20 research outputs found
Invariant measures concentrated on countable structures
Let L be a countable language. We say that a countable infinite L-structure M
admits an invariant measure when there is a probability measure on the space of
L-structures with the same underlying set as M that is invariant under
permutations of that set, and that assigns measure one to the isomorphism class
of M. We show that M admits an invariant measure if and only if it has trivial
definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary
finite tuple of M fixes no additional points. When M is a Fraisse limit in a
relational language, this amounts to requiring that the age of M have strong
amalgamation. Our results give rise to new instances of structures that admit
invariant measures and structures that do not.Comment: 46 pages, 2 figures. Small changes following referee suggestion
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
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