Exponential random graphs are important to model the structure of real-world complex networks. Here we solve two-star with degree-degree correlations in sparse regime. The constraints average correlation between degrees adjacent nodes (nearest neighbors) and at end-points two-stars (next nearest neighbors). We compute exactly network free energy show that this undergoes a first-order transition...

In this paper, an analytic solution to the scattering of water wave by submerged permeable breakwaters in infinite-depth is presented. The velocity field around breakwater under linear train deep condition sought formulating a nonhomogeneous Riemann-Hilbert problem. To deal with nonlinearity due permeability plates, perturbation method applied terms small parameter representing permeability. We...

in this article, we demonstrate an analytic-numeric solution for some dynamical systems using the multistagehomotopy perturbation method. the method yields results that are in complete agreement with their numericalcounterparts.