Abstract
In this article, we investigate how the choice of the attenuation factor in an extended version of Katz centrality influences the centrality of the nodes in evolving communication networks. For given snapshots of a network, observed over a period of time, recently developed communicability indices aim to identify the best broadcasters and listeners (receivers) in the network. Here we explore the attenuation factor constraint, in relation to the spectral radius (the largest eigenvalue) of the network at any point in time and its computation in the case of large networks. We compare three different communicability measures: standard, exponential, and relaxed (where the spectral radius bound on the attenuation factor is relaxed and the adjacency matrix is normalised, in order to maintain the convergence of the measure). Furthermore, using a vitality-based measure of both standard and relaxed communicability indices, we look at the ways of establishing the most important individuals for broadcasting and receiving of messages related to community bridging roles. We compare those measures with the scores produced by an iterative version of the PageRank algorithm and illustrate our findings with three examples of real-life evolving networks: the MIT reality mining data set, consisting of daily communications between 106 individuals over the period of 1 year, a UK Twitter mentions network, constructed from the direct tweets between \(12.4\) k individuals during 1 week, and a subset of the Enron email data set.
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Notes
The matrix exponential is defined for \(A\in \mathbb {C}^{n\times n}\) by \(e^{A}=I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots .\) (Higham 2008).
If A sends a message to B today and B sends a message to C tomorrow, then A could possibly reach C with its message via B, but the opposite path from C to A is impossible, as it would go against the time-flow
The Bonacich \(\alpha -\)centrality matrix is given by \(C(\alpha , \beta , n)=\beta A +\beta \alpha _1 A^2+\cdots +\beta \prod _{k=1}^n{\alpha _k A^{n+1}}\), where A is adjacency matrix. The normalised centrality matrix is then defined as \(NC(\alpha , \beta , n \rightarrow \infty )=\frac{C(\alpha , \beta , n \rightarrow \infty )}{\sum _{ij}C_{ij}(\alpha , \beta , n \rightarrow \infty )}\)
For a real-valued function \(f\) on G, a vitality measure based on \(f\) quantifies the difference between the value \(f(G\backslash v)\) and \(f(G)\) for each \(v \in G\). See Brandes and Erlebach (2005)
Note that \(\sum _{k=0}^{\infty }k\alpha ^{k-1}=\frac{1}{(1-\alpha )^2}\).
For another extension of standard communicability that adds a penalty with regard to time see Grindrod and Higham (2013)
For more details, see https://www.cs.cmu.edu/~enron/
In this subsection, by “correlation” (also denoted by \(\text{ corr }\)) we mean the Pearson’s correlation coefficient.
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Acknowledgments
This work is funded by the RCUK Digital Economy programme via EPSRC Grant EP/G065802/1 ‘The Horizon Hub’. We would like to thank Datasift for providing us with the Twitter data set, and Colin Singleton (CountingLab) and Des Higham (University of Strathclyde) for providing us with the Enron email data set.
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Greetham, D.V., Stoyanov, Z. & Grindrod, P. On the radius of centrality in evolving communication networks. J Comb Optim 28, 540–560 (2014). https://doi.org/10.1007/s10878-014-9726-0
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DOI: https://doi.org/10.1007/s10878-014-9726-0