# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## 3 Results
### Network Diagrams of Fibrosis Interactions  
In this section, we discuss two types of effective networks (of most important PPI links) obtained from the two matrices \(G_{\rm R}\) and \(G_{\rm rr}+G_{\rm qr}^{\rm(nd)}\), the latter containing the \"interesting\" links without the uniform background generated by the component \(G_{\rm pr}\) (and without self-interactions). We remind the reader that the value of a matrix element \(g(a,b)\) (with \(g\) being either \(G_{\rm R}\) or \(G_{\rm rr}+G_{\rm qr}^{\rm(nd)}\)) corresponds to the strength of the link \(b\to a\). If this value is sufficiently high, we say that \(a\) is a \"friend\" of \(b\) and \(b\) is a \"follower\" of \(a\). This distinction allows one to construct for each matrix two types of effective networks by choosing a few number of \"top nodes\" and adding a certain number of the strongest friends (or followers) according to the values of \(|g(a,b)|\) and repeating this procedure for a modest number of depth levels. Figure 3: Color density plot \(G_{\rm rr}+G_{\rm qr}^{\rm(nd)}\) for the group of Table 1. The matrix element at \((45,16)\) (\((49,3)\)) has been reduced from 0.849861 (0.240632) to the value 0.121433 at \((29,24)\); a few matrix elements of \(G_{\rm rr}+G_{\rm qr}^{\rm(nd)}\) have negative values visible as cyan color (see text). The numbers in the color bar correspond to \(\mathrm{sgn}(g)\sqrt{|g|/g_{\rm max}}\), with \(g\) being the value of the matrix element and \(g_{\rm max}\) being the maximum value. In Figure 4, we show four graphical representations of such effective networks for the two cases of friend or follower networks and the two matrices \(G_{\text{R}}\) and \(G_{\text{rr}}+G_{\text{qr}}^{\text{(nd)}}\) visible in Figures 2 and 3. In these figures and the remainder of this subsection, we use the short notations \(Tj,Ul,Dj\) or \(Xj\) for a protein/node where \(j=1,2,\ldots\) is the integer value of the subgroup index \(K_{t},K_{u},K_{d}\) or \(K_{x}\), respectively, with real protein names given in Table 1.  
To construct the effective network for a matrix component \(g\) (with \(g\) being either \(G_{\text{R}}\) or \(G_{\text{rr}}+G_{\text{qr}}^{\text{(nd)}}\)), we first choose five initial top nodes/proteins corresponding to \(U1,U2\) (ADAMTS16, FGF21), \(D1,D2\) (CLEC3B, SCARA5), and \(X9\) (MMP-14). \(U1,U2\) (\(D1,D2\)) have the strongest positive (negative) TGF-\(\beta\) response observed experimentally in [5]. The node corresponding to \(X9\) (MMP-14) produces the strongest sensitivity \(D_{ab}\) (among those elements \(D_{ab}\) where \(a\) is an up- or down protein and \(b\) is a TGF-\(\beta\) or X-protein; see next subsection for details on this). These five proteins form the set of level-0 nodes which are placed on a large circle. We attribute the color red to the combined subgroups of 10 external X-proteins (\(K_{x}=1,\ldots,10\)) and 4 TGF-\(\beta\) proteins (\(K_{t}=1,2,3,4\)). The transitions inside this red group are not taken into account since we are mainly interested in the influence of this group on the other up- and down-proteins. We attribute two colors to the up-proteins (olive green to \(U1\), green to \(U2\)) and two colors to the down-proteins (cyan to \(D1\), blue to \(D2\)). Inside the group of up-proteins, we attribute the color olive green to a protein \(Uj\) if \(Uj\) is a stronger follower of \(U1\) than of \(U2\) with respect to \(g=G_{\text{rr}}+G_{\text{qr}}^{\text{(nd)}}\), i.e., if \(g(K_{u}=1,K_{u}=j)>g(K_{u}=2,K_{u}=j)\), and green otherwise. In other words, we compare the strength of the links \(Uj\to U1\) and \(Uj\to U2\) to determine if \(Uj\) has the color olive green of \(U1\) or green of \(U2\). In a similar way, by comparing the strength of the two links from a \(Dj\) protein to either \(D1\) or \(D2\), we attribute the two colors cyan and blue to down-proteins.