On dualities of random graphs and branching processes with disasters to Piecewise deterministic Markov processes Abstract: This thesis is based on the interconnection between three classes of stochastic processes:<br>As links between branching processes and random graphs frequently appear in the literature we add to that a connection between birth death processes with disasters and duplication based random graphs with edge deletion. For the third class of processes i.e. a class of piecewise deterministic Markov processes we call p jump processes duality relations to a general class of branching processes with disasters as well as to duplication based random graphs will be shown. Precise limit results we obtain for p jump processes will be transferred to the other classes. Several possible generalisations of these dualities will be discussed.<br>Furthermore limit results for degrees cliques and star like sub graphs of the duplication graph model will be derived using martingale methods. Lastly we outline an approach for the analysis of the sub graph of non isolated vertices via p jump processes