Pathways of diffusion observed in real-world systems often require stochastic processes going beyond first-order Markov models, as implicitly assumed in network theory. In this work, we focus on second-order Markov models, and derive an analytical ex
pression for the effect of memory on the spectral gap and thus, equivalently, on the characteristic time needed for the stochastic process to asymptotically reach equilibrium. Perturbation analysis shows that standard first-order Markov models can either overestimate or underestimate the diffusion rate of flows across the modular structure of a system captured by a second-order Markov network. We test the theoretical predictions on a toy example and on numerical data, and discuss their implications for network theory, in particular in the case of temporal or multiplex networks.
The newly released Orange D4D mobile phone data base provides new insights into the use of mobile technology in a developing country. Here we perform a series of spatial data analyses that reveal important geographic aspects of mobile phone use in Co
te dIvoire. We first map the locations of base stations with respect to the population distribution and the number and duration of calls at each base station. On this basis, we estimate the energy consumed by the mobile phone network. Finally, we perform an analysis of inter-city mobility, and identify high-traffic roads in the country.
We focus on the statistics of word occurrences and of the waiting times between such occurrences in Blogs. Due to the heterogeneity of words frequencies, the empirical analysis is performed by studying classes of frequently-equivalent words, i.e. by
grouping words depending on their frequencies. Two limiting cases are considered: the dilute limit, i.e. for those words that are used less than once a day, and the dense limit for frequent words. In both cases, extreme events occur more frequently than expected from the Poisson hypothesis. These deviations from Poisson statistics reveal non-trivial time correlations between events that are associated with bursts of activities. The distribution of waiting times is shown to behave like a stretched exponential and to have the same shape for different sets of words sharing a common frequency, thereby revealing universal features.
