In some systems, the connecting probability (and thus the percolation process) between two sites depends on the geometric distance between them. To understand such process, we propose gravitationally correlated percolation models for link-adding networks on the two-dimensional lattice G with two strategies S max and S min , to add a link l i,j to connect site i and site j with mass m i and m j , respectively; m i and m j are sizes of the clusters which contain site i and site j, respectively.The probability to add the link l i,j is related to the generalized gravity g ij ≡ m i m j /r d ij , where r ij is the geometric distance between i and j, and d is an adjustable decaying exponent. In the beginning of the simulation, all sites of G are occupied and there is no link. In the simulation process, two inter-cluster links l i,j and l k,n are randomly chosen and the generalized gravities g ij and g kn are computed. In the strategy S max , the link with larger generalized gravity is added. In the strategy S min , the link with smaller generalized gravity is added, which include percolation on the Erdős-Rényi random graph and the Achlioptas process of explosive percolation as the limiting cases, d → ∞ and d → 0, respectively. Adjustable strategies facilitate or inhibit the network percolation in a generic view. We calculate percolation thresholds T c and critical exponents β by numerical simulations. We also obtain various finite-size scaling functions for the node fractions in percolating clusters or arrival of saturation length with different intervening strategies.

The scale-free features widely concerned by previous weighted network models seemed to not be the best choice to describe the link weight distributions of passenger air transportation networks. In this paper, by introducing an exponent α to the weight evolution rule of the earliest weighted model, and considering the spatial constraints of two-dimensional distances on linking probability, we have conducted rigorous analyses leading to the exponentially truncated power-law distribution for weights: P ( w ) = A ⋅ w − α ⋅ e − B w 1 − α , where A and B are parameters, α  <  1. The theoretical expression is consistent with the empirical results of the American domestic passenger flight records from 1995 to 2020. The model would be applicable to different air transportation systems and to problems in other fields.

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Air transportation systems are a foundational infrastructure for the human’s society. The lack of systematic and detailed investigation on a large amount of records for air flights has blocked seriously the deep understanding of the systems. By using the American domestic passenger flight records from 1995 to 2020, we constructed the air transportation networks and calculated the betweenness and the eigenvector centralities for the airports. It is found that in terms of eigenvector centrality, 15–30% airports in the unweighted and undirected networks behave anomalous. The anomalies disappear after considering the information of link weights or directionalites. Five widely used models for air transportation networks are evaluated, results for which tell us that the spatial constraints are required to eliminate the anomalies detected by the eigenvector centrality, and provide us some references for selecting the parameters in the models. We hope the empirical benchmarks reported in this paper can stimulate much more works on theoretical models for air transportation systems.