14 research outputs found
Adaptive Dynamics of Realistic Small-World Networks
Continuing in the steps of Jon Kleinberg's and others celebrated work on
decentralized search in small-world networks, we conduct an experimental
analysis of a dynamic algorithm that produces small-world networks. We find
that the algorithm adapts robustly to a wide variety of situations in realistic
geographic networks with synthetic test data and with real world data, even
when vertices are uneven and non-homogeneously distributed.
We investigate the same algorithm in the case where some vertices are more
popular destinations for searches than others, for example obeying power-laws.
We find that the algorithm adapts and adjusts the networks according to the
distributions, leading to improved performance. The ability of the dynamic
process to adapt and create small worlds in such diverse settings suggests a
possible mechanism by which such networks appear in nature
The Internet AS-Level Topology: Three Data Sources and One Definitive Metric
We calculate an extensive set of characteristics for Internet AS topologies
extracted from the three data sources most frequently used by the research
community: traceroutes, BGP, and WHOIS. We discover that traceroute and BGP
topologies are similar to one another but differ substantially from the WHOIS
topology. Among the widely considered metrics, we find that the joint degree
distribution appears to fundamentally characterize Internet AS topologies as
well as narrowly define values for other important metrics. We discuss the
interplay between the specifics of the three data collection mechanisms and the
resulting topology views. In particular, we show how the data collection
peculiarities explain differences in the resulting joint degree distributions
of the respective topologies. Finally, we release to the community the input
topology datasets, along with the scripts and output of our calculations. This
supplement should enable researchers to validate their models against real data
and to make more informed selection of topology data sources for their specific
needs.Comment: This paper is a revised journal version of cs.NI/050803
Systematic Topology Analysis and Generation Using Degree Correlations
We present a new, systematic approach for analyzing network topologies. We
first introduce the dK-series of probability distributions specifying all
degree correlations within d-sized subgraphs of a given graph G. Increasing
values of d capture progressively more properties of G at the cost of more
complex representation of the probability distribution. Using this series, we
can quantitatively measure the distance between two graphs and construct random
graphs that accurately reproduce virtually all metrics proposed in the
literature. The nature of the dK-series implies that it will also capture any
future metrics that may be proposed. Using our approach, we construct graphs
for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing
accuracy, important properties of measured and modeled Internet topologies. We
find that the d=2 case is sufficient for most practical purposes, while d=3
essentially reconstructs the Internet AS- and router-level topologies exactly.
We hope that a systematic method to analyze and synthesize topologies offers a
significant improvement to the set of tools available to network topology and
protocol researchers.Comment: Final versio
Networks become navigable as nodes move and forget
We propose a dynamical process for network evolution, aiming at explaining
the emergence of the small world phenomenon, i.e., the statistical observation
that any pair of individuals are linked by a short chain of acquaintances
computable by a simple decentralized routing algorithm, known as greedy
routing. Previously proposed dynamical processes enabled to demonstrate
experimentally (by simulations) that the small world phenomenon can emerge from
local dynamics. However, the analysis of greedy routing using the probability
distributions arising from these dynamics is quite complex because of mutual
dependencies. In contrast, our process enables complete formal analysis. It is
based on the combination of two simple processes: a random walk process, and an
harmonic forgetting process. Both processes reflect natural behaviors of the
individuals, viewed as nodes in the network of inter-individual acquaintances.
We prove that, in k-dimensional lattices, the combination of these two
processes generates long-range links mutually independently distributed as a
k-harmonic distribution. We analyze the performances of greedy routing at the
stationary regime of our process, and prove that the expected number of steps
for routing from any source to any target in any multidimensional lattice is a
polylogarithmic function of the distance between the two nodes in the lattice.
Up to our knowledge, these results are the first formal proof that navigability
in small worlds can emerge from a dynamical process for network evolution. Our
dynamical process can find practical applications to the design of spatial
gossip and resource location protocols.Comment: 21 pages, 1 figur
Recovering the Long Range Links in Augmented Graphs
The augmented graph model, as introduced by Kleinberg (STOC 2000), is an appealing model for analyzing navigability in social networks. Informally, this model is defined by a pair (H,phi), where H is a graph in which inter-node distances are supposed to be easy to compute or at least easy to estimate. This graph is "augmented" by links, called long range links, which are selected according to the probability distribution phi. The augmented graph model enables the analysis of greedy routing in augmented graphs G in (H,phi). In greedy routing, each intermediate node handling a message for a target t selects among all its neighbors in G the one that is the closest to t in H and forwards the message to it. This paper addresses the problem of checking whether a given graph G is an augmented graph. It answers part of the questions raised by Kleinberg in his Problem 9 (Int. Congress of Math. 2006). More precisely, given G in (H,phi), we aim at extracting the base graph H and the long range links R out of G. We prove that if H has a high clustering coefficient and bounded doubling dimension, then a simple algorithm enables to partition the edges of G into two sets H' and R' such that E(H) is included in H' and the edges in H'\E(H) are of small stretch, i.e., the map H is not perturbed too greatly by undetected long range links remaining in H'. The perturbation is actually so small that we can prove that the expected performances of greedy routing in G using the distances in H' are close to the expected performances of greedy routing in (H,phi). Although this latter result may appear intuitively straightforward, since H' is included in E(H), it is not, as we also show that routing with a map more precise than H may actually damage greedy routing significantly. Finally, we show that in absence of a hypothesis regarding the high clustering coefficient, any structural attempt to extract the long range links will miss the detection of at least long range links of stretch at least for any , and thus the map H cannot be recovered with good accuracy. To sum up, we solve Kleinberg's Problem 9 in the sense that we show that reconstructing augmented graphs is achievable if and only if the base graph has a high clustering coefficient
Distributed computing of efficient routing schemes in generalized chordal graphs
International audienceEfficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. Two of them are of particular interest: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we propose here a routing scheme that computes short routes in the class of -chordal graphs, i.e., graphs with no induced cycles of length more than . In the class of -chordal graphs, our routing scheme achieves an additive stretch of at most , i.e., for all pairs of nodes, the length of the route never exceeds their distance plus . In order to compute the routing tables of any -node graph with diameter we propose a distributed algorithm which uses messages of size and takes time. The corresponding routing scheme achieves the stretch of on -chordal graphs. We then propose a routing scheme that achieves a better additive stretch of in chordal graphs (notice that chordal graphs are 3-chordal graphs). In this case, the distributed computation of the routing tables takes time, where is the maximum degree of the graph. Our routing schemes use addresses of size bits and local memory of size bits per node of degree