# Centrality Application: What affects Diffusion?¶

• First contact points: let us examine how network positions of injection

• points matter

# Banerjee, Chandrasekhar, Duflo, Jackson, Diffusion of Microfinance (2012)¶

• 75 rural villages in Karnataka, relatively isolated from microfinance initially

• BSS entered 43 of them and offered microfinance

• We surveyed villages before entry, observed network structure and various demographics

• Tracked microfinance participation over time

# Background: 75 Indian Villages - Networks¶

• "Favor" Networks:

• both borrow and lend money
• both borrow and lend kero-rice
• "Social" Networks:

• both visit come and go
• friends (talk together most)
• Others (temple, medical help...)

# Data also include¶

• Microfinance participation by individual, time

• Number of households and their composition

• Demogratphics: age, gender, subcaste, religion, professfion, education level, family...

• Wealther variables: latrine, number rooms, roof

• Self Help Group participation rate, ration card, voting

• Caste: village fraction of "higher castes" (GM/FC and OBC, remainder are SC/ST)

# Degree Centrality¶

• Count how many links a node has
• node 7 and 6 would be the most central individuals in the village, and if you hit those individuals, you would expect to reach more just because they have higher degree.

# Hypothesis¶

• In villages where first contacted people have more connections, there should be a better spread of information about microfinance

• more people knowing should lead to higher participation

So it doesn't appear as if degree centrality really captures what's going on. So we need another centrality

Let's have a look at Eigenvector centrality, we realized that looking at degree doesn't tell a lot of the story because it doesn't capture how well you are positioned in a network. And so if we look at Eigenvector Centrality, where we have the centrality being proportional to the sum of the centralities of your neighbors, then we are getting something which reflects this better connectedness, as we talked about in the last lecture. Okay, so let's have a fiat and look and see if Eigenvector centrality does a better job.

### Eigenvector Centrality¶

• If centrality is proportional to the sum of neighbors' centralities, then we're defining eigenvector centrality. So:

$C_i$ proportional to $\sum_{j:friend.of.i}C_j$

$C_i = a\sum_j g_{ij}C_j$

• $g_{ij} = 1$, then $c_j$ being counted
• $g_{ij} = 0$, then $c_j$ not counted
• we're just counting the centrality's of the friends you're connected to.

Basically what we have is that the vector C is equal to sum a times the matrix g times the vector C. And so this is what's known as an Eigenvector: $$C = agC$$

# Hypothesis Revised¶

• In villages where first contacted people have higher eigenvector centrality, there should be a better spread of information about microfinance

• more people knowing should lead to higher participation

Now we get a significantly positive and strong relationship. So having better placed leaders in terms of eigenvector centrality does a reasonably good job of prediction the eventual mark microfinance participation, whereas the degree centrality didn't seem to pick things up.

the idea here is that, why's eigenvector centrality is working better? Because, you know this communication's a repeated process. You tell your friends. They have to tell their friends. And so forth. So if you have well-positioned friends, and they have well-positioned friends, that is good for diffusion. An eigenvector centrality is measuring that whereas degree centrality is not.

• MF Participation: Regress micro finance participation on a series of variables