Zipf's Law Leads to Heaps' Law: Analyzing Their Relation in Finite-Size Systems

Background: Zipf's law and Heaps' law are observed in disparate complex systems. Of particular interests, these two laws often appear together. Many theoretical models and analyses are performed to understand their co-occurrence in real systems, but it still lacks a clear picture about their relation. Methodology/Principal Findings: We show that the Heaps' law can be considered as a derivative phenomenon if the system obeys the Zipf's law. Furthermore, we refine the known approximate solution of the Heaps' exponent provided the Zipf's exponent. We show that the approximate solution is indeed an asymptotic solution for infinite systems, while in the finite-size system the Heaps' exponent is sensitive to the system size. Extensive empirical analysis on tens of disparate systems demonstrates that our refined results can better capture the relation between the Zipf's and Heaps' exponents. Conclusions/Significance: The present analysis provides a clear picture about the relation between the Zipf's law and Heaps' law without the help of any specific stochastic model, namely the Heaps' law is indeed a derivative phenomenon from Zipf's law. The presented numerical method gives considerably better estimation of the Heaps' exponent given the Zipf's exponent and the system size. Our analysis provides some insights and implications of real complex systems, for example, one can naturally obtained a better explanation of the accelerated growth of scale-free networks.Comment: 15 pages, 6 figures, 1 Tabl

The H-index of a network node and its relation to degree and coreness

Identifying influential nodes in dynamical processes is crucial in understanding network structure and function. Degree, H-index and coreness are widely used metrics, but previously treated as unrelated. Here we show their relation by constructing an operator , in terms of which degree, H-index and coreness are the initial, intermediate and steady states of the sequences, respectively. We obtain a family of H-indices that can be used to measure a node’s importance. We also prove that the convergence to coreness can be guaranteed even under an asynchronous updating process, allowing a decentralized local method of calculating a node’s coreness in large-scale evolving networks. Numerical analyses of the susceptible-infected-removed spreading dynamics on disparate real networks suggest that the H-index is a good tradeoff that in many cases can better quantify node influence than either degree or coreness.This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11205042, 11222543, 11075031, 61433014). L.L. acknowledges the research start-up fund of Hangzhou Normal University under Grant No. PE13002004039 and the EU FP7 Grant 611272 (project GROWTHCOM). The Boston University work was supported by NSF Grants CMMI 1125290, CHE 1213217 and PHY 1505000. (11205042 - National Natural Science Foundation of China; 11222543 - National Natural Science Foundation of China; 11075031 - National Natural Science Foundation of China; 61433014 - National Natural Science Foundation of China; PE13002004039 - research start-up fund of Hangzhou Normal University; 611272 - EU FP7 Grant (project GROWTHCOM); CMMI 1125290 - NSF; CHE 1213217 - NSF; PHY 1505000 - NSF)Published versio

The role of susceptible individuals in spreading dynamics

Exploring the internal mechanism of information spreading is critical for understanding and controlling the process. Traditional spreading models often assume individuals play the same role in the spreading process. In reality, however, individuals' diverse characteristics contribute differently to the spreading performance, leading to a heterogeneous infection rate across the system. To investigate network spreading dynamics under heterogeneous infection rates, we integrate two individual-level features -- influence (i.e., the ability to influence neighbors) and susceptibility (i.e., the extent to be influenced by neighbors) -- into the independent cascade model. Our findings reveal significant differences in spreading performance under heterogeneous and constant infection rates, with traditional structural centrality metrics proving more effective in the latter scenario. Additionally, we take the constant and heterogeneous infection rates into a state-of-the-art maximization algorithm, the well-known TIM algorithm, and find the seeds selected by heterogeneous infection rates are more dispersed compared to those under constant rates. Lastly, we find that both individuals' influence and susceptibility are vital to the spreading performance. Strikingly, susceptible individuals are particularly important to spreading when information is disseminated by social celebrities. By integrating influence and susceptibility into the spreading model, we gain a more profound understanding of the underlying mechanisms driving information spreading

Predicting missing links via local information

Missing link prediction in networks is of both theoretical interest and practical significance in modern science. In this paper, we empirically investigate a simple framework of link prediction on the basis of node similarity. We compare nine well-known local similarity measures on six real networks. The results indicate that the simplest measure, namely Common Neighbours, has the best overall performance, and the Adamic-Adar index performs second best. A new similarity measure, motivated by the resource allocation process taking place on networks, is proposed and shown to have higher prediction accuracy than common neighbours. It is found that many links are assigned the same scores if only the information of the nearest neighbours is used. We therefore design another new measure exploiting information on the next nearest neighbours, which can remarkably enhance the prediction accurac

Identifying influential spreaders by weighted LeaderRank

Identifying influential spreaders is crucial for understanding and controlling spreading processes on social networks. Via assigning degree-dependent weights onto links associated with the ground node, we proposed a variant to a recent ranking algorithm named LeaderRank (Lü et al., 2011). According to the simulations on the standard SIR model, the weighted LeaderRank performs better than LeaderRank in three aspects: (i) the ability to find out more influential spreaders; (ii) the higher tolerance to noisy data; and (iii) the higher robustness to intentional attacks

Analysis of the Vibration Type and Characteristics of the Electric Motor

Motor is a commonly used equipment in modern industrial production, vibration is an important basis to reflect the operation of the motor. The analysis of the vibration types and characteristics of the motor is of great significance to the performance, reliability and life of the motor. Through the analysis, it can effectively grasp the operation and performance status of the motor, and provide a basis for the maintenance work. Based on this, this paper analyzes the vibration types and characteristics of motors and proposes some effective prevention and treatment measures

A generalized simplicial model and its application

Higher-order structures, consisting of more than two individuals, provide a new perspective to reveal the missed non-trivial characteristics under pairwise networks. Prior works have researched various higher-order networks, but research for evaluating the effects of higher-order structures on network functions is still scarce. In this paper, we propose a framework to quantify the effects of higher-order structures (e.g., 2-simplex) and vital functions of complex networks by comparing the original network with its simplicial model. We provide a simplicial model that can regulate the quantity of 2-simplices and simultaneously fix the degree sequence. Although the algorithm is proposed to control the quantity of 2-simplices, results indicate it can also indirectly control simplexes more than 2-order. Experiments on spreading dynamics, pinning control, network robustness, and community detection have shown that regulating the quantity of 2-simplices changes network performance significantly. In conclusion, the proposed framework is a general and effective tool for linking higher-order structures with network functions. It can be regarded as a reference object in other applications and can deepen our understanding of the correlation between micro-level network structures and global network functions

Beyond network centrality: Individual-level behavioral traits for predicting information superspreaders in social media

Understanding the heterogeneous role of individuals in large-scale information spreading is essential to manage online behavior as well as its potential offline consequences. To this end, most existing studies from diverse research domains focus on the disproportionate role played by highly-connected “hub” individuals. However, we demonstrate here that information superspreaders in online social media are best understood and predicted by simultaneously considering two individual-level behavioral traits: influence and susceptibility. Specifically, we derive a nonlinear network-based algorithm to quantify individuals’ influence and susceptibility from multiple spreading event data. By applying the algorithm to large-scale data from Twitter and Weibo, we demonstrate that individuals’ estimated influence and susceptibility scores enable predictions of future superspreaders above and beyond network centrality, and reveal new insights on the network position of the superspreaders

Empirical analysis on a keyword-based semantic system

Keywords in scientific articles have found their significance in information filtering and classification. In this article, we empirically investigated statistical characteristics and evolutionary properties of keywords in a very famous journal, namely Proceedings of the National Academy of Science of the United States of America (PNAS), including frequency distribution, temporal scaling behavior, and decay factor. The empirical results indicate that the keyword frequency in PNAS approximately follows a Zipf's law with exponent 0.86. In addition, there is a power-low correlation between the cumulative number of distinct keywords and the cumulative number of keyword occurrences. Extensive empirical analysis on some other journals' data is also presented, with decaying trends of most popular keywords being monitored. Interestingly, top journals from various subjects share very similar decaying tendency, while the journals of low impact factors exhibit completely different behavior. Those empirical characters may shed some light on the in-depth understanding of semantic evolutionary behaviors. In addition, the analysis of keyword-based system is helpful for the design of corresponding recommender systems.Comment: 9 pages, 1 table and 4 figure