359 research outputs found
Functional centrality in graphs
In this paper we introduce the functional centrality as a generalization of
the subgraph centrality. We propose a general method for characterizing nodes
in the graph according to the number of closed walks starting and ending at the
node. Closed walks are appropriately weighted according to the topological
features that we need to measure
Properties of the Volume Operator in Loop Quantum Gravity I: Results
We analyze the spectral properties of the volume operator of Ashtekar and
Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the
classical volume expression for regions in three dimensional Riemannian space.
Our analysis considers for the first time generic graph vertices of valence
greater than four. Here we find that the geometry of the underlying vertex
characterizes the spectral properties of the volume operator, in particular the
presence of a `volume gap' (a smallest non-zero eigenvalue in the spectrum) is
found to depend on the vertex embedding. We compute the set of all
non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of
valence 5--7, and argue that these sets can be used to label spatial
diffeomorphism invariant states. We observe how gauge invariance connects
vertex geometry and representation properties of the underlying gauge group in
a natural way. Analytical results on the spectrum on 4-valent vertices are
included, for which the presence of a volume gap is proved. This paper presents
our main results; details are provided by a companion paper arXiv:0706.0382v1.Comment: 36 pages, 7 figures, LaTeX. See also companion paper
arXiv:0706.0382v1. Version as published in CQG in 2008. See arXiv:1003.2348
for important remarks regarding the sigma configurations. Subsequent
computations have revealed some minor errors, which do not change the
qualitative results but modify some of the numbers presented her
On Stability and Consensus of Signed Networks: A Self-loop Compensation Perspective
Positive semidefinite is not an inherent property of signed Laplacians, which
renders the stability and consensus of multi-agent system on undirected signed
networks intricate. Inspired by the correlation between diagonal dominance and
spectrum of signed Laplacians, this paper proposes a self-loop compensation
mechanism in the design of interaction protocol amongst agents and examines the
stability/consensus of the compensated signed networks. It turns out that
self-loop compensation acts as exerting a virtual leader on these agents that
are incident to negative edges, steering whom towards origin. Analytical
connections between self-loop compensation and the collective behavior of the
compensated signed network are established. Necessary and/or sufficient
conditions for predictable cluster consensus of signed networks via self-loop
compensation are provided. The optimality of self-loop compensation is
discussed. Furthermore, we extend our results to directed signed networks where
the symmetry of signed Laplacian is not free. Simulation examples are provided
to demonstrate the theoretical results
Diagonal Stability of Systems with Rank-1 Interconnections and Application to Automatic Generation Control in Power Systems
We study a class of matrices with a rank-1 interconnection structure, and derive a simple necessary and sufficient condition for diagonal stability. The underlying Lyapunov function is used to provide sufficient conditions for diagonal stability of approximately rank-1 interconnections. The main result is then leveraged as a key step in a larger stability analysis problem arising in power systems control. Specifically, we provide the first theoretical stability analysis of automatic generation control (AGC) in an interconnected nonlinear power system. Our analysis is based on singular perturbation theory, and provides theoretical justification for the conventional wisdom that AGC is stabilizing under the typical time-scales of operation. We illustrate how our main analysis results can be leveraged to provide further insight into the tuning and dynamic performance of AGC
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