The visibility graph algorithm proves to be a simple and efficient method to transform time series into complex network and has been widely used in time series analysis because it can inherit the dynamic characteristics of original time series in topological structure. Now, visibility graph analysis of univariate time series has become mature gradually. However, most of complex systems in real world are multi-dimensional, so the univariate analysis is difficult to describe the global characteristics when applied to multi-dimensional series. In this paper, a novel method of analyzing the multivariate time series is proposed. For patients with myocardial infarction and healthy subjects, the 12-lead electrocardiogram signals of each individual are considered as a multivariate time series, which is transformed into a multiplex visibility graph through visibility graph algorithm and then mapped to fully connected complex network. Each node of the network corresponds to a lead, and the inter-layer mutual information between visibility graphs of two leads represents the weight of edges. Owing to the fully connected network of different groups showing an identical topological structure, the dynamic characteristics of different individuals cannot be uniquely represented. Therefore, we reconstruct the fully connected network according to inter-layer mutual information, and when the value of inter-layer mutual information is less than the threshold we set, the edge corresponding to the inter-layer mutual information is deleted. We extract average weighted degree and average weighted clustering coefficient of reconstructed networks for recognizing the 12-lead ECG signals of healthy subjects and myocardial infarction patients. Moreover, multiscale weighted distribution entropy is also introduced to analyze the relation between the length of original time series and final recognition result. Owing to higher average weighted degree and average weighted clustering coefficient of healthy subjects, their reconstructed networks show a more regular structure, higher complexity and connectivity, and the healthy subjects can be distinguished from patients with myocardial infarction, whose reconstructed networks are sparser. Experimental results show that the identification accuracy of both parameters, average weighted degree and average weighted clustering coefficient, reaches 93.3%, which can distinguish between the 12-lead electrocardiograph signals of healthy people and patients with myocardial infarction, and realize the automatic detection of myocardial infarction.