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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Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work, we define a combinatorial distance for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions of this paper are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the Reeb graphs, measured by the edit distance, are as small as changes in the functions, measured by the maximum norm. The optimality result states that the edit distance discriminates Reeb graphs better than any other distance for Reeb graphs of surfaces satisfying the stability property. Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work we define a combinatorial metric for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions of this paper are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the functions, measured by the maximum norm, imply not greater changes in the corresponding Reeb graphs, measured by the edit distance. The optimality result states that our edit distance discriminates Reeb graphs better than any other metric for Reeb graphs of surfaces satisfying the stability property. The spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure. It has also been employed as a graph representation for pattern matching since it is invariant to the labelling of the graph. There are, however, a number of potential drawbacks in using the spectrum as a representation of a graph. Firstly, more than one graph may share the same spectrum. It is well known, for example, that very few trees can be uniquely specified by their spectrum. Secondly, the spectrum may change dramatically with a small change structure. There are a wide variety of graph matrix representations from which the spectrum can be extracted. Among these are the adjacency matrix, combinatorial Laplacian, normalised Laplacian and unsigned Laplacian. Spectra can also be derived from the heat kernel matrix and path length distribution matrix. The choice of matrix representation clearly has a large effect on the suitability of spectrum in a number of pattern recognition tasks. In this paper we investigate the performance of the spectra as a graph representation in a variety of situations. Firstly, we investigate the cospectrality of the various matrix representations over large graph and tree sets, extending the work of previous authors. We then show that the Euclidean distance between spectra tracks the edit distance between graphs over a wide range of edit costs, and we analyse the accuracy of this relationship. We then use the spectra to both cluster and classify the graphs and demonstrate the effect of the graph matrix formulation on error rates. These results are produced using both synthetic graphs and trees and graphs derived from shape and image data. We consider the problem of self-healing in reconfigurable networks (e.g. peer-to-peer and wireless mesh networks) that are under repeated attack by an omniscient adversary and propose a fully distributed algorithm, Xheal, that maintains good expansion and spectral properties of the network, also keeping the network connected. Moreover, Xheal does this while allowing only low stretch and degree increase per node. The algorithm heals global properties like expansion and stretch while only doing local changes and using only local information. We use a model similar to that used in recent work on self-healing. In our model, over a sequence of rounds, an adversary either inserts a node with arbitrary connections or deletes an arbitrary node from the network. The network responds by quick "repairs," which consist of adding or deleting edges in an efficient localized manner. These repairs preserve the edge expansion, spectral gap, and network stretch, after adversarial deletions, without increasing node degrees by too much, in the following sense. At any point in the algorithm, the expansion of the graph will be either 'better' than the expansion of the graph formed by considering only the adversarial insertions (not the adversarial deletions) or the expansion will be, at least, a constant. Also, the stretch i.e. the distance between any pair of nodes in the healed graph is no more than a O(log n) factor. Similarly, at any point, a node v whose degree would have been d in the graph with adversarial insertions only, will have degree at most O(κ d) in the actual graph, for a small parameter κ. We also provide bounds on the second smallest eigenvalue of the Laplacian which captures key properties such as mixing time, conductance, congestion in routing etc. Our distributed data structure has low amortized latency and bandwidth requirements. Our work improves over the self-healing algorithms Forgiving tree [PODC 2008] and Forgiving graph [PODC 2009] in that we are able to give guarantees on degree and stretch, while at the same time preserving the expansion and spectral properties of the network.