# Percolation and Internet Science
## 1 Introduction
Just below it, the average cluster size diverges, and the cluster size distribution becomes much flatter, changing from an exponential to a power law. This is reminiscent of the fluctuations observed in thermal systems (such as the Ising model) near a continuous phase transition. Indeed, one can introduce correlation functions that behave in a way similar to their thermal equivalent, allowing the evaluation of the critical exponents for such models. In principle, these results depend on the choice of the random numbers \(r_{i}\), and should be averaged over many realizations of the random field. However, in many cases, these systems show a property called \"selfa-veraging\", i.e., the quantities computed on a large enough system are the same as those averaged over many realizations. This is certainly true far from the critical value \(p_{c}\), since in these cases the correlation functions decrease exponentially with a well-defined correlation length. Patches that are farther away than this correlation length are uncorrelated and therefore act as two different samples. There are many variants of the percolation problem. For instance, one could also consider the case in which pores are all open and bonds or links are open or closed with a certain probability. While the first problem is denoted \"site percolation\", the second one receives the name of \"bond percolation\". One can be also interested in the dynamics of the percolation, i.e., how a percolation cluster is invaded starting from a given site or a given boundary. Other variants are possible. Many problems may be reformulated as percolation ones. For instance, if one considers the problems of avoiding the spreading of fire in a forest, one may be concerned with the problem of the minimum fraction of trees to be cut to avoid the percolation of the fire. A similar problem is that of the spreading of a disease: What is the maximum fraction of unvaccinated individuals in a population, given its networks of contacts, that still avoid the percolation of a disease? Many examples concern computer science and the Internet. Indeed, the birth of the Internet is just a percolation problem: a worldwide network arose from the sequential adding of links among nodes (computers) [15]. The resilience of the Internet to failure can be formulated as a percolation phase transition problem since its degree distribution implies the presence of heavily connected hubs [4]. Many routing protocols, which make possible dynamical communication among mobile devices [16], are based on the percolation concepts [17; 18]. In addition, the determination of the critical coverage of an area with sensors is essentially a percolation problem [19], with applications to healthcare topics. When considering the communication level on the Internet, more complex networks and processes arise. An example is the network given by the links among web pages, and the difficulty (and profitability) of exploring it [20]. Indeed, one of the main issue of the Internet of Things is the choice between hierarchic/centralized approaches and distributed/peer-to-peer ones [21]. From one side, the distributed approach is more scalable, but it requires special techniques for insuring the reachability of each node, the coverage of the whole domain, and the optimization of communications among different nodes. From the other side, a distributed approach implies the cooperation of the participating nodes. While this cooperation can be taken for granted when all sensors/devices belong to the same organization, it can break for heterogeneous devices or in the case of failure/hacking [22].