Nature of the Epidemic Threshold for the Susceptible-Infected-Susceptible Dynamics in Networks

We develop an analytical approach to the susceptible-infected-susceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks. We find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold. In particular, small-world random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit.

The Spatial Distribution of Spanish Transport Infrastructure between 1860 and 1930

The origin of Spanish regional economic divergence can be traced back at least until the seventeenth century, although its full definition took place during industrialisation. Historians have often included uneven regional infrastructure endowments among the factors that explain divergence among Spanish regions, although no systematic analysis of the spatial distribution of Spanish infrastructure and its determinants has been carried out so far. This paper aims at filling that gap, by offering a description of the regional distribution of the main Spanish transport infrastructure between the middle of the nineteenth century and the Civil War. In addition, it estimates a panel data model to search into the main reasons that explain the differences among the Spanish regional endowments of railways and roads during that period. The outcomes of that analysis indicate that both institutional factors and the physical characteristics of each area had a strong influence on the distribution of transport infrastructure among the Spanish regions.

Networks of noisy oscillators with correlated degree and frequency dispersion

We investigate how correlations between the diversity of the connectivity of networks and the dynamics at their nodes affect the macroscopic behavior. In particular, we study the synchronization transition of coupled stochastic phase oscillators that represent the node dynamics. Crucially in our work, the variability in the number of connections of the nodes is correlated with the width of the frequency distribution of the oscillators. By numerical simulations on Erdös-Rényi networks, where the frequencies of the oscillators are Gaussian distributed, we make the counterintuitive observation that an increase in the strength of the correlation is accompanied by an increase in the critical coupling strength for the onset of synchronization. We further observe that the critical coupling can solely depend on the average number of connections or even completely lose its dependence on the network connectivity. Only beyond this state, a weighted mean-field approximation breaks down. If noise is present, the correlations have to be stronger to yield similar observations.