# Fibroblast-Specific Protein Interactions for Myocardial Fibrosis from MetaCore Network
## 2 Data Sets and INFI Model Description
### Markov Chains, Google Matrix, PageRank and CheiRank  
In this section we remind some basic mathematical definitions of Markov chains and how the Google matrix and the related PageRank and CheiRank vectors for the MetaCorenetwork are constructed. For this we use a simple version of the MetaCore network where initial links are not weighted (see Refs. [15; 16] for some details about a more refined version with bi-functional links). First, one introduces the adjacency matrix with elements \(A_{ij}=1\) if a node \(j\) points to another node \(i\) (different from \(j\)) and \(A_{ij}=0\) if there is no link from \(j\) to \(i\) or if \(i=j\) (we use the same convention as in Ref. [16] where all diagonal elements are chosen to be \(A_{ij}=0\) even if there is a link from a node \(j\) to itself). Then we define a matrix \(\tilde{S}\) by \(\tilde{S}_{ij}=A_{ij}/\sum_{i}A_{ij}\) if \(\sum_{i}A_{ij}\
eq 0\) and \(\tilde{S}_{ij}=0\) otherwise (i.e., for columns \(j\) where all elements \(A_{ij}\) are \(0\)). This matrix does not yet describe a Markov process since it has potentially some zero columns \(j\) (such nodes \(j\) are also called dangling nodes). However, in this work we will use a Monte Carlo process based on this particular matrix (see next section).  
To obtain a proper stochastic matrix \(S\) one replaces \(S_{ij}=1/N\) for the dangling nodes \(j\) and \(S_{ij}=\tilde{S}_{ij}\) for the other nodes (dangling nodes have no outgoing links so that a column of dangling node \(j\) has only zero elements in the matrix \(\tilde{S}_{ij}\)). Here \(S_{ij}\) represents the transition probability for a random surfer from node \(j\) to \(i\) and the column sum normalization \(\sum_{i}S_{ij}=1\) ensures the conservation of probability.  
The Google matrix elements \(G_{ij}\) are typically defined by  
\[G_{ij}=\alpha S_{ij}+(1-\alpha)/N \tag{1}\]  
where \(\alpha=0.85\) is the usual damping factor [17; 18]. The Google matrix is also a proper stochastic matrix (column sum normalized) and here the random surfer jumps with probability \(\alpha\) on the network according to \(S\) and with a probability \((1-\alpha)\) to an arbitrary random node of the network. The damping factor modification helps to avoid possible isolated communities and ensures that the Markov process converges for long times rather quickly to a uniform stationary probability distribution. The latter defines the PageRank vector \(P\) which is actually the right eigenvector of the Google matrix \(G\) corresponding to the leading eigenvalue \(\lambda=1\), i.e., \(GP=P\). The elements \(P(j)\) of the PageRank vector correspond to the probability to find the random surfer on the node \(j\) in the stationary limit of the Markov process. The PageRank index \(K(j)\) is obtained by ordering the nodes with decreasing values of \(P(j)\), i.e., the highest (lowest) PageRank probability \(P(j)\) corresponds to \(K(j)=1\) (\(K(j)=N\)). In this work we will use the \(K\)-rank of a node to characterize this node in a unique way. The PageRank probability \(P(j)\) is typically related 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 \(j\).  
One can also consider the network obtained by the inversion of all link directions. The same construction as described above provides a Google matrix noted as \(G^{*}\) and the corresponding PageRank vector is called the CheiRank vector \(P^{*}\), defined by \(G^{*}P^{*}=P^{*}\). The CheiRank probability \(P^{*}(j)\) is typically related to the number of outgoing links weighted by the value \(P^{*}(i)\) of nodes \(i\) having a link from \(j\) to \(i\). The CheiRank index \(K^{*}(j)\) is obtained by ordering the CheiRank vector with decreasing values of \(P^{*}(j)\).  
For further details about the properties of Google matrix, PageRank and CheiRank vectors, with many different example networks (also with proteins networks), we refer to [15; 16; 18; 19].