# Fibroblast-Specific Protein Interactions for Myocardial Fibrosis from MetaCore Network
## 3 Results
### Numerical Results
The values of the color bar correspond to \(f_{r}(i)\) (i.e., red for \(f_{r}(i)=1\), green for \(f_{r}(i)\approx 0.06\) and blue for \(f_{r}(i)=0\)). Here small \(f_{r}(i)\) values have been amplified to improve the visibility (non-linear scale in the Colombo). Then we choose as example four \"optimal\" nodes with \(K=3,4,17,342\) (marked with an asterix in the first column of Table 2). The nodes \(K=4,17,342\) occupy indeed the top three places in the \(K_{f_{r}}\)-rank while the node \(K=3\) \"only\" corresponds to the position \(K_{f_{r}}=11\). The reason for this choice is related to the fact that these four nodes are more uniformly optimal if we also choose a small number of \(n_{ib}=2,\ldots,10\) initial blue nodes. In this case, using the data of Figure 4 (or the code to produce these data) it is actually possible to compute the conditional probability \(f_{rc}(i)\) of a red outcome when the node \(i\) is by chance selected by the random initial condition for \(n_{ib}>1\) (as one of the initial \(n_{ib}\) blue nodes). For example for \(n_{ib}=4\) with \(R=100,000\) different random initial conditions of 4 blues nodes out of 353 we have typically a bit more than 1000 realizations where an arbitrary fixed node \(i\) belongs to the random set of 4 initial nodes. This provides enough data for a reasonable average to compute the conditional probability of red outcome of the node \(i\). In this way, it is possible to compute more general \(K_{f_{r}}\)-rankings as in Table 2, also for modest values of \(n_{ib}>1\). It turns out that these rankings produce roughly the same sets of nodes in the first places (with possible permutations between different \(n_{ib}\)-values) and the four selected nodes \(K=3,4,17,342\) are indeed optimal as a group for the three values \(n_{ib}=2,3,4\) (and also some larger values). In Figure 6, we present (for \(n_{b}=6\)) the dependence of \(f_{r}\) on \(n_{i}\) for this small optimal set of 4 possible initial blue nodes (red square data points). Here the decrease of \(f_{r}\) with increasing \(n_{i}\) is indeed very strong with \(f_{r}\approx 0.035\) for \(n_{ib}=1\) and \(f_{r}\approx 0.00318\) for \(n_{ib}=4\). Furthermore, Figure 6 also provides the conditional probabilities of red outcome \(f_{rc}([K_{1},\ldots,K_{n_{ib}}]\) (black small circle data points) for specific configurations \([K_{1},\ldots,K_{n_{ib}}]\) with one configuration for \(n_{ib}=0,4\), four configurations for \(n_{ib}=1,3\) and six configu \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(K_{f_{r}}\) & \(K\) & \(K^{*}\) & \(f_{rc}\) & **Protein** \\ \hline \(*\) & 1 & 17 & 1 & 0.010691 & c-Myc \\ \(*\) & 2 & 4 & 6 & 0.029542 & p53 \\ \(*\) & 3 & 342 & 7 & 0.038899 & c-Fos \\  & 4 & 14 & 14 & 0.053526 & Androgen receptor \\  & 5 & 22 & 11 & 0.075088 & RelA (p65 NF-kB subunit) \\  & 6 & 94 & 34 & 0.07674 & HDAC1 \\  & 7 & 188 & 19 & 0.078157 & p300 \\  & 8 & 6272 & 3 & 0.078989 & IGF2 BP3 \\  & 9 & 232 & 9 & 0.081749 & SPI \\  & 10 & 64 & 29 & 0.083725 & HIF1A \\ \(*\) & 11 & 3 & 13 & 0.083939 & \(\beta\)-catenin \\  & 12 & 203 & 21 & 0.13861 & E2F1 \\  & 13 & 728 & 5 & 0.14694 & SOX9 \\  & 14 & 432 & 24 & 0.14808 & BRGI \\  & 15 & 8 & 115 & 0.15027 & EGFR \\  & 16 & 72 & 18 & 0.15149 & EZH2 \\  & 17 & 13 & 25 & 0.15649 & STAT3 \\  & 20 & 480 & 40 & 0.17501 & C/EBP\(\beta\) \\  & 21 & 38 & 82 & 0.20211 & PPAR-\(\gamma\) \\  & 24 & 298 & 33 & 0.20997 & ELAVL1 (HuR) \\  & 26 & 394 & 37 & 0.22213 & CREB1 \\  & 31 & 6370 & 17 & 0.28502 & PUM2 \\  & 32 & 17,711 & 8 & 0.29183 & CUX1 (p110) \\ \hline \end{tabular}
\end{table} 
Table 2: Table of selected fibrosis proteins (nodes) belonging to a smaller subset of the Erdös group such that either \(K\leq 40\) or \(K^{*}\leq 40\) where the \(K\) (\(K^{*}\)) index represents the PageRank (CheiRank) index for the global MetaCore network of \(N=40,079\) nodes.