8,730 research outputs found
On edge transitivity of directed graphs
We examine edge transitivity of directed graphs. The class of local comparability graphs is defined as the underlying graphs of locally edge transitive digraphs. The latter generalize edge transitive orientations, while local comparability graphs include comparability, anti-comparability and circle graphs. Recognizing local comperability graphs is NP-complete, however they are differences of comparability graphs. We define dimension so as to generalize that of an edge transitive digraph. Connect proper interval graphs are characterized as exaclty the class of local comparability graphs of dimension one. Finally, a characterization of circle graphs is given also in terms of edge transitivity.Examinamos transitividade em arestas de grafos direcionados. A classe dos grafos de comparabilidade local é definida como os grafos subjacentes dos dígrafos localmente transitivos em arestas. Estes últimos generalizam orientações transitivas em arestas, enquanto que grafos de comparabilidade local incluem os de comparabilidade, anti-comparabilidade e circulares. Reconhecer grafos de comparabilidade local é NP-completo, contudo, eles constituem diferenças de grafos de comparabilidade. Definimos dimensão de modo a generalizar a de um dígrafo transitivo em arestas. Os grafos conexos de intervalo próprio são caracterizados exatamente como a classe dos de comparabilidade local de dimensão um. Finalmente, uma caracterização dos grafos circulares é apresentada em termos de transitividade em arestas
On the notion of balance in social network analysis
The notion of "balance" is fundamental for sociologists who study social
networks. In formal mathematical terms, it concerns the distribution of triad
configurations in actual networks compared to random networks of the same edge
density. On reading Charles Kadushin's recent book "Understanding Social
Networks", we were struck by the amount of confusion in the presentation of
this concept in the early sections of the book. This confusion seems to lie
behind his flawed analysis of a classical empirical data set, namely the karate
club graph of Zachary. Our goal here is twofold. Firstly, we present the notion
of balance in terms which are logically consistent, but also consistent with
the way sociologists use the term. The main message is that the notion can only
be meaningfully applied to undirected graphs. Secondly, we correct the analysis
of triads in the karate club graph. This results in the interesting observation
that the graph is, in a precise sense, quite "unbalanced". We show that this
lack of balance is characteristic of a wide class of starlike-graphs, and
discuss possible sociological interpretations of this fact, which may be useful
in many other situations.Comment: Version 2: 23 pages, 4 figures. An extra section has been added
towards the end, to help clarify some things. Some other minor change
On semi-transitive orientability of Kneser graphs and their complements
An orientation of a graph is semi-transitive if it is acyclic, and for any
directed path either
there is no edge between and , or is an edge
for all . An undirected graph is semi-transitive if it admits
a semi-transitive orientation. Semi-transitive graphs include several important
classes of graphs such as 3-colorable graphs, comparability graphs, and circle
graphs, and they are precisely the class of word-representable graphs studied
extensively in the literature.
In this paper, we study semi-transitive orientability of the celebrated
Kneser graph , which is the graph whose vertices correspond to the
-element subsets of a set of elements, and where two vertices are
adjacent if and only if the two corresponding sets are disjoint. We show that
for , is not semi-transitive, while for , is semi-transitive. Also, we show computationally that a
subgraph on 16 vertices and 36 edges of , and thus itself
on 56 vertices and 280 edges, is non-semi-transitive. and are the
first explicit examples of triangle-free non-semi-transitive graphs, whose
existence was established via Erd\H{o}s' theorem by Halld\'{o}rsson et al. in
2011. Moreover, we show that the complement graph of
is semi-transitive if and only if
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
Wedge Sampling for Computing Clustering Coefficients and Triangle Counts on Large Graphs
Graphs are used to model interactions in a variety of contexts, and there is
a growing need to quickly assess the structure of such graphs. Some of the most
useful graph metrics are based on triangles, such as those measuring social
cohesion. Algorithms to compute them can be extremely expensive, even for
moderately-sized graphs with only millions of edges. Previous work has
considered node and edge sampling; in contrast, we consider wedge sampling,
which provides faster and more accurate approximations than competing
techniques. Additionally, wedge sampling enables estimation local clustering
coefficients, degree-wise clustering coefficients, uniform triangle sampling,
and directed triangle counts. Our methods come with provable and practical
probabilistic error estimates for all computations. We provide extensive
results that show our methods are both more accurate and faster than
state-of-the-art alternatives.Comment: Full version of SDM 2013 paper "Triadic Measures on Graphs: The Power
of Wedge Sampling" (arxiv:1202.5230
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