11,407 research outputs found
Perfect Elimination Orderings for Symmetric Matrices
We introduce a new class of structured symmetric matrices by extending the
notion of perfect elimination ordering from graphs to weighted graphs or
matrices. This offers a common framework capturing common vertex elimination
orderings of monotone families of chordal graphs, Robinsonian matrices and
ultrametrics. We give a structural characterization for matrices that admit
perfect elimination orderings in terms of forbidden substructures generalizing
chordless cycles in graphs.Comment: 16 pages, 3 figure
Condorcet Domains, Median Graphs and the Single Crossing Property
Condorcet domains are sets of linear orders with the property that, whenever
the preferences of all voters belong to this set, the majority relation has no
cycles. We observe that, without loss of generality, such domain can be assumed
to be closed in the sense that it contains the majority relation of every
profile with an odd number of individuals whose preferences belong to this
domain.
We show that every closed Condorcet domain is naturally endowed with the
structure of a median graph and that, conversely, every median graph is
associated with a closed Condorcet domain (which may not be a unique one). The
subclass of those Condorcet domains that correspond to linear graphs (chains)
are exactly the preference domains with the classical single crossing property.
As a corollary, we obtain that the domains with the so-called `representative
voter property' (with the exception of a 4-cycle) are the single crossing
domains.
Maximality of a Condorcet domain imposes additional restrictions on the
underlying median graph. We prove that among all trees only the chains can
induce maximal Condorcet domains, and we characterize the single crossing
domains that in fact do correspond to maximal Condorcet domains.
Finally, using Nehring's and Puppe's (2007) characterization of monotone
Arrowian aggregation, our analysis yields a rich class of strategy-proof social
choice functions on any closed Condorcet domain
Balanced Truncation of Networked Linear Passive Systems
This paper studies model order reduction of multi-agent systems consisting of
identical linear passive subsystems, where the interconnection topology is
characterized by an undirected weighted graph. Balanced truncation based on a
pair of specifically selected generalized Gramians is implemented on the
asymptotically stable part of the full-order network model, which leads to a
reduced-order system preserving the passivity of each subsystem. Moreover, it
is proven that there exists a coordinate transformation to convert the
resulting reduced-order model to a state-space model of Laplacian dynamics.
Thus, the proposed method simultaneously reduces the complexity of the network
structure and individual agent dynamics, and it preserves the passivity of the
subsystems and the synchronization of the network. Moreover, it allows for the
a priori computation of a bound on the approximation error. Finally, the
feasibility of the method is demonstrated by an example
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