Normal and anomalous diffusion are ubiquitous in many complex systems. In this talk, I will define a time and space generalized diffusion equation (GDE), which uses fractional-time derivatives and transformed d-path Laplacian operators on graphs/networks. I will find prove that the solution of this equation covers the regimes of normal, sub- and superdiffusion as a function of the two parameters of the model. I will also extend the GDE to consider a system with temporal alternancy of normal and anomalous diffusion, which is able to sucessfully model the diffusion of proteins along a DNA chain. Finally, I will briefly comment how a subdiffusive-superdiffusive alternant regime allows the diffusive particle to explore more slowly small regions of the chain with a faster global exploration.
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