We study the structure of supply chain trust networks (SCTNs) by analyzing the evolution of the networks. An SCTN here comprises enterprises in a fully competitive market connected through the preferential attachment mechanism. A Markov chain analysis is used to understand how various factors affect the structure of the SCTNs. The evolution of the SCTNs is also analyzed to identify the asymmetric conditions required for the degree distribution of the SCTNs to obey the power law distribution. The simulation results show that, when the degree of willingness to initiate a trust relationship and the attractiveness index of the supply chain networks meet certain criteria, the underlying network is of a scale-free nature.