Fig. 1 Hub microbes of maize phyllosphere microorganisms under the four different co-occurrence networks. In four different co-occurrence networks, hub microbes are divided based on degree centrality and closeness centrality, the right side of each network shows these two values of the operational taxonomic unit (OTU). The hub microbes are represented by different numbers in the network, the species composition corresponding to OTU is shown in Table 1, and finally the four interaction relationships are visualized using Gephi software.

Fig. 2 Six network indices of four co-occurrence networks of maize-phyllospheric microbiome. As shown in the figures above, the abscissa represents six network feature values from left to right, and the ordinate represents 230 different genotype maize samples, showing the difference in network feature values between different individuals and between different co-occurring networks. The color change of 0-1.0 on the right side of the figure represents the size of the difference, 0 represents no difference, and 1.0 represents the largest difference. The six types of network eigenvalues are: connectivity (Con), closeness (C(u)), betweenness (B(u)), eccentricity (E(u)), eigenvector (G(u)) and pagerank (P(u)).

Fig. 3 Comparison of General Linear Model (GLM), Mixed Linear Model (MLM), Fixed and Random Model Circulating Probability Unification (FarmCPU) under four interaction relationships. GLM, MLM, and FarmCPU models are obtained through network mapping and displayed with Quantile-Quantile diagrams. The indices of each network under the four interaction relationships are compared by three models, and the one closest to the middle red line is the optimal model.

Fig. 4 Significant single-nucleotide polymorphisms (SNPs) locus affecting phyllospheric microbiome of maize. Visualization based on the optimal model of each network eigenvalue in each relationship. The number of the outer circle represents the chromosome number, and the outermost circle represents the association result of the optimal model of connectivity (Con), closeness (C(u)), betweenness (B(u)), eccentricity (E(u)), eigenvector (G(u)), and pagerank (P(u)) from the inside to the outside, the significance threshold is set to -lg p ≥ 4, the red line is the threshold line, and the red point is the significant SNPs locus for the six network feature values under the four relationships.