Centrality and Community Scoring Functions in Incomplete Networks: Their Sensitivity, Robustness, and Reliability

One of the most elegant tools for understanding the behavior of a complex system of interacting entities is network analysis. Nevertheless, often such networks are incomplete because certain edges might be missing in the construction owing to limitations in data acquisition technologies. This is an ubiquitous problem for all application areas that use network analysis ranging from social networks to hyper-linked web networks to biological networks. As a result, an important question in analyzing such networks is how certain parameters get affected by varying levels of noise (i.e., percentage of missing edges). In this paper, we focus on two distinct types of parameters-community scoring functions and centrality measures and identify the effect of removal of edges in terms of (1) the sensitivity, that is how the parameter value changes as edges are removed, (2) the robustness, that is whether the network maintains certain structural features; specifically, we measure how well the change in structural features correlates with the change in the parameters, and (3) the reliability in the context of message spreading, that is how the time taken to broadcast a message changes as edges are removed; we measure how effective the parameters are for selecting the initiator node from which the message originates. We experiment using three noise models and various synthetic and real-world networks and test the effectiveness of the parameters; a majority of the outcomes are in favor of permanence thus making it the most effective metric. For the sensitivity experiments, permanence is the clear winner narrowly followed by closeness centrality. For robustness, permanence is highly correlated with both path based and spectral property based measures, which is remarkable considering its low computation cost compared to the other parameters. For the reliability experiments, closeness and betweenness centrality based initiator selection closely competes with permanence. Surprisingly permanence is a better parameter both in terms of sensitivity and reliability which are seemingly opposite in nature. This phenomena is due to a dual characteristic of permanence where the cumulative permanence over all vertices is sensitive to noise but the ids of the top-rank vertices, which are used to find initiators during message spreading remain relatively stable under noise. We discuss, in detail, how the joint community-like and centrality-like characteristic of permanence makes it an interesting metric for noisy graphs.