Abstract
Percolation, in its most general interpretation, refers to the “flow” of something (a physical agent, data or information) in a network, possibly accompanied by some nonlinear dynamical processes on the network nodes (sometimes denoted reaction–diffusion systems, voter or opinion formation models, etc.). Originated in the domain of theoretical and matter physics, it has many applications in epidemiology, sociology and, of course, computer and Internet sciences. In this review, we illustrate some aspects of percolation theory and its generalization, cellular automata and briefly discuss their relationship with equilibrium systems (Ising and Potts models). We present a model of opinion spreading, the role of the topology of the network to induce coherent oscillations and the influence (and advantages) of risk perception for stopping epidemics. The models and computational tools that are briefly presented here have an application to the filtering of tainted information in automatic trading. Finally, we introduce the open problem of controlling percolation and other processes on distributed systems.
Highlights
The Internet is naturally linked to the network concept, i.e., a set of nodes connected by links [1,2,3,4]
We present a model of opinion spreading, the role of the topology of the network to induce coherent oscillations and the influence of risk perception for stopping epidemics
We examine several examples in which the dynamics ranges from a simple “ferromagnetic” dynamics, i.e., a node tends to align to neighboring ones, to more complex dynamics, and the role of the topology in the global behavior
Summary
The Internet is naturally linked to the network concept, i.e., a set of nodes connected by links [1,2,3,4]. The birth of the Internet is just a percolation problem: a worldwide network arose from the sequential adding of links among nodes (computers) [15]. One possibility is the addition of nonlinear “reactions” on the nodes, as we discuss below Another extension is that of considering only clusters whose sites are connected with more than k links [29,30]. This “bootstrap” percolation [31] is strongly related to the robustness and resilience of a network structure. The main ingredients of this generalized percolation problems are the node dynamics and the connection network.
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