470 research outputs found
Pruning based Distance Sketches with Provable Guarantees on Random Graphs
Measuring the distances between vertices on graphs is one of the most
fundamental components in network analysis. Since finding shortest paths
requires traversing the graph, it is challenging to obtain distance information
on large graphs very quickly. In this work, we present a preprocessing
algorithm that is able to create landmark based distance sketches efficiently,
with strong theoretical guarantees. When evaluated on a diverse set of social
and information networks, our algorithm significantly improves over existing
approaches by reducing the number of landmarks stored, preprocessing time, or
stretch of the estimated distances.
On Erd\"{o}s-R\'{e}nyi graphs and random power law graphs with degree
distribution exponent , our algorithm outputs an exact distance
data structure with space between and
depending on the value of , where is the number of vertices. We
complement the algorithm with tight lower bounds for Erdos-Renyi graphs and the
case when is close to two.Comment: Full version for the conference paper to appear in The Web
Conference'1
Causal and homogeneous networks
Growing networks have a causal structure. We show that the causality strongly
influences the scaling and geometrical properties of the network. In particular
the average distance between nodes is smaller for causal networks than for
corresponding homogeneous networks. We explain the origin of this effect and
illustrate it using as an example a solvable model of random trees. We also
discuss the issue of stability of the scale-free node degree distribution. We
show that a surplus of links may lead to the emergence of a singular node with
the degree proportional to the total number of links. This effect is closely
related to the backgammon condensation known from the balls-in-boxes model.Comment: short review submitted to AIP proceedings, CNET2004 conference;
changes in the discussion of the distance distribution for growing trees,
Fig. 6-right change
Properties of stochastic Kronecker graphs
The stochastic Kronecker graph model introduced by Leskovec et al. is a
random graph with vertex set , where two vertices and
are connected with probability
independently of the presence or absence of any other edge, for fixed
parameters . They have shown empirically that the
degree sequence resembles a power law degree distribution. In this paper we
show that the stochastic Kronecker graph a.a.s. does not feature a power law
degree distribution for any parameters . In addition,
we analyze the number of subgraphs present in the stochastic Kronecker graph
and study the typical neighborhood of any given vertex.Comment: 37 pages, 2 figure
Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory
We define in this paper several Hopf algebras describing the combinatorics of
the so-called multi-scale renormalization in quantum field theory. After a
brief recall of the main mathematical features of multi-scale renormalization,
we define assigned graphs, that are graphs with appropriate decorations for the
multi-scale framework. We then define Hopf algebras on these assigned graphs
and on the Gallavotti-Nicol\`o trees, particular class of trees encoding the
supplementary informations of the assigned graphs. Several morphisms between
these combinatorial Hopf algebras and the Connes-Kreimer algebra are given.
Finally, scale dependent couplings are analyzed via this combinatorial
algebraic setting.Comment: 26 pages, 3 figures; the presentation of the results has been
reorganized. Several details of various proofs are given and some references
have been adde
- …