Can simple models explain Zipf’s law for all exponents?

H. Simon proposed a simple stochastic process for explaining Zipf’s law for word frequencies. Here we introduce two similar generalizations of Simon’s model that cover the same range of exponents as the standard Simon model. The mathematical approach followed minimizes the amount of mathematical background needed for deriving the exponent, compared to previous approaches to the standard Simon’s model. Reviewing what is known from other simple explanations of Zipf’s law, we conclude there is no single radically simple explanation covering the whole range of variation of the exponent of Zipf’s law in humans. The meaningfulness of Zipf’s law for word frequencies remains an open question.Peer ReviewedPostprint (published version

Opinion dynamics with disagreement and modulated information

Opinion dynamics concerns social processes through which populations or groups of individuals agree or disagree on specific issues. As such, modelling opinion dynamics represents an important research area that has been progressively acquiring relevance in many different domains. Existing approaches have mostly represented opinions through discrete binary or continuous variables by exploring a whole panoply of cases: e.g. independence, noise, external effects, multiple issues. In most of these cases the crucial ingredient is an attractive dynamics through which similar or similar enough agents get closer. Only rarely the possibility of explicit disagreement has been taken into account (i.e., the possibility for a repulsive interaction among individuals' opinions), and mostly for discrete or 1-dimensional opinions, through the introduction of additional model parameters. Here we introduce a new model of opinion formation, which focuses on the interplay between the possibility of explicit disagreement, modulated in a self-consistent way by the existing opinions' overlaps between the interacting individuals, and the effect of external information on the system. Opinions are modelled as a vector of continuous variables related to multiple possible choices for an issue. Information can be modulated to account for promoting multiple possible choices. Numerical results show that extreme information results in segregation and has a limited effect on the population, while milder messages have better success and a cohesion effect. Additionally, the initial condition plays an important role, with the population forming one or multiple clusters based on the initial average similarity between individuals, with a transition point depending on the number of opinion choices

Molecular Dynamics Simulation of Vascular Network Formation

Endothelial cells are responsible for the formation of the capillary blood vessel network. We describe a system of endothelial cells by means of two-dimensional molecular dynamics simulations of point-like particles. Cells' motion is governed by the gradient of the concentration of a chemical substance that they produce (chemotaxis). The typical time of degradation of the chemical substance introduces a characteristic length in the system. We show that point-like model cells form network resembling structures tuned by this characteristic length, before collapsing altogether. Successively, we improve the non-realistic point-like model cells by introducing an isotropic strong repulsive force between them and a velocity dependent force mimicking the observed peculiarity of endothelial cells to preserve the direction of their motion (persistence). This more realistic model does not show a clear network formation. We ascribe this partial fault in reproducing the experiments to the static geometry of our model cells that, in reality, change their shapes by elongating toward neighboring cells.Comment: 10 pages, 3 figures, 2 of which composite with 8 pictures each. Accepted on J.Stat.Mech. (2009). Appeared at the poster session of StatPhys23, Genoa, Italy, July 13 (2007

Complex delay dynamics on railway networks: from universal laws to realistic modelling

Railways are a key infrastructure for any modern country. The reliability and resilience of this peculiar transportation system may be challenged by different shocks such as disruptions, strikes and adverse weather conditions. These events compromise the correct functioning of the system and trigger the spreading of delays into the railway network on a daily basis. Despite their importance, a general theoretical understanding of the underlying causes of these disruptions is still lacking. In this work, we analyse the Italian and German railway networks by leveraging on the train schedules and actual delay data retrieved during the year 2015. We use {these} data to infer simple statistical laws ruling the emergence of localized delays in different areas of the network and we model the spreading of these delays throughout the network by exploiting a framework inspired by epidemic spreading models. Our model offers a fast and easy tool for the preliminary assessment of the {effectiveness of} traffic handling policies, and of the railway {network} criticalities.Comment: 32 pages (with appendix), 28 Figures (with appendix), 2 Table

The scale-free topology of market investments

We propose a network description of large market investments, where both stocks and shareholders are represented as vertices connected by weighted links corresponding to shareholdings. In this framework, the in-degree ($k_{in}$) and the sum of incoming link weights ($v$) of an investor correspond to the number of assets held (\emph{portfolio diversification}) and to the invested wealth (\emph{portfolio volume}) respectively. An empirical analysis of three different real markets reveals that the distributions of both $k_{in}$ and $v$ display power-law tails with exponents $\gamma$ and $\alpha$. Moreover, we find that $k_{in}$ scales as a power-law function of $v$ with an exponent $\beta$. Remarkably, despite the values of $\alpha$, $\beta$ and $\gamma$ differ across the three markets, they are always governed by the scaling relation $\beta=(1-\alpha)/(1-\gamma)$. We show that these empirical findings can be reproduced by a recent model relating the emergence of scale-free networks to an underlying Paretian distribution of `hidden' vertex properties.Comment: Final version accepted for publication on Physica

To what extent homophily and influencer networks explain song popularity

Forecasting the popularity of new songs has become a standard practice in the music industry and provides a comparative advantage for those that do it well. Considerable efforts were put into machine learning prediction models for that purpose. It is known that in these models, relevant predictive parameters include intrinsic lyrical and acoustic characteristics, extrinsic factors (e.g., publisher influence and support), and the previous popularity of the artists. Much less attention was given to the social components of the spreading of song popularity. Recently, evidence for musical homophily - the tendency that people who are socially linked also share musical tastes - was reported. Here we determine how musical homophily can be used to predict song popularity. The study is based on an extensive dataset from the last.fm online music platform from which we can extract social links between listeners and their listening patterns. To quantify the importance of networks in the spreading of songs that eventually determines their popularity, we use musical homophily to design a predictive influence parameter and show that its inclusion in state-of-the-art machine learning models enhances predictions of song popularity. The influence parameter improves the prediction precision (TP/(TP+FN)) by about 50% from 0.14 to 0.21, indicating that the social component in the spreading of music plays at least as significant a role as the artist's popularity or the impact of the genre.Comment: 7 pages, 3 figure