# Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network
## Appendix A Appendix A Additional Figures for REGOMAX Results
### Simple Estimate for the Sensitivity Matrix  
In the second part of this appendix we remind some details (see [34]) about the numerical computation of the sensitivity (8) and provide an analytic approximation based on a simplified model. Let \((a,b)\) be an arbitrary index pair and \(G_{\varepsilon}\) be the perturbed Google matrix obtained from a general unperturbed Google matrix \(G_{0}\) by multiplying its element \(G_{0}(a,b)\) at position \((a,b)\) by \((1+\varepsilon)\) and then sum-renormalizing the column \(b\) to unity. The elements in the other columns are not modified. In a more explicit formula we have:  
\[\forall_{c,d}\qquad G_{\varepsilon}(c,d)=\frac{\left(1+\varepsilon\,\delta_{ ca}\delta_{db}\right)G_{0}(c,d)}{1+\varepsilon\,\delta_{db}\,G_{0}(a,b)} \tag{11}\]where \(\delta_{ca}=1\) (or 0) if \(c=a\) (or \(c\
eq a\)). Note that the denominator is either 1 if \(d\
eq b\) or the modified column sum \(1+\varepsilon\,G_{0}(a,b)\) of column \(b\) if \(d=b\). Expanding (A1) up to first order in \(\varepsilon\) we obtain \(G_{\varepsilon}=G_{0}+\varepsilon\Delta G+\ldots\) with \(\Delta G\) having the elements:  
\[\forall_{c,d}\qquad\Delta G(c,d)=\delta_{ca}\delta_{db}\,G_{0}(c,d)-\delta_{db }\,G_{0}(a,b)\;G_{0}(c,d)\;.\] (A2)  
Let \(P_{\varepsilon}\) be the sum-normalized PageRank vector of \(G_{\varepsilon}\) determined by the conditions \(G_{\varepsilon}\,P_{\varepsilon}=P_{\varepsilon}\) and the normalization \(E^{T}\,P_{\varepsilon}=1\) where \(E^{T}=(1,\ldots,1)\) is a (row) vector with unit entries. Note that the column sum condition of \(G_{\varepsilon}\) can be written as \(E^{T}\,G_{\varepsilon}=E^{T}\) and of course for \(\varepsilon=0\) we also have \(G_{0}\,P_{0}=P_{0}\), \(E^{T}\,P_{0}=1\) and \(E^{T}\,G_{0}=E^{T}\). Furthermore we write the perturbed PageRank vector in the form \(P_{\varepsilon}=P_{0}+\varepsilon\Delta P+\ldots\) where the \(\Delta P\) must satisfy the condition \(E^{T}\,\Delta P=0\). Then the sensitivity (8) is directly related to \(\Delta P\) by:  
\[D_{(b\to a)}(j)=\frac{\Delta P(j)}{P_{0}(j)}\;.\] (A3)  
Expanding the PageRank equation \(G_{\varepsilon}\,P_{\varepsilon}=(G_{0}+\varepsilon\Delta G+\ldots)(P_{0}+ \varepsilon\Delta P+\ldots)=P_{\varepsilon}=P_{0}+\varepsilon\Delta P+\ldots\) to order one we first obtain the unperturbed PageRank equation \(G_{0}\,P_{0}=P_{0}\) and a further inhomogeneous equation :  
\[\Delta P=G_{0}\,\,\Delta P+\Delta G\,\,P_{0}\] (A4)  
which can be efficiently numerically solved by iteration (choosing initially \(\Delta P=0\) on the right hand side) once \(P_{0}\) has been computed (see [34] for details on this point). This provides a numerical precise scheme to compute the sensitivity in the limit \(\varepsilon\to 0\) without the need to take finite \(\varepsilon\)-differences. Now, we consider a particular very simple model where \(G_{0}\) has identical columns being the PageRank \(P_{0}\), i.e., \(G_{0}=P_{0}\,E^{T}\) or more explicitely \(G_{0}(c,d)=P_{0}(c)\) for all values of \(c,d\). Then we obtain from (A2)  
\[\forall_{c,d}\qquad\Delta G(c,d)=\delta_{ca}\delta_{db}\,P_{0}(c)-\delta_{db} \,P_{0}(a)\,P_{0}(c)\] (A5)  
and from (A4)  
\[\Delta P=(P_{0}\,E^{T})\,\Delta P+\Delta G\,P_{0}=\Delta G\,P_{0}\] (A6)  
since \(E^{T}\,\Delta P=0\). Inserting (A5) in (A6) we obtain (replacing \(c=j\) and performing the \(d\)-sum for the matrix vector product)  
\[\forall_{j}\qquad\Delta P(j)=\left[\delta_{ja}-P_{0}(a)\right]P_{0}(j)\,P_{0} (b)\] (A7)  
and from (A3)  
\[D_{(b\to a)}(j)=\left[\delta_{ja}-P_{0}(a)\right]P_{0}(b)\;.\] (A8)  
Choosing \(j=a\) this gives the sensitivity matrix  
\[D_{ab}=D_{(b\to a)}(a)=\left[1-P_{0}(a)\right]P_{0}(b)\approx P_{0}(b)\] (A9)  
where the last approximation holds if typically \(P_{0}(a)\ll 1\). This result is of course only valid for the simplified model of identical columns (being the PageRank vector) in \(G_{0}\).