<p>Let me use capital letter <span class="math-container">$W$</span> instead of small letter <span class="math-container">$w$</span> (as <span class="math-container">$W$</span> is actually a matrix).</p>
<p>So, let <span class="math-container">$W$</span> be the matrix that contains in row <span class="math-container">$i$</span> and column <span class="math-container">$j$</span> the probability that <span class="math-container">$j$</span> hears from <span class="math-container">$i$</span>. Then average number of pieces of information that <span class="math-container">$j$</span> gets from <span class="math-container">$i$</span> in one period is given by <span class="math-container">$W(i,j)$</span>.</p>
<p>The average number of information that <span class="math-container">$j$</span> gets from <span class="math-container">$i$</span> that takes 2 periods to travel is the information that goes first from <span class="math-container">$i$</span> to some intermediate person <span class="math-container">$k$</span> and then from <span class="math-container">$k$</span> to <span class="math-container">$j$</span>, via the path <span class="math-container">$i \to k \to j$</span>. The amount of information transmitted through this path is <span class="math-container">$W(i,k) \cdot W(k,j)$</span>. To see this, notice that <span class="math-container">$W(i,k)$</span> is the amount of info send to <span class="math-container">$k$</span> and of this a fraction <span class="math-container">$W(k,j)$</span> is sent forward to <span class="math-container">$j$</span>. Now, this has to be summed over all intermediate nodes <span class="math-container">$k$</span> (i.e. all paths of length 2). so we get:
<span class="math-container">$$
\sum_{k} W(i,k) \cdot W(k,j) = (W \times W)_{i,j} = (W^2)_{i,j}
$$</span>
Here <span class="math-container">$\times$</span> is matrix multiplication.</p>
<p>Let's now look at the information that takes 3 periods. This is the information that goes from <span class="math-container">$i$</span> to some intermediate <span class="math-container">$k$</span> then from <span class="math-container">$k$</span> to some intermediate <span class="math-container">$\ell$</span> and finally to <span class="math-container">$j$</span>, so the path <span class="math-container">$i \to k \to \ell \to j$</span>. The amount of information on this path is <span class="math-container">$W(i,k) \cdot W(k,\ell) \cdot W(\ell, j)$</span>. If we sum over all these paths of length 3, we get:
<span class="math-container">$$
\sum_k \sum_\ell W(i,k) \cdot W(k,\ell) \cdot W(\ell, j) = (W \times W \times W)_{i,j} = (W^3)_{i,j}.
$$</span>
If we continue, then we can generalize this to any path of length <span class="math-container">$T$</span>. The information sent from <span class="math-container">$i$</span> to <span class="math-container">$j$</span> through all such paths is given by:
<span class="math-container">$$
(W \times W \times \ldots W)_{i,j} = (W^T)_{i,j}.
$$</span></p>
<p>Now if info goes from <span class="math-container">$i$</span> to <span class="math-container">$j$</span> either takes 1 period, 2 periods, 3 periods, ... <span class="math-container">$T$</span> periods to travel. As such, to compute the total expected pieces of info that <span class="math-container">$j$</span> receives from <span class="math-container">$i$</span> in <span class="math-container">$T$</span> periods is given by the sum over all these periods:
<span class="math-container">$$
H(W, T) = \sum_{t = 1}^T (W^t)_{i,j}.
$$</span>
Now the total amount of info sent from <span class="math-container">$i$</span> and heard by someone can be obtained by taking the sum over all recipients <span class="math-container">$j$</span>.
<span class="math-container">$$
DC(W,T) = \sum_{j} (H(W,T))_{i,j} = (H(W,T)\times 1)_{i,j}
$$</span>
This amounts to adding up the rows of the matrix <span class="math-container">$H(W,T)$</span>.</p>
<h3>An example</h3>
<p>Let's take the example of three individuals <span class="math-container">$1,2$</span> and <span class="math-container">$3$</span>. Assume that <span class="math-container">$W$</span> is given by:
<span class="math-container">$$
W = \begin{bmatrix} 0 &amp; 0.3 &amp; 0.2\\ 0.2 &amp; 0 &amp; 0.6\\ 0 &amp; 0.4 &amp; 0\end{bmatrix}
$$</span>
This implies that (for example) the probability that 2 hears from 1 in one period equals 0.3.</p>
<p>Then
<span class="math-container">$$
W^2 = \begin{bmatrix} 0.06 &amp; 0.08 &amp; 0.18\\ 0 &amp; 0.3 &amp; 0.04\\ 0.08 &amp; 0 &amp; 0.24\end{bmatrix}
$$</span>
Here, for example, the amount received by 2 from 1 in two periods is <span class="math-container">$0.08 = 0.2 \times 0.4$</span></p>
<p>Next, multiplying <span class="math-container">$W^2$</span> once more by <span class="math-container">$W$</span> gives:
<span class="math-container">$$
W^3 = \begin{bmatrix} 0.016 &amp; 0.09 &amp; 0.06\\ 0.06 &amp; 0.016 &amp; 0.18\\ 0 &amp; 0.12 &amp; 0.016\end{bmatrix}
$$</span>
Then:
<span class="math-container">$$
H(W, 3) = W + W^2 + W^3 = \begin{bmatrix} 0.076 &amp; 0.47 &amp; 0.44\\ 0.26 &amp; 0.316 &amp; 0.82\\ 0.08 &amp; 0.52 &amp; 0.256\end{bmatrix}
$$</span>
So in three periods <span class="math-container">$2$</span> receives on average <span class="math-container">$0.47$</span> pieces of info from <span class="math-container">$1$</span></p>
<p>The total information sent from <span class="math-container">$1$</span> to someone is computed by taking the sum of the elements in row <span class="math-container">$1$</span>, which gives:
<span class="math-container">$$
0.076 + 0.47 + 0.44 = 0.986
$$</span>
In general:
<span class="math-container">$$
DC(W, 3) = \begin{bmatrix} 0.986\\1.396\\0.856\end{bmatrix}
$$</span></p>

CC BY-SA 4.0
2021-07-12T11:33:24.927Z
2021-07-12T11:33:24.927Z
tdm
<p>I am reading &quot;Using gossips to spread information: Theory and evidence from two Randomised Control Trials&quot; by Banerjee, Chandrashekhar, Duflo and Jackson, where they discuss the efficacy of gossips in spreading information regarding policies/ business offers.</p>
<p>In this paper, in page 21, they have introduced a generalised version of the network parameter &quot;Diffusion Centrality&quot;, which the same authors had defined earlier in a 2013 paper.</p>
<p>If <span class="math-container">$w$</span> is a directed and weighted adjacency matrix for a social network, then they define a 'hearing matrix' for <span class="math-container">$w$</span> and time duration <span class="math-container">$T \in N$</span> as:</p>
<p><span class="math-container">$$H(w, T) = \sum_{t=1}^T (w)^t$$</span></p>
<p>The <span class="math-container">$ij$</span>-th entry of <span class="math-container">$H$</span>, <span class="math-container">$H(w, T)_{ij}$</span> , is the expected number of times, within <span class="math-container">$T$</span> periods, that <span class="math-container">$j$</span> hears about a piece of information originating from <span class="math-container">$i$</span>.</p>
<p>Then Diffusion Centrality is defined as:</p>
<p><span class="math-container">$$DC(w, T) = H(w, T) \cdot 1$$</span></p>
<p>The paper says, &quot;<span class="math-container">$DC(w, T)_i$</span> is the expected total number of times that some piece of information
that originates from <span class="math-container">$i$</span> is heard by any of the members of the society during a <span class="math-container">$T$</span>-period
time interval&quot;.</p>
<p>I am not very familiar with network economics. I would like to understand why do <span class="math-container">$H$</span> and <span class="math-container">$DC$</span> match the verbal descriptions of them, that is, why is the <span class="math-container">$ij$</span>-th entry of <span class="math-container">$H$</span> the expected number of times <span class="math-container">$j$</span> hears something from <span class="math-container">$i$</span> and similarly for <span class="math-container">$DC$</span>. A proof directly written in the answer, or a reference to any good material to study this will help.</p>

<information><networks>
Definition of Diffusion Centrality in Banerjee et al. (2019)
Ishan Kashyap Hazarika
# Definition of Diffusion Centrality in Banerjee et al. (2019)

(Asked by: username_0 on 2021-07-12)

I am reading "Using gossips to spread information: Theory and evidence from two Randomised Control Trials" by Banerjee, Chandrashekhar, Duflo and Jackson, where they discuss the efficacy of gossips in spreading information regarding policies/ business offers.

In this paper, in page 21, they have introduced a generalised version of the network parameter "Diffusion Centrality", which the same authors had defined earlier in a 2013 paper.

If $w$ is a directed and weighted adjacency matrix for a social network, then they define a 'hearing matrix' for $w$ and time duration $T \in N$ as:

$$H(w, T) = \sum\_{t=1}^T (w)^t$$

The $ij$-th entry of $H$, $H(w, T)\_{ij}$ , is the expected number of times, within $T$ periods, that $j$ hears about a piece of information originating from $i$.

Then Diffusion Centrality is defined as:

$$DC(w, T) = H(w, T) \cdot 1$$

The paper says, "$DC(w, T)\_i$ is the expected total number of times that some piece of information
that originates from $i$ is heard by any of the members of the society during a $T$-period
time interval".

I am not very familiar with network economics. I would like to understand why do $H$ and $DC$ match the verbal descriptions of them, that is, why is the $ij$-th entry of $H$ the expected number of times $j$ hears something from $i$ and similarly for $DC$. A proof directly written in the answer, or a reference to any good material to study this will help.

**Tags:** information, networks


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## 1 answers

### Answer by username_1 on 2021-07-12. Score: 5 | Accepted answer

Let me use capital letter $W$ instead of small letter $w$ (as $W$ is actually a matrix).

So, let $W$ be the matrix that contains in row $i$ and column $j$ the probability that $j$ hears from $i$. Then average number of pieces of information that $j$ gets from $i$ in one period is given by $W(i,j)$.

The average number of information that $j$ gets from $i$ that takes 2 periods to travel is the information that goes first from $i$ to some intermediate person $k$ and then from $k$ to $j$, via the path $i \to k \to j$. The amount of information transmitted through this path is $W(i,k) \cdot W(k,j)$. To see this, notice that $W(i,k)$ is the amount of info send to $k$ and of this a fraction $W(k,j)$ is sent forward to $j$. Now, this has to be summed over all intermediate nodes $k$ (i.e. all paths of length 2). so we get:
$$
\sum\_{k} W(i,k) \cdot W(k,j) = (W \times W)\_{i,j} = (W^2)\_{i,j}
$$
Here $\times$ is matrix multiplication.

Let's now look at the information that takes 3 periods. This is the information that goes from $i$ to some intermediate $k$ then from $k$ to some intermediate $\ell$ and finally to $j$, so the path $i \to k \to \ell \to j$. The amount of information on this path is $W(i,k) \cdot W(k,\ell) \cdot W(\ell, j)$. If we sum over all these paths of length 3, we get:
$$
\sum\_k \sum\_\ell W(i,k) \cdot W(k,\ell) \cdot W(\ell, j) = (W \times W \times W)\_{i,j} = (W^3)\_{i,j}.
$$
If we continue, then we can generalize this to any path of length $T$. The information sent from $i$ to $j$ through all such paths is given by:
$$
(W \times W \times \ldots W)\_{i,j} = (W^T)\_{i,j}.
$$

Now if info goes from $i$ to $j$ either takes 1 period, 2 periods, 3 periods, ... $T$ periods to travel. As such, to compute the total expected pieces of info that $j$ receives from $i$ in $T$ periods is given by the sum over all these periods:
$$
H(W, T) = \sum\_{t = 1}^T (W^t)\_{i,j}.
$$
Now the total amount of info sent from $i$ and heard by someone can be obtained by taking the sum over all recipients $j$.
$$
DC(W,T) = \sum\_{j} (H(W,T))\_{i,j} = (H(W,T)\times 1)\_{i,j}
$$
This amounts to adding up the rows of the matrix $H(W,T)$.

### An example

Let's take the example of three individuals $1,2$ and $3$. Assume that $W$ is given by:
$$
W = \begin{bmatrix} 0 & 0.3 & 0.2\\ 0.2 & 0 & 0.6\\ 0 & 0.4 & 0\end{bmatrix}
$$
This implies that (for example) the probability that 2 hears from 1 in one period equals 0.3.

Then
$$
W^2 = \begin{bmatrix} 0.06 & 0.08 & 0.18\\ 0 & 0.3 & 0.04\\ 0.08 & 0 & 0.24\end{bmatrix}
$$
Here, for example, the amount received by 2 from 1 in two periods is $0.08 = 0.2 \times 0.4$

Next, multiplying $W^2$ once more by $W$ gives:
$$
W^3 = \begin{bmatrix} 0.016 & 0.09 & 0.06\\ 0.06 & 0.016 & 0.18\\ 0 & 0.12 & 0.016\end{bmatrix}
$$
Then:
$$
H(W, 3) = W + W^2 + W^3 = \begin{bmatrix} 0.076 & 0.47 & 0.44\\ 0.26 & 0.316 & 0.82\\ 0.08 & 0.52 & 0.256\end{bmatrix}
$$
So in three periods $2$ receives on average $0.47$ pieces of info from $1$

The total information sent from $1$ to someone is computed by taking the sum of the elements in row $1$, which gives:
$$
0.076 + 0.47 + 0.44 = 0.986
$$
In general:
$$
DC(W, 3) = \begin{bmatrix} 0.986\\1.396\\0.856\end{bmatrix}
$$


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