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Higher-order networks represent complex multinode interactions beyond what traditional pairwise models can capture, offering deeper insights into many real-world system dynamics. Synchronization of complex networks is essential for achieving coordinated behavior, promoting efficient communication, stability, and overall functionality across various domains, including neuroscience. Extreme events, marked by abrupt and significant deviations from typical behavior that surpass a statistical threshold, can severely affect system stability and performance. Understanding and managing these events are vital for improving resilience and preventing catastrophic failures within complex networks and dynamical systems. This paper explores synchronization and extreme events in the synchronous solutions of a higher-order Chialvo neuron map network model, utilizing inner-linking functions and chemical synapses to represent pairwise and nonpairwise interactions, respectively. By applying Master Stability Functions and assessing synchronization error, we derive synchronization criteria and investigate the potential for extreme events within stable synchronous regions. Our findings indicate that the emergence of extreme events is contingent on the values of the higher-order coupling parameter, while the pairwise coupling parameter is responsible for sustaining synchronization. In response to these events, we develop a control strategy to suppress extreme events and chaotic behavior. Our results demonstrate that connecting all nodes to an external reference source is necessary to stabilize the network and maintain synchronization, as partial control proved insufficient.

Studies have indicated that focusing solely on pairwise interactions between two nodes disregards the associativity among multi-nodes in the network’s local structure. This associativity can be seen as dependencies among nodes, where certain edges’ presence depends on the path leading to it. Examinations on diverse datasets have approved that the variable order of chained dependencies allows for the preservation of structure information, which enables the reconstruction of the original network into a Higher-Order Network (HON) with improved quality of network representation. This paper proposes a Density-based Higher-Order Network Embedding (DHONE) algorithm, which integrates the concept of higher-order density into the network-embedding process in order to classify the contribution of different orders of dependencies. Through the construction of a novel and effective higher-order adjacency matrix, DHONE steadily improves the accuracy of network representation learning. Experimental results demonstrate DHONEs proficiency in improving embedding accuracy and overall algorithm robustness. Furthermore, grounded in the concept of higher-order density proposed herein, numerous dependencies have been discerned within the network generated from trajectories, potentially indicating the role of multi-node structures in networks.

This paper explores the law of emotion transmission on the social Internet. By applying higher-order network theory, ER random networks are reconstructed and higher-order network structures with multiple interaction points are generated. The I-state of the SI model is extended to set up the propagation rules of emotions based on higher-order networks. Concepts such as sentiment recession rate are introduced to make sentiment propagate through different simplexes, so as to obtain the propagation rules of higher-order networks. Qualitative analysis of emotional changes during the propagation process verifies the phase transition phenomenon of emotions in the network. By comparing simulated results with actual data, we demonstrate the accuracy of our higher-order network-based emotion propagation model in reflecting real-world trends and outcomes. The study also explores the effects of factors such as the initial proportion of extreme emotions on emotion propagation. This study provides valuable insights into understanding the operation of complex systems and offers important implications for government predictions of opinion development.