327 research outputs found
The Manhattan product of digraphs
We give a formal definition of a new product of bipartite digraphs, the Manhattan product, and we study some of its main properties. It is shown that when all the factors of the above product are (directed) cycles, then the obtained digraph is the Manhattan street network. To this respect, it is proved that many properties of such
networks, such as high symmetries and the presence of Hamiltonian cycles, are shared by the Manhattan product of some digraphs
The multidimensional Manhattan networks
The -dimensional Manhattan network ---a special case of
-regular digraph---is formally defined and some of its structural
properties are studied. In particular, it is shown that is a
Cayley digraph, which can be seen as a subgroup of the -dim
version of the wallpaper group . These results induce a useful
new presentation of , which can be applied to design a
(shortest-path) local routing algorithm and to study some other
metric properties. Also it is shown that the -dim Manhattan
networks are Hamiltonian and, in the standard case (that is,
dimension two), they can be decomposed in two arc-disjoint
Hamiltonian cycles. Finally, some results on the connectivity and
distance-related parameters of , such as the distribution of
the node distances and the diameter are presented
The spectra of Manhattan street networks
AbstractThe multidimensional Manhattan street networks constitute a family of digraphs with many interesting properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity
The spectra of Manhattan street networks
The multidimensional Manhattan street networks constitute a family of digraphs
with many interesting properties, such as vertex symmetry (in fact they are Cayley
digraphs), easy routing, Hamiltonicity, and modular structure. From the known
structural properties of these digraphs, we determine their spectra, which always
contain the spectra of hypercubes. In particular, in the standard (two-dimensional)
case it is shown that their line digraph structure imposes the presence of the zero
eigenvalue with a large multiplicity
The Quadratic Cycle Cover Problem: special cases and efficient bounds
The quadratic cycle cover problem is the problem of finding a set of
node-disjoint cycles visiting all the nodes such that the total sum of
interaction costs between consecutive arcs is minimized. In this paper we study
the linearization problem for the quadratic cycle cover problem and related
lower bounds.
In particular, we derive various sufficient conditions for the quadratic cost
matrix to be linearizable, and use these conditions to compute bounds. We also
show how to use a sufficient condition for linearizability within an iterative
bounding procedure. In each step, our algorithm computes the best equivalent
representation of the quadratic cost matrix and its optimal linearizable matrix
with respect to the given sufficient condition for linearizability. Further, we
show that the classical Gilmore-Lawler type bound belongs to the family of
linearization based bounds, and therefore apply the above mentioned iterative
reformulation technique. We also prove that the linearization vectors resulting
from this iterative approach satisfy the constant value property.
The best among here introduced bounds outperform existing lower bounds when
taking both quality and efficiency into account
Consensus of Multi-Agent Networks in the Presence of Adversaries Using Only Local Information
This paper addresses the problem of resilient consensus in the presence of
misbehaving nodes. Although it is typical to assume knowledge of at least some
nonlocal information when studying secure and fault-tolerant consensus
algorithms, this assumption is not suitable for large-scale dynamic networks.
To remedy this, we emphasize the use of local strategies to deal with
resilience to security breaches. We study a consensus protocol that uses only
local information and we consider worst-case security breaches, where the
compromised nodes have full knowledge of the network and the intentions of the
other nodes. We provide necessary and sufficient conditions for the normal
nodes to reach consensus despite the influence of the malicious nodes under
different threat assumptions. These conditions are stated in terms of a novel
graph-theoretic property referred to as network robustness.Comment: This report contains the proofs of the results presented at HiCoNS
201
Product Structures of Networks and Their Spectra
Tohoku University尾畑伸
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