# From Color-Avoiding to Color-Favored Percolation in Diluted Lattices
## 2 Percolation and Directed Percolation  
Let us recall some definitions. We denote by \(s_{i}(t)\in\{0,1\}\) the state of site in position \(i\) (\(1\leq i\leq N\)) at time \(t\) (\(0\leq t\leq T\)), where one means wet or infected and zero dry or healthy. The index \(i\) identifies a site in a given network, connected to other sites \(j\) as specified by the adjacency matrix \(a\): \(a_{ij}=1\) if there is a link from site \(j\) to site \(i\) and zero otherwise. We denote by \(k_{i}=\sum_{j}a_{ij}\) the (input) connectivity of site \(i\). Let us suppose for the moment that the lattice has constant connectivity \(k\), and take the simplest case \(k=2\), with each site \(s_{i}\) connected to its neighboring two sites \(s_{i-1}\) and \(s_{i+1}\). In classical percolation, the evolution of the state of the whole system is given by the parallel application of the rule  
\[s_{i}(t+1)=[r_{i}<p]\wedge\big{(}s_{i-1}(t)\lor s_{i+1}(t)\big{)}\lor s_{i}(t), \tag{1}\]  
where \(r_{i}\) is a random number uniformly distributed in the interval \([0,1)\), \(\lor\) is the OR operation, \(\wedge\) the AND one (equivalent to the multiplication) and \([\cdot]\) is the truth function which takes the value one if \(\cdot\) is true, and zero otherwise. Equation (1) essentially says that if a site is wet, it stays wet, and that a dry site can become wet with probability \(p\) if at least one of neighbors is so (\([r_{i}<p]\) gives the probability that the \"pore\" \(s_{i}\) is open). The connected wet sites form a percolation cluster. When the probability \(p\) goes beyond a critical value \(p_{c}\), a giant percolation cluster appears, spanning the systems, while small isolated clusters are still present. This is the percolation transition. For modeling a disease spreading, one needs a modification, since the infection is a random event that can occur at every time step, so we modify Equation (1) as  
\[s_{i}(t+1)=[r_{i}(t)<p]\wedge\big{(}s_{i-1}(t)\lor s_{i+1}(t)\big{)}\lor s_{i}( t), \tag{2}\]where now \(r_{i}(t)\) is independently extracted for each site and each time step. In the language of disease spreading, it is an SIR (Susceptible-Infected-Recovered) model. Since in this and the following cases the \"properties\" of the lattice (the random numbers) are changing in time, they are also denoted \"directed\" percolation models [19] (time is the \"directed\" direction). A given set of random numbers \(r_{i}(t)\) is also called a \"realization\" of the disorder in the lattice. Finally, as happens in some disease and in many computer cases, the \"ineffectiveness\" or misbehaving can be only transient, so after some time a node returns to the initial state, i.e., a SIS (Susceptible-Infected-Susceptible) model. This has the following implementation,  
\[s_{i}(t)=[r_{i}(t)<p]\wedge\big{(}s_{i-1}(t-1)\lor s_{i+1}(t-1)\big{)}. \tag{3}\]  
We shall deal here essentially with similar SIS models. They can be generalized, for instance, one can have an infection probability that increases with the number of infected neighbors, or even cases in which there is an interference effect among them, such as in the Domany-Kinzel (DK) model [20; 21],  
\[s_{i}(t)=\big{[}r_{i}(t)<\tau\big{(}s_{i-1}(t-1)+s_{i+1}(t-1)\big{)}\big{]} \tag{4}\]  
where now the probabilities \(\tau(n)\), \(n=0,1,2\) (number of ones in the neighborhood) define the model. In the example of Equation (3), \(\tau(0)=0\); \(\tau(1)=\tau(2)=p\). In general, one assumes \(\tau(0)=0\), so that the configuration \(s_{i}=0\)\(\forall i\) becomes an absorbing state [22]. In the DK model, one defines \(\tau(1)=p\) and \(\tau(2)=q\) (Figure 1). The order parameter is the asymptotic \"density\" \(\rho=\lim_{t\to\infty}\rho(t)\) of active (non-zero) (\(s_{i}(t)=1\)) sites, where  
\[\rho(t)=\frac{1}{N}\sum_{i}s_{i}(t),\]  
which is zero below \(p_{c}\) and greater than zero above. Let us introduce some useful terms.