I just wanted to go to a recent video and tell you that you are a very important person to me. Your video on Ramsey theory popped into my feed 3 and a half years ago, and it inspired me greatly. I spent months searching for R(5,5) and teaching myself combinatorics and graph theory. I took some courses, and I was hooked. Now, I'm looking into PhD programs and I can absolutely say I found my passion in it. I spend every day thinking about graphs and combinatorics and you helped me find it.
I feel like talking through the construction a bit more would be enlightening. Closeness = distance, sure! So we examine the distances from our central vertex to every other vertex in the graph. Then, we need a way to collect this data into a single statistic. Summing seems like a natural option as it allows each value to contribute its weight to the final answer. But wait! If we sum, then the vertices that are FARTHER from others will have a higher centrality measure! We want to invert this - allowing vertices that are CLOSER to have higher centrality measures. So, we take 1/sum to order it as we wish. A natural question might be "Why don't we invert the distances and then sum them?". This is a case where order of operations matters quite a bit to interpretation! If we invert first, then we are saying that we want the larger distances to contribute less to the final statistic than the smaller distances. So, our final sum will now represent the contribution of these inverted distances. If this seems interesting to you, take a look at this follow on: If we use Sum(1/d(u,v)) rather than 1/Sum(d(u,v)), does the order change? If so, what causes the differences in orders?
I just wanted to go to a recent video and tell you that you are a very important person to me.
Your video on Ramsey theory popped into my feed 3 and a half years ago, and it inspired me greatly.
I spent months searching for R(5,5) and teaching myself combinatorics and graph theory. I took some courses, and I was hooked.
Now, I'm looking into PhD programs and I can absolutely say I found my passion in it. I spend every day thinking about graphs and combinatorics and you helped me find it.
Very direct and informative; I appreciate the upload!
I feel like talking through the construction a bit more would be enlightening.
Closeness = distance, sure! So we examine the distances from our central vertex to every other vertex in the graph.
Then, we need a way to collect this data into a single statistic. Summing seems like a natural option as it allows each value to contribute its weight to the final answer.
But wait! If we sum, then the vertices that are FARTHER from others will have a higher centrality measure! We want to invert this - allowing vertices that are CLOSER to have higher centrality measures. So, we take 1/sum to order it as we wish.
A natural question might be "Why don't we invert the distances and then sum them?". This is a case where order of operations matters quite a bit to interpretation! If we invert first, then we are saying that we want the larger distances to contribute less to the final statistic than the smaller distances. So, our final sum will now represent the contribution of these inverted distances. If this seems interesting to you, take a look at this follow on: If we use Sum(1/d(u,v)) rather than 1/Sum(d(u,v)), does the order change? If so, what causes the differences in orders?
This is really great comment! I have not thought about sum(1/d(u,v)), very interesting. Let me do a little analysis and see what I find!
Harmonic Centrality !