The phenomena of community structure in real-world networks has been central to network science's success. While much attention has been paid to community detectionalgorithms and evaluation criteria, a related question has been overlooked: what would the change in modularity be if a node were to be deleted from a network ? In this work, we first show that the answer to this question can be calculated for all nodes in the network in O(m) time. Then ,  we show that this quantity has implications in three research areas within network science: community analysis, network robustness, and community deception. Modularity-deltas are useful in community analysis as they are a semi-global measure , scalable like local measures but using global information through the network's partition.In robustness, modularity maximizing attack strategies are more efficient than betweenness-based ones, and are shown to damage the US Powergrid and European Road networks more effectively than degree based attacks . Finally, we show that inverting the robustness method to select modularity minimizing nodes solves the community deception problem in a scalable manner .
Group-centrality is an emerging area in Network Science which seeks to quantify the importance of nodes within and between communities. Previous group-centrality measures either have strong theoretical underpinnings in community detection , or scale well, but not both. We propose an extremely scalable measure, modularity-impact, that is directly derived from Newman's modularity. Unlike other measures, modularity-impact is signed, providing additional information about each node's role: hub within its community, or bridge between communities. We analyze our results in a network robustness framework, as it is a generalized version of immunization under SIR models, which prior works have used for evaluation. We show that our measure out-performs previous measures for immunizing strategies in generated networks with strong community structure. Additionally, we show that our measure gives the most effective attack strategy for a million-node infrastructure network, fragmenting it while removing less than 2\% of nodes. This is over 8 times more fragile than can be seen with previous group-centralities. We further find that the social media communication network demonstrates extreme robustness to group-centrality attacks, suggesting that user-based interventions to mitigate misinformation diffusion will be ineffective . Finally, we show that our group-centrality measure can be used as a greedy solution to the community deception problem .
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The phenomena of community structure in real-world networks has been central to network science's success. While much attention has been paid to community detectionalgorithms and evaluation criteria, a related question has been overlooked: what would the change in modularity be if a node were to be deleted from a network ? In this work, we first show that the answer to this question can be calculated for all nodes in the network in O(m) time. Then
Group-centrality is an emerging area in Network Science which seeks to quantify the importance of nodes within and between communities. Previous group-centrality measures either have strong theoretical underpinnings in community detection
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or scale well, but not both. We propose an extremely scalable measure, modularity-impact, that is directly derived from Newman's modularity. Unlike other measures, modularity-impact is signed, providing additional information about each node's role: hub within its community, or bridge between communities. We analyze our results in a network robustness framework, as it is a generalized version of immunization under SIR models, which prior works have used for evaluation. We show that our measure out-performs previous measures for immunizing strategies in generated networks with strong community structure. Additionally,
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this quantity has implications in three research areas within network science: community analysis, network robustness, and community deception. Modularity-deltas are useful in community analysis as they are a semi-global measure , scalable like local measures but using global information through the network's partition.In robustness, modularity maximizing attack strategies are more efficient than betweenness-based ones, and are shown to damage the US Powergrid and European Road networks more effectively than degree based attacks
our measure gives the most effective attack strategy for a million-node infrastructure network, fragmenting it while removing less than 2\% of nodes. This is over 8 times more fragile than can be seen with previous group-centralities. We further find that the social media communication network demonstrates extreme robustness to group-centrality attacks, suggesting that user-based interventions to mitigate misinformation diffusion will be ineffective
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inverting the robustness method to select modularity minimizing nodes solves
our group-centrality measure can be used as a greedy solution to
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in a scalable manner