Since network motifs are an important property of networks and some networks have the behaviors of rewiring or reducing or adding edges between old vertices before new vertices entering the networks, we construct our non-randomized model N(t) and randomized model N'(t) that have the predicated fixed subgraphs like motifs and satisfy both properties of growth and preferential attachment by means of the recursive algorithm from the lower levels of the so-called bound growing network models. To show the scale-free property of the randomized model N'(t), we design a new method, called edge-cumulative distribution, and democrat two edge-cumulative distributions of N(t) and N'(t) are equivalent to each other.