# From Color-Avoiding to Color-Favored Percolation in Diluted Lattices
## 1 Introduction  
Percolation theory concerns the flow and the diffusion of some quantity on lattices or networks, for instance a disease on a human network or a message in a communication one [1; 2; 3]. It has many applications in the Internet science [4], for instance in the problem of robustness under attack [5], or the resilience after a random failure of nodes and/or links [6]. Classical percolation theory studies the penetration of a scalar quantity on a lattice, but it may be extended to more complex situations [7]. Percolation models have been also widely used to model the spreading of a disease [8]. In such models, nodes correspond to individuals that can be in several states. The simplest situation is with two states, susceptible (healthy) and infectious (ill). In this model infectious individuals recover after some time and become susceptible again (susceptible-infectious-susceptible or SIS model). Alternatively, one can introduce a recovered (or dead) state, which does not spread the disease nor can be infected again (susceptible-infectious-recovered or SIR model), many other possibilities exists [9; 10; 11]. Links between nodes represent contacts capable of transmitting the disease among individuals: a susceptible individual connected to an infectious one can became infected with a certain probability. The percolation transition represents the outbreak of an epidemic. Similar approaches can be used to model the propagation of computer viruses [12; 13]. In many real-world networks, one can identify different classes of nodes that share the same vulnerabilities. For example, one can classify Internet routers that run different software versions in separate classes. Routers that are vulnerable to a certain bug may simultaneously fail and suddenly isolate parts of the network. In communication networks, vertices can be associated with servers controlled by different companies. Some of them could be interested in eavesdropping information that passes through theirservers, which therefore must be avoided for a secure communication. Even if all the companies are dangerous, a secure communication between the source and the receiver can be achieved by splitting messages in pieces by secret-sharing, and sending them through multiple paths, each one avoiding one of the vulnerable classes of nodes. The color-avoiding percolation (CAP) theory is a recent generalization of the percolation theory, useful to treat these problems [14; 15; 16; 17]. Every node of the network is marked with a one or more colors, associated with the possible vulnerabilities. The goal of CAP is to find whether one or more paths exist between couples of nodes avoiding one or more colors. Each color can be associated with a layer in a multi-graph, so that the problem of color-avoidance is that of deciding is there exists a path that avoids all nodes or links that are \"occupied\" in some of the layers. On the other side, one could profit of nodes that act as gateways towards different companies or alternative networks, seen as connections among layers, and so that in case of failure of one layer they could route the traffic to another one. Associating colors to layers, one could consider the problem of color-favored percolation (CFP), i.e., percolation with the possibility of passing from one layer to another in case of co-occupation. In principle, the two problems could be considered to be dual, by reverting the interpretation of occupation/vacancy of a node for a given color. In this paper, we show how CAP/CFP problems on direct lattices can be mapped into a self-organized critical (SOC) problem. We study the critical properties of CAP in a direct lattice using different SOC methods, in particular with the fragment method [18]. We also show that the interference among colors in color-favored percolation allows the presence of a percolating cluster that \"jumps\" among layers where the percolation of a single color is impossible. We shall deal here with regular lattices with dilution, i.e., with a fraction of links removed.