A fractal-cluster-based analytical model for spatial pattern of congestion

Research has shown that spatial patterns of congestion is neither compact as expected by typical model of cascade dynamics nor purely random as in percolation theory. Analyzing spatial patterns of congestion is critical for mining spatial-temporal characteristics of congestion evolution. Spatial patterns of congestion are the result of congestion interaction, which appears as the dependency relationship of the adjacent edges and the dependency relationship of the non-adjacent edges with a certain range in the network. Previous models which analyze spatial patterns of congestion mainly considers the dependency relationship of the directly connected edges, but lack the consideration of the dependency relationship of the indirectly connected edges. Therefore, this paper presents a fractal-cluster-based analytical model considering the dependency relationship of the indirectly connected edges to describe the dominant mechanism governing the formation and evolution of spatial pattern of congestion. First, we introduce the edge dependency coefficient to quantitatively describe the dependency strength of the adjacent edges. Next, we regard the basic fractal element of the network as a cluster and introduce the cluster dependency coefficient to quantitatively describe the dependency relationship of the non-adjacent edges with a certain range in the network. Finally, we construct a weighted network in which the weight of edges represents the congestion level of edges and introduce a novel load transfer mechanism to describe the results of congestion interaction. Based on this, a fractal-cluster-based congestion evolution model is established to analyze spatial patterns of congestion. To quantify spatial pattern, we use the fractal dimension of the weighted network dB (a measurement of objects' irregularity). The simulation comparison results have verified the feasibility of this indicator. Furthermore, simulation results have shown that our proposed model is more in line with the observed congestion propagation process, which verifies the effectiveness of our proposed model. This work can give precious hints on which step of the process is responsible for the congestion duo to the its mechanistic analysis of spatial patterns.