Abstract
In graph theory and network analysis, various measures of centrality are used to characterize the importance of vertices in a graph. Although different measures of centrality have been invented to suit the nature and requirements of different underlying problem domains, their application is restricted to explicit graphs. In this paper, we first define implicit graphs that involve auxiliary vertices in addition to the pertinent vertices. We then generalize the various measures of centrality on explicit graphs to corresponding measures of projected centrality on implicit graphs. We also propose a unifying framework for approximately, but very efficiently computing the top-K pertinent vertices in implicit graphs for various measures of projected centrality. Our framework is based on FastMap, a graph embedding algorithm that embeds a given undirected graph into a Euclidean space in near-linear time such that the pairwise Euclidean distances between vertices approximate a desired graph-based distance function between them. Using FastMap’s ability to facilitate geometric interpretations and analytical procedures in Euclidean space, we show that the top-K vertices for many popularly used measures of centrality—and their generalizations to projected centrality—can be computed very efficiently in our framework.
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Notes
- 1.
The clique on the pertinent vertices is a mere conceptualization. Constructing it explicitly may be prohibitively expensive for large graphs since it requires the computation of the graph-based distance between every pair of the pertinent vertices.
- 2.
The edit distance between two strings is the minimum number of insertions, deletions, or substitutions that are needed to transform one to the other.
- 3.
unless |E| is O(|V|), in which case the complexity is near-linear in the size of the input because of the \(\log |V|\) factor.
- 4.
For disconnected graphs, we usually consider the measures of centrality and projected centrality on each connected component separately.
- 5.
It is also conceivable to use the square-root of the PASPD function.
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- 7.
- 8.
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Acknowledgments
This work at the University of Southern California is supported by DARPA under grant number HR001120C0157 and by NSF under grant number 2112533. The views, opinions, and/or findings expressed are those of the author(s) and should not be interpreted as representing the official views or policies of the sponsoring organizations, agencies, or the U.S. Government. This research is also partially funded by the Australian Government through the Australian Research Council Industrial Transformation Training Centre in Optimisation Technologies, Integrated Methodologies, and Applications (OPTIMA), Project ID IC200100009.
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Li, A., Stuckey, P., Koenig, S., Kumar, T.K.S. (2024). A FastMap-Based Framework for Efficiently Computing Top-K Projected Centrality. In: Nicosia, G., Ojha, V., La Malfa, E., La Malfa, G., Pardalos, P.M., Umeton, R. (eds) Machine Learning, Optimization, and Data Science. LOD 2023. Lecture Notes in Computer Science, vol 14505. Springer, Cham. https://doi.org/10.1007/978-3-031-53969-5_13
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