12,019 research outputs found
Distance-Dependent Kronecker Graphs for Modeling Social Networks
This paper focuses on a generalization of stochastic
Kronecker graphs, introducing a Kronecker-like operator and
defining a family of generator matrices H dependent on distances
between nodes in a specified graph embedding. We prove
that any lattice-based network model with sufficiently small
distance-dependent connection probability will have a Poisson
degree distribution and provide a general framework to prove
searchability for such a network. Using this framework, we focus
on a specific example of an expanding hypercube and discuss
the similarities and differences of such a model with recently
proposed network models based on a hidden metric space. We
also prove that a greedy forwarding algorithm can find very short
paths of length O((log log n)^2) on the hypercube with n nodes,
demonstrating that distance-dependent Kronecker graphs can
generate searchable network models
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
On the Fiedler value of large planar graphs
The Fiedler value , also known as algebraic connectivity, is the
second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler
value among all planar graphs with vertices, denoted by
, and we show the bounds . We also provide bounds on the maximum
Fiedler value for the following classes of planar graphs: Bipartite planar
graphs, bipartite planar graphs with minimum vertex degree~3, and outerplanar
graphs. Furthermore, we derive almost tight bounds on for two
more classes of graphs, those of bounded genus and -minor-free graphs.Comment: 21 pages, 4 figures, 1 table. Version accepted in Linear Algebra and
Its Application
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
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