The early stage evolution of local atomic structures in a multicomponent metallic glass during its crystallization process has been investigated via molecular dynamics simulation. It is found that the initial thermal stability and earliest stage evolution of the local atomic clusters show no strong correlation with their initial short-range orders, and this leads to an observation of a novel symmetry convergence phenomenon, which can be understood as an atomic structure manifestation of the ergodicity. Furthermore, in our system we have quantitatively proved that the crucial factor for the thermal stability against crystallization exhibited by the metallic glass, is not the total amount of icosahedral clusters, but the degree of global connectivity among them.