6,116 research outputs found
Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering
Graph clustering, or community detection, is the task of identifying groups
of closely related objects in a large network. In this paper we introduce a new
community-detection framework called LambdaCC that is based on a specially
weighted version of correlation clustering. A key component in our methodology
is a clustering resolution parameter, , which implicitly controls the
size and structure of clusters formed by our framework. We show that, by
increasing this parameter, our objective effectively interpolates between two
different strategies in graph clustering: finding a sparse cut and forming
dense subgraphs. Our methodology unifies and generalizes a number of other
important clustering quality functions including modularity, sparsest cut, and
cluster deletion, and places them all within the context of an optimization
problem that has been well studied from the perspective of approximation
algorithms. Our approach is particularly relevant in the regime of finding
dense clusters, as it leads to a 2-approximation for the cluster deletion
problem. We use our approach to cluster several graphs, including large
collaboration networks and social networks
Novel Multidimensional Models of Opinion Dynamics in Social Networks
Unlike many complex networks studied in the literature, social networks
rarely exhibit unanimous behavior, or consensus. This requires a development of
mathematical models that are sufficiently simple to be examined and capture, at
the same time, the complex behavior of real social groups, where opinions and
actions related to them may form clusters of different size. One such model,
proposed by Friedkin and Johnsen, extends the idea of conventional consensus
algorithm (also referred to as the iterative opinion pooling) to take into
account the actors' prejudices, caused by some exogenous factors and leading to
disagreement in the final opinions.
In this paper, we offer a novel multidimensional extension, describing the
evolution of the agents' opinions on several topics. Unlike the existing
models, these topics are interdependent, and hence the opinions being formed on
these topics are also mutually dependent. We rigorous examine stability
properties of the proposed model, in particular, convergence of the agents'
opinions. Although our model assumes synchronous communication among the
agents, we show that the same final opinions may be reached "on average" via
asynchronous gossip-based protocols.Comment: Accepted by IEEE Transaction on Automatic Control (to be published in
May 2017
Consensus in the Presence of Multiple Opinion Leaders: Effect of Bounded Confidence
The problem of analyzing the performance of networked agents exchanging
evidence in a dynamic network has recently grown in importance. This problem
has relevance in signal and data fusion network applications and in studying
opinion and consensus dynamics in social networks. Due to its capability of
handling a wider variety of uncertainties and ambiguities associated with
evidence, we use the framework of Dempster-Shafer (DS) theory to capture the
opinion of an agent. We then examine the consensus among agents in dynamic
networks in which an agent can utilize either a cautious or receptive updating
strategy. In particular, we examine the case of bounded confidence updating
where an agent exchanges its opinion only with neighboring nodes possessing
'similar' evidence. In a fusion network, this captures the case in which nodes
only update their state based on evidence consistent with the node's own
evidence. In opinion dynamics, this captures the notions of Social Judgment
Theory (SJT) in which agents update their opinions only with other agents
possessing opinions closer to their own. Focusing on the two special DS
theoretic cases where an agent state is modeled as a Dirichlet body of evidence
and a probability mass function (p.m.f.), we utilize results from matrix
theory, graph theory, and networks to prove the existence of consensus agent
states in several time-varying network cases of interest. For example, we show
the existence of a consensus in which a subset of network nodes achieves a
consensus that is adopted by follower network nodes. Of particular interest is
the case of multiple opinion leaders, where we show that the agents do not
reach a consensus in general, but rather converge to 'opinion clusters'.
Simulation results are provided to illustrate the main results.Comment: IEEE Transactions on Signal and Information Processing Over Networks,
to appea
Recurrent Averaging Inequalities in Multi-Agent Control and Social Dynamics Modeling
Many multi-agent control algorithms and dynamic agent-based models arising in
natural and social sciences are based on the principle of iterative averaging.
Each agent is associated to a value of interest, which may represent, for
instance, the opinion of an individual in a social group, the velocity vector
of a mobile robot in a flock, or the measurement of a sensor within a sensor
network. This value is updated, at each iteration, to a weighted average of
itself and of the values of the adjacent agents. It is well known that, under
natural assumptions on the network's graph connectivity, this local averaging
procedure eventually leads to global consensus, or synchronization of the
values at all nodes. Applications of iterative averaging include, but are not
limited to, algorithms for distributed optimization, for solution of linear and
nonlinear equations, for multi-robot coordination and for opinion formation in
social groups. Although these algorithms have similar structures, the
mathematical techniques used for their analysis are diverse, and conditions for
their convergence and differ from case to case. In this paper, we review many
of these algorithms and we show that their properties can be analyzed in a
unified way by using a novel tool based on recurrent averaging inequalities
(RAIs). We develop a theory of RAIs and apply it to the analysis of several
important multi-agent algorithms recently proposed in the literature
A Bayesian alternative to mutual information for the hierarchical clustering of dependent random variables
The use of mutual information as a similarity measure in agglomerative
hierarchical clustering (AHC) raises an important issue: some correction needs
to be applied for the dimensionality of variables. In this work, we formulate
the decision of merging dependent multivariate normal variables in an AHC
procedure as a Bayesian model comparison. We found that the Bayesian
formulation naturally shrinks the empirical covariance matrix towards a matrix
set a priori (e.g., the identity), provides an automated stopping rule, and
corrects for dimensionality using a term that scales up the measure as a
function of the dimensionality of the variables. Also, the resulting log Bayes
factor is asymptotically proportional to the plug-in estimate of mutual
information, with an additive correction for dimensionality in agreement with
the Bayesian information criterion. We investigated the behavior of these
Bayesian alternatives (in exact and asymptotic forms) to mutual information on
simulated and real data. An encouraging result was first derived on
simulations: the hierarchical clustering based on the log Bayes factor
outperformed off-the-shelf clustering techniques as well as raw and normalized
mutual information in terms of classification accuracy. On a toy example, we
found that the Bayesian approaches led to results that were similar to those of
mutual information clustering techniques, with the advantage of an automated
thresholding. On real functional magnetic resonance imaging (fMRI) datasets
measuring brain activity, it identified clusters consistent with the
established outcome of standard procedures. On this application, normalized
mutual information had a highly atypical behavior, in the sense that it
systematically favored very large clusters. These initial experiments suggest
that the proposed Bayesian alternatives to mutual information are a useful new
tool for hierarchical clustering
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