Although extensive optimization of encoding and decoding schemes for joint source-channel coding (JSCC) systems has been conducted, efficient optimization schemes are still required for designing and optimizing the linking matrix between variable nodes of the source code and check nodes of the channel code. A scheme has been proposed for design and optimization of linking matrix with multi-edges by analyzing the performance of the JSCC system using the joint protograph extrinsic information transfer algorithm to calculate decoding thresholds. The proposed scheme incorporates structural constraints and is effective in designing and optimizing the multi-edges in linking matrix for the JSCC system. Experimental results have demonstrated that the designed and optimized linking matrix significantly improves the performance of the JSCC system. Furthermore, the proposed scheme reduces the complexity of the solution space for the optimized example.