165,766 research outputs found
Random planar maps and graphs with minimum degree two and three
We find precise asymptotic estimates for the number of planar maps and graphs
with a condition on the minimum degree, and properties of random graphs from
these classes. In particular we show that the size of the largest tree attached
to the core of a random planar graph is of order c log(n) for an explicit
constant c. These results provide new information on the structure of random
planar graphs.Comment: 32 page
Further results on random cubic planar graphs
We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.Peer ReviewedPostprint (author's final draft
Logical properties of random graphs from small addable classes
We establish zero-one laws and convergence laws for monadic second-order
logic (MSO) (and, a fortiori, first-order logic) on a number of interesting
graph classes. In particular, we show that MSO obeys a zero-one law on the
class of connected planar graphs, the class of connected graphs of tree-width
at most and the class of connected graphs excluding the -clique as a
minor. In each of these cases, dropping the connectivity requirement leads to a
class where the zero-one law fails but a convergence law for MSO still holds
Asymptotic study of subcritical graph classes
International audienceWe present a unified general method for the asymptotic study of graphs from the so-called subcritical graph classes, which include the classes of cacti graphs, outerplanar graphs, and series-parallel graphs. This general method works both in the labelled and unlabelled framework. The main results concern the asymptotic enumeration and the limit laws of properties of random graphs chosen from subcritical classes. We show that the number (resp. ) of labelled (resp. unlabelled) graphs on vertices from a subcritical graph class {\cG}=\cup_n {\cG_n} satisfies asymptotically the universal behaviour for computable constants , e.g. for unlabelled series-parallel graphs, and that the number of vertices of degree ( fixed) in a graph chosen uniformly at random from \cG_n, converges (after rescaling) to a normal law as
LASAGNE: Locality And Structure Aware Graph Node Embedding
In this work we propose Lasagne, a methodology to learn locality and
structure aware graph node embeddings in an unsupervised way. In particular, we
show that the performance of existing random-walk based approaches depends
strongly on the structural properties of the graph, e.g., the size of the
graph, whether the graph has a flat or upward-sloping Network Community Profile
(NCP), whether the graph is expander-like, whether the classes of interest are
more k-core-like or more peripheral, etc. For larger graphs with flat NCPs that
are strongly expander-like, existing methods lead to random walks that expand
rapidly, touching many dissimilar nodes, thereby leading to lower-quality
vector representations that are less useful for downstream tasks. Rather than
relying on global random walks or neighbors within fixed hop distances, Lasagne
exploits strongly local Approximate Personalized PageRank stationary
distributions to more precisely engineer local information into node
embeddings. This leads, in particular, to more meaningful and more useful
vector representations of nodes in poorly-structured graphs. We show that
Lasagne leads to significant improvement in downstream multi-label
classification for larger graphs with flat NCPs, that it is comparable for
smaller graphs with upward-sloping NCPs, and that is comparable to existing
methods for link prediction tasks
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