EigenfactorTM Score and Article Influence TM
Score: Detailed methods∗
Here we describe the methods used to compute the EigenfactorTM scoresand Article InfluenceTM scores featured at www.eigenfactor.org1. The
purpose of these metrics is to estimate the relative influence of reference
items based on cross-citation data. Like Thomson Scientific’s Impact Factor metric, the EigenfactorTM metrics measure the number of times thatarticles published during a census period provide citations to papers published during an earlier target window. While Impact Factor has a one year
census period and uses the two previous years for the target window, theEigenfactor metrics have a one year census period and use the five previousyears for the target window.
In principle the EigenfactorTM algorithms can be applied to any crosscitation dataset at any level: journals, institutions, authors, articles, etc. In
this document, we will describe the application of the Eigenfactor algorithm
to the cross citation data provided in the 2006 edition (released Summer2007) of Thomson Scientific’s Journal Citation Reports (JCR).

Calculating Eigenfactor Score and Article Influence Score
The 2006 JCR indexes the citations from 7611 “source” journals for theSciences and Social Sciences. From these data, we extract a 5-year crosscitation matrix Z. For 2006, the entries of this matrix would be:
Zij = Citations from journal j in 2006 to articles published injournal i during 2001–2005.
When constructing Z, we omit all self-citations2 , setting all of the diagonal
elements of this matrix to 0. We normalize Z by the column sums (i.e.,by the total number of outgoing citations from each journal) to create acolumn-stochastic matrix H:Zij
Hij = Pk Zkj
Methods version 2.01, November 2008. The EigenfactorTM Metrics, including theEigenfactorTM Score and the Article Influence TM Score, and the Eigenfactor.orgTM website
were initially developed by Jevin West, Ben Althouse, Martin Rosvall, and Carl Bergstrom
at the University of Washington and Ted Bergstrom at the University of California SantaBarbara.
∗1

We also compute an article vector a, where ai is the number of articles
published by journal i over the five-year target window, divided by the totalnumber of articles published by all source journals over the same five-year
window. Notice that a is thus normalized to sum to 1 and it’s i-th entry
specifies the fraction of all published articles that come from journal i.
Some of the journals listed in the H matrix will be dangling nodes —journals that do not cite any other journals. Any column of the H matrix
that has all 0 entries is a dangling node; we replace all such columns in H
with the a vector to produce a new modified matrix H′ .
Following Google’s PageRank approach, we define a new stochastic matrix P as follows3 :P = αH′ + (1 − α)a.eT .
Here eT is a row vector of 1’s and thus A = a.eT is a matrix with identicalcolumns each equal to the article vector a.
We define the journal influence vector π ∗ as the leading eigenvector of P.
This gives us the weights we will use in weighting citations value; under thestochastic process interpretation the π ∗ vector corresponds to steady-state
fraction of time spent at each journal represented in P. The Eigenfactor
Score EFi of journal i is defined as the percentage of the total weighted
citations that journal i receives from our 7611 source items. We can write
the vector of Eigenfactor Scores as
EF = 100 PH π∗.
∗i [H π ]i

Notice that the equation above uses the matrix H without the dangling nodecolumns replaced by the article vector a.
The Article Influence Score AIi for each journal i is a measure of theper-article citation influence of the journal4 . The Article Influence Score iscalculated as
AIi = 0.01

EFi,aiwhere EFi is the Eigenfactor Score for journal i and ai is the i-th entry of
the normalized article vector.

2

Notes1

Pseudocode for computing Eigenfactor scores and Article Influence scores is available at http://www.eigenfactor.org/EF pseudocode.pdf. Mathematica source code isavailable at http://www.eigenfactor.org/efcode compressed.pdf/.
2
We ignore self-citations for several reasons. First, we want to avoid over-inflating
journals that engage in the practice of opportunistic self-citation and to minimize theincentive that our measure provides for such practice. Second, we do not have self-citation
information for the journals not listed in the JCR. Considering self-citations for JCR-listed
but not non-listed journals would systematically over-value the journals in the former set
relative to the latter. Third, if self-citations are included, some small journals with unusual
citation patterns appear as nearly-dangling nodes, and this can bias their eigenfactor scores
upward. The tendency of the JCR data set to list some outgoing citations under a single
composite item “others” — which we cannot use our calculations because we do not know
where they are directed — exacerbates this problem.
3Under a stochastic process interpretation, the matrix H′ corresponds to a random
walk on the citation network with dangling nodes replaced by fully-connected nodes. Thematrix P corresponds to the Markov process which with probability α follows a random
walk on the journal citation network, and which with probability (1 − α) “teleports” to arandom journal in proportion to the number of articles published by each journal. Rather
than using the leading eigenvector of H′ for our journal weights, we compute the leading
eigenvector of the matrix P. We do so for a two of reasons.
First, the stochastic matrix H′ may be non-irreducible or periodic. Adding the teleport probability 1 − α ensures that P is both irreducible and aperiodic, and therefore hasa unique leading eigenvector by the Perron-Frobenius theorem.
Second, even if the network is irreducible, without teleporting rankings can be unreliable and highly volatile when some components are extremely sparsely connected.
Suppose, for example, that a citation network comprises two fields are connected only by
the citations of two journals, one in each field. The relative weight of each field would thenbe set solely by the relative frequencies with which these two journals cited the other field.
Similarly, teleporting keeps the system from getting trapped in small nearly-dangling clusters. If a small clique of journals are occasionally cited from outside but rarely cite out ofclique itself, the Markov process characteried by H′ can become trapped in this portionof the citation network for a very long period in time, effectively overvaluing the journals
in this clique. Teleporting corrects this problem by reducing the expected duration of astay in these small cliques.
We teleport to a journal with probability proportional to the number of articles published by that journal in order to avoid over-inflating the influence of small journals andunder-inflating the influence of large ones. This is important because the journals in thesocial sciences are much smaller, on average than the journals in the sciences. As a result,an unweighted teleportation process, in which one teleports to each journal with equal
probability, overestimates influence of articles in social science journals relative to sciencejournals because the teleportation process.
4The Eigenfactor Score provides a measure of the total influence that a journal provides,
rather than a measure of influence per article. Impact factor, by contrast, measures theper-article influence of a given journal. To make our results comparable to impact factor,
we need to divide the journal influence by the number of articles published. Doing soyields the Article Influence Score.

3