# Percolation and Internet Science
## 4 Network Influence and Small World Transitions  
Up to now, we have treated either regular lattices or the mean-field approach, which implies the absence of correlations and therefore is equivalent to reshuffling the sites in a lattice at each time step. Actually, for quenched disordered connections, the prediction of the mean-field approximation are also generally quite good. It is possible to investigate what happens when we partially disorder an otherwise regular lattice. An interesting procedure was proposed by [PERSON] and [PERSON] [75], and consists in replacing a fraction \(p\) of neighboring connections with random links (see Figure 10). In general, what happens with this procedure is that global observables like the density of \"wet\" sites assume the mean-field values also for moderate values of \(p\), while local observables such as the probability of having small-length loops are still similar to the value they take in the regular lattice. For this, it is called the \"small world\" effect\" if we consider diffusion of people or percolation of a disease among humans, both are quite slow processes in a regular lattice (or in a disordered lattice with local connections), but, as soon as long-range travel become possible (by boat or by airplane), the diffusion properties change, even if at the local level almost nothing has changed. An interesting effect arises when the mean-field approximation gives chaotic oscillations, as happens in the long-range opinion model of the previous Section. The plot of the mean-field approximation of the rule of Equation (8) is reported in Figure 11[74]. For sufficient large negative values of \(J\) and large connectivity \(k\), the map becomes chaotic. This chaotic character does not depend on the presence of absorbing states, and indeed they have been removed by inserting a small probability of opposing to the strong majority. Notice altogether that this chaos has nothing to do with the \"microscopic\" chaos already seen: the mean-field approximation refers to average quantities, so it implies a coherent oscillation of the whole, synchronized, network. The simplest mean-field description of the model described in Equation (8) is given by  
\[c^{\prime}=\sum_{S=0}^{k}\binom{k}{S}c^{av}(1-c)^{k-S}\tau\left(1|S\right), \tag{9}\]  
with \(c^{\prime}=c(t+1)\) and \(c=c(t)\). In Figure 11, we show some graphs of this map by varying \(J\) and \(k\). The bifurcation digram of this map after varying \(J\) is shown in Figure 12 (left). The doubling bifurcation route to chaos ends at \(J=J_{c}\). For \(0>J\geq J_{2}\) and \(J_{3}>J\geq 6\) there is only one attractor (blue, darker dots). For \(J_{2}>J\geq J_{c}\) there are two, one corresponding to the lower  
Figure 10: Rewiring in the Watts–Strogatz model [75]. (**Left**) \(p=0\); (**Middle**) \(p=0.2\); and (**Right**) \(p=0.9\). branches that bifurcate up to \(J_{c}\) (red, lighter dots), and the other one to the upper branches (blue, darker dots). For \(J_{c}>J\geq J_{3}\), there are two chaotic attractors, one corresponding to the lower branches (blue, darker dots), and the other to the top branches (red, lighter dots). For every value of \(J\), the dots are 64 iterates of the map after a transient of \(10^{3}\) time steps. For values of \(J\) with only one basin of attraction, the orbits do not depend on the initial average opinion \(c(t=0)\). For values of \(J\) that correspond to two attractors, one of them is found with \(c(0)=0.1\), the other one with \(c(0)=0.9\). By varying the long-range probability \(p\), we observe the transition towards the mean-field behavior, as reported in Figure 13. This induces a stochastic bifurcation diagram by varying \(p\) (Figure 12, right) that is quite similar to that obtained in the mean-field approximation by varying \(J\) (Figure 12, left). For \(p\lesssim p_{0}\), there are almost periodic orbits of period one and for \(p_{0}\lesssim p\lesssim p_{1}\) of period two.