174,508 research outputs found
The Role of Weight on Community Structure of Networks
The role of weight on the weighted networks is investigated by studying the
effect of weight on community structures. We use weighted modularity to
evaluate the partitions and Weighted Extremal Optimization algorithm to detect
communities. Starting from idealized and empirical weighted networks, the
distribution or matching between weights and edges are disturbed. Using
dissimilarity function to distinguish the difference between community
structures, it is found that the redistribution of weights does strongly affect
the community structure especially in dense networks. This indicates that the
community structure in networks is a suitable property to reflect the role of
weight.Comment: 10 pages, 6 figure
Beating ratio 0.5 for weighted oblivious matching problems
ESA 2016 is organized in collaboration with the European Association for Theoretical Computer Science (EATCS) and is a part of ALGO 2016We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem. Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs). (1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). (2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as matching coverage, in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied. Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5.published_or_final_versio
Algorithms for Vertex-Weighted Matching in Graphs
A matching M in a graph is a subset of edges such that no two edges in M are incident on the same vertex. Matching is a fundamental combinatorial problem that has applications in many contexts: high-performance computing, bioinformatics, network switch design, web technologies, etc. Examples in the first context include sparse linear systems of equations, where matchings are used to place large matrix elements on or close to the diagonal, to compute the block triangular decomposition of sparse matrices, to construct sparse bases for the null space or column space of under-determined matrices, and to coarsen graphs in multi-level graph partitioning algorithms. In the first part of this thesis, we develop exact and approximation algorithms for vertex weighted matchings, an under-studied variant of the weighted matching problem. We propose three exact algorithms, three half approximation algorithms, and a two-third approximation algorithm. We exploit inherent properties of this problem such as lexicographical orders, decomposition into sub-problems, and the reachability property, not only to design efficient algorithms, but also to provide simple proofs of correctness of the proposed algorithms. In the second part of this thesis, we describe work on a new parallel half-approximation algorithm for weighted matching. Algorithms for computing optimal matchings are not amenable to parallelism, and hence we consider approximation algorithms here. We extend the existing work on a parallel half approximation algorithm for weighted matching and provide an analysis of its time complexity. We support the theoretical observations with experimental results obtained with MatchBoxP, toolkit designed and implemented in C++ and MPI using modern software engineering techniques. The work in this thesis has resulted in better understanding of matching theory, a functional public-domain software toolkit, and modeling of the sparsest basis problem as a vertex-weighted matching problem
Balancing Score Adjusted Targeted Minimum Loss-based Estimation
Adjusting for a balancing score is sufficient for bias reduction when estimating causal effects including the average treatment effect and effect among the treated. Estimators that adjust for the propensity score in a nonparametric way, such as matching on an estimate of the propensity score, can be consistent when the estimated propensity score is not consistent for the true propensity score but converges to some other balancing score. We call this property the balancing score property, and discuss a class of estimators that have this property. We introduce a targeted minimum loss-based estimator (TMLE) for a treatment specific mean with the balancing score property that is additionally locally efficient and doubly robust. We investigate the new estimator\u27s performance relative to other estimators, including another TMLE, a propensity score matching estimator, an inverse probability of treatment weighted estimator, and a regression based estimator in simulation studies
Unimodal Mono-Partite Matching in a Bandit Setting
We tackle a new emerging problem, which is finding an optimal monopartite
matching in a weighted graph. The semi-bandit version, where a full matching is
sampled at each iteration, has been addressed by \cite{ADMA}, creating an
algorithm with an expected regret matching
with players, iterations and a minimum reward gap . We reduce
this bound in two steps. First, as in \cite{GRAB} and \cite{UniRank} we use the
unimodality property of the expected reward on the appropriate graph to design
an algorithm with a regret in . Secondly, we show
that by moving the focus towards the main question `\emph{Is user better
than user ?}' this regret becomes
, where \Tilde{\Delta} > \Delta
derives from a better way of comparing users. Some experimental results finally
show these theoretical results are corroborated in practice
UniRank: Unimodal Bandit Algorithm for Online Ranking
We tackle a new emerging problem, which is finding an optimal monopartite
matching in a weighted graph. The semi-bandit version, where a full matching is
sampled at each iteration, has been addressed by \cite{ADMA}, creating an
algorithm with an expected regret matching
with players, iterations and a minimum reward gap . We reduce
this bound in two steps. First, as in \cite{GRAB} and \cite{UniRank} we use the
unimodality property of the expected reward on the appropriate graph to design
an algorithm with a regret in . Secondly, we show
that by moving the focus towards the main question `\emph{Is user better
than user ?}' this regret becomes
, where \Tilde{\Delta} > \Delta
derives from a better way of comparing users. Some experimental results finally
show these theoretical results are corroborated in practice
Graph matching: relax or not?
We consider the problem of exact and inexact matching of weighted undirected
graphs, in which a bijective correspondence is sought to minimize a quadratic
weight disagreement. This computationally challenging problem is often relaxed
as a convex quadratic program, in which the space of permutations is replaced
by the space of doubly-stochastic matrices. However, the applicability of such
a relaxation is poorly understood. We define a broad class of friendly graphs
characterized by an easily verifiable spectral property. We prove that for
friendly graphs, the convex relaxation is guaranteed to find the exact
isomorphism or certify its inexistence. This result is further extended to
approximately isomorphic graphs, for which we develop an explicit bound on the
amount of weight disagreement under which the relaxation is guaranteed to find
the globally optimal approximate isomorphism. We also show that in many cases,
the graph matching problem can be further harmlessly relaxed to a convex
quadratic program with only n separable linear equality constraints, which is
substantially more efficient than the standard relaxation involving 2n equality
and n^2 inequality constraints. Finally, we show that our results are still
valid for unfriendly graphs if additional information in the form of seeds or
attributes is allowed, with the latter satisfying an easy to verify spectral
characteristic
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