Abstract: A lattice network is a special type of network in
which all nodes have the same number of links, and its boundary
conditions are periodic. The most basic lattice network is the ring, a
one-dimensional network with periodic border conditions. In contrast,
the Cartesian product of d rings forms a d-dimensional lattice
network. An analytical expression currently exists for the clustering
coefficient in this type of network, but the theoretical value is valid
only up to certain connectivity value; in other words, the analytical
expression is incomplete. Here we obtain analytically the clustering
coefficient expression in d-dimensional lattice networks for any link
density. Our analytical results show that the clustering coefficient for
a lattice network with density of links that tend to 1, leads to the
value of the clustering coefficient of a fully connected network. We
developed a model on criminology in which the generalized clustering
coefficient expression is applied. The model states that delinquents
learn the know-how of crime business by sharing knowledge, directly
or indirectly, with their friends of the gang. This generalization shed
light on the network properties, which is important to develop new
models in different fields where network structure plays an important
role in the system dynamic, such as criminology, evolutionary game
theory, econophysics, among others.
Abstract: In this paper we developed the Improved Runge-Kutta Nystrom (IRKN) method for solving second order ordinary differential equations. The methods are two step in nature and require lower number of function evaluations per step compared with the existing Runge-Kutta Nystrom (RKN) methods. Therefore, the methods are computationally more efficient at achieving the higher order of local accuracy. Algebraic order conditions of the method are obtained and the third and fourth order method are derived with two and three stages respectively. The numerical results are given to illustrate the efficiency of the proposed method compared to the existing RKN methods.