Diffusion processes in social networks often cause the emergence of global phenomena from individual behavior within a society. The study of those global phenomena and the simulation of those diffusion processes frequently require a good model of the global network. However, survey data and data from online sources are often restricted to single social groups or features, such as age groups, single schools, companies, or interest groups. Hence, a modeling approach is required that extrapolates the locally restricted data to a global network model. We tackle this Missing Data Problem using Link-Prediction techniques from social network research, network generation techniques from the area of Social Simulation, as well as a combination of both. We found that techniques employing less information may be more adequate to solve this problem, especially when data granularity is an issue. We validated the network models created with our techniques on a number of real-world networks, investigating degree distributions as well as the likelihood of links given the geographical distance between two nodes.
We consider an optimal shape design problem for the plate equation, where the variable thickness of the plate is the design function. This problem can be formulated as a control in the coefficient PDE-constrained optimal control problem with additional control and state constraints. The state constraints are treated with a Moreau-Yosida regularization of a dual problem. Variational discretization is employed for discrete approximation of the optimal control problem. For discretization of the state in the mixed formulation we compare the standard continuous piecewise linear ansatz with a piecewise constant one based on the lowest-order Raviart-Thomas mixed finite element. We derive bounds for the discretization and regularization errors and also address the coupling of the regularization parameter and finite element grid size. The numerical solution of the optimal control problem is realized with a semismooth Newton algorithm. Numerical examples show the performance of the method.
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