Objectives: This paper focuses on a privacy preservation technique which is applied on graphs to preserve sensitive information present as shortest paths. Methods/Statistical Analysis: Provide privacy in Social Network Analysis by protecting the sensitive edge weights with the help of preserving the nearest shortest path lengths as well as shortest paths so that individual confidential information can be protected from multiple type of attacks. This research work provides more privacy then greedy perturbation technique in social network analysis. Findings: Privacy preservation has a tradeoffbetween the utility of data and preservation of sensitive information. This is achieved by modification of shortest path length in graphs and also maintains the structure of the graph. This procedure enhances the privacy of sensitive information with minimal concerns to utility. Application/Improvements: The privacy preservation technique of edge weight perturbation is applied to social graphs in a small user group to preserve sensitive information when data is shared within the group members. The edge weight perturbation algorithm can be improved by combing the algorithm with the preservation techniques for the user nodes.