Privacy in social networks has been a vast active area of research due to the enormous increase in privacy concerns with social networking services. Social networks contain sensitive information of individuals, which could be leaked due to insecure data sharing. To enable a secure social network data publication, several privacy schemes were proposed and built upon the anonymity of users. In this paper, we incorporate unlinkability in the context of weighted network data publication, which has not been addressed in prior work. Two key privacy models are defined, namely edge weight unlinkability and node unlinkability to obviate the linking of auxiliary information to a targeted individual with high probability. Two new schemes satisfying these unlinkability notions, namely MinSwap and δ-MinSwapX are proposed to address edge weight disclosure, link disclosure and identity disclosure problems in publishing weighted network data. The edge weight is modified based on minimum value change in order to preserve the usefulness and properties of the edge weight data. In addition, edge randomization is performed to minimally modify the structural information of a user. Experimental results on real data sets show that our schemes efficiently achieve data utility preservation and privacy protection simultaneously.