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Both natural and engineered networks are often modular. Whether a network node interacts with only nodes from its own module or nodes from multiple modules provides insight into its functional role. The participation coefficient (PC) is typically used to measure this attribute although its value also depends on the size and connectedness of the module it belongs to and may lead to non-intuitive identification of highly connected nodes. Here, we develop a normalized PC that reduces the influence of intra-modular connectivity compared to the conventional PC. Using brain, C.elegans, airport and simulated networks, we show that our measure of participation is not influenced by the size or connectedness of modules, while preserving conceptual and mathematical properties, of the classic formulation of PC. Unlike the conventional PC, we identify London and New York as high participators in the air traffic network and demonstrate stronger associations with working memory in human brain networks, yielding new insights into nodal participation across network modules.

Author Summary

It is challenging to reliably quantify how single elements (i.e., nodes) in a network are connected to different sub-components (i.e., modules) of a network; this is important as inter modular connectivity contribute to efficient and distributed information processing. Participation coefficient PC calculates how distributed nodes are across modules. But PC is influenced by modularity algorithms that tends to favour large modules with strong intra module connectivity, that in turn generate low participation coefficient values, even if a node has strong inter-module connectivity. We use a network randomization approach and show that by reducing the influence of intra-modular connectivity, we obtain node participation results unaffected by size and connectedness of modules. This provides the network scientist with new insights into the inter-modular connectivity configurations of complex networks.

Mangor Pedersen*
The Florey Institute of Neuroscience and Mental Health, The University of Melbourne, Melbourne, VIC, Australia
Florey Department of Neuroscience and Mental Health, The University of Melbourne, Melbourne, Victoria, Australia
Amir Omidvarnia
The Florey Institute of Neuroscience and Mental Health, The University of Melbourne, Melbourne, VIC, Australia
Florey Department of Neuroscience and Mental Health, The University of Melbourne, Melbourne, Victoria, Australia
James M. Shine
Brain and Mind Center, The University of Sydney, Sydney, New South Wales, Australia
Graeme D. Jackson
The Florey Institute of Neuroscience and Mental Health, The University of Melbourne, Melbourne, VIC, Australia
Florey Department of Neuroscience and Mental Health, The University of Melbourne, Melbourne, Victoria, Australia
Department of Neurology, Austin Health, Melbourne, VIC, Australia
Andrew Zalesky
Department of Psychiatry, Melbourne Neuropsychiatry Centre, The University of Melbourne, VIC, Australia
* Corresponding Author:

Both natural and engineered networks are often modular. Whether a network node interacts with only nodes from its own module or nodes from multiple modules provides insight into its functional role. The participation coefficient (PC) is typically used to measure this attribute although its value also depends on the size and connectedness of the module it belongs to and may lead to non-intuitive identification of highly connected nodes. Here, we develop a normalized PC that reduces the influence of intra-modular connectivity compared to the conventional PC. Using brain, C.elegans, airport and simulated networks, we show that our measure of participation is not influenced by the size or connectedness of modules, while preserving conceptual and mathematical properties, of the classic formulation of PC. Unlike the conventional PC, we identify London and New York as high participators in the air traffic network and demonstrate stronger associations with working memory in human brain networks, yielding new insights into nodal participation across network modules.

Author Summary

It is challenging to reliably quantify how single elements (i.e., nodes) in a network are connected to different sub-components (i.e., modules) of a network; this is important as inter modular connectivity contribute to efficient and distributed information processing. Participation coefficient PC calculates how distributed nodes are across modules. But PC is influenced by modularity algorithms that tends to favour large modules with strong intra module connectivity, that in turn generate low participation coefficient values, even if a node has strong inter-module connectivity. We use a network randomization approach and show that by reducing the influence of intra-modular connectivity, we obtain node participation results unaffected by size and connectedness of modules. This provides the network scientist with new insights into the inter-modular connectivity configurations of complex networks.

Mangor Pedersen*
The Florey Institute of Neuroscience and Mental Health, The University of Melbourne, Melbourne, VIC, Australia
Florey Department of Neuroscience and Mental Health, The University of Melbourne, Melbourne, Victoria, Australia
Amir Omidvarnia
The Florey Institute of Neuroscience and Mental Health, The University of Melbourne, Melbourne, VIC, Australia
Florey Department of Neuroscience and Mental Health, The University of Melbourne, Melbourne, Victoria, Australia
James M. Shine
Brain and Mind Center, The University of Sydney, Sydney, New South Wales, Australia
Graeme D. Jackson
The Florey Institute of Neuroscience and Mental Health, The University of Melbourne, Melbourne, VIC, Australia
Florey Department of Neuroscience and Mental Health, The University of Melbourne, Melbourne, Victoria, Australia
Department of Neurology, Austin Health, Melbourne, VIC, Australia
Andrew Zalesky
Department of Psychiatry, Melbourne Neuropsychiatry Centre, The University of Melbourne, VIC, Australia
* Corresponding Author: