214,341 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
Randomness in topological models
p. 914-925There are two aspects of randomness in topological models. In the first one, topological
idealization of random patterns found in the Nature can be regarded as planar
representations of three-dimensional lattices and thus reconstructed in the space. Another aspect of randomness is related to graphs in which some properties are determined in a random way. For example, combinatorial properties of graphs: number of vertices, number of edges, and connections between them can be regarded as events in the defined probability space. Random-graph theory deals with a question: at what connection probability a particular property reveals. Combination of probabilistic description of planar graphs and their spatial reconstruction creates new opportunities in structural form-finding, especially in the inceptive, the most creative, stage.Tarczewski, R.; Bober, W. (2010). Randomness in topological models. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/695
Metrics for Graph Comparison: A Practitioner's Guide
Comparison of graph structure is a ubiquitous task in data analysis and
machine learning, with diverse applications in fields such as neuroscience,
cyber security, social network analysis, and bioinformatics, among others.
Discovery and comparison of structures such as modular communities, rich clubs,
hubs, and trees in data in these fields yields insight into the generative
mechanisms and functional properties of the graph.
Often, two graphs are compared via a pairwise distance measure, with a small
distance indicating structural similarity and vice versa. Common choices
include spectral distances (also known as distances) and distances
based on node affinities. However, there has of yet been no comparative study
of the efficacy of these distance measures in discerning between common graph
topologies and different structural scales.
In this work, we compare commonly used graph metrics and distance measures,
and demonstrate their ability to discern between common topological features
found in both random graph models and empirical datasets. We put forward a
multi-scale picture of graph structure, in which the effect of global and local
structure upon the distance measures is considered. We make recommendations on
the applicability of different distance measures to empirical graph data
problem based on this multi-scale view. Finally, we introduce the Python
library NetComp which implements the graph distances used in this work
Graphical Markov models, unifying results and their interpretation
Graphical Markov models combine conditional independence constraints with
graphical representations of stepwise data generating processes.The models
started to be formulated about 40 years ago and vigorous development is
ongoing. Longitudinal observational studies as well as intervention studies are
best modeled via a subclass called regression graph models and, especially
traceable regressions. Regression graphs include two types of undirected graph
and directed acyclic graphs in ordered sequences of joint responses. Response
components may correspond to discrete or continuous random variables and may
depend exclusively on variables which have been generated earlier. These
aspects are essential when causal hypothesis are the motivation for the
planning of empirical studies.
To turn the graphs into useful tools for tracing developmental pathways and
for predicting structure in alternative models, the generated distributions
have to mimic some properties of joint Gaussian distributions. Here, relevant
results concerning these aspects are spelled out and illustrated by examples.
With regression graph models, it becomes feasible, for the first time, to
derive structural effects of (1) ignoring some of the variables, of (2)
selecting subpopulations via fixed levels of some other variables or of (3)
changing the order in which the variables might get generated. Thus, the most
important future applications of these models will aim at the best possible
integration of knowledge from related studies.Comment: 34 Pages, 11 figures, 1 tabl
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