4,645 research outputs found
Signless Laplacian spectral radius for a k-extendable graph
Let and be two nonnegative integers with (mod 2), and let
be a graph of order with a 1-factor. Then is said to be
-extendable for if every matching in of size
can be extended to a 1-factor. In this paper, we first establish a lower
bound on the signless Laplacian spectral radius of to ensure that is
-extendable. Then we create some extremal graphs to claim that all the
bounds derived in this article are sharp.Comment: 11 page
A Tensor Approach to Learning Mixed Membership Community Models
Community detection is the task of detecting hidden communities from observed
interactions. Guaranteed community detection has so far been mostly limited to
models with non-overlapping communities such as the stochastic block model. In
this paper, we remove this restriction, and provide guaranteed community
detection for a family of probabilistic network models with overlapping
communities, termed as the mixed membership Dirichlet model, first introduced
by Airoldi et al. This model allows for nodes to have fractional memberships in
multiple communities and assumes that the community memberships are drawn from
a Dirichlet distribution. Moreover, it contains the stochastic block model as a
special case. We propose a unified approach to learning these models via a
tensor spectral decomposition method. Our estimator is based on low-order
moment tensor of the observed network, consisting of 3-star counts. Our
learning method is fast and is based on simple linear algebraic operations,
e.g. singular value decomposition and tensor power iterations. We provide
guaranteed recovery of community memberships and model parameters and present a
careful finite sample analysis of our learning method. As an important special
case, our results match the best known scaling requirements for the
(homogeneous) stochastic block model
New and Provable Results for Network Inference Problems and Multi-agent Optimization Algorithms
abstract: Our ability to understand networks is important to many applications, from the analysis and modeling of biological networks to analyzing social networks. Unveiling network dynamics allows us to make predictions and decisions. Moreover, network dynamics models have inspired new ideas for computational methods involving multi-agent cooperation, offering effective solutions for optimization tasks. This dissertation presents new theoretical results on network inference and multi-agent optimization, split into two parts -
The first part deals with modeling and identification of network dynamics. I study two types of network dynamics arising from social and gene networks. Based on the network dynamics, the proposed network identification method works like a `network RADAR', meaning that interaction strengths between agents are inferred by injecting `signal' into the network and observing the resultant reverberation. In social networks, this is accomplished by stubborn agents whose opinions do not change throughout a discussion. In gene networks, genes are suppressed to create desired perturbations. The steady-states under these perturbations are characterized. In contrast to the common assumption of full rank input, I take a laxer assumption where low-rank input is used, to better model the empirical network data. Importantly, a network is proven to be identifiable from low rank data of rank that grows proportional to the network's sparsity. The proposed method is applied to synthetic and empirical data, and is shown to offer superior performance compared to prior work. The second part is concerned with algorithms on networks. I develop three consensus-based algorithms for multi-agent optimization. The first method is a decentralized Frank-Wolfe (DeFW) algorithm. The main advantage of DeFW lies on its projection-free nature, where we can replace the costly projection step in traditional algorithms by a low-cost linear optimization step. I prove the convergence rates of DeFW for convex and non-convex problems. I also develop two consensus-based alternating optimization algorithms --- one for least square problems and one for non-convex problems. These algorithms exploit the problem structure for faster convergence and their efficacy is demonstrated by numerical simulations.
I conclude this dissertation by describing future research directions.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201
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