Universality of Zipf's Law

Zipf's law is the most common statistical distribution displaying scaling behavior. Cities, populations or firms are just examples of this seemingly universal law. Although many different models have been proposed, no general theoretical explanation has been shown to exist for its universality. Here we show that Zipf's law is, in fact, an inevitable outcome of a very general class of stochastic systems. Borrowing concepts from Algorithmic Information Theory, our derivation is based on the properties of the symbolic sequence obtained through successive observations over a system with an unbounded number of possible states. Specifically, we assume that the complexity of the description of the system provided by the sequence of observations is the one expected for a system evolving to a stable state between order and disorder. This result is obtained from a small set of mild, physically relevant assumptions. The general nature of our derivation and its model-free basis would explain the ubiquity of such a law in real systems.Comment: 11 Pages, 2 figure

Inertial measurent units and their application to mobile robot localization

Resumen del trabajo presentado al seminario celebrado en el Instituto de Robótica e Informática Industrial (IRII-CSIC-UPC) el 6 de marzo de 2014.Peer Reviewe

Detection of the elite structure in a virtual multiplex social system by means of a generalized KK-core

Elites are subgroups of individuals within a society that have the ability and means to influence, lead, govern, and shape societies. Members of elites are often well connected individuals, which enables them to impose their influence to many and to quickly gather, process, and spread information. Here we argue that elites are not only composed of highly connected individuals, but also of intermediaries connecting hubs to form a cohesive and structured elite-subgroup at the core of a social network. For this purpose we present a generalization of the $K$-core algorithm that allows to identify a social core that is composed of well-connected hubs together with their `connectors'. We show the validity of the idea in the framework of a virtual world defined by a massive multiplayer online game, on which we have complete information of various social networks. Exploiting this multiplex structure, we find that the hubs of the generalized $K$-core identify those individuals that are high social performers in terms of a series of indicators that are available in the game. In addition, using a combined strategy which involves the generalized $K$-core and the recently introduced $M$-core, the elites of the different 'nations' present in the game are perfectly identified as modules of the generalized $K$-core. Interesting sudden shifts in the composition of the elite cores are observed at deep levels. We show that elite detection with the traditional $K$-core is not possible in a reliable way. The proposed method might be useful in a series of more general applications, such as community detection.Comment: 13 figures, 3 tables, 19 pages. Accepted for publication in PLoS ON

Map-based localization for urban service mobile robotics

Mobile robotics research is currently interested on exporting autonomous navigation results achieved in indoor environments, to more challenging environments, such as, for instance, urban pedestrian areas. Developing mobile robots with autonomous navigation capabilities in such urban environments supposes a basic requirement for a upperlevel service set that could be provided to an users community. However, exporting indoor techniques to outdoor urban pedestrian scenarios is not evident due to the larger size of the environment, the dynamism of the scene due to pedestrians and other moving obstacles, the sunlight conditions, and the high presence of three dimensional elements such as ramps, steps, curbs or holes. Moreover, GPS-based mobile robot localization has demonstrated insufficient performance for robust long-term navigation in urban environments. One of the key modules within autonomous navigation is localization. If localization supposes an a priori map, even if it is not a complete model of the environment, localization is called map-based. This assumption is realistic since current trends of city councils are on building precise maps of their cities, specially of the most interesting places such as city downtowns. Having robots localized within a map allows for a high-level planning and monitoring, so that robots can achieve goal points expressed on the map, by following in a deliberative way a previously planned route. This thesis deals with the mobile robot map-based localization issue in urban pedestrian areas. The thesis approach uses the particle filter algorithm, a well-known and widely used probabilistic and recursive method for data fusion and state estimation. The main contributions of the thesis are divided on four aspects: (1) long-term experiments of mobile robot 2D and 3D position tracking in real urban pedestrian scenarios within a full autonomous navigation framework, (2) developing a fast and accurate technique to compute on-line range observation models in 3D environments, a basic step required by the real-time performance of the developed particle filter, (3) formulation of a particle filter that integrates asynchronous data streams and (4) a theoretical proposal to solve the global localization problem in an active and cooperative way, defining cooperation as either information sharing among the robots or planning joint actions to solve a common goal.Actualment, la recerca en robòtica mòbil té un interés creixent en exportar els resultats de navegació autònoma aconseguits en entorns interiors cap a d'altres tipus d'entorns més exigents, com, per exemple, les àrees urbanes peatonals. Desenvolupar capacitats de navegació autònoma en aquests entorns urbans és un requisit bàsic per poder proporcionar un conjunt de serveis de més alt nivell a una comunitat d'usuaris. Malgrat tot, exportar les tècniques d'interiors cap a entorns exteriors peatonals no és evident, a causa de la major dimensió de l'entorn, del dinamisme de l'escena provocada pels peatons i per altres obstacles en moviment, de la resposta de certs sensors a la il.luminació natural, i de la constant presència d'elements tridimensionals tals com rampes, escales, voreres o forats. D'altra banda, la localització de robots mòbils basada en GPS ha demostrat uns resultats insuficients de cara a una navegació robusta i de llarga durada en entorns urbans. Una de les peces clau en la navegació autònoma és la localització. En el cas que la localització consideri un mapa conegut a priori, encara que no sigui un model complet de l'entorn, parlem d'una localització basada en un mapa. Aquesta assumpció és realista ja que la tendència actual de les administracions locals és de construir mapes precisos de les ciutats, especialment dels llocs d'interés tals com les zones més cèntriques. El fet de tenir els robots localitzats en un mapa permet una planificació i una monitorització d'alt nivell, i així els robots poden arribar a destinacions indicades sobre el mapa, tot seguint de forma deliberativa una ruta prèviament planificada. Aquesta tesi tracta el tema de la localització de robots mòbils, basada en un mapa i per entorns urbans peatonals. La proposta de la tesi utilitza el filtre de partícules, un mètode probabilístic i recursiu, ben conegut i àmpliament utilitzat per la fusió de dades i l'estimació d'estats. Les principals contribucions de la tesi queden dividides en quatre aspectes: (1) experimentació de llarga durada del seguiment de la posició, tant en 2D com en 3D, d'un robot mòbil en entorns urbans reals, en el context de la navegació autònoma, (2) desenvolupament d'una tècnica ràpida i precisa per calcular en temps d'execució els models d'observació de distàncies en entorns 3D, un requisit bàsic pel rendiment del filtre de partícules a temps real, (3) formulació d'un filtre de partícules que integra conjunts de dades asíncrones i (4) proposta teòrica per solucionar la localització global d'una manera activa i cooperativa, entenent la cooperació com el fet de compartir informació, o bé com el de planificar accions conjuntes per solucionar un objectiu comú

Understanding scaling through history-dependent processes with collapsing sample space

History-dependent processes are ubiquitous in natural and social systems. Many such stochastic processes, especially those that are associated with complex systems, become more constrained as they unfold, meaning that their sample-space, or their set of possible outcomes, reduces as they age. We demonstrate that these sample-space reducing (SSR) processes necessarily lead to Zipf's law in the rank distributions of their outcomes. We show that by adding noise to SSR processes the corresponding rank distributions remain exact power-laws, $p(x)\sim x^{-\lambda}$, where the exponent directly corresponds to the mixing ratio of the SSR process and noise. This allows us to give a precise meaning to the scaling exponent in terms of the degree to how much a given process reduces its sample-space as it unfolds. Noisy SSR processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from $\alpha = 2$ to $\infty$. We discuss several applications showing how SSR processes can be used to understand Zipf's law in word frequencies, and how they are related to diffusion processes in directed networks, or ageing processes such as in fragmentation processes. SSR processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative, preferential, or self-organised critical processes.Comment: 7 pages, 5 figures in Proceedings of the National Academy of Sciences USA (published ahead of print April 13, 2015

Extreme robustness of scaling in sample space reducing processes explains Zipf's law in diffusion on directed networks

It has been shown recently that a specific class of path-dependent stochastic processes, which reduce their sample space as they unfold, lead to exact scaling laws in frequency and rank distributions. Such Sample Space Reducing processes (SSRP) offer an alternative new mechanism to understand the emergence of scaling in countless processes. The corresponding power law exponents were shown to be related to noise levels in the process. Here we show that the emergence of scaling is not limited to the simplest SSRPs, but holds for a huge domain of stochastic processes that are characterized by non-uniform prior distributions. We demonstrate mathematically that in the absence of noise the scaling exponents converge to $-1$ (Zipf's law) for almost all prior distributions. As a consequence it becomes possible to fully understand targeted diffusion on weighted directed networks and its associated scaling laws law in node visit distributions. The presence of cycles can be properly interpreted as playing the same role as noise in SSRPs and, accordingly, determine the scaling exponents. The result that Zipf's law emerges as a generic feature of diffusion on networks, regardless of its details, and that the exponent of visiting times is related to the amount of cycles in a network could be relevant for a series of applications in traffic-, transport- and supply chain management.Comment: 11 pages, 5 figure

Sample space reducing cascading processes produce the full spectrum of scaling exponents

Sample Space Reducing (SSR) processes are simple stochastic processes that offer a new route to understand scaling in path-dependent processes. Here we define a cascading process that generalises the recently defined SSR processes and is able to produce power laws with arbitrary exponents. We demonstrate analytically that the frequency distributions of states are power laws with exponents that coincide with the multiplication parameter of the cascading process. In addition, we show that imposing energy conservation in SSR cascades allows us to recover Fermi's classic result on the energy spectrum of cosmic rays, with the universal exponent -2, which is independent of the multiplication parameter of the cascade. Applications of the proposed process include fragmentation processes or directed cascading diffusion on networks, such as rumour or epidemic spreading.Comment: 10 pages, 6 figure