Abstract
We introduce a new class of power centrality measures, which we refer to as Dominance Centrality, for complex networks which may have weighted, directed edges, as well as weights associated with nodes. These measures are similar in spirit to the power-over-powerless class introduced by Bonacich decades ago in the context of eigenvector centrality, but are quite different in their roots and effect. Our Dominance Centrality measure, referred to as DONEX, is derived as a Pareto optimal solution to collective welfare maximisation problem that allocates values to nodes in the network on the basis of expected interaction strengths captured by edge weights between pairs of nodes, where the expectation is taken over the probability of there being an undirected or directed link between them. We show how this formulation yields a new centrality measure which captures the notion of a weighted degree-differential virtual flow along the edges in the network in such a manner that the sum of such flows at each node itself represents the notion of dominance over the neighbourhood. We develop a greedy propagation algorithm called NetProp which allows us to estimate the reach of dominant nodes, and network boundaries where the dominance-derived influence or attention are maximum and minimum. We also demonstrate how DONEX extends to multi-hop neighbourhoods taking account of both local and non-local effects. The new methods are illustrated extensively with synthetic and real networks.
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Notes
Even though the Configuration Model, as explained by Newman in [16], was proposed and examined by many authors in the context of network applications in physics, its variants have found wider utility in the context of community detection problems [60, 61] as a means for defining the notion of modularity [4, 5]. The central premise in the Newman model is that the probability with which a node with degree \(\alpha_{i}\) attaches its edge to another node in a fixed graph with n nodes, is proportional to its own degree, \(\alpha_{i}\), normalised to values < = 1, by dividing by 2 m =|E|, the number of edges in the graph, assuming that the probabilities for the two ends of a single edge are independent of each other. Since the probability that there exists an edge between node i and node j must be symmetric, it must be proportional to the product of their degrees.
This, as is evident, is quite different from deriving directions simply on the basis of degree-differential alone.
Note that the Fiedler vector, obtained from eigenspace of the Laplacian in (16) could also give us a sparse-cut into two clusters of nodes that the connected network could have been split into. But they would have been obtained only on the basis of degree connectivity, and not on the basis of flows which lead to dominance.
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Acknowledgements
The authors thank anonymous reviewers for comments and suggestions to improve the quality of this paper. The first author thanks Prof Srinath Srinivasa, and research students Raksha RP and Jayati Deshmukh of the Web Science Lab, IIIT, Bangalore for helping to explore many applications of the central ideas of dominance centrality set out in this work, including early versions of DON and DONEX measures. Thanks are also due to Prof Anjan Dasgupta, Department of Biochemistry, Calcutta University, for many illuminating discussions and close collaboration in the development of time-series analysis applications of the basic dominance metrics described here.
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Mandyam Kannappan, S., Sridhar, U. A Flow-Based Node Dominance Centrality Measure for Complex Networks. SN COMPUT. SCI. 3, 379 (2022). https://doi.org/10.1007/s42979-022-01270-2
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DOI: https://doi.org/10.1007/s42979-022-01270-2