6,399 research outputs found
Degree Distribution of Competition-Induced Preferential Attachment Graphs
We introduce a family of one-dimensional geometric growth models, constructed
iteratively by locally optimizing the tradeoffs between two competing metrics,
and show that this family is equivalent to a family of preferential attachment
random graph models with upper cutoffs. This is the first explanation of how
preferential attachment can arise from a more basic underlying mechanism of
local competition. We rigorously determine the degree distribution for the
family of random graph models, showing that it obeys a power law up to a finite
threshold and decays exponentially above this threshold.
We also rigorously analyze a generalized version of our graph process, with
two natural parameters, one corresponding to the cutoff and the other a
``fertility'' parameter. We prove that the general model has a power-law degree
distribution up to a cutoff, and establish monotonicity of the power as a
function of the two parameters. Limiting cases of the general model include the
standard preferential attachment model without cutoff and the uniform
attachment model.Comment: 24 pages, one figure. To appear in the journal: Combinatorics,
Probability and Computing. Note, this is a long version, with complete
proofs, of the paper "Competition-Induced Preferential Attachment"
(cond-mat/0402268
A Network Model characterized by a Latent Attribute Structure with Competition
The quest for a model that is able to explain, describe, analyze and simulate
real-world complex networks is of uttermost practical as well as theoretical
interest. In this paper we introduce and study a network model that is based on
a latent attribute structure: each node is characterized by a number of
features and the probability of the existence of an edge between two nodes
depends on the features they share. Features are chosen according to a process
of Indian-Buffet type but with an additional random "fitness" parameter
attached to each node, that determines its ability to transmit its own features
to other nodes. As a consequence, a node's connectivity does not depend on its
age alone, so also "young" nodes are able to compete and succeed in acquiring
links. One of the advantages of our model for the latent bipartite
"node-attribute" network is that it depends on few parameters with a
straightforward interpretation. We provide some theoretical, as well
experimental, results regarding the power-law behaviour of the model and the
estimation of the parameters. By experimental data, we also show how the
proposed model for the attribute structure naturally captures most local and
global properties (e.g., degree distributions, connectivity and distance
distributions) real networks exhibit. keyword: Complex network, social network,
attribute matrix, Indian Buffet processComment: 34 pages, second version (date of the first version: July, 2014).
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Scale-free behavior of networks with the copresence of preferential and uniform attachment rules
Complex networks in different areas exhibit degree distributions with heavy
upper tail. A preferential attachment mechanism in a growth process produces a
graph with this feature. We herein investigate a variant of the simple
preferential attachment model, whose modifications are interesting for two main
reasons: to analyze more realistic models and to study the robustness of the
scale free behavior of the degree distribution. We introduce and study a model
which takes into account two different attachment rules: a preferential
attachment mechanism (with probability 1-p) that stresses the rich get richer
system, and a uniform choice (with probability p) for the most recent nodes.
The latter highlights a trend to select one of the last added nodes when no
information is available. The recent nodes can be either a given fixed number
or a proportion (\alpha n) of the total number of existing nodes. In the first
case, we prove that this model exhibits an asymptotically power-law degree
distribution. The same result is then illustrated through simulations in the
second case. When the window of recent nodes has constant size, we herein prove
that the presence of the uniform rule delays the starting time from which the
asymptotic regime starts to hold. The mean number of nodes of degree k and the
asymptotic degree distribution are also determined analytically. Finally, a
sensitivity analysis on the parameters of the model is performed
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