660 research outputs found
On Computing the Hyperbolicity of Real-World Graphs
International audienceThe (Gromov) hyperbolicity is a topological property of a graph, which has been recently applied in several different contexts, such as the design of routing schemes, network security, computational biology, the analysis of graph algorithms, and the classification of complex networks. Computing the hyperbolicity of a graph can be very time consuming: indeed, the best available algorithm has running-time O(n^{3.69}), which is clearly prohibitive for big graphs. In this paper, we provide a new and more efficient algorithm: although its worst-case complexity is O(n^4), in practice it is much faster, allowing, for the first time, the computation of the hyperbolicity of graphs with up to 200,000 nodes. We experimentally show that our new algorithm drastically outperforms the best previously available algorithms, by analyzing a big dataset of real-world networks. Finally, we apply the new algorithm to compute the hyperbolicity of random graphs generated with the Erdös-Renyi model, the Chung-Lu model, and the Configuration Model
Hyperbolicity Measures "Democracy" in Real-World Networks
We analyze the hyperbolicity of real-world networks, a geometric quantity
that measures if a space is negatively curved. In our interpretation, a network
with small hyperbolicity is "aristocratic", because it contains a small set of
vertices involved in many shortest paths, so that few elements "connect" the
systems, while a network with large hyperbolicity has a more "democratic"
structure with a larger number of crucial elements.
We prove mathematically the soundness of this interpretation, and we derive
its consequences by analyzing a large dataset of real-world networks. We
confirm and improve previous results on hyperbolicity, and we analyze them in
the light of our interpretation.
Moreover, we study (for the first time in our knowledge) the hyperbolicity of
the neighborhood of a given vertex. This allows to define an "influence area"
for the vertices in the graph. We show that the influence area of the highest
degree vertex is small in what we define "local" networks, like most social or
peer-to-peer networks. On the other hand, if the network is built in order to
reach a "global" goal, as in metabolic networks or autonomous system networks,
the influence area is much larger, and it can contain up to half the vertices
in the graph. In conclusion, our newly introduced approach allows to
distinguish the topology and the structure of various complex networks
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
Ricci Curvature of the Internet Topology
Analysis of Internet topologies has shown that the Internet topology has
negative curvature, measured by Gromov's "thin triangle condition", which is
tightly related to core congestion and route reliability. In this work we
analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin,
etc. Ricci curvature measures whether local distances diverge or converge. It
is a more local measure which allows us to understand the distribution of
curvatures in the network. We show by various Internet data sets that the
distribution of Ricci cuvature is spread out, suggesting the network topology
to be non-homogenous. We also show that the Ricci curvature has interesting
connections to both local measures such as node degree and clustering
coefficient, global measures such as betweenness centrality and network
connectivity, as well as auxilary attributes such as geographical distances.
These observations add to the richness of geometric structures in complex
network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201
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