  Community identification of network components enables us to understand the
mesoscale clustering structure of networks. A number of algorithms have been
developed to determine the most likely community structures in networks. Such a
probabilistic or stochastic nature of this problem can naturally involve the
ambiguity in resultant community structures. More specifically, stochastic
algorithms can result in different community structures for each realization in
principle. In this study, instead of trying to "solve" this community
degeneracy problem, we turn the tables by taking the degeneracy as a chance to
quantify how strong companionship each node has with other nodes. For that
purpose, we define the concept of companionship inconsistency that indicates
how inconsistently a node is identified as a member of a community regarding
the other nodes. Analyzing model and real networks, we show that companionship
inconsistency discloses unique characteristics of nodes, thus we suggest it as
a new type of node centrality. In social networks, for example, companionship
inconsistency can classify outsider nodes without firm community membership and
promiscuous nodes with multiple connections to several communities. In
infrastructure networks such as power grids, it can diagnose how the connection
structure is evenly balanced in terms of power transmission. Companionship
inconsistency, therefore, abstracts individual nodes' intrinsic property on its
relationship to a higher-order organization of the network.
