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 $\lambda$, which assumes the value $\lambda=0$ (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 $\lambda \ne 0$, the patterns are fractal on the small length scales, but homogeneous on the large ones. We evaluated the mean density of particles $\bar{\rho}$ in the region defined by a circle of radius $r$ centered at the initial seed. As a function of $r$, $\bar{\rho}$ reaches the asymptotic value $\rho_0(\lambda)$ following a power law $\bar{\rho}=\rho_0+Ar^{-\gamma}$ with a universal exponent $\gamma=0.46(2)$, independent of $\lambda$. The asymptotic value has the behavior $\rho_0\sim|1-\lambda|^\beta$, where $\beta= 0.26(1)$. The characteristic crossover length that determines the transition from DLA- to BA-like scaling regimes is given by $\xi\sim|1-\lambda|^{-\nu}$, where $\nu=0.61(1)$, while the cluster mass at the crossover follows a power law $M_\xi\sim|1 -\lambda|^{-\alpha}$, where $\alpha=0.97(2)$. We deduce the scaling relations \beta=\n u\gamma and $\beta=2\nu-\alpha$ 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 $x$, grows by an amount $\alpha x$ 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 $c(x,t)\sim t^{-\beta}\phi(x/t^z)$ accompanied by the emergence of fractal of dimension $d_f={{1}\over{1+2\alpha}}$. One of the remarkable feature of this model is that it is governed by a non-trivial conservation law, namely, the $d_f^{th}$ moment of $c(x,t)$ 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 $\beta$ and $z$ are locked with the fractal dimension $d_f$ via a generalized scaling relation $\beta=(1+d_f)z$.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 ($\eta$) added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, $| {\bf v}_a | =0$) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous since $| {\bf v}_a |$ is found to scale as $(\eta_c-\eta)^\beta$ with $\beta\simeq 0.45$