# Fibroblast-Specific Protein Interactions for Myocardial Fibrosis from MetaCore Network
## 3 Results
### Numerical Results
For each value of \(n_{ib}\) the initial condition corresponds to \(n_{ib}\) initial blue nodes (\(\sigma_{i}(\tau=0)=-1\)) in this reduced set of 38 nodes (with random positions for each of the \(R=100{,}000\) pathway realizations). All other variable nodes have initial white values (\(\sigma_{i}(\tau=0)=0\)). The first data point at \(n_{ib}=0\) has the same value as in Figure 2 and it has also artificially been placed at a finite position below \(n_{ib}=1\) for practical reasons. In Figure 5, we show in a color plot the dependence of \(f_{r}(i)\) for the 54 nodes \(i\) belonging to the set of Table 1 on the index \(n_{g}\) which is a monotonic function of \(n_{ib}\) (essentially linear for \(n_{ib}\leq 10\) and logarithmic for \(n_{ib}>10\); see caption of Figure 5 and Appendix Figure A1 for details). The data of Figure 5 correspond to the data of Figure 4 using the Erdos set with \(N_{E}=353\) as potential initial blue nodes (i.e., for each value of \(n_{ib}\in\{0,\ldots,353\}\) we have \(n_{ib}\) initial blue nodes at random positions in this set). In Figure 5, we can of course identify the 10 fixed red nodes (\(K_{g}=1,\ldots,10\)) and the \(n_{b}=6\) fixed blue nodes (\(K_{g}=25,\ldots,30\)) with either \(f_{r}(i)=1\) or \(f_{r}(i)=0\) respectively for all values of \(n_{g}\). The other nodes follow quite closely the decrease of Figure 4 for the global average of \(f_{r}\). However, for certain specific nodes (\(K_{g}=17,23,36,38,41,44\)) the decay of \(f_{r}\) with \(n_{g}\) (or \(n_{ib}\)) is less pronounced for \(n_{g}>10\). Apparently, these nodes are less likely to be blocked by a modest number of initial blue nodes. We have also analyzed the data of Figure 2 (using all \(N_{v}\) variable nodes as potential initial blue nodes) and Figure 3 (using the remaining set of 38 non-fixed nodes of Table 1 as potential initial blue nodes) with similar color plots and in both cases we observe the same qualitative behavior as in Figure 5: identification of fixed red/blue nodes, similar decrease of \(f_{r}(i)\) with increasing \(n_{ib}\) for the other nodes and less pronounced decrease for the 6 specific nodes mentioned above. The question arises which of the nodes of the Erdos set, or more generally, which configurations of few selected nodes of this set, are most effective to reduce the red outcome when selected as initial blue nodes. To answer this question, we compute for each node \(i\) of the Erdos set the red outcome \(f_{rc}(i)\) averaged over the full network when this specific node \(i\) is selected as single initial blue node, i.e., with \(n_{ib}=1\) but now with different cases of given fixed positions (instead of random positions). Note that this quantity is different from \(f_{r}(i)\) used in the histogram of Figure 1 which is the probability of red outcome of node \(i\) (not averaged over the network) with full white initial condition. The nodes of the Erdos set can be ordered with increasing values of the new quantity \(f_{rc}(i)\) which provides a specific ranking index \(K_{f_{r}}\) in this set. In Table 2, we present in \(K_{f_{r}}\)-order the nodes (proteins) of the Erdos set which have either \(K\leq 40\) or \(K^{*}\leq 40\), either low \(K\)- or \(K^{*}\)-rank. It turns out these nodes also provide the lowest values of \(f_{rc}\) and \(K_{f_{r}}\) (with some holes for \(K_{f_{r}}>17\)). Figure 5: Color plot of \(f_{r}(i)\) for the 54 nodes of the set of Table 1 for the case of Figure 4 (i.e., \(n_{b}=6\), using the Erdős group with \(N_{r}=353\) nodes as potential initial blue nodes and \(n_{ib}\) being the number of initial blue nodes at random positions in the Erdös group). The \(x\)-axis corresponds to the index \(K_{g}\) of Table 1 and the \(y\)-axis represents to the coarse-grained index \(n_{g}=0,\ldots,31\) which corresponds to \(n_{ib}=n_{g}\) for \(n_{g}\leq 10\) (linear scale) and \(n_{ib}\approx 10\times 1.2^{n_{g}-10}\) for \(n_{g}>10\) (logarithmic scale). See also Figure A2 which shows the link between \(n_{ib}\) and \(n_{g}\).