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What is the notation in this formula that causes it to produce a vector? I found it in a paper and I just cannot figure it out.

  • $\mathbf{g}$ is a $N\times N$ matrix. (graph adjacency matrix)
  • $q$ is a value between zero and one (a probability)
  • $T$ is a value greater than zero (time)

Text reads....

We define the diffusion centrality of a node i in a network with an adjacency matrix $\mathbf{g}$, passing probability $q$, and iterations $T$, as the $i$-th entry of the vector: $$ DC(\mathbf{g};q,T) := \left [ \sum_{t=1}^{T}(q\mathbf{g})^{t} \right ]\cdot 1 $$

(For complete info and a great paper: Banerjee, Chandrasekhar, Duflo,& Jackson, The diffusion of microfinance. Science, 2013)

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    What is $1$? A vector(!) of 1s, possibly? – 2017-01-04
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    That makes sense thanks! – 2017-01-05

2 Answers 2

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The sum in the brackets is clearly an $N\times N$ matrix (same size as matrix $g$). So the ultimate value becoming a vector has something to do with that "$\cdot1$" in the end. You should check the original reference for the authors' notation. But I'm gonna go on a limb here and assume that this "$1$" may stand for the $N\times1$ vector whose all components are ones: $$\mathbf{1}=\begin{bmatrix}1\\1\\\vdots\\1\end{bmatrix}$$

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    In which case you’re summing across the rows of the $N\times N$ matrix. – 2017-01-04
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The $1$ here is probably the vector $\mathbf{1}=(1,1,\dots,1)$, so that the product is between a $N\times N$ matrix and a $N\times 1$ vector, resulting in a $N\times 1$ vector. Of course, one would need the text to confirm this.