66 research outputs found
Understanding frequency distributions of path-dependent processes with non-multinomial maximum entropy approaches
Path-dependent stochastic processes are often non-ergodic and observables can no longer be computed within the ensemble picture. The resulting mathematical difficulties pose severe limits to the analytical understanding of path-dependent processes. Their statistics is typically non-multinomial in the sense that the multiplicities of the occurrence of states is not a multinomial factor. The maximum entropy principle is tightly related to multinomial processes, non-interacting systems, and to the ensemble picture; it loses its meaning for path-dependent processes. Here we show that an equivalent to the ensemble picture exists for path-dependent processes, such that the non-multinomial statistics of the underlying dynamical process, by construction, is captured correctly in a functional that plays the role of a relative entropy. We demonstrate this for self-reinforcing Pólya urn processes, which explicitly generalize multinomial statistics. We demonstrate the adequacy of this constructive approach towards non-multinomial entropies by computing frequency and rank distributions of Pólya urn processes. We show how microscopic update rules of a path-dependent process allow us to explicitly construct a non-multinomial entropy functional, that, when maximized, predicts the time-dependent distribution function
Universality in percolation of arbitrary Uncorrelated Nested Subgraphs
The study of percolation in so-called {\em nested subgraphs} implies a
generalization of the concept of percolation since the results are not linked
to specific graph process. Here the behavior of such graphs at criticallity is
studied for the case where the nesting operation is performed in an
uncorrelated way. Specifically, I provide an analyitic derivation for the
percolation inequality showing that the cluster size distribution under a
generalized process of uncorrelated nesting at criticality follows a power law
with universal exponent . The relevance of the result comes from
the wide variety of processes responsible for the emergence of the giant
component that fall within the category of nesting operations, whose outcome is
a family of nested subgraphs.Comment: 5 pages, no figures. Mistakes found in early manuscript have been
remove
How driving rates determine the statistics of driven non-equilibrium systems with stationary distributions
Sample space reducing (SSR) processes offer a simple analytical way to understand the origin and ubiquity of power-laws in many path-dependent complex systems. SRR processes show a wide range of applications that range from fragmentation processes, language formation to search and cascading processes. Here we argue that they also offer a natural framework to understand stationary distributions of generic driven non-equilibrium systems that are composed of a driving- and a relaxing process. We show that the statistics of driven non-equilibrium systems can be derived from the understanding of the nature of the underlying driving process. For constant driving rates exact power-laws emerge with exponents that are related to the driving rate. If driving rates become state-dependent, or if they vary across the life-span of the process, the functional form of the state-dependence determines the statistics. Constant driving rates lead to exact power-laws, a linear state-dependence function yields exponential or Gamma distributions, a quadratic function produces the normal distribution. Logarithmic and power-law state dependence leads to log-normal and stretched exponential distribution functions, respectively. Also Weibull, Gompertz and Tsallis-Pareto distributions arise naturally from simple state-dependent driving rates. We discuss a simple physical example of consecutive elastic collisions that exactly represents a SSR process
On the origin of ambiguity in efficient communication
This article studies the emergence of ambiguity in communication through the
concept of logical irreversibility and within the framework of Shannon's
information theory. This leads us to a precise and general expression of the
intuition behind Zipf's vocabulary balance in terms of a symmetry equation
between the complexities of the coding and the decoding processes that imposes
an unavoidable amount of logical uncertainty in natural communication.
Accordingly, the emergence of irreversible computations is required if the
complexities of the coding and the decoding processes are balanced in a
symmetric scenario, which means that the emergence of ambiguous codes is a
necessary condition for natural communication to succeed.Comment: 28 pages, 2 figure
Rigidity percolation uncovers a structural basis for embryonic tissue phase transitions
Embryo morphogenesis is impacted by dynamic changes in tissue material properties, which have been proposed to occur via processes akin to phase transitions (PTs). Here, we show that rigidity percolation provides a simple and robust theoretical framework to predict material/structural PTs of embryonic tissues from local cell connectivity. By using percolation theory, combined with directly monitoring dynamic changes in tissue rheology and cell contact mechanics, we demonstrate that the zebrafish blastoderm undergoes a genuine rigidity PT, brought about by a small reduction in adhesion-dependent cell connectivity below a critical value. We quantitatively predict and experimentally verify hallmarks of PTs, including power-law exponents and associated discontinuities of macroscopic observables. Finally, we show that this uniform PT depends on blastoderm cells undergoing meta-synchronous divisions causing random and, consequently, uniform changes in cell connectivity. Collectively, our theoretical and experimental findings reveal the structural basis of material PTs in an organismal context
Robustness of the European power grids under intentional attack
The power grid defines one of the most important technological networks of
our times and sustains our complex society. It has evolved for more than a
century into an extremely huge and seemingly robust and well understood system.
But it becomes extremely fragile as well, when unexpected, usually minimal,
failures turn into unknown dynamical behaviours leading, for example, to sudden
and massive blackouts. Here we explore the fragility of the European power grid
under the effect of selective node removal. A mean field analysis of fragility
against attacks is presented together with the observed patterns. Deviations
from the theoretical conditions for network percolation (and fragmentation)
under attacks are analysed and correlated with non topological reliability
measures.Comment: 7 pages, 4 figure
Multiplex core-periphery organization of the human connectome
The behavior of many complex systems is determined by a core of densely
interconnected units. While many methods are available to identify the core of
a network when connections between nodes are all of the same type, a principled
approach to define the core when multiple types of connectivity are allowed is
still lacking. Here we introduce a general framework to define and extract the
core-periphery structure of multi-layer networks by explicitly taking into
account the connectivity of the nodes at each layer. We show how our method
works on synthetic networks with different size, density, and overlap between
the cores at the different layers. We then apply the method to multiplex brain
networks whose layers encode information both on the anatomical and the
functional connectivity among regions of the human cortex. Results confirm the
presence of the main known hubs, but also suggest the existence of novel brain
core regions that have been discarded by previous analysis which focused
exclusively on the structural layer. Our work is a step forward in the
identification of the core of the human connectome, and contributes to shed
light to a fundamental question in modern neuroscience.Comment: Main text (12 pages, 5 figures) + Supplementary material (6 pages, 5
figures, 1 table
Comparing the hierarchy of keywords in on-line news portals
The tagging of on-line content with informative keywords is a widespread
phenomenon from scientific article repositories through blogs to on-line news
portals. In most of the cases, the tags on a given item are free words chosen
by the authors independently. Therefore, relations among keywords in a
collection of news items is unknown. However, in most cases the topics and
concepts described by these keywords are forming a latent hierarchy, with the
more general topics and categories at the top, and more specialised ones at the
bottom. Here we apply a recent, cooccurrence-based tag hierarchy extraction
method to sets of keywords obtained from four different on-line news portals.
The resulting hierarchies show substantial differences not just in the topics
rendered as important (being at the top of the hierarchy) or of less interest
(categorised low in the hierarchy), but also in the underlying network
structure. This reveals discrepancies between the plausible keyword association
frameworks in the studied news portals
Topological reversibility and causality in feed-forward networks
Systems whose organization displays causal asymmetry constraints, from
evolutionary trees to river basins or transport networks, can be often
described in terms of directed paths (causal flows) on a discrete state space.
Such a set of paths defines a feed-forward, acyclic network. A key problem
associated with these systems involves characterizing their intrinsic degree of
path reversibility: given an end node in the graph, what is the uncertainty of
recovering the process backwards until the origin? Here we propose a novel
concept, \textit{topological reversibility}, which rigorously weigths such
uncertainty in path dependency quantified as the minimum amount of information
required to successfully revert a causal path. Within the proposed framework we
also analytically characterize limit cases for both topologically reversible
and maximally entropic structures. The relevance of these measures within the
context of evolutionary dynamics is highlighted.Comment: 9 pages, 3 figure
Dynamics on expanding spaces: modeling the emergence of novelties
Novelties are part of our daily lives. We constantly adopt new technologies,
conceive new ideas, meet new people, experiment with new situations.
Occasionally, we as individuals, in a complicated cognitive and sometimes
fortuitous process, come up with something that is not only new to us, but to
our entire society so that what is a personal novelty can turn into an
innovation at a global level. Innovations occur throughout social, biological
and technological systems and, though we perceive them as a very natural
ingredient of our human experience, little is known about the processes
determining their emergence. Still the statistical occurrence of innovations
shows striking regularities that represent a starting point to get a deeper
insight in the whole phenomenology. This paper represents a small step in that
direction, focusing on reviewing the scientific attempts to effectively model
the emergence of the new and its regularities, with an emphasis on more recent
contributions: from the plain Simon's model tracing back to the 1950s, to the
newest model of Polya's urn with triggering of one novelty by another. What
seems to be key in the successful modelling schemes proposed so far is the idea
of looking at evolution as a path in a complex space, physical, conceptual,
biological, technological, whose structure and topology get continuously
reshaped and expanded by the occurrence of the new. Mathematically it is very
interesting to look at the consequences of the interplay between the "actual"
and the "possible" and this is the aim of this short review.Comment: 25 pages, 10 figure
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