process as follows: each web page is viewed as a state, and after the random walk hits a state it moves on randomly to one of the states that the current state gives them a link. The ranking of a web page is defined as the long-run proportion of time that the process spends in a given state. Since this value depends not only on the number of incoming links but also on the proportion of time the system spends on the states that send these links, the induced ranking indeed grants more weight to links from higher-ranked pages.
We propose a different approach to ranking that consists of constructing an economy based on the network of links and deriving the ranks from the competitive equilibrium prices.5 Our approach employs a neoclassical pure-exchange economy. Each web page is represented by one consumer, who is initially endowed with one unit of a specific good. The market price of his good becomes his budget, which in turn serves to buy other goods. The consumer derives utility from consuming the specific goods provided by exactly those consumers that he or she sends a link to. For example, if page i has links only to pages j and k, then consumer i has utility where are the quantities that i consumes from goods j and k.
The main idea of this paper is to use the competitive prices of this economy as a ranking system. That is, the ranking of a web page is defined as the price of the corresponding good. In this pure-exchange economy, higher-ranked pages correspond to more expensive goods. Moreover, since the budget of a consumer equals the price of the good he initially owns, the initial owners of highly demanded goods are rich. These owners demand larger quantities of the goods they like, thus pushing their prices higher. Hence, those web pages that are pointed to by highly ranked web pages, are highly ranked themselves.
The specific ranking obtained depends on the modeler's choice of utility functions. We assume throughout that all consumers, although consuming different goods, have the same type of utility function. We first consider the Cobb–Douglas utility function which is perhaps the most widely used in economic modeling. We show that when all consumers have Cobb–Douglas preferences, the resulting vector of competitive equilibrium prices coincides with the PageRank ranking system.
We next show that the citation count cannot be derived from our economy. The reason is that, in this type of economy, the value of a paper is identified with the budget of the corresponding consumer, which in turn is his reviewing power. However, in citation count, the reviewing power of a paper is a fixed constant. This impossibility extends also to the ‘Normalized’ Citation Count, in which the value of a citation from an article is inversely proportional to the number of citations the article makes.
A ranking system based solely on the network information6 actually considers each item (e.g., web page, article) both as a reviewer, whose judgment (link, references) determines the rank of others, and as a refereed, whose assessed quality depends on the links it obtained from others. The following economic metaphor can sharpen this distinction: the value of an agent as refereed is the market price of its good. Its power as a reviewer equals its budget. In the formulation we employed so far, these two powers are, by definition, the same, since the budget comes from selling the agent's specific good.
The citation count, that does not make any connection between the value of a paper and its refereeing power seems to lack the desirable property of allocating greater weight to citations originating from more important papers. However, there are many cases in which also PageRank, or in fact any ranking system derived from an exchange economy, fails to produce meaningful results. Consider, for example, a sequence of papers published sequentially, each at a distinct time. As citations can only refer to earlier work, all these papers will have zero value. This happens precisely because the value of an item as a reviewer is equated with its value as a refereed entity. The latest article has no incoming links, implying that its price and budget are 0. In turn, it has 0 demand for the articles it has links to. Thus, the penultimate article has 0 value as well, and so forth. In similar cases the citation count – where the reviewing power of all items are the same and independent of their power as refereed items – may perform better than the PageRank.
One can combine the advantages of these two ranking systems by developing the economic metaphor a bit further. We add to the exchange economy a taxation scheme, thus allowing to disentangle the value of an item as a reviewer and its quality as determined by others. Each consumer pays a proportion, say α, of his income as a tax. The tax revenue is then equally redistributed between all the consumers. When , a consumer's budget equals the price of his specific good, leading us back to the original model. With a 100% tax (i.e., ), the budgets of all consumers are equal. As a result, their reviewing power is equal regardless of the prices of their goods. The competitive equilibrium prices in this case could serve as a ranking system that bears the spirit of the citation count.7 By choosing a tax rate between the two extremes, one can control the extent to which the reviewing power of an item depends on its quality (as assessed by the ranking system). We show examples in which the ranking system derived from an intermediate tax seems to do better than both PageRank and the citation count.
Related literature. In the 1960s, Garfield introduced the first citation index for papers published in academic journals: the Science Citation Index (SCI). This index was followed by the Social Sciences Citation Index (SSCI) and later by the Arts and Humanities Citation Index (AHCI). In 1972 Garfield established a ranking of scientific journals, known as Impact Factor. Liebowitz and Palmer (1984) analyzed the impact factors of economic journals by using an iteration (impact adjusted) method, without making the mechanism explicit.
The Markov-chain approach behind PageRank has been originally proposed by Wei (1952) and Kendall (1955). The approach was applied to create PageRank by Brin and Page who describe the algorithm in detail in Brin and Page (1998).
Another major direction that was applied in the literature has been the axiomatic approach. By postulating a number of desired requirements that the ranking system must satisfy, one attempts to identify a specific ranking. This approach was adopted by Palacios-Huerta and Volij (2004) and Altman and Tennenholtz (2005) who axiomatized PageRank. Demange (2011) also takes an axiomatic approach and derives an index based on treating members in the network both as referees and being refereed at the same time.
Posner (2000) criticizes many ranking methods and states that ‘citation analysis is not an inherently economic methodology’ due to its lack of theoretical or empirical grounding. Amir (2002) studies properties of various indices. The influence model of Demange (2011) describes a dynamics whereby the ranking of journals affect the intensities of citations.
Structure of the paper. Section 2 introduces the model of an exchange economy induced by a network and derives the ranking scheme. In Section 3 we show that if an exchange economy is governed by Cobb–Douglas utility maximizers, then the resulting ranking coincides with PageRank. Section 4 studies the citation count and shows that it cannot be derived from an exchange economy. Section 5 introduces taxation into the economy and deals with the resulting distinction between quality and refereeing power. Final remarks appear in Section 6.
Section snippets
The graph of citations
Our initial data is a directed graph G with the set of nodes and the set of directed edges . Each node (vertex) represents a web page (or an article) and each directed edge represents a link (or citation) from web page i to web page j. We refer to as ‘i has a link to j’, ‘i cites j’, ‘j is cited by i’ and alike. It is also useful to define the coincidence matrix in which if and , otherwise. Denote by the set of nodes
Cobb–Douglas utility yields Google's ranking
In this section we illustrate the main idea using the Cobb–Douglas utility function and show the connection with Google's PageRank.
The citation count method
A common ranking method of a scientific paper is the citation count (CC), which simply counts the number of citations it received from other papers. This method does not take into consideration the ranking of the citing paper nor its venue. One can also define a modified version of this method, in which a citation from an article that makes n citations is counted as rather than 1. Thus, each article has the same reviewing power independently of the number of citation it makes. We call this
Quality and refereeing power
Any ranking systems induced by an exchange economy equates, by definition, the refereeing power of a paper with its quality (its rank). The reason is that the price of good i, which is identified as i's rank, becomes the budget of consumer i. This budget is then split between the papers that i cites, so it is exactly i's refereeing power. By contrast, the citation count disregards completely the rank of an article when it determines its refereeing power. In the CC, a citation from any article
Non-connected graphs
Throughout the paper we have made the assumption that the graph of links is connected. When this is not the case, the ranking we obtain is not unique. Each connected component of the graph is in fact an isolated economy, an “island”. Within each island the relative prices are indeed determined by the equilibrium; however, there are no restrictions on the intra-island relative prices.
More formally, assume that the graph G is the union of m connected components , and let be a vector
References (16)
- et al.
Ranking systems: the PageRank axioms
Impact-adjusted citations as a measure of journal quality
(2002)- et al.
The anatomy of a large-scale hypertextual web search engine
Comput. Netw. ISDN Syst.
(1998) Topological methods in cardinal utility
New concepts and techniques for equilibrium analysis
Int. Econ. Rev.
(1962)- Demange, G., 2011. On the influence of rankings,...
Finite solution of pure trade markets with Cobb–Douglas utilities
Math. Program. Stud.
(1985)Citation analysis as a tool in journal evaluation
Science
(1972)
Cited by (12)
The degree measure as utility function over positions in graphs and digraphs
2022, European Journal of Operational ResearchCitation Excerpt :Various ranking methods are characterized in the literature, in particular, methods based on directed networks, see e.g. Rubinstein (1980) for the ranking by outdegree on the class of tournaments, see also Henriet (1985) and Bouyssou (1992) for the ranking by Copeland score (Bouyssou & Perny, 1992; van den Brink & Gilles, 2003; Copeland, 1951) for the ranking by outdegree for arbitrary directed networks, van den Brink & Rusinowska (2021) for the degree ratio ranking method for directed networks, and van den Brink & Gilles (2009) for the outflow ranking method for weighted directed networks. Du, Lehrer, & Pauzner (2015) investigates ranking of items in a network determined by choice of utility function. More precisely, the network is transformed into an exchange economy and the competitive equilibrium prices of the network nodes are used for the ranking.
The degree ratio ranking method for directed graphs
2021, European Journal of Operational ResearchCitation Excerpt :We obtain the peers setting when the experts coincide with the items (e.g., webpages linking to other pages or journals citing each other). Du, Lehrer, and Pauzner (2015) studies ranking of items in a graph determined by a choice of utility function. Demange (2014b, 2017) characterizes ranking methods based on evaluations or citations which consider one-sided settings and two-sided settings, respectively.
Power method tâtonnements for Cobb–Douglas economies
2018, Journal of Mathematical EconomicsCitation Excerpt :Applying recent results by Nesterov and Nemirovski (2015) on finding stationary states of Markov chains, we get insights on how economic equilibria can be easily computed in reasonable time. A complementary line of research has been recently suggested by Du et al. (2015) on how to obtain Google rankings by using market mechanisms. For other tâtonnements with guaranteed worst-case efficiency, we refer to Cole and Fleischer (2008) and Cheung (2014).
Mutual rankings
2017, Mathematical Social SciencesCitation Excerpt :Several axiomatizations of eigencentrality have been provided (Palacios-Huerta and Volij, 2004, Slutzki and Volij, 2006 for its intensity-invariant version, Altman and Tennenholtz, 2005 for its ordinal version, and Kitti, 2016). Also Du, Lehrer and Pauzner (Du et al., 2015) propose a ‘market’ approach to rank Web pages and obtain a variety of methods that includes an eigencentrality method. The mutual centrality method is related to eigenvalue centrality since it is built on the dominant eigenvector of a matrix (it does not coincide with it due to the independence of the rankings between the two sides).
Economic Networks: Theory and Computation
2022, arXivA network approach to public goods
2021, arXiv
- ☆
This paper contains the results obtained independently by two groups and replaces “Ranking Via Arrow–Debreu Equilibrium” by Du. The authors wish to thank Elchanan Ben-Porath, Eddie Dekel, Ignacio Palacios-Huerta, David Schmeidler, Daniel Seidman, Orit Tykocinski, Asher Wolinsky and Tim van Zandt for their valuable comments. We are specially indebted to Dudu Lagziel for his help with the simulations.
- 1
Part of the work was done when Du was a PhD student in the University of Michigan.
- 3
Pauzner worked on this project also while visiting the Institute for Advanced Studies at the Hebrew University of Jerusalem, and while he was also teaching at the Interdisciplinary Center, Herzlia. His research was supported in part by the Pinhas Sapir Center for Development.