CheiRank

The CheiRank is an eigenvector with a maximal real eigenvalue of the Google matrix G ∗ {displaystyle G^{*}} constructed for a directed network with the inverted directions of links. It is similar to the PageRank vector, which ranks the network nodes in average proportionally to a number of incoming links being the maximal eigenvector of the Google matrix G {displaystyle G} with a given initial direction of links. Due to inversion of link directions the CheiRank ranks the network nodes in average proportionally to a number of outgoing links. Since each node belongs both to CheiRank and PageRank vectors the ranking of information flow on a directed network becomes two-dimensional. The CheiRank is an eigenvector with a maximal real eigenvalue of the Google matrix G ∗ {displaystyle G^{*}} constructed for a directed network with the inverted directions of links. It is similar to the PageRank vector, which ranks the network nodes in average proportionally to a number of incoming links being the maximal eigenvector of the Google matrix G {displaystyle G} with a given initial direction of links. Due to inversion of link directions the CheiRank ranks the network nodes in average proportionally to a number of outgoing links. Since each node belongs both to CheiRank and PageRank vectors the ranking of information flow on a directed network becomes two-dimensional. For a given directed network the Google matrix is constructed in the way described in the article Google matrix. The PageRank vector is the eigenvector with the maximal real eigenvalue λ = 1 {displaystyle lambda =1} . It was introduced in and is discussed in the article PageRank. In a similar way the CheiRank is the eigenvector with the maximal real eigenvalue of the matrix G ∗ {displaystyle G^{*}} built in the same way as G {displaystyle G} but using inverted direction of links in the initially given adjacency matrix. Both matrices G {displaystyle G} and G ∗ {displaystyle G^{*}} belong to the class of Perron–Frobenius operators and according to the Perron–Frobenius theorem the CheiRank P i ∗ {displaystyle P_{i}^{*}} and PageRank P i {displaystyle P_{i}} eigenvectors have nonnegative components which can be interpreted as probabilities. Thus all N {displaystyle N} nodes i {displaystyle i} of the network can be ordered in a decreasing probability order with ranks K i ∗ , K i {displaystyle K_{i}^{*},K_{i}} for CheiRank and PageRank P i ∗ , P i {displaystyle P_{i}^{*},P_{i}} respectively. In average the PageRank probability P i {displaystyle P_{i}} is proportional to the number of ingoing links with P i ∝ 1 / K i β {displaystyle P_{i}propto 1/{K_{i}}^{eta }} . For the World Wide Web (WWW) network the exponent β = 1 / ( ν − 1 ) ≈ 0.9 {displaystyle eta =1/( u -1)approx 0.9} where ν ≈ 2.1 {displaystyle u approx 2.1} is the exponent for ingoing links distribution. In a similar way the CheiRank probability is in average proportional to the number of outgoing links with P i ∗ ∝ 1 / K i ∗ β ∗ {displaystyle P_{i}^{*}propto 1/{K_{i}^{*}}^{eta ^{*}}} with β ∗ = 1 / ( ν ∗ − 1 ) ≈ 0.6 {displaystyle eta ^{*}=1/( u ^{*}-1)approx 0.6} where ν ∗ ≈ 2.7 {displaystyle u ^{*}approx 2.7} is the exponent for outgoing links distribution of the WWW. The CheiRank was introduced for the procedure call network of Linux Kernel software in, the term itself was used in Zhirov. While the PageRank highlights very well known and popular nodes, the CheiRank highlights very communicative nodes. Top PageRank and CheiRank nodes have certain analogy to authorities and hubs appearing in the HITS algorithm but the HITS is query dependent while the rank probabilities P i {displaystyle P_{i}} and P i ∗ {displaystyle P_{i}^{*}} classify all nodes of the network. Since each node belongs both to CheiRank and PageRank we obtain a two-dimensional ranking of network nodes. There had been early studies of PageRank in networks with inverted direction of links but the properties of two-dimensional ranking had not been analyzed in detail. An example of nodes distribution in the plane of PageRank and CheiRank is shown in Fig.1 for the procedure call network of Linux Kernel software. The dependence of P , P ∗ {displaystyle P,P^{*}} on K , K ∗ {displaystyle K,K^{*}} for the network of hyperlink network of Wikipedia English articles is shown in Fig.2 from Zhirov. The distribution of these articles in the plane of PageRank and CheiRank is shown in Fig.3 from Zhirov. The difference between PageRank and CheiRank is clearly seen from the names of Wikipedia articles (2009) with highest rank. At the top of PageRank we have 1.United States, 2.United Kingdom, 3.France while for CheiRank we find 1.Portal:Contents/Outline of knowledge/Geography and places, 2.List of state leaders by year, 3.Portal:Contents/Index/Geography and places. Clearly PageRank selects first articles on a broadly known subject with a large number of ingoing links while CheiRank selects first highly communicative articles with many outgoing links. Since the articles are distributed in 2D they can be ranked in various ways corresponding to projection of 2D set on a line. The horizontal and vertical lines correspond to PageRank and CheiRank, 2DRank combines properties of CheiRank and PageRank as it is discussed in Zhirov. It gives top Wikipedia articles 1.India, 2.Singapore, 3.Pakistan.