74 research outputs found
Localization theorems for nonlinear eigenvalue problems
Let T : \Omega \rightarrow \bbC^{n \times n} be a matrix-valued function
that is analytic on some simply-connected domain \Omega \subset \bbC. A point
is an eigenvalue if the matrix is singular.
In this paper, we describe new localization results for nonlinear eigenvalue
problems that generalize Gershgorin's theorem, pseudospectral inclusion
theorems, and the Bauer-Fike theorem. We use our results to analyze three
nonlinear eigenvalue problems: an example from delay differential equations, a
problem due to Hadeler, and a quantum resonance computation.Comment: Submitted to SIMAX. 22 pages, 11 figure
How Bad is Forming Your Own Opinion?
The question of how people form their opinion has fascinated economists and
sociologists for quite some time. In many of the models, a group of people in a
social network, each holding a numerical opinion, arrive at a shared opinion
through repeated averaging with their neighbors in the network. Motivated by
the observation that consensus is rarely reached in real opinion dynamics, we
study a related sociological model in which individuals' intrinsic beliefs
counterbalance the averaging process and yield a diversity of opinions.
By interpreting the repeated averaging as best-response dynamics in an
underlying game with natural payoffs, and the limit of the process as an
equilibrium, we are able to study the cost of disagreement in these models
relative to a social optimum. We provide a tight bound on the cost at
equilibrium relative to the optimum; our analysis draws a connection between
these agreement models and extremal problems that lead to generalized
eigenvalues. We also consider a natural network design problem in this setting:
which links can we add to the underlying network to reduce the cost of
disagreement at equilibrium
Network Density of States
Spectral analysis connects graph structure to the eigenvalues and
eigenvectors of associated matrices. Much of spectral graph theory descends
directly from spectral geometry, the study of differentiable manifolds through
the spectra of associated differential operators. But the translation from
spectral geometry to spectral graph theory has largely focused on results
involving only a few extreme eigenvalues and their associated eigenvalues.
Unlike in geometry, the study of graphs through the overall distribution of
eigenvalues - the spectral density - is largely limited to simple random graph
models. The interior of the spectrum of real-world graphs remains largely
unexplored, difficult to compute and to interpret.
In this paper, we delve into the heart of spectral densities of real-world
graphs. We borrow tools developed in condensed matter physics, and add novel
adaptations to handle the spectral signatures of common graph motifs. The
resulting methods are highly efficient, as we illustrate by computing spectral
densities for graphs with over a billion edges on a single compute node. Beyond
providing visually compelling fingerprints of graphs, we show how the
estimation of spectral densities facilitates the computation of many common
centrality measures, and use spectral densities to estimate meaningful
information about graph structure that cannot be inferred from the extremal
eigenpairs alone.Comment: 10 pages, 7 figure
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