Abstract
A fundamental concept of social network analysis is centrality. Many analyses represent the network topology in terms of concept transmission/variation, e.g., influence, social structure, community or other aggregations. Even when the temporal nature of the network is considered, analysis is conducted at discrete points along a continuous temporal scale. Unfortunately, well-studied metrics of centrality do not take varying probabilities into account. The assumption that social and other networks that may be physically stationary, e.g., hard wired, are conceptually static in terms of information diffusion or conceptual aggregation (communities, etc.) can lead to incorrect conclusions. Our findings illustrate, both mathematically and experimentally, that if the notion of network topology is not stationary or fixed in terms of the concept, e.g., groups, belonging, community or other aggregations, centrality should be viewed probabilistically. We show through some surprising examples that study of transmission behavior based solely on a graph’s topological and degree properties is lacking when it comes to modeling network propagation or conceptual (vs. physical) structure.
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Notes
In the number of links sense, since in point set topology the network is either connected, or it is not. Of course, homological considerations can consider the extra link structures.
Please note though that even though the heuristic has its strengths and is, on the whole, a good model of virus spread, it also has its weaknesses (e.g., Lewis 2009).
We will use similar notation in the rest of the paper. If we do not specify a node it is understood it is \(n_{\mathrm{src}}\) that is under consideration.
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Acknowledgments
We acknowledge our students I. Ray, R. Kang and G. Whelan and our colleagues P. Cotae, P. Safier and M. Kang. Special thanks go to Swmbo Heilizer and Andre Harrison and to the reviewers for their very helpful suggestions and corrections.
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Moskowitz, I.S., Hyden, P. & Russell, S. Network topology and mean infection times. Soc. Netw. Anal. Min. 6, 34 (2016). https://doi.org/10.1007/s13278-016-0338-9
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DOI: https://doi.org/10.1007/s13278-016-0338-9