# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## 2 Datasets and Methods
### Without Formulas: Methods, Characteristics, and Expected Network Results  
Here, we present qualitative explanations without formulas of the mathematical methods and characteristics described in the next subsections. Our aim here is to give a global view of our approach for a common reader. We use the MetaCore directed network [8] which represents an action of a protein A on protein B in a form of a directed link (edge) for \(N=40,079\) proteins forming the network nodes (proteins). Such links are obtained on the basis of careful and detailed analysis of scientific literature about thousands of experiments of various research groups that allowed collection of information about PPI and thus generated a network database with \(N=40,079\) nodes and \(N_{\ell}=292,191\) links. The universal mathematical methods to analyze such networks are generic and based on the concept of Markov chains [18] and Google matrix [15; 16; 17]. The validity of these methods has been confirmed for various directed networks from various fields of science. Therefore, since the Google matrix analysis is based on a generic mathematical foundation, we expect that this analysis will also work efficiently for PPI networks. The Google matrix of the global MetaCore PPI network \(G\) is constructed with specific rules described in [15; 16; 17], and the mathematical aspects of this construction are given in Section 2.3. The important property of \(G\) is that its application (multiplication) to an initial vector \(v\) preserves the probability and the normalization of this vector (sum of all vector elements) remains constant (taken to be unity). As a result of multiple multiplications of \(v\) by \(G\), any initial vector converges in the long time limit to the steady-state distribution given by the PageRank vector \(P\). The components of this vector represent the probabilities of each node (protein) in this limit. The nodes with the highest probabilities are the most influential nodes of the network (all nodes are monotonically ordered by decreasing values of the PageRank components which provides the \"PageRank index\" \(K\) such \(K(j)=1,2,\ldots\) for nodes \(j\) with largest values \(P(j)\)). These nodes have typically many ingoing links and it is likely that some of these ingoing links come from other nodes that also have large PageRank values. It is also useful to consider the same network but with the inversed direction of links. For this inverse network, the corresponding PageRank is called CheiRank vector \(P^{*}\)[17] with the highest probabilities \(P^{*}(j)\) for nodes \(j\) with the CheiRank index \(K^{*}(j)=1,2,\ldots\) being the most communicative nodes with typically many outgoing links. If we are interested in a specific selected, typically rather small, group of \(N_{r}\) nodes (\(N_{r}\ll N\)), then the reduced Google matrix (REGOMAX) algorithm (described in Section 2.4 and Equations (2)-(5)) allows us to obtain a \"reduced Google matrix\" \(G_{\rm R}\) which describes effective interactions between these \(N_{r}\) nodes, taking into account both direct links but also all indirect links due to pathways through the complementary network of the other \(N-N_{r}\gg N_{r}\) nodes. In our study, the group of 44 nodes, given in Table 1, is selected on the basis of the experimental results for fibrosis responses obtained in [5]. In addition to these 44 fibrosis internal proteins (\(1\leq K_{g}\leq 44\) in Table 1), we determine a special group of 10 external proteins (\(45\leq K_{g}\leq 54\) in Table 1). These external proteins are found numerically with the following procedure: outside of the 44 proteins, we take those proteins which have at least one ingoing link to the top five positive response proteins (\(5\leq K_{g}\leq 9\)\(1\leq K_{u}\leq 5\)) and the top five negative response proteins (\(25\leq K_{g}\leq 29\), \(1\leq K_{d}\leq 5\)). There are 122 such external proteins, so that in total we have a group of \(44+122=166\) proteins (44 internal and 122 external ones).