In this article we propose a growing network model based on an optimal policy involving both topological and geographical measures. In this model, at each time step, a node, having randomly assigned coordinates in a 1x1 square, is added and connected to a previously existing node i, which minimizes the quantity ri2/kialpha, where ri is the geographical distance, ki the degree, and alpha a free parameter. The degree distribution obeys a power-law form when alpha=1, and an exponential form when alpha=0. When alpha is in the interval (0, 1), the network exhibits a stretched exponential distribution. We prove that the average topological distance increases in a logarithmic scale of the network size, indicating the existence of the small-world property. Furthermore, we obtain the geographical edge length distribution, the total geographical length of all edges, and the average geographical distance of the whole network. Interestingly, we found that the total edge length will sharply increase when alpha exceeds the critical value alphac=1, and the average geographical distance has an upper bound independent of the network size. All the results are obtained analytically with some reasonable approximations, which are well verified by simulations.

The community structure and motif-modular-network hierarchy are of great importance for understanding the relationship between structures and functions. In this paper, we investigate the distribution of clique-degree, which is an extension of degree and can be used to measure the density of cliques in networks. The empirical studies indicate the extensive existence of power-law clique-degree distributions in various real networks, and the power-law exponent decreases with the increasing of clique size.

Numerical simulation plays an important role in the study of structure formation of the universe. However, the mass resolution in current simulations is still poor. Due to technical difficulties, it is necessary to use both greatly reduced number density of particles and greatly raised unit particle mass. Consequently, the particle masses used in cosmological simulations are about $10^{70}$ times larger than the $Gev$ candidates in particle physics. This is a huge physical bias that cannot be neglected when interpreting the results of the simulations. Here we discuss how such a bias affects Cold Dark Matter (CDM) cosmological simulations. We find that the small scale properties of the CDM particle system are changed in two aspects. 1) An upper limit is imposed on the spatial resolution of the simulation results. 2) Most importantly, an unexpected short mean free path is produced, and the corresponding two body scattering cross section is close to the value expected in the Self-Interaction Dark Matter(SIDM) model. Since the mean free path of real CDM particle systems is much longer than that in the simulations, our results imply 1) that there is probably no 'cusp problem' in real CDM halos, and 2) that a much longer time is needed to form new virialized halos in real CDM particle systems than in the simulations. This last result can help us understand the 'substructure problem'. Our discussion can also explain why the massive halos in the simulations may have smaller concentration coefficients.Comment: 4 pages,no figures. Submitted to ApJ

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