2b for large n. In addition, Figure 2c also shows that the heavy tail will disappear with increasing system size. Our model can be easily extended to a general case with more layers. BOYER, T. & Jonard, N. (2014). In the price space dimension of innovation diffusion and for a network of infinite size, the percolation threshold is given by, The effects of social networks can be expressed in welfare-theoretical terms. The model evolves in generations with the result of the present percolation depending on the previous state, and thus is history-dependent. First, while individual intermediate generations can have their direct real counterparts, they are overlooked by focusing on the infinite (steady-state) generation. By definition, a lost consumer has a reservation price in the range between 1 and \(p\). ), rather than an accurate description of various diffusion processes that exist in reality. 3a). Its average path-length is almost the same of a Poisson network, but the clustering coefficient is very large and comparable to the original regular network. By this construction, both layers are represented by weighted complete networks. The percolation process is initiated by 10 seeds (n = 10). This also suggests that the universality class is the same for all finite generations of the iterated percolation process. Concluding, scale-free networks do not depart much from Poisson networks in terms of their diffusion performance. The results indicate that the first generation and also all finite generations of iterative percolation demonstrate a continuous percolation transition. For a better understanding of this, we further show the distribution of |$\psi ^n_c$| obtained in each individual realization. Hence, for a given society with N members, the welfare loss due to network inefficiency is solely a function of the price. Therefore, Fig. CENTOLA, D. (2010). In P. Carrington, J. Scott & S. Wasserman (Eds. See the last section of the Supplementary Information for a detailed description of the data and the MRI data preprocessing strategy. We can distinguish between two different regimes: a diffusion regime, where information spreads throughout the whole operational network, and diffusion is full or almost-full. (c) The distribution of the order parameters obtained in each individual realization |$f(\psi _c^n)$| for generation n = 10. [doi:10.1103/PhysRevE.64.026118]. The percolation threshold is defined as the critical value of nodes’ activation probability where a phase transition occurs in the size of connected components of active nodes, the so-called percolating cluster. Consider for instance a Poisson network. Technological Forecasting and Social Change, 76, 487–496. Whenever there is more weight on larger reservation prices, the realised demand increases, with a larger threshold, but not as much as for the Poisson network and the difference with the potential demand amplifies. Cambridge, USA: Cambridge University Press. For simplicity and without loss of generality, we consider here only the case of a multiplex network with two layers, which we refer to as layers A and B, respectively (see Fig. The structure of interconnected systems and its impact on the system dynamics is a much-studied cross-disciplinary topic. The configuration C1 induced by layer A is connected. [doi:10.1038/30918], Keywords: Our understanding of percolation properties of networks has expanded significantly in recent years. Although these works consider a similar mechanism (i.e. As a topological phase transition, there is no free energy that can be used to determine the type of the percolation transition. If the system is large enough, this process can be done to any number of generations. (a) ps for different generations n; the network size is N = 220. This fact is actually a manifestation of a well defined mathematical property, which is a relatively short average path-length. Information about the innovative product is local (Huang et al.
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