Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric graphs in the context of comparison, we are interested in determining the relative discriminative capabilities of two topology-based distances between a pair of arbitrary finite metric graphs: the persistence distortion distance and the intrinsic Cech distance. We explicitly show how to compute the intrinsic Cech distance between two metric graphs based solely on knowledge of the shortest systems of loops for the graphs. Our main theorem establishes an inequality between the intrinsic Cech and persistence distortion distances in the case when one of the graphs is a bouquet graph and the other is arbitrary. The relationship also holds when both graphs are constructed via wedge sums of cycles and edges.
Well-known methods for comparing graphs using distance measures include combinatorial (e.g., graph edit distance @cite_14 ) and spectral (e.g., eigenvalue decomposition @cite_4 ) approaches. Graph edit distance minimizes the cost of transforming one graph to another via a set of elementary operators such as node edge insertions deletions, while spectral approaches optimize objective functions based on properties of the graph spectra.
@cite_14 @cite_4
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Graph data have become ubiquitous and manipulating them based on similarity is essential for many applications. Graph edit distance is one of the most widely accepted measures to determine similarities between graphs and has extensive applications in the fields of pattern recognition, computer vision etc. Unfortunately, the problem of graph edit distance computation is NP-Hard in general. Accordingly, in this paper we introduce three novel methods to compute the upper and lower bounds for the edit distance between two graphs in polynomial time. Applying these methods, two algorithms AppFull and AppSub are introduced to perform different kinds of graph search on graph databases. Comprehensive experimental studies are conducted on both real and synthetic datasets to examine various aspects of the methods for bounding graph edit distance. Result shows that these methods achieve good scalability in terms of both the number of graphs and the size of graphs. The effectiveness of these algorithms also confirms the usefulness of using our bounds in filtering and searching of graphs. In this paper we generalize and improve the multiscale organization of graphs by introducing a new measure that quantifies the "closeness" between two nodes. The calculation of the measure is linear in the number of edges in the graph and involves just a small number of relaxation sweeps. A similar notion of distance is then calculated and used at each coarser level. We demonstrate the use of this measure in multiscale methods for several important combinatorial optimization problems and discuss the multiscale graph organization.