The initial theoretical connections between Leontief input-output models and Markov chains were established back in 1950s. However, considering the wide variety of mathematical properties of Markov chains, there has not been a full investigation of evolving world economic networks with Markov chain formalism. Using the recently available world input-output database, we modeled the evolution of the world economic network from 1995 to 2011 through analysis of a series of finite Markov chains. We assessed different aspects of this evolving system via different properties of the Markov chains such as mixing time, Kemeny constant, steady state probabilities and perturbation analysis of the transition matrices. First, we showed how the time series of mixing times and Kemeny constants could be used as an aggregate index of globalization. Next, we focused on the steady state probabilities as a measure of structural power of the economies that are comparable to GDP shares of economies as the traditional index of economies. Further, we introduced two measures of systemic risk, called systemic influence and systemic fragility, where the former is the ratio of number of influenced nodes to the total number of nodes, caused by a shock in the activity of a node and the latter is based on the number of times a specific economic node is affected by a shock in the activity of any of the other nodes. Finally, focusing on Kemeny constant as a global indicator of monetary flow across the network, we showed that there is a paradoxical effect of a change in activity levels of economic nodes on the overall flow of the network. While the economic slowdown of the majority of nodes with high structural power results to a slower average monetary flow over the network, there are some nodes, where their slowdowns improve the overall quality of the network in terms of connectivity and the average monetary flow.