170,553 research outputs found
Random Regular Graphs are not Asymptotically Gromov Hyperbolic
In this paper we prove that random --regular graphs with have
traffic congestion of the order where is the number
of nodes and geodesic routing is used. We also show that these graphs are not
asymptotically --hyperbolic for any non--negative almost
surely as .Comment: 6 pages, 2 figure
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
Entangled networks, synchronization, and optimal network topology
A new family of graphs, {\it entangled networks}, with optimal properties in
many respects, is introduced. By definition, their topology is such that
optimizes synchronizability for many dynamical processes. These networks are
shown to have an extremely homogeneous structure: degree, node-distance,
betweenness, and loop distributions are all very narrow. Also, they are
characterized by a very interwoven (entangled) structure with short average
distances, large loops, and no well-defined community-structure. This family of
nets exhibits an excellent performance with respect to other flow properties
such as robustness against errors and attacks, minimal first-passage time of
random walks, efficient communication, etc. These remarkable features convert
entangled networks in a useful concept, optimal or almost-optimal in many
senses, and with plenty of potential applications computer science or
neuroscience.Comment: Slightly modified version, as accepted in Phys. Rev. Let
The geodesic flow on nilmanifolds associated to graphs
In this work we study the geodesic flow on nilmanifolds associated to graphs.
We are interested in the construction of first integrals to show complete
integrability on some compact quotients. Also examples of integrable geodesic
flows and of non-integrable ones are shown.Comment: 22 page
Traffic Analysis in Random Delaunay Tessellations and Other Graphs
In this work we study the degree distribution, the maximum vertex and edge
flow in non-uniform random Delaunay triangulations when geodesic routing is
used. We also investigate the vertex and edge flow in Erd\"os-Renyi random
graphs, geometric random graphs, expanders and random -regular graphs.
Moreover we show that adding a random matching to the original graph can
considerably reduced the maximum vertex flow.Comment: Submitted to the Journal of Discrete Computational Geometr
Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts
A skew-symmetric graph is a directed graph with an
involution on the set of vertices and arcs. In this paper, we
introduce a separation problem, -Skew-Symmetric Multicut, where we are given
a skew-symmetric graph , a family of of -sized subsets of
vertices and an integer . The objective is to decide if there is a set
of arcs such that every set in the family has a vertex
such that and are in different connected components of
. In this paper, we give an algorithm for
this problem which runs in time , where is the
number of arcs in the graph, the number of vertices and the length
of the family given in the input.
Using our algorithm, we show that Almost 2-SAT has an algorithm with running
time and we obtain algorithms for {\sc Odd Cycle Transversal}
and {\sc Edge Bipartization} which run in time and
respectively. This resolves an open problem posed by Reed,
Smith and Vetta [Operations Research Letters, 2003] and improves upon the
earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010].
We also show that Deletion q-Horn Backdoor Set Detection is a special case of
3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor
Set Detection which runs in time . This gives the first
fixed-parameter tractable algorithm for this problem answering a question posed
in a paper by a superset of the authors [STACS, 2013]. Using this result, we
get an algorithm for Satisfiability which runs in time where
is the size of the smallest q-Horn deletion backdoor set, with being
the length of the input formula
Achieving the Optimal Steaming Capacity and Delay Using Random Regular Digraphs in P2P Networks
In earlier work, we showed that it is possible to achieve
streaming delay with high probability in a peer-to-peer network, where each
peer has as little as four neighbors, while achieving any arbitrary fraction of
the maximum possible streaming rate. However, the constant in the
delay term becomes rather large as we get closer to the maximum streaming rate.
In this paper, we design an alternative pairing and chunk dissemination
algorithm that allows us to transmit at the maximum streaming rate while
ensuring that all, but a negligible fraction of the peers, receive the data
stream with delay with high probability. The result is established
by examining the properties of graph formed by the union of two or more random
1-regular digraphs, i.e., directed graphs in which each node has an incoming
and an outgoing node degree both equal to one
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