302,257 research outputs found
Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups
We examine two criteria for balance of a gain graph, one based on binary
cycles and one on circles. The graphs for which each criterion is valid depend
on the set of allowed gain groups. The binary cycle test is invalid, except for
forests, if any possible gain group has an element of odd order. Assuming all
groups are allowed, or all abelian groups, or merely the cyclic group of order
3, we characterize, both constructively and by forbidden minors, the graphs for
which the circle test is valid. It turns out that these three classes of groups
have the same set of forbidden minors. The exact reason for the importance of
the ternary cyclic group is not clear.Comment: 19 pages, 3 figures. Format: Latex2e. Changes: minor. To appear in
Journal of Graph Theor
Renormalization of noncommutative phi 4-theory by multi-scale analysis
In this paper we give a much more efficient proof that the real Euclidean phi
4-model on the four-dimensional Moyal plane is renormalizable to all orders. We
prove rigorous bounds on the propagator which complete the previous
renormalization proof based on renormalization group equations for non-local
matrix models. On the other hand, our bounds permit a powerful multi-scale
analysis of the resulting ribbon graphs. Here, the dual graphs play a
particular r\^ole because the angular momentum conservation is conveniently
represented in the dual picture. Choosing a spanning tree in the dual graph
according to the scale attribution, we prove that the summation over the loop
angular momenta can be performed at no cost so that the power-counting is
reduced to the balance of the number of propagators versus the number of
completely inner vertices in subgraphs of the dual graph.Comment: 34 page
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
Six signed Petersen graphs, and their automorphisms
Up to switching isomorphism there are six ways to put signs on the edges of
the Petersen graph. We prove this by computing switching invariants, especially
frustration indices and frustration numbers, switching automorphism groups,
chromatic numbers, and numbers of proper 1-colorations, thereby illustrating
some of the ideas and methods of signed graph theory. We also calculate
automorphism groups and clusterability indices, which are not invariant under
switching. In the process we develop new properties of signed graphs,
especially of their switching automorphism groups.Comment: 39 pp., 7 fi
Experimental analysis of the accessibility of drawings with few segments
The visual complexity of a graph drawing is defined as the number of
geometric objects needed to represent all its edges. In particular, one object
may represent multiple edges, e.g., one needs only one line segment to draw two
collinear incident edges. We study the question if drawings with few segments
have a better aesthetic appeal and help the user to asses the underlying graph.
We design an experiment that investigates two different graph types (trees and
sparse graphs), three different layout algorithms for trees, and two different
layout algorithms for sparse graphs. We asked the users to give an aesthetic
ranking on the layouts and to perform a furthest-pair or shortest-path task on
the drawings.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Regularized spectral methods for clustering signed networks
We study the problem of -way clustering in signed graphs. Considerable
attention in recent years has been devoted to analyzing and modeling signed
graphs, where the affinity measure between nodes takes either positive or
negative values. Recently, Cucuringu et al. [CDGT 2019] proposed a spectral
method, namely SPONGE (Signed Positive over Negative Generalized Eigenproblem),
which casts the clustering task as a generalized eigenvalue problem optimizing
a suitably defined objective function. This approach is motivated by social
balance theory, where the clustering task aims to decompose a given network
into disjoint groups, such that individuals within the same group are connected
by as many positive edges as possible, while individuals from different groups
are mainly connected by negative edges. Through extensive numerical
simulations, SPONGE was shown to achieve state-of-the-art empirical
performance. On the theoretical front, [CDGT 2019] analyzed SPONGE and the
popular Signed Laplacian method under the setting of a Signed Stochastic Block
Model (SSBM), for equal-sized clusters, in the regime where the graph is
moderately dense.
In this work, we build on the results in [CDGT 2019] on two fronts for the
normalized versions of SPONGE and the Signed Laplacian. Firstly, for both
algorithms, we extend the theoretical analysis in [CDGT 2019] to the general
setting of unequal-sized clusters in the moderately dense regime.
Secondly, we introduce regularized versions of both methods to handle sparse
graphs -- a regime where standard spectral methods underperform -- and provide
theoretical guarantees under the same SSBM model. To the best of our knowledge,
regularized spectral methods have so far not been considered in the setting of
clustering signed graphs. We complement our theoretical results with an
extensive set of numerical experiments on synthetic data.Comment: 55 pages, 5 figure
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