# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## 2 Datasets and Methods
### Google Matrix Construction, PageRank and CheiRank  
First, we construct the Google matrix \(G\) of the MetaCore network for the simple case where the bifunctional nature of links is neglected. Furthermore, the directed links are nonweighted. First, one defines an adjacency matrix with elements \(A_{ij}\) being equal to 1 if node \(j\) points to node \(i\), and equal to 0 otherwise. In the next step, the stochastic matrix \(S\) describing the node-to-node Markov transitions is obtained by normalizing each column sum of the matrix \(A\) elements to unity. For dangling nodes \(j\) corresponding to zero columns of \(A\), i.e., \(A_{ij}=0\) for all nodes \(i\), the corresponding elements of \(S\) are defined by \(S_{ij}=1/N\). The stochastic matrix \(S\) describes a Markov process on the network: a random surfer jumps from node \(j\) to node \(i\) with the probability \(S_{ij}\), therefore following the directed links. The column sum normalization \(\sum_{i}S_{ij}=1\) ensures the conservation of probability. The elements of the Google matrix \(G\) are then defined by the standard form  
\[G_{ij}=\alpha S_{ij}+(1-\alpha)/N \tag{1}\]  
where \(\alpha=0.85\) is the usual damping factor [15; 16]. The Google matrix is also column sum normalized and now the random surfer jumps on the network in accordance with the stochastic matrix \(S\) with a probability \(\alpha\) and with a complementary probability \((1-\alpha)\), to an arbitrary random node of the network. The damping factor allows escape from possible isolated communities and ensures that the Markov process converges for long times rather quickly to a uniform stationary probability distribution. The latter is given by the PageRank vector \(P\), which is the right eigenvector of the Google matrix \(G\) corresponding to the leading eigenvalue, here, \(\lambda=1\). The corresponding eigenvalue equation is then \(GP=P\). According to the Perron-Frobenius theorem, the PageRank vector \(P\) has positive elements and their sum is normalized to unity. The PageRank vector element \(P(j)\) gives the probability to find the random surfer on the node \(j\) at the stationary state of the Markov process. Thus, all nodes can be ranked by a monotonically decreasing PageRank probability. The PageRank index \(K(j)\) gives the rank of the node \(j\) with the highest (lowest) PageRank probability \(P(j)\) corresponding to \(K(j)=1\) (\(K(j)=N\)). The PageRank probability \(P(j)\) is proportional, on average, to the number of ingoing links pointing to node \(j\). However, it also takes into account the \"importance\" (i.e., PageRank probability) of the nodes having a direct link to \(j\).  
We note that multiple checks, described in [16; 17; 23] and carried out for a variety of directed networks, including PPI networks [21; 22], showed that the PageRank probabilities are stable with respect to variation of \(\alpha\) in the range \((0.5,0.95)\). Here, we use the traditional value \(\alpha=0.85\) used in [16; 21; 15; 22].  
It is also useful to consider a network obtained by the inversion of all link directions. For this inverted network, the corresponding Google matrix is denoted \(G^{*}\) and the corresponding PageRank vector, called the CheiRank vector \(P^{*}\), is defined such as \(G^{*}P^{*}=P^{*}\). A detailed statistical analysis of the CheiRank vector can be found in [28; 29] (see also [17]). Similarly to the PageRank vector, the CheiRank probability \(P^{*}(j)\) is proportional, on average, to the number of outgoing links going out from node \(j\). The CheiRank index \(K^{*}(j)\) is also defined as the rank of the node \(j\) according to decreasing values of the CheiRank probability \(P^{*}(j)\).