The Immense Impact of Reverse Edges on Large Hierarchical Networks

Hierarchical networks are frequently encountered in animal groups,gene networks,and artificial engineering systems such as multiple robots,unmanned vehicle systems,smart grids,wind farm networks,and so forth.The structure of a large directed hierarchical network is often strongly influenced by reverse edges from lower-to higher-level nodes,such as lagging birds’howl in a flock or the opinions of lowerlevel individuals feeding back to higher-level ones in a social group.This study reveals that,for most large-scale real hierarchical networks,the majority of the reverse edges do not affect the synchronization process of the entire network;the synchronization process is influenced only by a small part of these reverse edges along specific paths.More surprisingly,a single effective reverse edge can slow down the synchronization of a huge hierarchical network by over 60%.The effect of such edges depends not on the network size but only on the average in-degree of the involved subnetwork.The overwhelming majority of active reverse edges turn out to have some kind of“bunching”effect on the information flows of hierarchical networks,which slows down synchronization processes.This finding refines the current understanding of the role of reverse edges in many natural,social,and engineering hierarchical networks,which might be beneficial for precisely tuning the synchronization rhythms of these networks.Our study also proposes an effective way to attack a hierarchical network by adding a malicious reverse edge to it and provides some guidance for protecting a network by screening out the specific small proportion of vulnerable nodes.

The insight of why:Causal inference in Earth system science

The utilization of big Earth data has provided insights into the planet we inhabit in unprecedented dimensions and scales.Unraveling the concealed causal connections within intricate data holds paramount importance for attaining a profound comprehension of the Earth system.Statistical methods founded on correlation have predominated in Earth system science(ESS)for a long time.Nevertheless,correlation does not imply causation,especially when confronted with spurious correlations resulting from big data.Consequently,traditional correlation and regression methods are inadequate for addressing causation related problems in the Earth system.In recent years,propelled by advancements in causal theory and inference methods,particularly the maturity of causal discovery and causal graphical models,causal inference has demonstrated vigorous vitality in various research directions in the Earth system,such as regularities revealing,processes understanding,hypothesis testing,and physical models improving.This paper commences by delving into the origins,connotations,and development of causality,subsequently outlining the principal frameworks of causal inference and the commonly used methods in ESS.Additionally,it reviews the applications of causal inference in the main branches of the Earth system and summarizes the challenges and development directions of causal inference in ESS.In the big Earth data era,as an important method of big data analysis,causal inference,along with physical model and machine learning,can assist the paradigm transformation of ESS from a model-driven paradigm to a paradigm of integration of both mechanism and data.Looking forward,the establishment of a meticulously structured and normalized causal theory can act as a foundational cornerstone for fostering causal cognition in ESS and propel the leap from fragmented research towards a comprehensive understanding of the Earth system.