Author
Leonard M. Sander
Bio: Leonard M. Sander is an academic researcher from University of Michigan. The author has contributed to research in topics: Diffusion-limited aggregation & Scaling. The author has an hindex of 11, co-authored 18 publications receiving 4969 citations.
Papers
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Abstract: A model for random aggregates is studied by computer simulation The model is applicable to a metal-particle aggregation process whose correlations have been measured previously Density correlations within the model aggregates fall off with distance with a fractional power law, like those of the metal aggregates The radius of gyration of the model aggregates has power-law behavior The model is a limit of a model of dendritic growth
4,248 citations
TL;DR: The current state of knowledge about the DLA model, its applications and theoretical analysis of the results are reviewed.
Abstract: Diffusion-limited aggregation (DLA) is a model which represents noisy growth limited by diffusion. This process is quite common in nature and the simple algorithm gives a good representation of the large-scale structure of many natural objects. The clusters grown in the computer and the real objects in question are tenuous and approximately self-similar. A good deal is known about the algorithm, but a complete theory is not yet available. I review the current state of knowledge about the model, its applications and theoretical analysis of the results.
487 citations
Abstract: We offer an example of a network model with a power-law degree distribution, P(k) approximately k(-alpha), for nodes, but which nevertheless has a well-defined geography and a nonzero threshold percolation probability for alpha>2, the range of real-world contact networks. This is different from p(c)=0 for alpha<3 results for the original well-mixed scale-free networks. In our lattice-based scale-free network, individuals link to nearby neighbors on a lattice. Even considerable additional small-world links do not change our conclusion of nonzero thresholds. When applied to disease propagation, these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering.
122 citations
TL;DR: It is pointed out that with disorder in the strength of contacts between individuals patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the disorder because the critical region of the corresponding percolation model is broadened.
Abstract: Spatial models for spread of an epidemic may be mapped onto bond percolation. We point out that with disorder in the strength of contacts between individuals patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the disorder because the critical region of the corresponding percolation model is broadened. In some networks the percolation threshold is zero if another kind of disorder is present, namely divergent fluctuations in the number of contacts. We give an example, a network with a well-defined geography, where this is not necessarily so, and discuss whether real infection networks are likely to have this property.
70 citations
TL;DR: In this paper, the scaling properties of fractal clusters by diffusion limited aggregation (DLA) are studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov.
Abstract: The creation of fractal clusters by diffusion limited aggregation (DLA) is studied by using iterated stochastic conformal maps following the method proposed recently by Hastings and Levitov. The object of interest is the function {Phi}{sup (n)} which conformally maps the exterior of the unit circle to the exterior of an {ital n}-particle DLA. The map {Phi}{sup (n)} is obtained from {ital n} stochastic iterations of a function {phi} that maps the unit circle to the unit circle with a bump. The scaling properties usually studied in the literature on DLA appear in a new light using this language. The dimension of the cluster is determined by the linear coefficient in the Laurent expansion of {Phi}{sup (n)}, which asymptotically becomes a deterministic function of {ital n}. We find new relationships between the generalized dimensions of the harmonic measure and the scaling behavior of the Laurent coefficients. {copyright} {ital 1999} {ital The American Physical Society}
48 citations
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TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
17,647 citations
TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
9,441 citations
TL;DR: In this paper, the correlation exponent v is introduced as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise, and algorithms for extracting v from the time series of a single variable are proposed.
5,239 citations
TL;DR: A coherent and comprehensive review of the vast research activity concerning epidemic processes is presented, detailing the successful theoretical approaches as well as making their limits and assumptions clear.
Abstract: Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.
3,173 citations
TL;DR: Analysis is given of ''elementary'' cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to p definite rules involving the values of its nearest neighbors.
Abstract: Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the values of its nearest neighbors. With simple initial configurations, the cellular automata either tend to homogeneous states, or generate self-similar patterns with fractal dimensions \ensuremath{\simeq} 1.59 or \ensuremath{\simeq} 1.69. With "random" initial configurations, the irreversible character of the cellular automaton evolution leads to several self-organization phenomena. Statistical properties of the structures generated are found to lie in two universality classes, independent of the details of the initial state or the cellular automaton rules. More complicated cellular automata are briefly considered, and connections with dynamical systems theory and the formal theory of computation are discussed.
2,860 citations