690 research outputs found
Mixing patterns and community structure in networks
Common experience suggests that many networks might possess community
structure - division of vertices into groups, with a higher density of edges
within groups than between them. Here we describe a new computer algorithm that
detects structure of this kind. We apply the algorithm to a number of
real-world networks and show that they do indeed possess non-trivial community
structure. We suggest a possible explanation for this structure in the
mechanism of assortative mixing, which is the preferential association of
network vertices with others that are like them in some way. We show by
simulation that this mechanism can indeed account for community structure. We
also look in detail at one particular example of assortative mixing, namely
mixing by vertex degree, in which vertices with similar degree prefer to be
connected to one another. We propose a measure for mixing of this type which we
apply to a variety of networks, and also discuss the implications for network
structure and the formation of a giant component in assortatively mixed
networks.Comment: 21 pages, 9 postscript figures, 2 table
Optimal design, robustness, and risk aversion
Highly optimized tolerance is a model of optimization in engineered systems,
which gives rise to power-law distributions of failure events in such systems.
The archetypal example is the highly optimized forest fire model. Here we give
an analytic solution for this model which explains the origin of the power
laws. We also generalize the model to incorporate risk aversion, which results
in truncation of the tails of the power law so that the probability of
disastrously large events is dramatically lowered, giving the system more
robustness.Comment: 11 pages, 2 figure
Modularity and community structure in networks
Many networks of interest in the sciences, including a variety of social and
biological networks, are found to divide naturally into communities or modules.
The problem of detecting and characterizing this community structure has
attracted considerable recent attention. One of the most sensitive detection
methods is optimization of the quality function known as "modularity" over the
possible divisions of a network, but direct application of this method using,
for instance, simulated annealing is computationally costly. Here we show that
the modularity can be reformulated in terms of the eigenvectors of a new
characteristic matrix for the network, which we call the modularity matrix, and
that this reformulation leads to a spectral algorithm for community detection
that returns results of better quality than competing methods in noticeably
shorter running times. We demonstrate the algorithm with applications to
several network data sets.Comment: 7 pages, 3 figure
Mixture models and exploratory analysis in networks
Networks are widely used in the biological, physical, and social sciences as
a concise mathematical representation of the topology of systems of interacting
components. Understanding the structure of these networks is one of the
outstanding challenges in the study of complex systems. Here we describe a
general technique for detecting structural features in large-scale network data
which works by dividing the nodes of a network into classes such that the
members of each class have similar patterns of connection to other nodes. Using
the machinery of probabilistic mixture models and the expectation-maximization
algorithm, we show that it is possible to detect, without prior knowledge of
what we are looking for, a very broad range of types of structure in networks.
We give a number of examples demonstrating how the method can be used to shed
light on the properties of real-world networks, including social and
information networks.Comment: 8 pages, 4 figures, two new examples in this version plus minor
correction
A Simple Model of Epidemics with Pathogen Mutation
We study how the interplay between the memory immune response and pathogen
mutation affects epidemic dynamics in two related models. The first explicitly
models pathogen mutation and individual memory immune responses, with contacted
individuals becoming infected only if they are exposed to strains that are
significantly different from other strains in their memory repertoire. The
second model is a reduction of the first to a system of difference equations.
In this case, individuals spend a fixed amount of time in a generalized immune
class. In both models, we observe four fundamentally different types of
behavior, depending on parameters: (1) pathogen extinction due to lack of
contact between individuals, (2) endemic infection (3) periodic epidemic
outbreaks, and (4) one or more outbreaks followed by extinction of the epidemic
due to extremely low minima in the oscillations. We analyze both models to
determine the location of each transition. Our main result is that pathogens in
highly connected populations must mutate rapidly in order to remain viable.Comment: 9 pages, 11 figure
Stochastic blockmodels with growing number of classes
We present asymptotic and finite-sample results on the use of stochastic
blockmodels for the analysis of network data. We show that the fraction of
misclassified network nodes converges in probability to zero under maximum
likelihood fitting when the number of classes is allowed to grow as the root of
the network size and the average network degree grows at least
poly-logarithmically in this size. We also establish finite-sample confidence
bounds on maximum-likelihood blockmodel parameter estimates from data
comprising independent Bernoulli random variates; these results hold uniformly
over class assignment. We provide simulations verifying the conditions
sufficient for our results, and conclude by fitting a logit parameterization of
a stochastic blockmodel with covariates to a network data example comprising a
collection of Facebook profiles, resulting in block estimates that reveal
residual structure.Comment: 12 pages, 3 figures; revised versio
External Periodic Driving of Large Systems of Globally Coupled Phase Oscillators
Large systems of coupled oscillators subjected to a periodic external drive
occur in many situations in physics and biology. Here the simple, paradigmatic
case of equal-strength, all-to-all sine-coupling of phase oscillators subject
to a sinusoidal external drive is considered. The stationary states and their
stability are determined. Using the stability information and numerical
experiments, parameter space phase diagrams showing when different types of
system behavior apply are constructed, and the bifurcations marking transitions
between different types of behavior are delineated. The analysis is supported
by results of direct numerical simulation of an ensemble of oscillators
Finding and evaluating community structure in networks
We propose and study a set of algorithms for discovering community structure
in networks -- natural divisions of network nodes into densely connected
subgroups. Our algorithms all share two definitive features: first, they
involve iterative removal of edges from the network to split it into
communities, the edges removed being identified using one of a number of
possible "betweenness" measures, and second, these measures are, crucially,
recalculated after each removal. We also propose a measure for the strength of
the community structure found by our algorithms, which gives us an objective
metric for choosing the number of communities into which a network should be
divided. We demonstrate that our algorithms are highly effective at discovering
community structure in both computer-generated and real-world network data, and
show how they can be used to shed light on the sometimes dauntingly complex
structure of networked systems.Comment: 16 pages, 13 figure
Communities as Well Separated Subgraphs With Cohesive Cores: Identification of Core-Periphery Structures in Link Communities
Communities in networks are commonly considered as highly cohesive subgraphs
which are well separated from the rest of the network. However, cohesion and
separation often cannot be maximized at the same time, which is why a
compromise is sought by some methods. When a compromise is not suitable for the
problem to be solved it might be advantageous to separate the two criteria. In
this paper, we explore such an approach by defining communities as well
separated subgraphs which can have one or more cohesive cores surrounded by
peripheries. We apply this idea to link communities and present an algorithm
for constructing hierarchical core-periphery structures in link communities and
first test results.Comment: 12 pages, 2 figures, submitted version of a paper accepted for the
7th International Conference on Complex Networks and Their Applications,
December 11-13, 2018, Cambridge, UK; revised version at
http://141.20.126.227/~qm/papers
On metastable configurations of small-world networks
We calculate the number of metastable configurations of Ising small-world
networks which are constructed upon superimposing sparse Poisson random graphs
onto a one-dimensional chain. Our solution is based on replicated
transfer-matrix techniques. We examine the denegeracy of the ground state and
we find a jump in the entropy of metastable configurations exactly at the
crossover between the small-world and the Poisson random graph structures. We
also examine the difference in entropy between metastable and all possible
configurations, for both ferromagnetic and bond-disordered long-range
couplings.Comment: 9 pages, 4 eps figure
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