57,725 research outputs found
Spectral distances on graphs
By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using Lp Wasserstein distances between probability measures, we define the corresponding spectral distances dp on the set of all graphs. This approach can even be extended to measuring the distances between infinite graphs. We prove that the diameter of the set of graphs, as a pseudo-metric space equipped with d1, is one. We further study the behavior of d1 when the size of graphs tends to infinity by interlacing inequalities aiming at exploring large real networks. A monotonic relation between d1 and the evolutionary distance of biological networks is observed in simulations
Euclidean Distances, soft and spectral Clustering on Weighted Graphs
We define a class of Euclidean distances on weighted graphs, enabling to
perform thermodynamic soft graph clustering. The class can be constructed form
the "raw coordinates" encountered in spectral clustering, and can be extended
by means of higher-dimensional embeddings (Schoenberg transformations).
Geographical flow data, properly conditioned, illustrate the procedure as well
as visualization aspects.Comment: accepted for presentation (and further publication) at the ECML PKDD
2010 conferenc
Pattern vectors from algebraic graph theory
Graphstructures have proven computationally cumbersome for pattern analysis. The reason for this is that, before graphs can be converted to pattern vectors, correspondences must be established between the nodes of structures which are potentially of different size. To overcome this problem, in this paper, we turn to the spectral decomposition of the Laplacian matrix. We show how the elements of the spectral matrix for the Laplacian can be used to construct symmetric polynomials that are permutation invariants. The coefficients of these polynomials can be used as graph features which can be encoded in a vectorial manner. We extend this representation to graphs in which there are unary attributes on the nodes and binary attributes on the edges by using the spectral decomposition of a Hermitian property matrix that can be viewed as a complex analogue of the Laplacian. To embed the graphs in a pattern space, we explore whether the vectors of invariants can be embedded in a low- dimensional space using a number of alternative strategies, including principal components analysis ( PCA), multidimensional scaling ( MDS), and locality preserving projection ( LPP). Experimentally, we demonstrate that the embeddings result in well- defined graph clusters. Our experiments with the spectral representation involve both synthetic and real- world data. The experiments with synthetic data demonstrate that the distances between spectral feature vectors can be used to discriminate between graphs on the basis of their structure. The real- world experiments show that the method can be used to locate clusters of graphs
Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy
The Laplacian energy of a graph is the sum of the distances of the
eigenvalues of the Laplacian matrix of the graph to the graph's average degree.
The maximum Laplacian energy over all graphs on nodes and edges is
conjectured to be attained for threshold graphs. We prove the conjecture to
hold for graphs with the property that for each there is a threshold graph
on the same number of nodes and edges whose sum of the largest Laplacian
eigenvalues exceeds that of the largest Laplacian eigenvalues of the graph.
We call such graphs spectrally threshold dominated. These graphs include split
graphs and cographs and spectral threshold dominance is preserved by disjoint
unions and taking complements. We conjecture that all graphs are spectrally
threshold dominated. This conjecture turns out to be equivalent to Brouwer's
conjecture concerning a bound on the sum of the largest Laplacian
eigenvalues
Just SLaQ When You Approximate: Accurate Spectral Distances for Web-Scale Graphs
Graph comparison is a fundamental operation in data mining and information
retrieval. Due to the combinatorial nature of graphs, it is hard to balance the
expressiveness of the similarity measure and its scalability. Spectral analysis
provides quintessential tools for studying the multi-scale structure of graphs
and is a well-suited foundation for reasoning about differences between graphs.
However, computing full spectrum of large graphs is computationally
prohibitive; thus, spectral graph comparison methods often rely on rough
approximation techniques with weak error guarantees. In this work, we propose
SLaQ, an efficient and effective approximation technique for computing spectral
distances between graphs with billions of nodes and edges. We derive the
corresponding error bounds and demonstrate that accurate computation is
possible in time linear in the number of graph edges. In a thorough
experimental evaluation, we show that SLaQ outperforms existing methods,
oftentimes by several orders of magnitude in approximation accuracy, and
maintains comparable performance, allowing to compare million-scale graphs in a
matter of minutes on a single machine.Comment: To appear at TheWebConf (WWW) 202
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
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