587 research outputs found
Multifractal Network Generator
We introduce a new approach to constructing networks with realistic features.
Our method, in spite of its conceptual simplicity (it has only two parameters)
is capable of generating a wide variety of network types with prescribed
statistical properties, e.g., with degree- or clustering coefficient
distributions of various, very different forms. In turn, these graphs can be
used to test hypotheses, or, as models of actual data. The method is based on a
mapping between suitably chosen singular measures defined on the unit square
and sparse infinite networks. Such a mapping has the great potential of
allowing for graph theoretical results for a variety of network topologies. The
main idea of our approach is to go to the infinite limit of the singular
measure and the size of the corresponding graph simultaneously. A very unique
feature of this construction is that the complexity of the generated network is
increasing with the size. We present analytic expressions derived from the
parameters of the -- to be iterated-- initial generating measure for such major
characteristics of graphs as their degree, clustering coefficient and
assortativity coefficient distributions. The optimal parameters of the
generating measure are determined from a simple simulated annealing process.
Thus, the present work provides a tool for researchers from a variety of fields
(such as biology, computer science, biology, or complex systems) enabling them
to create a versatile model of their network data.Comment: Preprint. Final version appeared in PNAS
Microscopic model of diffusion limited aggregation and electrodeposition in the presence of levelling molecules
A microscopic model of the effect of unbinding in diffusion limited
aggregation based on a cellular automata approach is presented. The geometry
resembles electrochemical deposition - ``ions'' diffuse at random from the top
of a container until encountering a cluster in contact with the bottom, to
which they stick. The model exhibits dendritic (fractal) growth in the
diffusion limited case. The addition of a field eliminates the fractal nature
but the density remains low. The addition of molecules which unbind atoms from
the aggregate transforms the deposit to a 100% dense one (in 3D). The molecules
are remarkably adept at avoiding being trapped. This mimics the effect of
so-called ``leveller'' molecules which are used in electrochemical deposition
Stability of glassy hierarchical networks
The structure of interactions in most animal and human societies can be best represented by complex hierarchical networks. In order to maintain close-to-optimal function both stability and adaptability are necessary. Here we investigate the stability of hierarchical networks that emerge from the simulations of an organization type with an efficiency function reminiscent of the Hamiltonian of spin glasses. Using this quantitative approach we find a number of expected (from everyday observations) and highly non-trivial results for the obtained locally optimal networks, including, for example: (i) stability increases with growing efficiency and level of hierarchy; (ii) the same perturbation results in a larger change for more efficient states; (iii) networks with a lower level of hierarchy become more efficient after perturbation; (iv) due to the huge number of possible optimal states only a small fraction of them exhibit resilience and, finally, (v) 'attacks' targeting the nodes selectively (regarding their position in the hierarchy) can result in paradoxical outcomes
A cluster mode-coupling approach to weak gelation in attractive colloids
Mode-coupling theory (MCT) predicts arrest of colloids in terms of their
volume fraction, and the range and depth of the interparticle attraction. We
discuss how effective values of these parameters evolve under cluster
aggregation. We argue that weak gelation in colloids can be idealized as a
two-stage ergodicity breaking: first at short scales (approximated by the bare
MCT) and then at larger scales (governed by MCT applied to clusters). The
competition between arrest and phase separation is considered in relation to
recent experiments. We predict a long-lived `semi-ergodic' phase of mobile
clusters, showing logarithmic relaxation close to the gel line.Comment: 4 pages, 3 figure
Correlation length by measuring empty space in simulated aggregates
We examine the geometry of the spaces between particles in diffusion-limited
cluster aggregation, a numerical model of aggregating suspensions. Computing
the distribution of distances from each point to the nearest particle, we show
that it has a scaled form independent of the concentration phi, for both two-
(2D) and three-dimensional (3D) model gels at low phi. The mean remoteness is
proportional to the density-density correlation length of the gel, xi, allowing
a more precise measurement of xi than by other methods. A simple analytical
form for the scaled remoteness distribution is developed, highlighting the
geometrical information content of the data. We show that the second moment of
the distribution gives a useful estimate of the permeability of porous media.Comment: 4 page
Morphological transition between diffusion-limited and ballistic aggregation growth patterns
In this work, the transition between diffusion-limited and ballistic
aggregation models was revisited using a model in which biased random walks
simulate the particle trajectories. The bias is controlled by a parameter
, which assumes the value (1) for ballistic
(diffusion-limited) aggregation model. Patterns growing from a single seed were
considered. In order to simulate large clusters, a new efficient algorithm was
developed. For , the patterns are fractal on the small length
scales, but homogeneous on the large ones. We evaluated the mean density of
particles in the region defined by a circle of radius centered
at the initial seed. As a function of , reaches the asymptotic
value following a power law
with a universal exponent , independent of . The
asymptotic value has the behavior , where . The characteristic crossover length that determines the transition
from DLA- to BA-like scaling regimes is given by ,
where , while the cluster mass at the crossover follows a power
law , where . We deduce the
scaling relations \beta=\n u\gamma and between these
exponents.Comment: 7 pages, 8 figure
Percolation, Morphogenesis, and Burgers Dynamics in Blood Vessels Formation
Experiments of in vitro formation of blood vessels show that cells randomly
spread on a gel matrix autonomously organize to form a connected vascular
network. We propose a simple model which reproduces many features of the
biological system. We show that both the model and the real system exhibit a
fractal behavior at small scales, due to the process of migration and dynamical
aggregation, followed at large scale by a random percolation behavior due to
the coalescence of aggregates. The results are in good agreement with the
analysis performed on the experimental data.Comment: 4 pages, 11 eps figure
Emergence of fractal behavior in condensation-driven aggregation
We investigate a model in which an ensemble of chemically identical Brownian
particles are continuously growing by condensation and at the same time undergo
irreversible aggregation whenever two particles come into contact upon
collision. We solved the model exactly by using scaling theory for the case
whereby a particle, say of size , grows by an amount over the
time it takes to collide with another particle of any size. It is shown that
the particle size spectra of such system exhibit transition to dynamic scaling
accompanied by the emergence of fractal of
dimension . One of the remarkable feature of this
model is that it is governed by a non-trivial conservation law, namely, the
moment of is time invariant regardless of the choice of the
initial conditions. The reason why it remains conserved is explained by using a
simple dimensional analysis. We show that the scaling exponents and
are locked with the fractal dimension via a generalized scaling relation
.Comment: 8 pages, 6 figures, to appear in Phys. Rev.
Novel type of phase transition in a system of self-driven particles
A simple model with a novel type of dynamics is introduced in order to
investigate the emergence of self-ordered motion in systems of particles with
biologically motivated interaction. In our model particles are driven with a
constant absolute velocity and at each time step assume the average direction
of motion of the particles in their neighborhood with some random perturbation
() added. We present numerical evidence that this model results in a
kinetic phase transition from no transport (zero average velocity, ) to finite net transport through spontaneous symmetry breaking of the
rotational symmetry. The transition is continuous since is
found to scale as with
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