# From Color-Avoiding to Color-Favored Percolation in Diluted Lattices
## 5 Applications
Considering a random variable \(r\), associated with and edge connecting two nodes, and a color \(c\), the variable \(\mathcal{C}_{c}(r)\) is equal to \(1\) with probability \(p(c)\) and equal to \(0\) with probability \(1-p(c)\), thus allowing us to determine if the edge is colored with color \(i\), or another color. In addition, if \(\mathcal{C}_{c}(r)=1\) for a certain value of \(r\), i.e., the edge has the color \(c\), then \(\mathcal{C}_{j}(r)=0\) for every other color \(j\
eq c\), hence the edge cannot have two different colors at the same time. Therefore, for a generic network with \(C\) exclusive colors, and some color \(c\) to be avoided, the bond color-avoiding percolation can be studied with the following local evolution rule  
\[s_{i}^{(\overline{c})}(t+1)=\bigvee_{j=a_{ij}=1}[\phi>\eta_{ij}(t)]\overline{ \mathcal{C}_{c}}(r_{ij}(t))s_{j}^{(\overline{c})}(t+1)\quad\forall c, \tag{9}\]  
where  
\[\overline{\mathcal{C}_{c}}(r)=1\oplus\mathcal{C}_{c}(r) \tag{10}\]  
is the negation of \(\mathcal{C}_{c}(r)\). As for the case of the bond CAP with two exclusive colors, the random variable \(\eta_{ij}(t)\) is confronted with the dilution parameter \(\phi\) to determine whether the connection between node \(i\) and \(j\) is removed or not, while \(r_{ij}(t)\) determines the color of the edge. We can consider again the product of the states \(s_{i}^{(\overline{c})}\) associated with all the colors \(c\) to be avoided, which is equal to one only if node \(i\) can be reached avoiding every untrusted color, and the fraction of nodes for which this product is equal to one, denoted as  
\[\mathcal{M}(t)=\frac{1}{N}\sum_{i}\left(\prod_{c}s_{i}^{(\overline{c})}(t) \right). \tag{11}\]  
Imposing as initial condition \(s_{i}^{(\overline{c})}(0)=1\) for all nodes and all colors, the asymptotic value \(\lim_{t\to\infty}\mathcal{M}(t)\) can be interpreted as the relative size of the CAGC. Figure 2: Phase planes of the bond CAP for two exclusive colors (denoted color \(1\) and \(2\)). assigned to sites with probability \(p(1)\) and \(p(2)=1-p(1)\) and different dilutions (different connectivities). Using Equation (9), we studied the bond CAP in the case of three exclusive colors, and obtained the phase planes shown in Figures 3 and 4. The phase planes were obtained with a single simulation of the evolution of the fragments in a lattice with \(N=10{,}000\), at the asymptotic time \(t=100{,}000\). For the phase planes shown in Figure 3 we considered the case in which the tree colors (\(1-3\)) are untrusted, and we studied the values of \(\mathcal{M}\) in the parameter space, that can be represented as the set of allowed values of the color probabilities \(p(1)\) and \(p(2)\) such that \(\sum_{c=1}^{3}p(c)=1\), and \(p(c)\geq 0\quad\forall c\) (the probability for the third color, \(p(3)\), is fixed given the values of \(p(1)\) and \(p(2)\) since \(p(3)=1-p(1)-p(2)\)), in networks with average connectivity \(k=5\), at different values of the dilution parameter \(\phi\). As it can be seen, \(\mathcal{M}=0\) near the three vertices of the \"triangle\" of the allowed values of \(p(1)\) and \(p(2)\), that corresponds to the situations in which one of the three colors prevails in the network and is hard to avoid, and the other two are less present, as for example for \(p(1)\sim 0\), \(p(2)\sim 0\), while \(p(3)\sim 1\). Far from the vertices, the system can be in the active phase with \(\mathcal{M}\
eq 0\), and its size in the phase plane depends on the connectivity of the network, as it can be seen considering different values of the dilution parameter. The size of the active phase decreases lowering the value of \(\phi\), and for small enough values as \(\phi=0.5\), if one color is nearly absent from the network and easy to avoid, as for example for \(p(1)\sim 0\), the other colors become harder to avoid, and \(\mathcal{M}=0\) for all the values of \(p(2)\) and \(p(3)\). We also studied a different problem for the phase planes, considering networks with one trusted color and two untrusted ones.