386,999 research outputs found
Condensation of degrees emerging through a first-order phase transition in classical random graphs
Due to their conceptual and mathematical simplicity, Erd\"os-R\'enyi or
classical random graphs remain as a fundamental paradigm to model complex
interacting systems in several areas. Although condensation phenomena have been
widely considered in complex network theory, the condensation of degrees has
hitherto eluded a careful study. Here we show that the degree statistics of the
classical random graph model undergoes a first-order phase transition between a
Poisson-like distribution and a condensed phase, the latter characterized by a
large fraction of nodes having degrees in a limited sector of their
configuration space. The mechanism underlying the first-order transition is
discussed in light of standard concepts in statistical physics. We uncover the
phase diagram characterizing the ensemble space of the model and we evaluate
the rate function governing the probability to observe a condensed state, which
shows that condensation of degrees is a rare statistical event akin to similar
condensation phenomena recently observed in several other systems. Monte Carlo
simulations confirm the exactness of our theoretical results.Comment: 8 pages, 6 figure
Condensation of degrees emerging through a first-order phase transition in classical random graphs
Due to their conceptual and mathematical simplicity, Erdös-Rényi or classical random graphs remain as a fundamental paradigm to model complex interacting systems in several areas. Although condensation phenomena have been widely considered in complex network theory, the condensation of degrees has hitherto eluded a careful study. Here we show that the degree statistics of the classical random graph model undergoes a first-order phase transition between a Poisson-like distribution and a condensed phase, the latter characterized by a large fraction of nodes having degrees in a limited sector of their configuration space. The mechanism underlying the first-order transition is discussed in light of standard concepts in statistical physics. We uncover the phase diagram characterizing the ensemble space of the model, and we evaluate the rate function governing the probability to observe a condensed state, which shows that condensation of degrees is a rare statistical event akin to similar condensation phenomena recently observed in several other systems. Monte Carlo simulations confirm the exactness of our theoretical result
Null Models of Economic Networks: The Case of the World Trade Web
In all empirical-network studies, the observed properties of economic
networks are informative only if compared with a well-defined null model that
can quantitatively predict the behavior of such properties in constrained
graphs. However, predictions of the available null-model methods can be derived
analytically only under assumptions (e.g., sparseness of the network) that are
unrealistic for most economic networks like the World Trade Web (WTW). In this
paper we study the evolution of the WTW using a recently-proposed family of
null network models. The method allows to analytically obtain the expected
value of any network statistic across the ensemble of networks that preserve on
average some local properties, and are otherwise fully random. We compare
expected and observed properties of the WTW in the period 1950-2000, when
either the expected number of trade partners or total country trade is kept
fixed and equal to observed quantities. We show that, in the binary WTW,
node-degree sequences are sufficient to explain higher-order network properties
such as disassortativity and clustering-degree correlation, especially in the
last part of the sample. Conversely, in the weighted WTW, the observed sequence
of total country imports and exports are not sufficient to predict higher-order
patterns of the WTW. We discuss some important implications of these findings
for international-trade models.Comment: 39 pages, 46 figures, 2 table
Tailored graph ensembles as proxies or null models for real networks I: tools for quantifying structure
We study the tailoring of structured random graph ensembles to real networks,
with the objective of generating precise and practical mathematical tools for
quantifying and comparing network topologies macroscopically, beyond the level
of degree statistics. Our family of ensembles can produce graphs with any
prescribed degree distribution and any degree-degree correlation function, its
control parameters can be calculated fully analytically, and as a result we can
calculate (asymptotically) formulae for entropies and complexities, and for
information-theoretic distances between networks, expressed directly and
explicitly in terms of their measured degree distribution and degree
correlations.Comment: 25 pages, 3 figure
A Relational Event Approach to Modeling Behavioral Dynamics
This chapter provides an introduction to the analysis of relational event
data (i.e., actions, interactions, or other events involving multiple actors
that occur over time) within the R/statnet platform. We begin by reviewing the
basics of relational event modeling, with an emphasis on models with piecewise
constant hazards. We then discuss estimation for dyadic and more general
relational event models using the relevent package, with an emphasis on
hands-on applications of the methods and interpretation of results. Statnet is
a collection of packages for the R statistical computing system that supports
the representation, manipulation, visualization, modeling, simulation, and
analysis of relational data. Statnet packages are contributed by a team of
volunteer developers, and are made freely available under the GNU Public
License. These packages are written for the R statistical computing
environment, and can be used with any computing platform that supports R
(including Windows, Linux, and Mac).
Graph Annotations in Modeling Complex Network Topologies
The coarsest approximation of the structure of a complex network, such as the
Internet, is a simple undirected unweighted graph. This approximation, however,
loses too much detail. In reality, objects represented by vertices and edges in
such a graph possess some non-trivial internal structure that varies across and
differentiates among distinct types of links or nodes. In this work, we
abstract such additional information as network annotations. We introduce a
network topology modeling framework that treats annotations as an extended
correlation profile of a network. Assuming we have this profile measured for a
given network, we present an algorithm to rescale it in order to construct
networks of varying size that still reproduce the original measured annotation
profile.
Using this methodology, we accurately capture the network properties
essential for realistic simulations of network applications and protocols, or
any other simulations involving complex network topologies, including modeling
and simulation of network evolution. We apply our approach to the Autonomous
System (AS) topology of the Internet annotated with business relationships
between ASs. This topology captures the large-scale structure of the Internet.
In depth understanding of this structure and tools to model it are cornerstones
of research on future Internet architectures and designs. We find that our
techniques are able to accurately capture the structure of annotation
correlations within this topology, thus reproducing a number of its important
properties in synthetically-generated random graphs
Postmortem Analysis of Decayed Online Social Communities: Cascade Pattern Analysis and Prediction
Recently, many online social networks, such as MySpace, Orkut, and
Friendster, have faced inactivity decay of their members, which contributed to
the collapse of these networks. The reasons, mechanics, and prevention
mechanisms of such inactivity decay are not fully understood. In this work, we
analyze decayed and alive sub-websites from the StackExchange platform. The
analysis mainly focuses on the inactivity cascades that occur among the members
of these communities. We provide measures to understand the decay process and
statistical analysis to extract the patterns that accompany the inactivity
decay. Additionally, we predict cascade size and cascade virality using machine
learning. The results of this work include a statistically significant
difference of the decay patterns between the decayed and the alive
sub-websites. These patterns are mainly: cascade size, cascade virality,
cascade duration, and cascade similarity. Additionally, the contributed
prediction framework showed satisfactory prediction results compared to a
baseline predictor. Supported by empirical evidence, the main findings of this
work are: (1) the decay process is not governed by only one network measure; it
is better described using multiple measures; (2) the expert members of the
StackExchange sub-websites were mainly responsible for the activity or
inactivity of the StackExchange sub-websites; (3) the Statistics sub-website is
going through decay dynamics that may lead to it becoming fully-decayed; and
(4) decayed sub-websites were originally less resilient to inactivity decay,
unlike the alive sub-websites
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