145,203 research outputs found
Networks with small stretch number
Abstract In a previous work, the authors introduced the class of graphs with bounded induced distance of order k (BID(k) for short), to model non-reliable interconnection networks. A network modeled as a graph in BID(k) can be characterized as follows: if some nodes have failed, as long as two nodes remain connected, the distance between these nodes in the faulty graph is at most k times the distance in the non-faulty graph. The smallest k such that G∈BID(k) is called stretch number of G. We show an odd characteristic of the stretch numbers: every rational number greater or equal 2 is a stretch number, but only discrete values are admissible for smaller stretch numbers. Moreover, we give a new characterization of classes BID(2−1/i), i⩾1, based on forbidden induced subgraphs. By using this characterization, we provide a polynomial time recognition algorithm for graphs belonging to these classes, while the general recognition problem is Co-NP-complete
Similarity-Aware Spectral Sparsification by Edge Filtering
In recent years, spectral graph sparsification techniques that can compute
ultra-sparse graph proxies have been extensively studied for accelerating
various numerical and graph-related applications. Prior nearly-linear-time
spectral sparsification methods first extract low-stretch spanning tree from
the original graph to form the backbone of the sparsifier, and then recover
small portions of spectrally-critical off-tree edges to the spanning tree to
significantly improve the approximation quality. However, it is not clear how
many off-tree edges should be recovered for achieving a desired spectral
similarity level within the sparsifier. Motivated by recent graph signal
processing techniques, this paper proposes a similarity-aware spectral graph
sparsification framework that leverages efficient spectral off-tree edge
embedding and filtering schemes to construct spectral sparsifiers with
guaranteed spectral similarity (relative condition number) level. An iterative
graph densification scheme is introduced to facilitate efficient and effective
filtering of off-tree edges for highly ill-conditioned problems. The proposed
method has been validated using various kinds of graphs obtained from public
domain sparse matrix collections relevant to VLSI CAD, finite element analysis,
as well as social and data networks frequently studied in many machine learning
and data mining applications
Compact Routing on Internet-Like Graphs
The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing
scheme delivering a nearly optimal local memory upper bound. Using both direct
analysis and simulation, we calculate the stretch distribution of this routing
scheme on random graphs with power-law node degree distributions, . We find that the average stretch is very low and virtually
independent of . In particular, for the Internet interdomain graph,
, the average stretch is around 1.1, with up to 70% of paths
being shortest. As the network grows, the average stretch slowly decreases. The
routing table is very small, too. It is well below its upper bounds, and its
size is around 50 records for -node networks. Furthermore, we find that
both the average shortest path length (i.e. distance) and width of
the distance distribution observed in the real Internet inter-AS graph
have values that are very close to the minimums of the average stretch in the
- and -directions. This leads us to the discovery of a unique
critical quasi-stationary point of the average TZ stretch as a function of
and . The Internet distance distribution is located in a
close neighborhood of this point. This observation suggests the analytical
structure of the average stretch function may be an indirect indicator of some
hidden optimization criteria influencing the Internet's interdomain topology
evolution.Comment: 29 pages, 16 figure
Navigability of temporal networks in hyperbolic space
Information routing is one of the main tasks in many complex networks with a
communication function. Maps produced by embedding the networks in hyperbolic
space can assist this task enabling the implementation of efficient navigation
strategies. However, only static maps have been considered so far, while
navigation in more realistic situations, where the network structure may vary
in time, remain largely unexplored. Here, we analyze the navigability of real
networks by using greedy routing in hyperbolic space, where the nodes are
subject to a stochastic activation-inactivation dynamics. We find that such
dynamics enhances navigability with respect to the static case. Interestingly,
there exists an optimal intermediate activation value, which ensures the best
trade-off between the increase in the number of successful paths and a limited
growth of their length. Contrary to expectations, the enhanced navigability is
robust even when the most connected nodes inactivate with very high
probability. Finally, our results indicate that some real networks are
ultranavigable and remain highly navigable even if the network structure is
extremely unsteady. These findings have important implications for the design
and evaluation of efficient routing protocols that account for the temporal
nature of real complex networks.Comment: 10 pages, 4 figures. Includes Supplemental Informatio
Efficient Computation of Distance Sketches in Distributed Networks
Distance computation is one of the most fundamental primitives used in
communication networks. The cost of effectively and accurately computing
pairwise network distances can become prohibitive in large-scale networks such
as the Internet and Peer-to-Peer (P2P) networks. To negotiate the rising need
for very efficient distance computation, approximation techniques for numerous
variants of this question have recently received significant attention in the
literature. The goal is to preprocess the graph and store a small amount of
information such that whenever a query for any pairwise distance is issued, the
distance can be well approximated (i.e., with small stretch) very quickly in an
online fashion. Specifically, the pre-processing (usually) involves storing a
small sketch with each node, such that at query time only the sketches of the
concerned nodes need to be looked up to compute the approximate distance. In
this paper, we present the first theoretical study of distance sketches derived
from distance oracles in a distributed network. We first present a fast
distributed algorithm for computing approximate distance sketches, based on a
distributed implementation of the distance oracle scheme of [Thorup-Zwick, JACM
2005]. We also show how to modify this basic construction to achieve different
tradeoffs between the number of pairs for which the distance estimate is
accurate and other parameters. These tradeoffs can then be combined to give an
efficient construction of small sketches with provable average-case as well as
worst-case performance. Our algorithms use only small-sized messages and hence
are suitable for bandwidth-constrained networks, and can be used in various
networking applications such as topology discovery and construction, token
management, load balancing, monitoring overlays, and several other problems in
distributed algorithms.Comment: 18 page
Search in Complex Networks : a New Method of Naming
We suggest a method for routing when the source does not posses full
information about the shortest path to the destination. The method is
particularly useful for scale-free networks, and exploits its unique
characteristics. By assigning new (short) names to nodes (aka labelling) we are
able to reduce significantly the memory requirement at the routers, yet we
succeed in routing with high probability through paths very close in distance
to the shortest ones.Comment: 5 pages, 4 figure
Self-organized Emergence of Navigability on Small-World Networks
This paper mainly investigates why small-world networks are navigable and how
to navigate small-world networks. We find that the navigability can naturally
emerge from self-organization in the absence of prior knowledge about
underlying reference frames of networks. Through a process of information
exchange and accumulation on networks, a hidden metric space for navigation on
networks is constructed. Navigation based on distances between vertices in the
hidden metric space can efficiently deliver messages on small-world networks,
in which long range connections play an important role. Numerical simulations
further suggest that high cluster coefficient and low diameter are both
necessary for navigability. These interesting results provide profound insights
into scalable routing on the Internet due to its distributed and localized
requirements.Comment: 3 figure
An Experimental Investigation of Hyperbolic Routing with a Smart Forwarding Plane in NDN
Routing in NDN networks must scale in terms of forwarding table size and
routing protocol overhead. Hyperbolic routing (HR) presents a potential
solution to address the routing scalability problem, because it does not use
traditional forwarding tables or exchange routing updates upon changes in
network topologies. Although HR has the drawbacks of producing sub-optimal
routes or local minima for some destinations, these issues can be mitigated by
NDN's intelligent data forwarding plane. However, HR's viability still depends
on both the quality of the routes HR provides and the overhead incurred at the
forwarding plane due to HR's sub-optimal behavior. We designed a new forwarding
strategy called Adaptive Smoothed RTT-based Forwarding (ASF) to mitigate HR's
sub-optimal path selection. This paper describes our experimental investigation
into the packet delivery delay and overhead under HR as compared with
Named-Data Link State Routing (NLSR), which calculates shortest paths. We run
emulation experiments using various topologies with different failure
scenarios, probing intervals, and maximum number of next hops for a name
prefix. Our results show that HR's delay stretch has a median close to 1 and a
95th-percentile around or below 2, which does not grow with the network size.
HR's message overhead in dynamic topologies is nearly independent of the
network size, while NLSR's overhead grows polynomially at least. These results
suggest that HR offers a more scalable routing solution with little impact on
the optimality of routing paths
Greedy Forwarding in Dynamic Scale-Free Networks Embedded in Hyperbolic Metric Spaces
We show that complex (scale-free) network topologies naturally emerge from
hyperbolic metric spaces. Hyperbolic geometry facilitates maximally efficient
greedy forwarding in these networks. Greedy forwarding is topology-oblivious.
Nevertheless, greedy packets find their destinations with 100% probability
following almost optimal shortest paths. This remarkable efficiency sustains
even in highly dynamic networks. Our findings suggest that forwarding
information through complex networks, such as the Internet, is possible without
the overhead of existing routing protocols, and may also find practical
applications in overlay networks for tasks such as application-level routing,
information sharing, and data distribution
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