Narrow Results

We comment on the derivation of the main equation in the bounded confidence model of opinion dynamics. In the original work, the equation is derived using an ad-hoc counting method. We point that the original derivation does contain some small mistake. The mistake does not have a large qualitative impact, but it reveals the danger of the ad-hoc counting method. We show how a more systematic approach, which we call micro to macro, can avoid such mistakes, without adding any significant complexity.

We study Facebook networks at 40 American universities, with focus on the comparison of their degree distributions and mechanism governing their evolution. We find that the heterogeneity indexes of these networks are all small compared with scale-free networks, and different from real-world social networks 5 Facebook networks show significant degree disassortativity; the exponent γ for the power-law model of the degree distributions is large for the networks, indicating obvious homogeneity of network structure. We calculate the goodness-of-fit between the data and power law and find that the p-values are larger than threshold 0.1 for 20 networks, implying that power law is a plausible hypothesis; we compare the power-law model with 4 alternative competing distributions and find that power-law model gives the best fit for all 40 networks. However in wider interval of degrees some other distributions, such as log-normal or stretched exponential, can give the best fit. Further based on the homogeneity of Facebook we propose an analyzable model that integrates the introduction of new vertices and edges. The edges can be established either between new vertices and old vertices or between old vertices. The model captures the real evolution processes of Facebook networks and can well reproduce their degree distributions.

This study offers a high-resolution model of nationwide water supply. The model is sufficiently detailed to represent all main water sources in an economy, the principal segments of the conveyance system, urban, industrial and agricultural demand regions, and various water types, including fresh, saline and recycled. Calibrated for Israeli 2010 data, we find that the optimal extraction of fresh water is only 2% larger than the total observed supply from those sources. However, for some specific sources, the deviation between optimal and observed quantities is significant. Assuming average constant recharge, the optimal aggregated desalination is 57% of the 2010 desalination capacity and only 33% of the present desalination capacity. Even with an assumed 40% decline in recharge (for example, due to climate change), the model uses only 50% of the present desalination capacity. This may suggest that the construction of desalination facilities in Israel, which began in 2005, could have been delayed. The model establishes a comprehensive system of pumping levies and user fees that support the optimal allocation. However, due to considerable scale economies, the average cost is almost 50% larger than the marginal cost. The implications are that the welfare cost of the recent Israeli Balanced Budget Water Economy legislation is more than 100 million USD per year, about 10% of the water economy share of the GDP.