Conservation laws of some lattice equations

We derive infinitely many conservation laws for some multi-dimensionally consistent lattice equations from their Lax pairs. These lattice equations are the Nijhoff-Quispel-Capel equation, lattice Boussinesq equation, lattice nonlinear Schr\"{o}dinger equation, modified lattice Boussinesq equation, Hietarinta's Boussinesq-type equations, Schwarzian lattice Boussinesq equation and Toda-modified lattice Boussinesq equation

Tracing the Attention of Moving Citizens

With the widespread use of mobile computing devices in contemporary society, our trajectories in the physical space and virtual world are increasingly closely connected. Using the anonymous smartphone data of $1 \times 10^5$ users in 30 days, we constructed the mobility network and the attention network to study the correlations between online and offline human behaviours. In the mobility network, nodes are physical locations and edges represent the movements between locations, and in the attention network, nodes are websites and edges represent the switch of users between websites. We apply the box-covering method to renormalise the networks. The investigated network properties include the size of box $l_B$ and the number of boxes $N(l_B)$. We find two universal classes of behaviours: the mobility network is featured by a small-world property, $N(l_B) \simeq e^{-l_B}$, whereas the attention network is characterised by a self-similar property $N(l_B) \simeq l_B^{-\gamma}$. In particular, with the increasing of the length of box $l_B$, the degree correlation of the network changes from positive to negative which indicates that there are two layers of structure in the mobility network. We use the results of network renormalisation to detect the community and map the structure of the mobility network. Further, we located the most relevant websites visited in these communities, and identified three typical location-based behaviours, including the shopping, dating, and taxi-calling. Finally, we offered a revised geometric network model to explain our findings in the perspective of spatial-constrained attachment.Comment: 15 pages, 8 figure