1,402 research outputs found
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Quantum walks with infinite hitting times
Hitting times are the average time it takes a walk to reach a given final
vertex from a given starting vertex. The hitting time for a classical random
walk on a connected graph will always be finite. We show that, by contrast,
quantum walks can have infinite hitting times for some initial states. We seek
criteria to determine if a given walk on a graph will have infinite hitting
times, and find a sufficient condition, which for discrete time quantum walks
is that the degeneracy of the evolution operator be greater than the degree of
the graph. The set of initial states which give an infinite hitting time form a
subspace. The phenomenon of infinite hitting times is in general a consequence
of the symmetry of the graph and its automorphism group. Using the irreducible
representations of the automorphism group, we derive conditions such that
quantum walks defined on this graph must have infinite hitting times for some
initial states. In the case of the discrete walk, if this condition is
satisfied the walk will have infinite hitting times for any choice of a coin
operator, and we give a class of graphs with infinite hitting times for any
choice of coin. Hitting times are not very well-defined for continuous time
quantum walks, but we show that the idea of infinite hitting-time walks
naturally extends to the continuous time case as well.Comment: 28 pages, 3 figures in EPS forma
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