3,654 research outputs found
Connectivity for bridge-alterable graph classes
A collection of graphs is called bridge-alterable if, for each graph G with a
bridge e, G is in the class if and only if G-e is. For example the class of
forests is bridge-alterable. For a random forest sampled uniformly from
the set of forests on vertex set {1,..,n}, a classical result of Renyi (1959)
shows that the probability that is connected is .
Recently Addario-Berry, McDiarmid and Reed (2012) and Kang and Panagiotou
(2013) independently proved that, given a bridge-alterable class, for a random
graph sampled uniformly from the graphs in the class on {1,..,n}, the
probability that is connected is at least . Here we give
a more straightforward proof, and obtain a stronger non-asymptotic form of this
result, which compares the probability to that for a random forest. We see that
the probability that is connected is at least the minimum over of the probability that is connected.Comment: Amplified the discussion on raising the lower bound 2/5 to 1/
Random graphs from a weighted minor-closed class
There has been much recent interest in random graphs sampled uniformly from
the n-vertex graphs in a suitable minor-closed class, such as the class of all
planar graphs. Here we use combinatorial and probabilistic methods to
investigate a more general model. We consider random graphs from a
`well-behaved' class of graphs: examples of such classes include all
minor-closed classes of graphs with 2-connected excluded minors (such as
forests, series-parallel graphs and planar graphs), the class of graphs
embeddable on any given surface, and the class of graphs with at most k
vertex-disjoint cycles. Also, we give weights to edges and components to
specify probabilities, so that our random graphs correspond to the random
cluster model, appropriately conditioned.
We find that earlier results extend naturally in both directions, to general
well-behaved classes of graphs, and to the weighted framework, for example
results concerning the probability of a random graph being connected; and we
also give results on the 2-core which are new even for the uniform (unweighted)
case.Comment: 46 page
Connectivity for random graphs from a weighted bridge-addable class
There has been much recent interest in random graphs sampled uniformly from
the n-vertex graphs in a suitable structured class, such as the class of all
planar graphs. Here we consider a general 'bridge-addable' class of graphs - if
a graph is in the class and u and v are vertices in different components then
the graph obtained by adding an edge (bridge) between u and v must also be in
the class.
Various bounds are known concerning the probability of a random graph from
such a class being connected or having many components, sometimes under the
additional assumption that bridges can be deleted as well as added. Here we
improve or amplify or generalise these bounds. For example, we see that the
expected number of vertices left when we remove a largest component is less
than 2. The generalisation is to consider 'weighted' random graphs, sampled
from a suitable more general distribution, where the focus is on the bridges.Comment: 23 page
Modularity of regular and treelike graphs
Clustering algorithms for large networks typically use modularity values to
test which partitions of the vertex set better represent structure in the data.
The modularity of a graph is the maximum modularity of a partition. We consider
the modularity of two kinds of graphs.
For -regular graphs with a given number of vertices, we investigate the
minimum possible modularity, the typical modularity, and the maximum possible
modularity. In particular, we see that for random cubic graphs the modularity
is usually in the interval , and for random -regular graphs
with large it usually is of order . These results help to
establish baselines for statistical tests on regular graphs.
The modularity of cycles and low degree trees is known to be close to 1: we
extend these results to `treelike' graphs, where the product of treewidth and
maximum degree is much less than the number of edges. This yields for example
the (deterministic) lower bound mentioned above on the modularity of
random cubic graphs.Comment: 25 page
Random graphs from a block-stable class
A class of graphs is called block-stable when a graph is in the class if and
only if each of its blocks is. We show that, as for trees, for most -vertex
graphs in such a class, each vertex is in at most blocks, and each path passes through at most blocks.
These results extend to `weakly block-stable' classes of graphs
Anonymous network access using the digital marketplace
With increasing usage of mobile telephony, and the trend towards additional mobile Internet usage, privacy and anonymity become more and more important. Previously-published anonymous communication schemes aim to obscure their users' network addresses, because real-world identity can be easily be derived from this information. We propose modifications to a novel call-management architecture, the digital marketplace, which will break this link, therefore enabling truly anonymous network access
Random graphs with few disjoint cycles
The classical Erd\H{o}s-P\'{o}sa theorem states that for each positive
integer k there is an f(k) such that, in each graph G which does not have k+1
disjoint cycles, there is a blocker of size at most f(k); that is, a set B of
at most f(k) vertices such that G-B has no cycles. We show that, amongst all
such graphs on vertex set {1,..,n}, all but an exponentially small proportion
have a blocker of size k. We also give further properties of a random graph
sampled uniformly from this class; concerning uniqueness of the blocker,
connectivity, chromatic number and clique number. A key step in the proof of
the main theorem is to show that there must be a blocker as in the
Erd\H{o}s-P\'{o}sa theorem with the extra `redundancy' property that B-v is
still a blocker for all but at most k vertices v in B
Random Perfect Graphs
We investigate the asymptotic structure of a random perfect graph
sampled uniformly from the perfect graphs on vertex set . Our
approach is based on the result of Pr\"omel and Steger that almost all perfect
graphs are generalised split graphs, together with a method to generate such
graphs almost uniformly.
We show that the distribution of the maximum of the stability number
and clique number is close to a concentrated
distribution which plays an important role in our generation method. We
also prove that the probability that contains any given graph as an
induced subgraph is asymptotically or or . Further we show
that almost all perfect graphs are -clique-colourable, improving a result of
Bacs\'o et al from 2004; they are almost all Hamiltonian; they almost all have
connectivity equal to their minimum degree; they are almost all
in class one (edge-colourable using colours, where is the
maximum degree); and a sequence of independently and uniformly sampled perfect
graphs of increasing size converges almost surely to the graphon
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