20,391 research outputs found
The largest strongly connected component in Wakeley et al's cyclical pedigree model
We establish a link between Wakeley et al's (2012) cyclical pedigree model
from population genetics and a randomized directed configuration model (DCM)
considered by Cooper and Frieze (2004). We then exploit this link in
combination with asymptotic results for the in-degree distribution of the
corresponding DCM to compute the asymptotic size of the largest strongly
connected component (where is the population size) of the DCM resp.
the pedigree. The size of the giant component can be characterized explicitly
(amounting to approximately of the total populations size) and thus
contributes to a reduced `pedigree effective population size'. In addition, the
second largest strongly connected component is only of size .
Moreover, we describe the size and structure of the `domain of attraction' of
. In particular, we show that with high probability for any individual the
shortest ancestral line reaches after generations, while
almost all other ancestral lines take at most generations.Comment: 21 pages, 2 figure
Ramsey games with giants
The classical result in the theory of random graphs, proved by Erdos and
Renyi in 1960, concerns the threshold for the appearance of the giant component
in the random graph process. We consider a variant of this problem, with a
Ramsey flavor. Now, each random edge that arrives in the sequence of rounds
must be colored with one of R colors. The goal can be either to create a giant
component in every color class, or alternatively, to avoid it in every color.
One can analyze the offline or online setting for this problem. In this paper,
we consider all these variants and provide nontrivial upper and lower bounds;
in certain cases (like online avoidance) the obtained bounds are asymptotically
tight.Comment: 29 pages; minor revision
Random graphs containing arbitrary distributions of subgraphs
Traditional random graph models of networks generate networks that are
locally tree-like, meaning that all local neighborhoods take the form of trees.
In this respect such models are highly unrealistic, most real networks having
strongly non-tree-like neighborhoods that contain short loops, cliques, or
other biconnected subgraphs. In this paper we propose and analyze a new class
of random graph models that incorporates general subgraphs, allowing for
non-tree-like neighborhoods while still remaining solvable for many fundamental
network properties. Among other things we give solutions for the size of the
giant component, the position of the phase transition at which the giant
component appears, and percolation properties for both site and bond
percolation on networks generated by the model.Comment: 12 pages, 6 figures, 1 tabl
Sampling Geometric Inhomogeneous Random Graphs in Linear Time
Real-world networks, like social networks or the internet infrastructure,
have structural properties such as large clustering coefficients that can best
be described in terms of an underlying geometry. This is why the focus of the
literature on theoretical models for real-world networks shifted from classic
models without geometry, such as Chung-Lu random graphs, to modern
geometry-based models, such as hyperbolic random graphs.
With this paper we contribute to the theoretical analysis of these modern,
more realistic random graph models. Instead of studying directly hyperbolic
random graphs, we use a generalization that we call geometric inhomogeneous
random graphs (GIRGs). Since we ignore constant factors in the edge
probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic
cosines), while preserving the qualitative behaviour of hyperbolic random
graphs, and we suggest to replace hyperbolic random graphs by this new model in
future theoretical studies.
We prove the following fundamental structural and algorithmic results on
GIRGs. (1) As our main contribution we provide a sampling algorithm that
generates a random graph from our model in expected linear time, improving the
best-known sampling algorithm for hyperbolic random graphs by a substantial
factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in
{\Omega}(1), (3) we prove that GIRGs have small separators, i.e., it suffices
to delete a sublinear number of edges to break the giant component into two
large pieces, and (4) we show how to compress GIRGs using an expected linear
number of bits.Comment: 25 page
The k-core and branching processes
The k-core of a graph G is the maximal subgraph of G having minimum degree at
least k. In 1996, Pittel, Spencer and Wormald found the threshold
for the emergence of a non-trivial k-core in the random graph ,
and the asymptotic size of the k-core above the threshold. We give a new proof
of this result using a local coupling of the graph to a suitable branching
process. This proof extends to a general model of inhomogeneous random graphs
with independence between the edges. As an example, we study the k-core in a
certain power-law or `scale-free' graph with a parameter c controlling the
overall density of edges. For each k at least 3, we find the threshold value of
c at which the k-core emerges, and the fraction of vertices in the k-core when
c is \epsilon above the threshold. In contrast to , this
fraction tends to 0 as \epsilon tends to 0.Comment: 30 pages, 1 figure. Minor revisions. To appear in Combinatorics,
Probability and Computin
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