# Percolation and Internet Science
## 5 Criticality and Self-Organized Criticality  
The topic of phase transitions has fascinated physicists for many years, originally for computations related to high-energy physics and then to continuous phase transitions in statistical mechanics (Ising) models [77]. The idea is that, near the phase transition, the correlation length diverges, and clusters of all sizes are present, thus one can perform a coarse-graining (renormalisation) of the lattice, and exactly at the boundary of the phase transition this operation gives an invariant pattern. Another signature of this invariance is the power-law distribution of many quantities, all related to the correlation length. While this is a powerful technique for getting the relevant parameters at the transition, the chance of observing power-laws originating from a phase transition in nature seems to be quite low, due to the necessity of carefully adjusting control parameters. However, in many cases (earthquakes, waves, brain dynamics, and social organizations), fluctuations with a power-law distribution can be observed [78]. As illustrated by the Bak-Tang-Wiesenfeld model [79], a slowly driven system may behave in this way, giving origin to the expression \"Self-Organized Criticality\" (SOC) [80]. A similar behavior is also soown by systems whose evolution is given by some extremal value, such as the maxima of some quantity over all the lattice [81]. Finally, one can try to control the location of the critical point by replacing the random processes with correlated ones, as in explosive percolation [82; 83]. Actually, a method for mapping a percolation problem into a SOC one had been discovered before, and is named invasion percolation [84]. First, extract all needed random numbers, attaching them to the sites of the space-time lattice as a quenched random field. Then, suppose to slowly increase the percolation probability, marking all sites (or links, according with the problem) with a random number greater than \(p\). If the percolation probability is a monotonous function of the number of \"wet\" neighbors, the clusters will keep growing with increasing values of \(p\), until a percolative cluster appears. If one now starts from a given set of wet sites, for instance one boundary of the system, and looks for the minimum value of the random numbers linked to connecting sites, this is the value of \(p\) for which this site will be added to the existing cluster. After adding this site, one should also add all  
Figure 13: Return map of the average opinion \(c\) on small-world networks for several values of the long-range connection probability \(p\) with \(J=-6\), \(k=20\), \(N=10^{3}\), and a transient of \(10^{3}\) time steps. The following 200 iterations are shown as (blue, darker) dots. The (red, lighter) continuous curve is Equation (9). (**a**) \(p=0.0\), (**b**) \(p=0.5\), (**c**) \(p=0.6\), and (**d**) \(p=1.0\). Figures from [74]. connected sites with random number less than the established value of \(p\). When no more sites can be added, a new search for the minimum on the boundary is performed (this minimum is necessarily larger than \(p\)), and the process is repeated until the opposite side is reached. This procedure gives the minimum number of connected sites for percolation, what is called the \"backbone\" of the percolation cluster. To illustrate the basic idea with a working procedure in case of processing of information, let us consider a simple bond percolation problem, with a given fixed infection probability [85]. There is a set of \(N\) nodes, \(x_{i}\), that can stay in two states: 0 for \"healthy\" and 1 for \"tainted\" (or contaminated). Node \(i\) processes information coming from another node \(j\), defined by an adjacency matrix \(a_{ij}=1(0)\) for connected (disconnected) nodes. We assume that node \(i\) is tainted and it can \"infect\" other nodes with a probability \(\tau\).