51,079 research outputs found
On the rational spectra of graphs with abelian singer groups
AbstractLet G be a finite abelian group. We investigate those graphs G admitting G as a sharply 1-transitive automorphism group and all of whose eigenvalues are rational. The study is made via the rational algebra P(G) of rational matrices with rational eigenvalues commuting with the regular matrix representation of G. In comparing the spectra obtainable for graphs in P(G) for various G's, we relate subschemes of a related association scheme, subalgebras of P(G), and the lattice of subgroups of G. One conclusion is that if the order of G is fifth-power-free, any graph with rational eigenvalues admitting G has a cospectral mate admitting the abelian group of the same order with prime-order elementary divisors
Representation and generation of plans using graph spectra
Numerical comparison of spaces with one another is often achieved with set scalar
measures such as global and local integration, connectivity, etc., which capture a
particular quality of the space but therefore lose much of the detail of its overall
structure. More detailed methods such as graph edit distance are difficult to calculate,
particularly for large plans. This paper proposes the use of the graph spectrum, or the
ordered eigenvalues of a graph adjacency matrix, as a means to characterise the space
as a whole. The result is a vector of high dimensionality that can be easily measured
against others for detailed comparison.
Several graph types are investigated, including boundary and axial representations, as
are several methods for deriving the spectral vector. The effectiveness of these is
evaluated using a genetic algorithm optimisation to generate plans to match a given
spectrum, and evolution is seen to produce plans similar to the initial targets, even in
very large search spaces. Results indicate that boundary graphs alone can capture the
gross topological qualities of a space, but axial graphs are needed to indicate local
relationships. Methods of scaling the spectra are investigated in relation to both global
local changes to plan arrangement. For all graph types, the spectra were seen to
capture local patterns of spatial arrangement even as global size is varied
Representation and generation of plans using graph spectra
Numerical comparison of spaces with one another is often achieved with set scalar measures such as global and local integration, connectivity, etc., which capture a particular quality of the space but therefore lose much of the detail of its overall structure. More detailed methods such as graph edit distance are difficult to calculate, particularly for large plans. This paper proposes the use of the graph spectrum, or the ordered eigenvalues of a graph adjacency matrix, as a means to characterise the space as a whole. The result is a vector of high dimensionality that can be easily measured
against others for detailed comparison.
Several graph types are investigated, including boundary and axial representations, as are several methods for deriving the spectral vector. The effectiveness of these is evaluated using a genetic algorithm optimisation to generate plans to match a given spectrum, and evolution is seen to produce plans similar to the initial targets, even in very large search spaces. Results indicate that boundary graphs alone can capture the
gross topological qualities of a space, but axial graphs are needed to indicate local relationships. Methods of scaling the spectra are investigated in relation to both global local changes to plan arrangement. For all graph types, the spectra were seen to capture local patterns of spatial arrangement even as global size is varied
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
- âŠ