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The notion of structural heterogeneity is pervasive in real networks, and their community organization is no exception. Still, a vast majority of community detection methods assume neatly hierarchically organized communities of a characteristic scale for a given hierarchical level. In this work, we demonstrate that the reality of scale-dependent community reorganization is convoluted with simultaneous processes of community splitting and merging, challenging the conventional understanding of community-scale adjustment. We provide the mathematical argument on the modularity function, the results from the real-network analysis, and a simple network model for a comprehensive understanding of the nontrivial community reorganization process characterized by a local dip in the number of communities as the resolution parameter varies. This study suggests a need for a paradigm shift in the study of network communities, which emphasizes the importance of considering scale-dependent reorganization to better understand the genuine structural organization of networks., Comment: 6 pages, 5 figures

Effective frequency control in power grids has become increasingly important with the increasing demand for renewable energy sources. Here, we propose a novel strategy for resolving this challenge using graph convolutional proximal policy optimization (GC-PPO). The GC-PPO method can optimally determine how much power individual buses dispatch to reduce frequency fluctuations across a power grid. We demonstrate its efficacy in controlling disturbances by applying the GC-PPO to the power grid of the UK. The performance of GC-PPO is outstanding compared to the classical methods. This result highlights the promising role of GC-PPO in enhancing the stability and reliability of power systems by switching lines or decentralizing grid topology., Comment: 11 pages, 6 figures

Ballet, a mainstream performing art predominantly associated with women, exhibits significant gender imbalances in leading positions. However, the collaboration's structural composition on gender representation in the field remains unexplored. Our study investigates the gendered labor force composition and collaboration patterns in ballet creations. Our findings reveal gender disparities in ballet creations aligned with gendered collaboration patterns and women occupying more peripheral network positions respect to men. Productivity disparities show women accessing 20-25\% of ballet creations compared to men. Mathematically derived perception errors show the underestimation of women artists' representation within ballet collaboration networks, potentially impacting women's careers in the field. Our study highlights the structural disadvantages that women face in ballet and emphasizes the need for a more inclusive and equal professional environment to improve the career development of women in the ballet industry. These insights contribute to a broader understanding of structural gender imbalances in artistic domains and can inform cultural organizations about potential affirmative actions towards a better representation of women leaders in ballet., Comment: 29 pages, 2 tables, 5 figures, 7 supplementary figures

We study a way to set the natural frequency of a newly added oscillator in a growing network to enhance synchronization. Population growth is one of the typical features of many oscillator systems for which synchronization is required to perform their functions properly. Despite this significance, little has been known about synchronization in growing systems. We suggest effective growing schemes to enhance synchronization as the network grows under a predetermined rule. Specifically, we find that a method based on a link-wise order parameter outperforms that based on the conventional global order parameter. With simple solvable examples, we verify that the results coincide with intuitive expectations. The numerical results demonstrate that the approximate optimal values from the suggested method show a larger synchronization enhancement in comparison to other naive strategies. The results also show that our proposed approach outperforms others over a wide range of coupling strengths., Comment: 19 pages, 4 figures

The gravity model, inspired by Newton's law of universal gravitation, has long served as a primary tool for interpreting trade flows between countries, using a country's economic `mass' as a key determinant. Despite its wide application, the definition of `mass' within this model remains ambiguous. It is often approximated using indicators like GDP, which may not accurately reflect a country's true trade potential. Here, we introduce a data-driven, self-consistent numerical approach that redefines `mass' from a static proxy to a dynamic attribute inferred directly from flow data. We infer mass distribution and interaction nature through our method, mirroring Newton's approach to understanding gravity. Our methodology accurately identifies predefined embeddings and reconstructs system attributes when applied to synthetic flow data, demonstrating its strong predictive power and adaptability. Further application to real-world trade networks yields critical insights, revealing the spatial spectrum of trade flows and the economic mass of countries, two key features unexplored in depth by existing models. Our methodology not only enables accurate reconstruction of the original flow but also allows for a deep understanding of the unique capabilities of each node within the network. This study marks a significant shift in the understanding and application of the gravity model, providing a more comprehensive tool for analyzing complex systems and uncovering new insights into various fields, including global trade, traffic engineering, epidemic disease prevention, and infrastructure design., Comment: 18 pages, 4 figures, 1 table

Complex network analyses have provided clues to improve power-grid stability with the help of numerical models. The high computational cost of numerical simulations, however, has inhibited the approach, especially when it deals with the dynamic properties of power grids such as frequency synchronization. In this study, we investigate machine learning techniques to estimate the stability of power-grid synchronization. We test three different machine learning algorithms -- random forest, support vector machine, and artificial neural network -- training them with two different types of synthetic power grids consisting of homogeneous and heterogeneous input-power distribution, respectively. We find that the three machine learning models better predict the synchronization stability of power-grid nodes when they are trained with the heterogeneous input-power distribution than the homogeneous one. With the real-world power grids of Great Britain, Spain, France, and Germany, we also demonstrate that the machine learning algorithms trained on synthetic power grids are transferable to the stability prediction of the real-world power grids, which implies the prospective applicability of machine learning techniques on power-grid studies., Comment: 10 pages, 6 figures, 4 tables

The synchronization stability has been analyzed as one of the important dynamical characteristics of power grids. In this study, we bring the operational perspective to the synchronization stability analysis by counting not only full but also partial synchronization between nodes. To do so, we introduce two distinct measures that estimate the operational resilience of power-grid nodes: functional stability and functional resistance. We demonstrate the practical applicability of the measures in a model network motif and an IEEE test power grid. As a case study of German power grid, we reveal that the modular structure of a power grid and particular unidirectional current flow determine the distribution of the operational resilience of power-grid nodes. Reproducing our finding on clustered benchmark networks, we validate the modular effect on power grid stability and confirm that our measures can be the insightful tools to understand the power grids' synchronization dynamics.

The concept of community detection has long been used as a key device for handling the mesoscale structures in networks. Suitably conducted community detection reveals various embedded informative substructures of network topology. However, regarding the practical usage of community detection, it has always been a tricky problem to assign a reasonable community resolution for networks of interest. Because of the absence of the unanimously accepted criterion, most of the previous studies utilized rather ad hoc heuristics to decide the community resolution. In this work, we harness the concept of consistency in community structures of networks to provide the overall community resolution landscape of networks, which we eventually take to quantify the reliability of detected communities for a given resolution parameter. More precisely, we exploit the ambiguity in the results of stochastic detection algorithms and suggest a method that denotes the relative validity of community structures in regard to their stability of global and local inconsistency measures using multiple detection processes. Applying our framework to synthetic and real networks, we confirm that it effectively displays insightful fundamental aspects of community structures., Comment: 12 pages, 11 figures

In green plants, chlorophyll-a and chlorophyll-b are the predominant pigments bound to light-harvesting proteins. While the individual characteristics of these chlorophylls are well understood, the advantages of their coexistence remain unclear. In this study, we establish a method to simulate excitation energy transfer within the entire photosystem II supercomplex by employing network analysis integrated with quantum dynamic calculations. We then investigate the effects of the coexistence of chlorophyll-a and chlorophyll-b by comparing various chlorophyll compositions. Our results reveal that the natural chlorophyll composition allows the excited energy to preferentially flow through specific domains that act as safety valves, preventing downstream overflow. Our findings suggest that the light-harvesting proteins in a photosystem II supercomplex achieve evolutionary advantages with the natural chlorophyll-a/b ratio, capturing light energy efficiently and safely across various light intensities. Using our framework, one can better understand how green plants harvest light energy and adapt to changing environmental conditions., Comment: 20 pages, 4 figures

Multiple organs in a living system respond to environmental changes, and the signals from the organs regulate the physiological environment. Inspired by this biological feedback, we propose a simple autonomous system of active rotators to explain how multiple units are synchronized under a fluctuating environment. We find that the feedback via an environment can entrain rotators to have synchronous phases for specific conditions. This mechanism is markedly different from the simple entrainment by a common oscillatory external stimulus that is not interacting with systems. We theoretically examine how the phase synchronization depends on the interaction strength between rotators and environment. Furthermore, we successfully demonstrate the proposed model by realizing an analog electric circuit with microelectronic devices. This bio-inspired platform can be used as a sensor for monitoring varying environments, and as a controller for amplifying signals by their feedback-induced synchronization., Comment: 8 pages, 6 figures

In electric power systems delivering alternating current, it is essential to maintain its synchrony of the phase with the rated frequency. The synchronization stability that quantifies how well the power-grid system recovers its synchrony against perturbation depends on various factors. As an intrinsic factor that we can design and control, the transmission capacity of the power grid affects the synchronization stability. Therefore, the transition pattern of the synchronization stability with the different levels of transmission capacity against external perturbation provides the stereoscopic perspective to understand the synchronization behavior of power grids. In this study, we extensively investigate the factors affecting the synchronization stability transition by using the concept of basin stability as a function of the transmission capacity. For a systematic approach, we introduce the integrated basin instability, which literally adds up the instability values as the transmission capacity increases. We first take simple 5-node motifs as a case study of building blocks of power grids, and a more realistic IEEE 24-bus model to highlight the complexity of decisive factors. We find that both structural properties such as gate keepers in network topology and dynamical properties such as large power input/output at nodes cause synchronization instability. The results suggest that evenly distributed power generation and avoidance of bottlenecks can improve the overall synchronization stability of power-grid systems., Comment: 11 pages, 5 figures, 3 tables

Community identification of network components enables us to understand the mesoscale clustering structure of networks. A number of algorithms have been developed to determine the most likely community structures in networks. Such a probabilistic or stochastic nature of this problem can naturally involve the ambiguity in resultant community structures. More specifically, stochastic algorithms can result in different community structures for each realization in principle. In this study, instead of trying to "solve" this community degeneracy problem, we turn the tables by taking the degeneracy as a chance to quantify how strong companionship each node has with other nodes. For that purpose, we define the concept of companionship inconsistency that indicates how inconsistently a node is identified as a member of a community regarding the other nodes. Analyzing model and real networks, we show that companionship inconsistency discloses unique characteristics of nodes, thus we suggest it as a new type of node centrality. In social networks, for example, companionship inconsistency can classify outsider nodes without firm community membership and promiscuous nodes with multiple connections to several communities. In infrastructure networks such as power grids, it can diagnose how the connection structure is evenly balanced in terms of power transmission. Companionship inconsistency, therefore, abstracts individual nodes' intrinsic property on its relationship to a higher-order organization of the network., Comment: 10 pages, 5 figures, 2 tables

Power grids sustain modern society by supplying electricity and thus their stability is a crucial factor for our civilization. The dynamic stability of a power grid is usually quantified by the probability of its nodes' recovery to phase synchronization of the alternating current it carries, in response to external perturbation. Intuitively, the stability of nodes in power grids is supposed to become more robust as the coupling strength between the nodes increases. However, we find a counterintuitive range of coupling strength values where the synchronization stability suddenly droops as the coupling strength increases, on a number of simple graph structures. Since power grids are designed to fulfill both local and long-range power demands, such simple graph structures or graphlets for local power transmission are indeed relevant in reality. We show that the observed nonmonotonic behavior is a consequence of transitions in multistability, which are related to changes in stability of the \emph{unsynchronized} states. Therefore, our findings suggest that a comprehensive understanding of changes in multistability are necessary to prevent the unexpected catastrophic instability in the building blocks of power grids.

Given a power grid and a transmission (coupling) strength, basin stability is a measure of synchronization stability for individual nodes. Earlier studies have focused on the basin stability's dependence of the position of the nodes in the network for single values of transmission strength. Basin stability grows from zero to one as transmission strength increases, but often in a complex, nonmonotonous way. In this study, we investigate the entire functional form of the basin stability's dependence on transmission strength. To be able to perform a systematic analysis, we restrict ourselves to small networks. We scan all isomorphically distinct networks with an equal number of power producers and consumers of six nodes or less. We find that the shapes of the basin stability fall into a few, rather well-defined classes, that could be characterized by the number of edges and the betweenness of the nodes, whereas other network positional quantities matter less., Comment: 7 pages, 6 figures

The synchrony of electric power systems is important in order to maintain stable electricity supply. Recently, the measure basin stability was introduced to quantify a node's ability to recover its synchronization when perturbed. In this work, we focus on how basin stability depends on the coupling strength between nodes. We use the Chilean power grid as a case study. In general, basin stability goes from zero to one as coupling strength increases. However, this transition does not happen at the same value for different nodes. By understanding the transition for individual nodes, we can further characterize their role in the power-transmission dynamics. We find that nodes with an exceptionally large transition window also have a low community consistency. In other words, they are hard to classify to one community when applying a community detection algorithm. This also gives an efficient way to identify nodes with a long transition window (which is computationally time consuming). Finally, to corroborate these results, we present a stylized example network with prescribed community structures that captures the mentioned characteristics of basin stability transition and recreates our observations.