Complex Markov models is a widely used and powerful predictive tool to analyze stochastic biochemical processes. When, however, the network of states is unknown, it is necessary to extract information from the data to partially build the network and to give estimates about the rates. The short-time behavior of the first-passage time distributions between two states in linear chains has been shown recently to behave as a power of time with an exponent equal to the number of intermediate states. For a general Markov network system we derive here the complete Taylor expansion of the first passage time distribution in terms of absorption times. By combining algebraic methods and graph theoretical approaches it is shown that the first term of the Taylor expansion is determined by the shortest path from the initial state to the absorbing state. When this path is unique, we prove that the coefficient of the first term can be written in terms of the product of the transition rates along the path. It is argued that the application of our results to first-return times may be used to estimate the dependence of rates from external parameters in experimentally measured time distributions.