In mobile communication systems, congestion is related to high-traffic events (HTEs) that occur in the coverage areas of base stations. Understanding, recognizing, and predicting these HTEs and researching their occurrence rules provides theoretical and decision-making support for preventing system congestion. Communication sectors are regarded as nodes, and if HTEs occur synchronously among sectors, then the corresponding nodes are connected. The total number of synchronous HTEs determines the edge weights. The mobile-communication spatiotemporal data are mapped to a weighted network, with the occurrence locations of HTEs as the basic elements. Network analysis provides a structure for representing the interaction of HTEs. By analyzing the topological features of the event synchronization network, the associations among the occurrence times of HTEs can be mined. We find that the event synchronization network is a small-world network, the cumulative strength distribution is exponential, and the edge weight obeys a power law. Moreover, the node clustering coefficient is negatively correlated with the node degree. A congestion coefficient based on several topological parameters is proposed, and the system congestion is visualized. The congestion coefficient contains information about the synchronous occurrence of HTEs between a sector and its neighbors and information about the synchronous occurrence of HTEs among its neighbors. For the mobile communication system considered in this study, the congestion coefficient of a large number of sectors is small and the risk of system congestion is low.

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The original article unfortunately was published with an error. In Table 1 of original article, the data "M-correlation network" should had been move down to be aligned with "0.4218". The original article has been corrected.Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The problem of synchronicity quantification, based on event occurrence time, has become the research focus in different fields. Methods of synchrony measurement provide an effective way to explore spatial propagation characteristics of extreme events. Using the synchrony measurement method of event coincidence analysis, we construct a directed weighted network and innovatively explore the direction of correlations between event sequences. Based on trigger event coincidence, the synchrony of traffic extreme events of base stations is measured. Analyzing topology characteristics of the network, we study the spatial propagation characteristics of traffic extreme events in the communication system, including the propagation area, propagation influence, and spatial aggregation. This study provides a framework of network modeling to quantify the propagation characteristics of extreme events, which is helpful for further research on the prediction of extreme events. In particular, our framework is effective for events that occurred in time aggregation. In addition, from the perspective of a directed network, we analyze differences between the precursor event coincidence and the trigger event coincidence and the impact of event aggregation on the synchrony measurement methods. The precursor event coincidence and the trigger event coincidence are consistent when identifying event synchronization, while there are differences when measuring the event synchronization extent. Our study can provide a reference for the analysis of extreme climatic events such as rainstorms, droughts, and others in the climate field.

The question of the power-law exponent of exponential growth networks is studied here. In a discrete case, the degree distribution is defined as the probability distribution of the discrete variable. Based on this, the degree distribution of the pseudofractal scale-free web, an exponential growth network, is obtained. The power-law exponent ln3/ln2 is analyzed according to the maximum likelihood principle. It satisfies consistency and is good for small generations of the network. For many exponential growth networks, their power-law exponent needs to be tested. The work provides a new view on the power-law exponent of an exponential growth network.