Diffusion Centrality

Diffusion centrality (Banerjee et al. 2013) counts the walks of length 1 to $T$ that start at a node, each discounted by $q$ per step: $$DC(A; q, T) = \sum_{t=1}^{T} (qA)^t \mathbf{1}.$$ Walks may revisit nodes and return to the source, so the score counts repeated hearings of a message.

Usage

centrality_diffusion_centrality(x, diffusion_q = 1, diffusion_steps = 3, ...)

Arguments

x: Network input accepted by centrality.
diffusion_q: Discount $q$, between 0 and 1. Default 1.
diffusion_steps: Horizon $T$, a nonnegative integer. Default 3.
...: Further arguments to centrality. The measure uses weighted (default TRUE), loops (default TRUE) and normalized (default FALSE).

Value

A named numeric vector with one score per node, in input node order.

Details

$A$ holds the edge weights, or ones with weighted = FALSE. Directed edges carry information from source to target, so the score uses row sums and mode has no effect. Transposing the input gives incoming walks. Self-loops follow loops. Negative or non-finite weights raise an error. $T = 0$ gives 0 and $T = 1$ gives $q$ times the out-strength. When every entry of $qA$ lies between 0 and 1 the score is the expected number of times the information is heard (Banerjee et al. 2013), and it can exceed the number of nodes. The defaults $q = 1$ and $T = 3$ are package choices. This measure differs from centrality_diffusion.

References

Banerjee, A., Chandrasekhar, A. G., Duflo, E., & Jackson, M. O. (2013). The Diffusion of Microfinance. Science, 341, 1236498. doi:10.1126/science.1236498 .

Banerjee, A., Chandrasekhar, A. G., Duflo, E., & Jackson, M. O. (2019). Using Gossips to Spread Information: Theory and Evidence from Two Randomized Controlled Trials. Review of Economic Studies, 86, 2453-2490. doi:10.1093/restud/rdz008 .

Examples

centrality_diffusion_centrality(regulation_net)
#>    Explore       Plan    Monitor      Adapt    Reflect    Discuss Synthesize 
#>   1.265867   3.898775   1.365553   1.617645   0.399290   1.429347   1.124945 
#>   Evaluate     Create      Share 
#>   1.755606   2.225349   2.720083