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

Star 5-edge-colorings of subcubic multigraphs

The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $\chi'_{s}(G)\le k$. Dvo\v{r}\'ak, Mohar and \v{S}\'amal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than $7/3$ is star list-$5$-edge-colorable. It is known that a graph with maximum average degree $14/5$ is not necessarily star $5$-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than $12/5$ is star $5$-edge-colorable.Comment: to appear in Discrete Mathematics. arXiv admin note: text overlap with arXiv:1701.0410

Solving the Cold-Start Problem in Recommender Systems with Social Tags

In this paper, based on the user-tag-object tripartite graphs, we propose a recommendation algorithm, which considers social tags as an important role for information retrieval. Besides its low cost of computational time, the experiment results of two real-world data sets, \emph{Del.icio.us} and \emph{MovieLens}, show it can enhance the algorithmic accuracy and diversity. Especially, it can obtain more personalized recommendation results when users have diverse topics of tags. In addition, the numerical results on the dependence of algorithmic accuracy indicates that the proposed algorithm is particularly effective for small degree objects, which reminds us of the well-known \emph{cold-start} problem in recommender systems. Further empirical study shows that the proposed algorithm can significantly solve this problem in social tagging systems with heterogeneous object degree distributions

Distinguish the charged black with scalar hair or not by the rings and images

In this paper, we investigate the optical properties of a charged black hole with scalar hair (CSH) within the context of four-dimensional Einstein-Maxwell-Dilaton gravity. To achieve this, we consider three distinct toy models of thin accretion disks. The presence of dilaton coupling allows us to express both the solutions of CSH and the Reissner-Nordstr\"om (RN) black hole in terms of their mass ($M$) and charge ($Q$). Our findings reveal differences in the effective potentials $V_{eff}$, photon sphere radii $r_{ph}$, and innermost stable circular orbit $r_{isco}$ between the CSH and RN black hole cases, which become increasingly pronounced as the charge parameter $Q$ increases. However, no noticeable distinctions are observed concerning the critical impact parameter $b_{ph}$. When the ratio of the photon ring band and the lensed ring band exceeds 0.1, it may suggest the presence of a charged black hole with scalar hair. Furthermore, our results underscore the significant influence of the charge parameter $Q$ on the brightness distributions of the direct, lensed ring, and photon ring for three standard emission functions. These findings emphasize the potential for distinguishing between CSH and RN black holes through an analysis of direct intensity and peak brightness in specific accretion disk models