11,877 research outputs found
Geometric Inhomogeneous Random Graphs
Real-world networks, like social networks or the internet infrastructure, have structural properties such as their large clustering coefficient 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. However, we do not directly study hyperbolic random graphs, but replace them by a more general model that we call \emph{geometric inhomogeneous random graphs} (GIRGs). Since we ignore constant factors in the edge probabilities, our model is 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 our new model in future theoretical studies. We prove the following fundamental structural and algorithmic results on GIRGs. (1) 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 factor , (2) we establish that GIRGs have a constant clustering coefficient, (3) we show 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
Maximal Cliques in Scale-Free Random Graphs
We investigate the number of maximal cliques, i.e., cliques that are not
contained in any larger clique, in three network models: Erd\H{o}s-R\'enyi
random graphs, inhomogeneous random graphs (also called Chung-Lu graphs), and
geometric inhomogeneous random graphs. For sparse and not-too-dense
Erd\H{o}s-R\'enyi graphs, we give linear and polynomial upper bounds on the
number of maximal cliques. For the dense regime, we give super-polynomial and
even exponential lower bounds. Although (geometric) inhomogeneous random graphs
are sparse, we give super-polynomial lower bounds for these models. This comes
form the fact that these graphs have a power-law degree distribution, which
leads to a dense subgraph in which we find many maximal cliques. These lower
bounds seem to contradict previous empirical evidence that (geometric)
inhomogeneous random graphs have only few maximal cliques. We resolve this
contradiction by providing experiments indicating that, even for large
networks, the linear lower-order terms dominate, before the super-polynomial
asymptotic behavior kicks in only for networks of extreme size
Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs
Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a power-law degree distribution with controllable exponent beta, and high clustering that can be controlled via the temperature T.
We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to T = 0. We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation.
Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input.
Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straight-forward inclusion does not hold in practice. However, the difference is negligible for most use cases
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
Geometric Inhomogeneous Random Graphs for Algorithm Engineering
The design and analysis of graph algorithms is heavily based on the worst case.
In practice, however, many algorithms perform much better than the worst case would suggest.
Furthermore, various problems can be tackled more efficiently if one assumes the input to be, in a sense, realistic.
The field of network science, which studies the structure and emergence of real-world networks, identifies locality and heterogeneity as two frequently occurring properties.
A popular model that captures these properties are geometric inhomogeneous random graphs (GIRGs), which is a generalization of hyperbolic random graphs (HRGs).
Aside from their importance to network science, GIRGs can be an immensely valuable tool in algorithm engineering.
Since they convincingly mimic real-world networks, guarantees about quality and performance of an algorithm on instances of the model can be transferred to real-world applications.
They have model parameters to control the amount of heterogeneity and locality, which allows to evaluate those properties in isolation while keeping the rest fixed.
Moreover, they can be efficiently generated which allows for experimental analysis.
While realistic instances are often rare, generated instances are readily available.
Furthermore, the underlying geometry of GIRGs helps to visualize the network, e.g.,~for debugging or to improve understanding of its structure.
The aim of this work is to demonstrate the capabilities of geometric inhomogeneous random graphs in algorithm engineering and establish them as routine tools to replace previous models like the Erd\H{o}s-R{\\u27e}nyi model, where each edge exists with equal probability.
We utilize geometric inhomogeneous random graphs to design, evaluate, and optimize efficient algorithms for realistic inputs.
In detail, we provide the currently fastest sequential generator for GIRGs and HRGs and describe algorithms for maximum flow, directed spanning arborescence, cluster editing, and hitting set.
For all four problems, our implementations beat the state-of-the-art on realistic inputs.
On top of providing crucial benchmark instances, GIRGs allow us to obtain valuable insights.
Most notably, our efficient generator allows us to
experimentally show sublinear running time of our flow algorithm,
investigate the solution structure of cluster editing,
complement our benchmark set of arborescence instances with a density for which there are no real-world networks available,
and generate networks with adjustable locality and heterogeneity to reveal the effects of these properties on our algorithms
Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs
In this paper we study weighted distances in scale-free spatial network
models: hyperbolic random graphs (HRG), geometric inhomogeneous random graphs
(GIRG) and scale-free percolation (SFP). In HRGs,
vertices are sampled independently from the hyperbolic disk with radius and
two vertices are connected either when they are within hyperbolic distance ,
or independently with a probability depending on the hyperbolic distance. In
GIRGs and SFP, each vertex is given an independent weight and location from an
underlying measured metric space and , respectively, and two
vertices are connected independently with a probability that is a function of
their distance and weights. We assign i.i.d. weights to the edges of the random
graphs and study the weighted distance between two uniformly chosen vertices.
In SFP, we study the weighted distance from the origin of vertex-sequences with
norm tending to infinity. In particular, we study the case when the parameters
are so that the degree distribution in the graph follows a power law with
exponent (infinite variance), and the edge-weight distribution
is such that it produces an explosive age-dependent branching process with
power-law offspring distribution. We show that in all three models, typical
distances within the giant/infinite component converge in distribution, solving
an open question in [Explosion and distances in scale-free percolation (2017)].
The main tools of our proof are to couple the models to infinite versions, to
follow the shortest paths to infinity and to connect these paths using
weight-dependent percolation on the graphs: delete edges attached to vertices
with higher weight with higher probability. We realise this using the
edge-weights: only short edges connected to high weight vertices will stay,
yielding arbitrarily short upper bounds for the connections.Comment: 49 pages, 4 figure
Cliques in High-Dimensional Geometric Inhomogeneous Random Graphs
A recent trend in the context of graph theory is to bring theoretical analyses closer to empirical observations, by focusing the studies on random graph models that are used to represent practical instances. There, it was observed that geometric inhomogeneous random graphs (GIRGs) yield good representations of complex real-world networks, by expressing edge probabilities as a function that depends on (heterogeneous) vertex weights and distances in some underlying geometric space that the vertices are distributed in. While most of the parameters of the model are understood well, it was unclear how the dimensionality of the ground space affects the structure of the graphs.
In this paper, we complement existing research into the dimension of geometric random graph models and the ongoing study of determining the dimensionality of real-world networks, by studying how the structure of GIRGs changes as the number of dimensions increases. We prove that, in the limit, GIRGs approach non-geometric inhomogeneous random graphs and present insights on how quickly the decay of the geometry impacts important graph structures. In particular, we study the expected number of cliques of a given size as well as the clique number and characterize phase transitions at which their behavior changes fundamentally. Finally, our insights help in better understanding previous results about the impact of the dimensionality on geometric random graphs
Efficiently generating geometric inhomogeneous and hyperbolic random graphs
Hyperbolic random graphs (HRGs) and geometric inhomogeneous random graphs (GIRGs) are two similar generative network models that were designed to resemble complex real-world networks. In particular, they have a power-law degree distribution with controllable exponent and high clustering that can be controlled via the temperature .
We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support non-zero temperatures. Though non-zero temperatures are crucial for many applications, most existing generators are restricted to . We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, that is, they involve no approximation.
Besides the generators themselves, we also provide an efficient algorithm to determine the non-trivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify the desired expected average degree as input.
Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straightforward inclusion does not hold in practice. However, the difference is negligible for most use cases
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