6,773 research outputs found
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
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
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
Clique topology reveals intrinsic geometric structure in neural correlations
Detecting meaningful structure in neural activity and connectivity data is
challenging in the presence of hidden nonlinearities, where traditional
eigenvalue-based methods may be misleading. We introduce a novel approach to
matrix analysis, called clique topology, that extracts features of the data
invariant under nonlinear monotone transformations. These features can be used
to detect both random and geometric structure, and depend only on the relative
ordering of matrix entries. We then analyzed the activity of pyramidal neurons
in rat hippocampus, recorded while the animal was exploring a two-dimensional
environment, and confirmed that our method is able to detect geometric
organization using only the intrinsic pattern of neural correlations.
Remarkably, we found similar results during non-spatial behaviors such as wheel
running and REM sleep. This suggests that the geometric structure of
correlations is shaped by the underlying hippocampal circuits, and is not
merely a consequence of position coding. We propose that clique topology is a
powerful new tool for matrix analysis in biological settings, where the
relationship of observed quantities to more meaningful variables is often
nonlinear and unknown.Comment: 29 pages, 4 figures, 13 supplementary figures (last two authors
contributed equally
Degree correlations in scale-free null models
We study the average nearest neighbor degree of vertices with degree
. In many real-world networks with power-law degree distribution
falls off in , a property ascribed to the constraint that any two vertices
are connected by at most one edge. We show that indeed decays in in
three simple random graph null models with power-law degrees: the erased
configuration model, the rank-1 inhomogeneous random graph and the hyperbolic
random graph. We consider the large-network limit when the number of nodes
tends to infinity. We find for all three null models that starts to
decay beyond and then settles on a power law , with the degree exponent.Comment: 21 pages, 4 figure
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
- …