## 1 Introduction  
Limit theory is a basic theoretical issue and has attracted wide interest across many fields. On the 100 th anniversary of its foundation, _Science_ raised 125 unresolved scientific questions, and many of these issues related to limit theory [1]. Link prediction predicts missing links in current networks and new or dissolution links in future networks [2]. With continuous improvement of link prediction methods and, the theoretical limit of link prediction has attracted considerable research interest [3].  
Considering structure or attribute features, link prediction methods based on classification have been proposed by computer science community [4; 5]. Subsequently, more insightful methods of network structure, such as similarity based methods [6], have become a focus, these methods pay more attention to the physical meaning. At the same time, similarity index fusion methods are springing up [7; 8]. Recent years, with the development of deep learning, some deep features extraction methods have been proposed [9; 10], the fusion of structure and attribute information has been attached importance again [11; 12; 13; 14]. These methods have strong consistency. We divide link prediction method into single and combination methods, based on whether they use multidimension information, and whether they define the relation of multidimension information directly. For example, single methods, such as RA index [15], which defines the relation of common neighbors and degree of nodes directly; and classification based methods, index fusion methods, fusion of structure and attribute information methods belong to link prediction combination methods.  
Most combination methods perform better than single methods that will be fused, and are robust to many network types. However, what is the reason for this improved accuracy and robustness, and is there a theoretical limit for combination methods? This paper proposes the mathematic description of combination methods, and obtains the necessary and sufficient conditions for theoretical limit. The limit theorem also has important practical application value. It reveals the ultimate goal of combination methods that is to estimate probability density functions of existing links and nonexistent links. Thus, an appropriate form of the transformation function could be selected from the complete set. Based on the limit theorem, a new combination method, theoretical limit fusion (TLF) method, is proposed. We use the Parzen kernel method [16] of destiny estimation in the TLF method. Simulations and empirical studies have shown that TLF method can achieve higher prediction accuracy.  
Section 2 introduces a mathematical description for the theoretical limit of combination methods and evaluation metrics for link prediction. Section 3 proposes and proves necessary and sufficient conditions for the theoretical limit of combination methods. Section 4 proposes a fusion link prediction method based on limit theorem (TLF method). Section 5 provides simulation examples for limit theorem and proposed TLF method with other combination methods, and gives comparison experiments in real networks. Sections 6 and 7 discuss some results and conclude the paper.