56,356 research outputs found
Limits of dense graph sequences
We show that if a sequence of dense graphs has the property that for every
fixed graph F, the density of copies of F in these graphs tends to a limit,
then there is a natural ``limit object'', namely a symmetric measurable
2-variable function on [0,1]. This limit object determines all the limits of
subgraph densities. We also show that the graph parameters obtained as limits
of subgraph densities can be characterized by ``reflection positivity'',
semidefiniteness of an associated matrix. Conversely, every such function
arises as a limit object. Along the lines we introduce a rather general model
of random graphs, which seems to be interesting on its own right.Comment: 27 pages; added extension of result (Sept 22, 2004
An theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions
We introduce and develop a theory of limits for sequences of sparse graphs
based on graphons, which generalizes both the existing theory
of dense graph limits and its extension by Bollob\'as and Riordan to sparse
graphs without dense spots. In doing so, we replace the no dense spots
hypothesis with weaker assumptions, which allow us to analyze graphs with power
law degree distributions. This gives the first broadly applicable limit theory
for sparse graphs with unbounded average degrees. In this paper, we lay the
foundations of the theory of graphons, characterize convergence, and
develop corresponding random graph models, while we prove the equivalence of
several alternative metrics in a companion paper.Comment: 44 page
Multigraph limit of the dense configuration model and the preferential attachment graph
The configuration model is the most natural model to generate a random multigraph with a given degree sequence. We use the notion of dense graph limits to characterize the special form of limit objects of convergent sequences of configuration models. We apply these results to calculate the limit object corresponding to the dense preferential attachment graph and the edge reconnecting model. Our main tools in doing so are (1) the relation between the theory of graph limits and that of partially exchangeable random arrays (2) an explicit construction of our random graphs that uses urn model
Multigraph limit of the dense configuration model and the preferential attachment graph
The configuration model is the most natural model to generate a random
multigraph with a given degree sequence.
We use the notion of dense graph limits to characterize the special form of
limit objects of convergent sequences of configuration models. We apply these
results to calculate the limit object corresponding to the dense preferential
attachment graph and the edge reconnecting model. Our main tools in doing so
are (1) the relation between the theory of graph limits and that of partially
exchangeable random arrays (2) an explicit construction of our random graphs
that uses urn models.Comment: Some of the results of this submission already appeared in an older
version of arXiv:0912.3904v3, "Time evolution of dense multigraph limits
under edge-conservative preferential attachment dynamics." Accepted for
publication in Acta Mathematica Hungaric
Measures on the square as sparse graph limits
We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on [0, 1](2). This leads to a generalization of dense graph limit theory to sparse graph sequences. (C) 2019 Elsevier Inc. All rights reserved
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