Evaluation of Authors and Journals

arXiv:0810.0852v1 [math.HO] 5 Oct 2008 Evaluation of Authors and Journals Joseph B. Keller Departments of Mathematics and Mechanical Engineering Stanford University Stanford, CA 94305 October 1985 Abstract A method is presented for evaluating authors on the basis of citations. It assigns to each author a citation score which depends upon the number of times he is cited, and upon the scores of the citers. The scores are found to be the components of an eigenvector of a normalized citation matrix. The same method can be applied to citation of journals by other journals, to evaluating teams in a league [1], etc. 1 Introduction One commonly used measure of the influence of an author is the number of times his work is referred to by others in a given period of time. For a scientific author, this number can be 1 found by counting the number of citations of his work listed in the Science Citation Index for that period. However, this measure fails to take into account who is doing the citing. A citation by an influential author ought to carry more weight than one by an unknown author, but a simple count of citations does not give it more weight. A more appropriate measure would be a weighted count of the citations, in which each citation is weighted by some measure of the influence of the citer. We shall show how to find such a measure, which we call a citation score, or just score for short. We begin by assuming that each author can be assigned a score, and that the weight of a citation is proportional to the score of the citer. Then we determine the score of an author by adding up the weights of all the citations of his work. The circularity of this method leads to a requirement of consistency among the scores, which determines them as the solutions of a system of linear equations. This method also applies to the evaluation of journals on the basis of citations of them in other journals. We have used this methods before [1] to evaluate teams in a league, with cij the number of times that team i beats team j. Other ranking methods are reviewed by Moon and Pullman [2]. 2 The citation matrix Let us consider N authors, numbered from 1 to N. We denote by c′ ij the number of times that j cited i, omitting self citations, so that c′ ii = 0. The c′ ij form an N by N square matrix C′, which we call the citation matrix. 2 The j-th column of C′ records all the citations by j of others, and the sum of the entries in that column is the total number of citations by j. If the sum is not zero, we normalize the column by dividing each entry by the column sum. Thus we define cij by cij = c′ ij  N X k=1 c′ kj. (1) If all the entries in column j are zero, we define cij = 0. We call the matrix with entries cij the normalized citation matrix C. From the definition (1) we see that cij is the fraction of j’s citations which refer to i. Furthermore, each column sum is unity, unless all the entries in the column are zero, in which case it is zero. Next we denote by xi the score of author i. According to the method mentioned in the Introduction, xi is given by xi = λ−1 N X j=1 cijxj. (2) Here λ−1 is a factor of proportionality. Thus xi is the sum of contributions from all citers j ̸= i. Each contribution is the product of the score of j times the fraction of j’s citations which refer to i, all times λ−1. Equation (2) is a system of N linear homogenous equations for the scores xi. In terms of the score vector x = (x1, . . . , xN), (2) can be written Cx = λx. (3) Thus the score vector x is an eigenvector of the normalized citation matrix C corresponding to the eigenvalue λ. 3 3 Eigenvectors of the citation matrix In order for x to be a score vector, its components must be non-negative numbers. Since all the entries cij of C are non-negative, Theorem 3 on p. 66 of Gantmacher [3] shows that C has a real non-negative eigenvalue λ with a non-negative eigenvector. This non-negative eigenvector x could be used to determine the scores if it were unique, aside from a constant factor. It will certainly be unique if C is irreducible, i.e., if it cannot be put in the following form by a permutation of the indices: C =      A 0 B D     . (4) Here A and D are square matrices. When C is irreducible, Frobenius’ theorem states that it has a real non-negative eigen- value λ which is larger than the modulus of any other eigenvalue. Furthermore the eigenvec- tor x corresponding to λ is unique up to a scalar factor, and all its components are positive. (Gantmacher, p. 53, theorem 2.) In this case the components of x, with some suitable normalization, can be used as the scores. The eigenvalue λ can be determined by first summing (2) over i. Then from the fact that C is normalized, it follows that the sum of cij over i is unity. The other possibility, that all the cij in one column vanish, cannot occur when C is irreducible. Thus we obtain from (2) N X i=1 xi = λ−1 N X j=1 N X i=1 cij ! xj = λ−1 N X j=1 xj. (5) Now (5) shows that λ = 1. 4 Since λ = 1, we can

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