# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## 3 Results
### Network Diagrams of Fibrosis Interactions
This attribution rule, using the strongest followers with respect to \(G_{\text{rr}}+G_{\text{qr}}^{\text{(nd)}}\) of the two top nodes inside a subgroup, ensures that for all colors there is a considerable number of proteins and it is the same for all four network diagrams (both matrices and both friend/follower cases). For each of the five level-0 proteins, noted \(a\), we first search the four strongest friends (followers), noted \(b\), with largest value of \(|g(b,a)|\) (or \(|g(a,b)|\)) corresponding the strongest link \(a\to b\) (or \(b\to a\)), where the matrix \(g\) is either \(G_{\text{R}}\) or \(G_{\text{rr}}+G_{\text{qr}}^{\text{(nd)}}\). The new nodes \(b\) (if not yet present in the set of level-0 nodes) form the set of new level-1 nodes and they are placed on medium-sized circles of level 1 around the corresponding \"parent\" node \(a\) of level-0. The links between the nodes \(a\) and \(b\) are drawn as thick black arrows with direction \(a\to b\) (\(b\to a\)) for the friend (follower) case. If a node \(b\) already belongs to the set of level-0 nodes, we also draw a thick black arrow but using its already existing position on the initial large circle. If a node \(b\) has several parent nodes \(a\), we place it only on one medium circle, preferably around a parent node of the same color if possible. This procedure is repeated once: for each level-1 protein we determine the four strongest level-2 friends (or followers) which are placed on smaller circles of level 2 around the corresponding level-1 protein, provided that they are not yet present in the former sets of level-0 or level-1 proteins. The links corresponding to this stage are drawn as thin red arrows with the same directions as in the first stage (we also draw thin arrows for selected nodes who were already previously selected and using their former positions). As already mentioned above, links where _both_ proteins (\(a\) and \(b\)) belong to the combined set of X- and TGF-\(\beta\) proteins are not taken into account (otherwise they would strongly dominate these diagrams). We limit ourselves to two stages of the procedure (i.e., three levels of nodes) because otherwise the diagrams would require still smaller circles and many nodes would be hidden by former nodes. We note that for the friend-\(G_{\text{R}}\) diagram, a further third stage would not add any new nodes since the strongest friends of level-2 are already in the network. For the other cases, additional further stages would only add a few number of new nodes with a quite rapid saturation of the network at some limit level where no new nodes are selected. Figure 4 shows diagrams of level-2 networks for the cases of friend (top row) and follower (bottom row) diagrams and the two matrices \(g=G_{\rm R}\) (left column) or \(g=G_{\rm rr}+G_{\rm qr}^{\rm(nd)}\) (right column). Concerning the two cases of \(g=G_{\rm rr}+G_{\rm qr}^{\rm(nd)}\), about 15% of the shown arrows correspond to negative values of the matrix element of \(g\) (link strength is determined by the modulus of the matrix element). For the friend network of \(G_{\rm R}\), there is a dominance of links (black arrows) \(U1,U2,D1,D2\to Xj\) for certain X-proteins \(Xj\) which can be understood by the fact that most \(Xj\) proteins have significantly higher PageRank probabilities than the other proteins. Furthermore, the total number of nodes in this diagram is quite small because the strongest friends of level-1 nodes (\(X1,X2,X3,U4,U5,D14\)) are mostly other level-1 nodes and there is only one new level-2 node (\(D20\)). This diagram is obviously dominated by the uniform background (of the component \(G_{\rm pr}\) contributing to \(G_{\rm R}\)) which tends to select mostly the \"same new friends\" at each level. For the friend case of \(G_{\rm rr}+G_{\rm qr}^{\rm(nd)}\), the network structure is significantly richer, since here, the global PageRank transitions (due to the uniform background of \(G_{\rm pr}\)) do not play a role. The group around \(U1\) includes \(T2,T3,T4\).