The optimization algorithms for solving multi-parameter inverse problem for the mathematical model of parabolic equations arising in social networks, epidemiology and economy are investigated. The data fitting is formulated as optimization of least squares misfit function. Firstly, the tensor train decomposition approach is presented as global convergence algorithm. The idea of proposed method is to extract the tensor structure of the optimized functional and use it for optimization. Then the inverse problem solution is reached by implementation of the local gradient approach. The evident formula for the gradient of the misfit function is obtained. The inverse problem for the diffusive logistic mathematical model described online social networks is solved by combination of tensor train optimization and local gradient methods. The numerical results are presented and discussed.