Rich-club Phenomenon Research Articles

The behavior of rich-club coefficient in scale-free networks

The rich-club phenomenon, which provides information about the association between nodes, is a useful method to study the hierarchy structure of networks. In this work, we explore the behavior of rich-club coefficient (RCC) in scale-free networks, and find that the degree-based RCC is a power function of degree centrality with power exponent β and the betweenness-based RCC is a linear function of betweenness centrality with slope θ. Moreover, we calculate the value of RCC in a BA network by deleting nodes and obtain a general expression for RCC as a function of node sequence. On this basis, the solution of RCC for centrality is also obtained, which shows that the curve of RCC is determined by the centrality distribution. In the numerical simulation, we observe: β and γ (the degree distribution exponent) increase together, θ increases with the average degree and decreases to convergence as γ increases.

P300 intention recognition based on phase lag index (PLI)-rich-club brain functional network.

Brain-computer interface (BCI) technology based on P300 signals has a broad application prospect in the assessment and diagnosis of clinical diseases and game control. The paper of selecting key electrodes to realize a wearable intention recognition system has become a hotspot for scholars at home and abroad. In this paper, based on the rich-club phenomenon that exists in the process of intention generation, a phase lag index (PLI)-rich-club-based intention recognition method for P300 is proposed. The rich-club structure is a network consisting of electrodes that are highly connected with other electrodes in the process of P300 generation. To construct the rich-club network, this paper uses PLI to construct the brain functional network, calculates rich-club coefficients of the network in the range of k degrees, initially identifies rich-club nodes based on the feature of node degree, and then performs a descending order of betweenness centrality and identifies the nodes with larger betweenness centrality as the specific rich-club nodes, extracts the non-linear features and frequency domain features of Rich-club nodes, and finally uses support vector machine for classification. The experimental results show that the range of rich-club coefficients is smaller with intent compared to that without intent. Validation was performed on the BCI Competition III dataset by reducing the number of channels to 17 and 16 for subject A and subject B, with recognition quasi-departure rates of 96.93% and 94.93%, respectively, and on the BCI Competition II dataset by reducing the number of channels to 17 for subjects, with a recognition accuracy of 95.50%.

Temporal rich club phenomenon and its formation mechanisms.

The temporal rich club (TRC) phenomenon is widespread in real systems, forming a tight and continuous collection of the prominent nodes that control the system. However, there is still a lack of sufficient understanding of the mechanisms of TRC formation. Here we use the international N-nutrient trade network as an example of an in-depth identification, analysis, and modeling of its TRC phenomenon. The system exhibits a statistically significant TRC phenomenon, with eight economies forming the cornerstone club. Our analysis reveals that node degree is the most influential factor in TRC formation compared to other variables. The mathematical evolution models we constructed propose that the TRC in the N-nutrient trade network arises from the coexistence of degree-homophily and path-dependence mechanisms. By comprehending these mechanisms, we introduce a different perspective on TRC formation. Although our analysis is limited to the international trade system, the methodology can be extended to analyze the mechanisms underlying TRC emergence in other systems.

Global land drought hubs confounded by teleconnection hotspots in equatorial oceans

Emerging data-driven techniques, such as Complex Networks (CNs), can identify spatial linkages between droughts on a global scale and can improve early warning systems. Recent studies used CNs to identify hotspots of global drought teleconnections as land drought hubs; however, these studies excluded the ocean regions in CN, an oversight that can upend the insights gained thus far. Here, using a comprehensive global CN analysis on drought onsets, we show that oceanic regions harbor significantly larger drought hubs than land regions. The Indo-Pacific Warm pool (IPWP) in the Maritime continent emerges as the most significant drought hub having the farthest teleconnections. We show that IPWP, together with a few sub-tropical land and ocean regions, exhibit a ‘rich club phenomenon’ in CN. Further, using a causal network learning algorithm, we demonstrate the confounding role of oceans in modulating drought onsets on land regions. Our study reveals insights on the spatiotemporal linkages of global drought onsets and highlights the role of oceans in driving global drought teleconnections and modulation of land drought hubs.

Determination of influential nodes based on the Communities’ structure to maximize influence in social networks

With the increasing development of social networks, they have turned into important research platforms. Influence maximization is one of the most important research issues in the field of social networks. This problem detects influential k-node with the greatest influence spread. The influence maximization faces two important challenges, time efficiency and optimal selection of seed nodes. In order to solve such challenges, we propose an algorithm based on optimal pruning and scoring adjustment, which is called IMBC for short. The IMBC (Influence Maximization Based on Community structure) algorithm uses optimal pruning and a minimum of dominating nodes to improve time efficiency. In addition, for optimal selection of seed nodes, the IMBC algorithm modulates the scores of nodes with a high Rich-Club coefficient. In order to select influential nodes, we first select an optimal set using the minimum dominating nodes and node scores, with the aim of optimal pruning in influence spread calculations. Because large-scale social networks have many nodes, optimal pruning reduces computational overhead. Then, the seed nodes are selected based on the scoring adjustment. Scoring adjustment is done to avoid the Rich Club phenomenon because avoiding this phenomenon causes a large amount of diffusion in social networks. The experimental results show that the proposed algorithm performs better than the algorithms presented in recent years in influence spread and runtime. Therefore, the IMBC algorithm is a balance between quality and efficiency. Also, in the PGP dataset results, the PHG algorithm with as much as a 5.08% increase in influence spread, and the runtime has decreased by 97%.

Higher-order rich-club phenomenon in collaborative research grant networks

Modern scientific work, including writing papers and submitting research grant proposals, increasingly involves researchers from different institutions. In grant collaborations, it is known that institutions involved in many collaborations tend to densely collaborate with each other, forming rich clubs. Here we investigate higher-order rich-club phenomena in networks of collaborative research grants among institutions and their associations with research impact. Using publicly available data from the National Science Foundation in the US, we construct a bipartite network of institutions and collaborative grants, which distinguishes among the collaboration with different numbers of institutions. By extending the concept and algorithms of the rich club for dyadic networks to the case of bipartite networks, we find rich clubs both in the entire bipartite network and the bipartite subnetwork induced by the collaborative grants involving a given number of institutions up to five. We also find that the collaborative grants within rich clubs tend to be more impactful in a per-dollar sense than the control. Our results highlight advantages of collaborative grants among the institutions in the rich clubs.

Global droughts connected by linkages between drought hubs

Quantifying the spatial and interconnected structure of regional to continental scale droughts is one of the unsolved global hydrology problems, which is important for understanding the looming risk of mega-scale droughts and the resulting water and food scarcity and their cascading impact on the worldwide economy. Using a Complex Network analysis, this study explores the topological characteristics of global drought events based on the self-calibrated Palmer Drought Severity Index. Event Synchronization is used to measure the strength of association between the onset of droughts at different spatial locations within the time lag of 1-3 months. The network coefficients derived from the synchronization network indicate a highly heterogeneous connectivity structure underlying global drought events. Drought hotspot regions such as Southern Europe, Northeast Brazil, Australia, and Northwest USA behave as drought hubs that synchronize regionally and with other hubs at inter-continental or even inter-hemispheric scale. This observed affinity among drought hubs is equivalent to the ‘rich-club phenomenon’ in Network Theory, where ‘rich’ nodes (here, drought hubs) are tightly interconnected to form a club, implicating the possibility of simultaneous large-scale droughts over multiple continents.

Does there exist rich-club phenomenon within two steps in complex networks?

Rich club is a special phenomenon in complex networks. It refers to the nodes with high degrees (i.e., rich nodes) that tend to connect with each other to form a “rich club”. This phenomenon exists in many real networks, especially in social networks. However, there are also some networks not having this phenomenon, for example, in many biological networks. Thus, a natural question comes up: do these rich nodes contact with each other through an “agent”? Two rich nodes do not connect directly, but they can establish a connection by an “agent”, i.e., they can connect with each other within two steps. If the rich nodes tend to connect with each other within two steps, we say that they form a “2-step rich club”. The traditional rich club is then called “1-step rich club”. In this paper, we aim to design one method to detect the “2-step rich club” phenomenon in real networks and check whether this new “2-step rich club” phenomenon commonly exists in real networks. We find that most networks with “1-step rich club” will also have “2-step rich club”. However, there also exist few networks having “2-step rich club” structure while not having “1-step rich club” structure. In particular, either 1-step rich-club phenomenon or 2-step rich-club phenomenon exists in most social networks.

Universal behaviour of the growth method and importance of local hubs in cascading failure

Abstract We introduce hub centrality and study the relation between hub centrality and the degree of each node in the networks. We discover and verify a universal relation between them in various networks generated by the growth method, but the relation is not applied to real-world networks due to the rich-club phenomenon and the presence of local hubs. Through the study of a targeted attack and overload cascading failure, we prove that hub centrality is a meaningful parameter that gives extra insight beyond degree in real-world networks. Especially, we show that the local hubs occupy key positions in real-world networks with higher probabilities to incur global cascading failure. Therefore, we conclude that networks generated by the growth method, which do not include local hubs, have inevitable limitations to describe real-world networks.

The temporal rich club phenomenon

Identifying the hidden organizational principles and relevant structures of complex networks is fundamental to understand their properties. To this end, uncovering the structures involving the prominent nodes in a network is an effective approach. In temporal networks, the simultaneity of connections is crucial for temporally stable structures to arise. Here, we propose a measure to quantitatively investigate the tendency of well-connected nodes to form simultaneous and stable structures in a temporal network. We refer to this tendency as the temporal rich club phenomenon, characterized by a coefficient defined as the maximal value of the density of links between nodes with a minimal required degree, which remain stable for a certain duration. We illustrate the use of this concept by analysing diverse data sets and their temporal properties, from the role of cohesive structures in relation to processes unfolding on top of the network to the study of specific moments of interest in the evolution of the network. Uncovering structures in temporal networks requires different tools than in their static counterparts. A metric now quantifies whether the nodes with a large number of connections also tend to stay simultaneously connected for longer times.

Multi-hop analysis method for rich-club phenomenon of influence maximization in social networks

Multi-hop analysis method for rich-club phenomenon of influence maximization in social networks

Unveiling the rich-club phenomenon in urban mobility networks through the spatiotemporal characteristics of passenger flow

With increasing passenger flows and rapid urbanization, the Mass Transit Railway (MTR) systems in Hong Kong have entered into an era of complex networking operation, which enables people to travel across 18 local districts and enhance sustainable transport accessibility. The urban mobility flow patterns typically encoded as origin–destination (OD) matrices inspire the constitution of the temporal urban mobility networks (TUMNs), which are represented as the directed weighted networks. The dynamics of the TUMNs have significant implications for transportation systems, urban traffic planning, and mobility pattern exploration. This paper provides a thorough analysis of the rich-club phenomenon of TUMNs in Hong Kong during different time periods. Additionally, the space–time characteristics of passenger flow are analyzed and based on which we build the directed weighted TUMNs where the nodes are the stations, and the weighted links represent the number of trips between nodes. By using the outgoing-strength and incoming-strength global rich-club coefficient, local rich-club coefficient, community detection, and assortativity coefficient, we quantitatively identify the rich-club phenomenon in the TUMNs during both the peak hours (8:00–9:00 and 18:00–19:00) and non-peak hours (10:00–11:00, 15:00–16:00, and 20:00–21:00). The findings uncover the rich-club phenomenon in the TUMNs, and the distribution of the rich-club members evolving over time. The findings of this research pave the way towards a richer spectrum of complex real-world networks and are useful for transport infrastructure planning.

Exploring the rich-club characteristic in internal migration: Evidence from Chinese Chunyun migration

Regional development differences are universal economic development process problems of countries around the world. Unbalanced development theories have been used to interpret population migration; conversely, population migration patterns can also reflect unbalanced regional development. Inspired by the growth pole and pole-axis theories, we infer that a rich-club phenomenon in internal migration caused by unbalanced regional development exists. We propose a progressive methodological framework combining complex network science and geographic information science applied to Chinese Chunyun migration to empirically research this potential rich-club phenomenon in an intercity population migration network and investigate the functions of prominent cities in intra/interregional migration. We use the global rich-club coefficient to examine whether the rich-club phenomenon exists in the population migration network. Then, the community detection method and assortativity coefficient are applied to further understand the connections between cities. Finally, the influence and functions of prominent cities are explored from multiple perspectives and scales. The results indicate that the rich-club phenomenon existed in intercity migration during the Chinese Chunyun period and presents the positive role of rich-club cities in intra/interregional migration. An imbalance in urban and regional development is also found. This paper can be used to guide urban planning, transportation planning, and population research.

Network Characterization of Goods Movement in Indonesia

Network characterization of goods movement is particularly important to estimate needs and accurate management for freight transportation. The present study aims at characterizing the network of goods movement in Indonesia and examining the effectiveness of network characterization for prediction purpose. A network approach using RStudio was applied. It was found that the network of goods movement in Indonesia is characterized by weak inter-provincial goods movement, scale-free network, disassortative, having rich-club phenomenon, and having a core-periphery structure in which Sumatera Barat (West Sumatera) and Jawa Barat (West Java) appears to be the cores of the network. The findings also demonstrate that the prediction model developed by characterization performs better and can explain, on average, 39% of the variances. The prediction models for chemicals, fuel, meat and livestock, fish, crude oil, and fertilizer, have even better capability to explain variances in the volumes of goods movement by more than 60%. Potential avenues for future research are also discussed.

Configuration and dynamics of dominant inspiratory multineuronal activity patterns during eupnea and gasping generation in vitro.

The pre-Bötzinger complex (preBötC), located within the ventral respiratory column, produces inspiratory bursts in varying degrees of synchronization/amplitude. This wide range of population burst patterns reflects the flexibility of the preBötC neurons, which is expressed in variations in the onset/offset times of their activations and their activity during the population bursts, with respiratory neurons exhibiting a large cycle-to-cycle timing jitter both at the population activity onset and at the population activity peak, suggesting that respiratory neurons are stochastically activated before and during the inspiratory bursts. However, it is still unknown whether this stochasticity is maintained while evaluating the coactivity of respiratory neuronal ensembles. Moreover, the preBötC topology also remains unknown. In this study, by simultaneously recording tens of preBötC neurons and using coactivation analysis during the inspiratory periods, we found that the preBötC has a scale-free configuration (mixture of not many highly connected nodes, hubs, with abundant poorly connected elements) exhibiting the rich-club phenomenon (hubs more likely interconnected with each other). PreBötC neurons also produce multineuronal activity patterns (MAPs) that are highly stable and change during the hypoxia-induced reconfiguration. Moreover, preBötC contains a coactivating core network shared by all its MAPs. Finally, we found a distinctive pattern of sequential coactivation of core network neurons at the beginning of the inspiratory periods, indicating that, when evaluated at the multicellular level, the coactivation of respiratory neurons seems not to be stochastic.NEW & NOTEWORTHY By means of multielectrode recordings of preBötC neurons, we evaluated their configuration in normoxia and hypoxia, finding that the preBötC exhibits a scale-free configuration with a rich-club phenomenon. preBötC neurons produce multineuronal activity patterns that are highly stable but change during hypoxia. The preBötC contains a coactivating core network that exhibit a distinctive pattern of coactivation at the beginning of inspirations. These results reveal some network basis of inspiratory rhythm generation and its reconfiguration during hypoxia.

The Complexity of Urban CO2 Emission Network: An Exploration of the Yangtze River Middle Reaches Megalopolis, China

With their focus on human production and consumption activities, cities incur massive energy consumption and CO2 emissions. An intercity connection is a typical complex system in which the interaction between cities is crucial for developing low-carbon outputs within the urban agglomeration. This paper presents the construction of the CO2 emission network of an urban agglomeration in the Yangtze River middle reaches megalopolis, based on the gravity model. Combined with social network analysis (SNA), a multilevel analysis framework is proposed to deal with the complexity, spatiality, and visualization of the CO2 emission network with reference to the network features, structural equivalence, and the rich-club phenomenon. The following results emerged: firstly, the spatial structure of the CO2 emissions was characterized by low robustness and compactness, indicating disunity among the studied cities. Secondly, there was found to be a strong correlation between regionalism and intercity connections, with geographically close cities playing a similar role in the network. Thirdly, the “rich-club” cities, including Wuhan, Changsha, Xiaogan, and Zhuzhou, dominated the connections, covering more than 87.1% of the network in the Yangtze River Middle Reaches Megalopolis.

Social network analysis in China’s hospital healthcare

The objective of this paper is to identify whether the interorganization of healthcare providers in China had the global property of an oligopoly market tendencies by analyzing the rich-club phenomenon of patient flow. This paper examines the properties of patient transfer networks in an entire province of China using social network analysis. Hypertension is one of the most prevalent chronic conditions in China, thus the study examined the transfer network of patients with hypertension. The rich-club coefficient, the assortativity coefficient, and community detection were used to quantitatively illustrate the rich-club tendency of the social network. The results reveal significant rich-club characteristics and regional clustering patterns in this transfer network. First, the global rich-club coefficients calculated based on both degree and weighted degree are larger than 1, suggesting a relatively noticeable rich-club effect. Second, 269 nodes (69.33%) attach only to rich nodes, suggesting that most nodes within the hypertensive-patient transfer network are highly influenced by rich-club members. Third, community detection found a significant pattern of regional clustering in the transfer network. Our findings are the first to demonstrate that the hospital system operated as an oligopoly market, in which the “elite” providers formed an “inner circle”, and other providers were only loosely interconnected. It might be valuable to patients to transfer this oligarchical system into a more pluralistic network.

TI-SC: top-k influential nodes selection based on community detection and scoring criteria in social networks

Influence maximization is a classic optimization problem to find a subset of seed nodes in a social network that has a maximum influence with respect to a propagation model. This problem suffers from the overlap of seed nodes and the lack of optimal selection of seed nodes. Kempe et al. have shown that this problem is an NP-hard problem, and the objective function is submodular. Therefore, some heuristic and greedy algorithms have been proposed to find a near-optimal solution. However, the greedy algorithm may not satisfy the accuracy of a given solution and high time-consuming problem. To overcome these problems, the TI-SC algorithm is proposed for the problem of influence maximization. The TI-SC algorithm selects the influential nodes by examining the relationships between the core nodes and the scoring ability of other nodes. After selecting each seed node, the scores are updated to reduce the overlap in selecting the seed nodes. This algorithm has efficient performance in high Rich-Club networks. The Rich-Club phenomenon causes overlapping of the influence spread among the seed nodes in most of the other methods so that the TI-SC algorithm reduces this overlapping. Furthermore, the discovered communities with low expansion are not considered in the seed node selection phase, and this is useful for reducing computational overhead. Experimental results on both synthetic and real datasets show that the proposed TI-SC algorithm significantly outperforms the state-of-the-art algorithms in terms of efficiency in both small and large-scale datasets.

Do Chinese hospital services constitute an oligopoly? Evidence of the rich-club phenomenon in a patient referral network

Research on medical practice that uses big data has attracted considerable attention recently. In this paper, we focused on a large set of patient referral data gathered in Fujian province, China, between 2009 and 2011. We built a directed weighted patient referral network. By using four metrics from network science, namely, the power-law distribution, global rich-club coefficient, local rich-club coefficient, and assortativity coefficient, we identified a significant rich-club phenomenon in this network. In addition, the community detection was also carried out to find the relationship between rich members and non-rich members. The findings indicate an oligopoly in which Class-III hospitals occupy an overwhelmingly dominant position over the competition. Also, the characteristic ‘significant regional clustering’ was inferred from the results.

A reversed node ranking approach for influence maximization in social networks

Influence maximization, i.e. to maximize the influence spread in a social network by finding a group of influential nodes as small as possible, has been studied widely in recent years. Many methods have been developed based on either explicit Monte Carlo simulation or scoring systems, among which the former perform well yet are very time-consuming while the latter ones are efficient but sensitive to different spreading models. In this paper, we propose a novel influence maximization algorithm in social networks, named Reversed Node Ranking (RNR). It exploits the reversed rank information of a node and the effects of its neighbours upon this node to estimate its influence power, and then iteratively selects the top node as a seed node once the ranking reaches stable. Besides, we also present two optimization strategies to tackle the rich-club phenomenon. Experiments on both Independent Cascade (IC) model and Weighted Cascade (WC) model show that our proposed RNR method exhibits excellent performance and outperforms other state-of-the-arts. As a by-product, our work also reveals that the IC model is more sensitive to the rich-club phenomenon than the WC model.