383 research outputs found
Degrees and distances in random and evolving Apollonian networks
This paper studies Random and Evolving Apollonian networks (RANs and EANs),
in d dimension for any d>=2, i.e. dynamically evolving random d dimensional
simplices looked as graphs inside an initial d-dimensional simplex. We
determine the limiting degree distribution in RANs and show that it follows a
power law tail with exponent tau=(2d-1)/(d-1). We further show that the degree
distribution in EANs converges to the same degree distribution if the
simplex-occupation parameter in the n-th step of the dynamics is q_n->0 and
sum_{n=0}^infty q_n =infty. This result gives a rigorous proof for the
conjecture of Zhang et al. that EANs tend to show similar behavior as RANs once
the occupation parameter q->0. We also determine the asymptotic behavior of
shortest paths in RANs and EANs for arbitrary d dimensions. For RANs we show
that the shortest path between two uniformly chosen vertices (typical
distance), the flooding time of a uniformly picked vertex and the diameter of
the graph after n steps all scale as constant times log n. We determine the
constants for all three cases and prove a central limit theorem for the typical
distances. We prove a similar CLT for typical distances in EANs
Self-similar disk packings as model spatial scale-free networks
The network of contacts in space-filling disk packings, such as the
Apollonian packing, are examined. These networks provide an interesting example
of spatial scale-free networks, where the topology reflects the broad
distribution of disk areas. A wide variety of topological and spatial
properties of these systems are characterized. Their potential as models for
networks of connected minima on energy landscapes is discussed.Comment: 13 pages, 12 figures; some bugs fixed and further discussion of
higher-dimensional packing
Characterizing the network topology of the energy landscapes of atomic clusters
By dividing potential energy landscapes into basins of attractions
surrounding minima and linking those basins that are connected by transition
state valleys, a network description of energy landscapes naturally arises.
These networks are characterized in detail for a series of small Lennard-Jones
clusters and show behaviour characteristic of small-world and scale-free
networks. However, unlike many such networks, this topology cannot reflect the
rules governing the dynamics of network growth, because they are static spatial
networks. Instead, the heterogeneity in the networks stems from differences in
the potential energy of the minima, and hence the hyperareas of their
associated basins of attraction. The low-energy minima with large basins of
attraction act as hubs in the network.Comparisons to randomized networks with
the same degree distribution reveals structuring in the networks that reflects
their spatial embedding.Comment: 14 pages, 11 figure
Exploring the origins of the power-law properties of energy landscapes: An egg-box model
Multidimensional potential energy landscapes (PELs) have a Gaussian
distribution for the energies of the minima, but at the same time the
distribution of the hyperareas for the basins of attraction surrounding the
minima follows a power-law. To explore how both these features can
simultaneously be true, we introduce an ``egg-box'' model. In these model
landscapes, the Gaussian energy distribution is used as a starting point and we
examine whether a power-law basin area distribution can arise as a natural
consequence through the swallowing up of higher-energy minima by larger
low-energy basins when the variance of this Gaussian is increased sufficiently.
Although the basin area distribution is substantially broadened by this
process,it is insufficient to generate power-laws, highlighting the role played
by the inhomogeneous distribution of basins in configuration space for actual
PELs.Comment: 7 pages, 8 figure
Locating the Source of Diffusion in Large-Scale Networks
How can we localize the source of diffusion in a complex network? Due to the
tremendous size of many real networks--such as the Internet or the human social
graph--it is usually infeasible to observe the state of all nodes in a network.
We show that it is fundamentally possible to estimate the location of the
source from measurements collected by sparsely-placed observers. We present a
strategy that is optimal for arbitrary trees, achieving maximum probability of
correct localization. We describe efficient implementations with complexity
O(N^{\alpha}), where \alpha=1 for arbitrary trees, and \alpha=3 for arbitrary
graphs. In the context of several case studies, we determine how localization
accuracy is affected by various system parameters, including the structure of
the network, the density of observers, and the number of observed cascades.Comment: To appear in Physical Review Letters. Includes pre-print of main
paper, and supplementary materia
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