Centrality measures have been widely used to capture the properties of different nodes in a social network, particularly when the edges are fully deterministic. Various models have also been proposed to calculate nodes' centrality in graphs where there might be some uncertainty in relation to the edges. Their common characteristic is that graph uncertainty is essentially embedded into the calculation of centrality to compute a single crisp value. However, as the degree of uncertainty may vary, centrality values may also vary. In this paper, making use of fuzzy set theory, we assume that a social network is modelled by a fuzzy graph and a fuzzy variable is used to describe the truth degree of an edge between two nodes. Based on this formulation, appropriate definitions are given to determine the truth degree of different centrality values for a node and thereby centrality as a fuzzy relation. Three well-known centrality measures, degree, h-index and k-shell, are extended to calculate the truth degree for the centrality of a node in a fuzzy graph. Experimental results demonstrate that the proposed centrality measures can determine the importance of nodes in a fuzzy graph more accurately than other fuzzy or deterministic centrality measures.(c) 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
