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The variability in the generation dispatch of the natural gas generation units will lead to fluctuation in natural gas demand profile that could further jeopardize the security of the natural gas network. The coordinated operation of electricity and natural gas infrastructure systems would help to improve the security and reliability measures in both infrastructure systems and mitigate the risk of demand curtailment. The electricity and natural gas network operation problems are non-convex mixed-integer nonlinear programming problems that are hard to solve in polynomial time. The non-convex feasible regions are formed by the Weymouth constraint and the introduced binary commitment decision variables in the natural gas and electricity network operation problems, respectively. This paper utilized a sparse semidefinite programming (SDP) relaxation to procure the optimal solution for the coordinated operation of electricity and natural gas networks. The presented algorithm leverages the sparseness of the natural gas network to construct several small matrices of lifting variables that are used to form a tight and traceable SDP relaxation. A set of valid constraints that tighten the relaxation ensures the exactness of the solution procured from the relaxed problem. The effectiveness of the presented approach is shown in case studies.