739 research outputs found
Discrete scale invariance and complex dimensions
We discuss the concept of discrete scale invariance and how it leads to
complex critical exponents (or dimensions), i.e. to the log-periodic
corrections to scaling. After their initial suggestion as formal solutions of
renormalization group equations in the seventies, complex exponents have been
studied in the eighties in relation to various problems of physics embedded in
hierarchical systems. Only recently has it been realized that discrete scale
invariance and its associated complex exponents may appear ``spontaneously'' in
euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples
are diffusion-limited-aggregation clusters, rupture in heterogeneous systems,
earthquakes, animals (a generalization of percolation) among many other
systems. We review the known mechanisms for the spontaneous generation of
discrete scale invariance and provide an extensive list of situations where
complex exponents have been found. This is done in order to provide a basis for
a better fundamental understanding of discrete scale invariance. The main
motivation to study discrete scale invariance and its signatures is that it
provides new insights in the underlying mechanisms of scale invariance. It may
also be very interesting for prediction purposes.Comment: significantly extended version (Oct. 27, 1998) with new examples in
several domains of the review paper with the same title published in Physics
Reports 297, 239-270 (1998
Nurturing Breakthroughs: Lessons from Complexity Theory
A general theory of innovation and progress in human society is outlined,
based on the combat between two opposite forces (conservatism/inertia and
speculative herding "bubble" behavior). We contend that human affairs are
characterized by ubiquitous ``bubbles'', which involve huge risks which would
not otherwise be taken using standard cost/benefit analysis. Bubbles result
from self-reinforcing positive feedbacks. This leads to explore uncharted
territories and niches whose rare successes lead to extraordinary discoveries
and provide the base for the observed accelerating development of technology
and of the economy. But the returns are very heterogeneous, very risky and may
not occur. In other words, bubbles, which are characteristic definitions of
human activity, allow huge risks to get huge returns over large scales. We
outline some underlying mathematical structure and a few results involving
positive feedbacks, emergence, heavy-tailed power laws, outliers/kings/black
swans, the problem of predictability and the illusion of control, as well as
some policy implications.Comment: 14 pages, Invited talk at the workshop Trans-disciplinary Research
Agenda for Societal Dynamics (http://www.uni-lj.si/trasd in Ljubljana),
organized by J. Rogers Hollingsworth, Karl H. Mueller, Ivan Svetlik, 24 - 25
May 2007, Ljubljana, Sloveni
A generic model of dyadic social relationships
We introduce a model of dyadic social interactions and establish its
correspondence with relational models theory (RMT), a theory of human social
relationships. RMT posits four elementary models of relationships governing
human interactions, singly or in combination: Communal Sharing, Authority
Ranking, Equality Matching, and Market Pricing. To these are added the limiting
cases of asocial and null interactions, whereby people do not coordinate with
reference to any shared principle. Our model is rooted in the observation that
each individual in a dyadic interaction can do either the same thing as the
other individual, a different thing or nothing at all. To represent these three
possibilities, we consider two individuals that can each act in one out of
three ways toward the other: perform a social action X or Y, or alternatively
do nothing. We demonstrate that the relationships generated by this model
aggregate into six exhaustive and disjoint categories. We propose that four of
these categories match the four relational models, while the remaining two
correspond to the asocial and null interactions defined in RMT. We generalize
our results to the presence of N social actions. We infer that the four
relational models form an exhaustive set of all possible dyadic relationships
based on social coordination. Hence, we contribute to RMT by offering an answer
to the question of why there could exist just four relational models. In
addition, we discuss how to use our representation to analyze data sets of
dyadic social interactions, and how social actions may be valued and matched by
the agents
Faults Self-Organized by Repeated Earthquakes in a Quasi-Static Antiplane Crack Model
We study a 2D quasi-static discrete {\it crack} anti-plane model of a
tectonic plate with long range elastic forces and quenched disorder. The plate
is driven at its border and the load is transfered to all elements through
elastic forces. This model can be considered as belonging to the class of
self-organized models which may exhibit spontaneous criticality, with four
additional ingredients compared to sandpile models, namely quenched disorder,
boundary driving, long range forces and fast time crack rules. In this
''crack'' model, as in the ''dislocation'' version previously studied, we find
that the occurrence of repeated earthquakes organizes the activity on
well-defined fault-like structures. In contrast with the ''dislocation'' model,
after a transient, the time evolution becomes periodic with run-aways ending
each cycle. This stems from the ''crack'' stress transfer rule preventing
criticality to organize in favor of cyclic behavior. For sufficiently large
disorder and weak stress drop, these large events are preceded by a complex
space-time history of foreshock activity, characterized by a Gutenberg-Richter
power law distribution with universal exponent . This is similar
to a power law distribution of small nucleating droplets before the nucleation
of the macroscopic phase in a first-order phase transition. For large disorder
and large stress drop, and for certain specific initial disorder
configurations, the stress field becomes frustrated in fast time : out-of-plane
deformations (thrust and normal faulting) and/or a genuine dynamics must be
introduced to resolve this frustration
Modeling the Stock Market prior to large crashes
We propose that the minimal requirements for a model of stock market price
fluctuations should comprise time asymmetry, robustness with respect to
connectivity between agents, ``bounded rationality'' and a probabilistic
description. We also compare extensively two previously proposed models of
log-periodic behavior of the stock market index prior to a large crash. We find
that the model which follows the above requirements outperforms the other with
a high statistical significance.Comment: 18 pages with 4 figures. Submitted to Eur.Phys.
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