Title: An Analysis of Composite Neural Network Performance from Function Composition Perspective. Abstract: This work investigates the performance of a composite neural network, which is composed of pre-trained neural network models and non-instantiated neural network models, connected to form a rooted directed graph. A pre-trained neural network model is generally a well trained neural network model targeted for a specific function. The advantages of adopting such a pre-trained model in a composite neural network are two folds. One is to benefit from other's intelligence and diligence and the other is saving the efforts in data preparation and resources and time in training. However, the overall performance of composite neural network is still not clear. In this work, we prove that a composite neural network, with high probability, performs better than any of its pre-trained components under certain assumptions. In addition, if an extra pre-trained component is added to a composite network, with high probability the overall performance will be improved. In the empirical evaluations, distinctively different applications support the above findings.   