47,233 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
On the Spectral Gap of a Quantum Graph
We consider the problem of finding universal bounds of "isoperimetric" or
"isodiametric" type on the spectral gap of the Laplacian on a metric graph with
natural boundary conditions at the vertices, in terms of various analytical and
combinatorial properties of the graph: its total length, diameter, number of
vertices and number of edges. We investigate which combinations of parameters
are necessary to obtain non-trivial upper and lower bounds and obtain a number
of sharp estimates in terms of these parameters. We also show that, in contrast
to the Laplacian matrix on a combinatorial graph, no bound depending only on
the diameter is possible. As a special case of our results on metric graphs, we
deduce estimates for the normalised Laplacian matrix on combinatorial graphs
which, surprisingly, are sometimes sharper than the ones obtained by purely
combinatorial methods in the graph theoretical literature
Super-expanders and warped cones
For a Banach space , we show that any family of graphs quasi-isometric to
levels of a warped cone is an expander with respect to
if and only if the induced -representation on has a
spectral gap. This provides examples of graphs that are an expander with
respect to all Banach spaces of non-trivial type.Comment: 15 pages; to appear in Ann. Inst. Fourier; exposition rewritten, main
result slightly generalised to accommodate local spectral gap
Superexpanders from group actions on compact manifolds
It is known that the expanders arising as increasing sequences of level sets
of warped cones, as introduced by the second-named author, do not coarsely
embed into a Banach space as soon as the corresponding warped cone does not
coarsely embed into this Banach space. Combining this with non-embeddability
results for warped cones by Nowak and Sawicki, which relate the
non-embeddability of a warped cone to a spectral gap property of the underlying
action, we provide new examples of expanders that do not coarsely embed into
any Banach space with nontrivial type. Moreover, we prove that these expanders
are not coarsely equivalent to a Lafforgue expander. In particular, we provide
infinitely many coarsely distinct superexpanders that are not Lafforgue
expanders. In addition, we prove a quasi-isometric rigidity result for warped
cones.Comment: 16 pages, to appear in Geometriae Dedicat
- …