Welcome to the course blog for MS&E 135 Networks at Stanford University. This course explores how diverse social, economic, and technological systems are built up from connections, and how the study of networks can help us understand these systems.

Enrolled undergraduate students will be writing regular posts on varied subject matters and current events related to the course. Topics include: networked markets, social networks, information networks, the aggregate behavior of crowds, information diffusion, the implications of popular concepts such as “six degrees of separation” and the “friendship paradox.”

The blog is visible to the public, however only students and course staff are able to post and comment. Students should refer to the course blog guide for more information.

The Influencer Pay Gap, Power, and Small World Core-Periphery Structure: Why Marketers Pay More for White Influence

A study by the PR company MSL and The Influencer League revealed that the pay gap between white and Black influencers is 35%. The study suggests that this gap is not only driven by Black influencers having less followers on average, but also on a lack of pay transparency between Black and white influencers with the same following. 

The basis for the influencer marketing industry is that advertisers are willing to pay to manufacture an information cascade through paying influential people with many (one directional) connections to recommend that their audience buys their product. Brands are especially willing to pay niche influencers, whose audiences represent dense communities, where there is a higher threshold for information cascades/new product adoption. Influencer marketing allows brands to mobilize small world phenomena and diffusion of information in order to increase their brand awareness and sales.

I’d argue that the reason a racial pay gap exists and persists in the influencer marketing industry is partly due to historical, systemic disadvantages placing Black influencers on the periphery of the influence industry’s small world, having less followers. The reason this racial pay gap exists, even between Black and white influencers of the same following, is because advertisers perceive the (usually white) audiences of white influencers as more valuable and less dense than the peripheral (Black) audiences of Black influencers. The idea behind core-periphery structure in small worlds is born out of the finding that Milgram-style small world targeted search performs best when the target person is affluent. In a core-periphery model, the core is a dense network of high-status folks, and the low-status people are organized in the more clustered and local periphery. Milgram-style targeted search between two low-status people usually travels into the core and back out. For that reason, brands are not likely to find it as valuable to pay for peripheral Black influencers to advertise their products to their even lower-status, more peripheral audiences. 

As suggested by MSL and the Influencer League, the clear solution here is to establish pay transparency across the industry and pull Black influencers into the core of the industry, rather than devalue them due to systemic disadvantages. This will not only allow for the prosperity of more Black influencers, but also for brands to more effectively penetrate the Black community, which has much more spending power than the pay gap suggests brands believe.

Article: https://www.prnewswire.com/news-releases/msl-study-reveals-racial-pay-gap-in-influencer-marketing-301437451.html?tc=eml_cleartime

Fortnite and Connections

Fortnite is a video game, similar in a sense to the fictional Hunger Games series. 100 people are dropped in a map in a battle royale, and a fight to the death ensues, in which the area that the players can travel is shortened over time, drawing them closer and closer and into contact with one another.

In the course of the game, players are eliminated by one another. I charted the course of 10 games and the end result of and each player to record the average connections.

On average, 8 players on average from each game were disconnected, so they were “alone” in a sense and had no connections. For the purpose of this experiment, I ignored these.

64 players on average were eliminated with zero kills. Thus, their only connection to the game was being eliminated. That means, a total of 28 people had on average at least a kill a game… and that the average kills per game of people that had at least 1 kill was 2.28 kills on average.
The most connections across the 10 games I watched were 23 — a player, over the course of 1 game, killed 23 separate players and did not die himself once.

The most common amount of connections is one — players that are killed without a kill themselves. Following that, the most common occurrence is two — a player that records one kill, and then suffers death at the hands of another player.

Here are the results of the average across the 10 games:
Disconnected (0 connections): 8
1 connection: 64
2 connections: 13
3 connections: 6
4+ connections: 9

The most connected person was only 3 connections away from any one player in their game mode, whereas the furthest apart in connections that I found was a connection that was 17 connections apart.

Elon Musk’s Hyperloops and Braess’s Paradox

Elon Musk’s Boring Company seeks to “solve traffic” through building loops and hyperloops. According to their website, these tunnels “provide high-throughput transportation in an economically viable way.” They claim the hyperloop “enables access to individualized, point to point high-speed transportation.” The company has already built a few R&D loops, including a tunnel underneath the Las Vegas Convention Center, which has already been experiencing considerable traffic congestion. Videos shared by attendees of the Consumer Electronics Show (CES) in January of this year show people stuck in traffic of about 90 Teslas congested in the single lane underground tunnel.

Braess’s paradox can explain why Boring Company’s tunnel networks may not actually “solve traffic.” Adding a new route to a transportation system can sometimes increase everyone’s traffic time in the new equilibrium. Specifically, when the perceived payoff of taking the new route makes taking the shortcut a dominant strategy for every driver, the new equilibrium will likely increase everyone’s travel time, rather than decrease it.

It is worth noting, though, that Musk and Boring Company’s ideal world would be one where the tunnels like those at CES are exclusively driven through by autonomous cars, which would ideally synchronize in order to eliminate traffic congestion. Especially in these controlled, single lane environments, it has been shown that autonomous vehicles are likely to speed up traffic flow significantly. Theoretically, a single-lane traffic route with only autonomous vehicles would be one with a travel time mostly insensitive to congestion. However, as long as there are humans behind the wheel, it is unlikely that Boring Company’s loops will eliminate traffic for us all.

Articles: https://mashable.com/article/ces-las-vegas-boring-tunnel-tesla-traffic , https://www.boringcompany.com/tunnels , https://www.traffictechnologytoday.com/news/autonomous-vehicles/new-study-finds-driverless-vehicles-could-create-more-traffic-congestion.html

A Network of Bots

As social media networks have grown and allowed people to connect digitally, measures like ‘Friends’ lists and follower counts have become visual representations of the vast networks each individual is a part of. However, among the “friend” and “follow” requests from real life acquaintances, there are accounts mimicking authentic, human accounts, attempting to infiltrate and pollute our networks. These are known as bots, and they can range from harmless fake follower bots to aggressive troll and roadblock bots.

In a recent article, “Did You Know There Are Billions of Social Media Bots Designed to Mimic Humans? Here’s How to Spot Them,” Mohammed Dahiru Lawal defines bots as “automated social media accounts which pose as real people.” Lawal cites research regarding the heavy prevalence of bots on social media and gives context for their varying purposes on the platforms. 

As mentioned in the title of the article, there have been found to be billions of bot accounts on social media platforms currently. Bots form groups of bots, creating their own networks of bots. They are created by programs that set up their intentions and missions. One of which is to follow accounts that are already growing. In this way, it is sort of like the rich-get-richer principle. These bots target accounts with higher followings, giving them an even higher follow count. 

In a 2018 study from University of Southern California, researchers found that “bots are responsible for as many as two-thirds of all tweets that link to popular websites”–another example of how they may contribute to a rich-get-richer effect.

Other bots, like roadblock and troll bots, set out to infiltrate comment sections and skew debates. There can be many bots programmed to tackle the same comment section, account, or hashtag. This is interesting when considering information cascades and herding. Bots can influence human thought and behavior. They can create echo chambers and instigate extremities.

The article mentions that Facebook removed approximately 3 billion bots over the span of 6 months, but the rate at which they can be created suggests that they are an enduring part of our networks. 

Article link: https://dubawa.org/did-you-know-there-are-billions-of-social-media-bots-designed-to-mimic-humans-heres-how-to-spot-them/

NFTs, Exclusivity, and Networks

A recent article, “​​NFTs as Micro-Social Networks: The Path to Crypto Adoption,” details the recent emergence and growth of the virtual network of digital owners. This article claims that NFTs are currently on a path of becoming a significant factor of social interactions, and are shaping a new type of social interaction in itself.

NFT stands for non-fungible tokens–”non-fungible” essentially means it is one of a kind, and “token” insinuates that it is a thing that holds value to be traded or collected. While NFTs can be music, recordings, tweets, etc., digital art NFTs tend to be the most well known. Digital art has found an opportunistic market through the popularization of NFTs, with some of the most valued pieces selling for millions of dollars (but bought in crypto). An interesting comparison can be made between the value of NFTs with the seemingly exclusive communities that bid on them, and the value of traditional–tangible–art and the networks that have had access to possessing these symbols of wealth and status.

Buzz about NFTs seems to have reached the mainstream only recently, though they have been around since 2014. The article insists that “the ideas forming in NFT communities highlight a new type of social interaction built on NFTs instead of follows and likes” (Caseres). 

However, their current popularity in mainstream networks seems contradictory to the apparent exclusivity within NFT networks. They have reached other platforms and the general public because of the average person’s fascination with the idea of NFTs, rather than the average person’s ability to penetrate NFT communities and join in on the collecting, trading, and profiting of this somewhat novel phenomenon.

NFTs seem to relate well to Networks; in further discussions, it would be an interesting topic for social networks, auctions, and perhaps the Rich-Get-Richer principle.



‘Get off my beach!’: The game theory behind the co-location of fast-food chains.

Have you ever wondered why fast-food chain restaurants like Taco Bell, McDonald’s, and Burger King are always located literally within 10 feet of each other? At first glance, it seems highly counter-intuitive that competing brands position their chains in such a manner, as one would assume this breeds more competition and negatively impacts their respective sales. Strangely, however, this phenomenon transcends the fast-food industry, with petrol stations, coffee shops, and even department stores all following the same trend of co-location.

In his article titled ‘Why Are Fast-Food Restaurants Always Found Next To Each Other?, ‘Ben the Trader’, an online blogger, trader, and investor helps make sense of this phenomenon using the Model of Spatial Competition. First postulated in 1929 by influential American economic theorist and mathematical statistician, Harold Hotelling, this theory uses the famed analogy of two ice-cream vendors selling ice cream on a beach to demystify this paradoxical anomaly.

This intelligibility of this analogy rests on four basic assumptions: 

  1. The beach is 2 km long.
  2. Customers are spread across the beach uniformly. 
  3. People like to eat ice cream.
  4. Vendor stalls are mobile. 

It all begins with a single ice cream vendor, Tom, selling ice cream on a beach. In this opening scenario, Tom has a monopoly over all the ice-cream customers on the beach and gets 100% of the ice-cream sales regardless of where he sets up his stall. However, to make the maximum walking distance to his cart smaller, Tom sets up his cart right in the middle of the beach, with the rest of the beach stretching one kilometer to the right and left of his cart.

Initial scenario: Tom as the only ice-cream vendor

After a flourishing period of successful sales, a new competitor named, Sam arrives unexpectedly, also deciding to set up shop on the beach. Tom, feeling threatened by this new competitor, walks up to Sam and decides to coordinate, forming a pact that will maximally reduce the damage done to his sales. They decide to split the beach in half, with each of them occupying the center position on each half of the beach. Essentially, they each occupy the first and third quarter marks of the full length of the beach, with half a kilometer between each of their stalls and the edges of the beach and one kilometer in between their carts. For customers at the beach, this is the best possible location of the two carts as the maximum walking distance to reach a cart would be just half a kilometer. This positioning is known as the socially optimal solution for the entire market. However, from each of the vendor’s perspectives, it is still possible to capture more of the market by repositioning their stall, albeit, their pact must be broken to achieve this.

Scenario #2: Sam arrives at the beach

Being the trouble maker that he is, Sam decides to break the pact by moving his stall directly beside Tom’s. In doing this, he leaves Tom with only 0.5km of beach space to sell his ice cream, that is, the space between him and the edge of the beach while Sam gets 1.5km of beach space to sell his ice cream. After realizing what Sam did, Tom decides to retaliate by moving his cart to the right of Tom, effectively reversing the tables and putting Sam at a disadvantage. Not wanting to be outdone, Sam, once again, retaliates. They keep going on and on until finally, they stumble on a pair positions whereby nothing can be done to capture more of beach space. In this special situation, changing your position will only result in a loss of market to you, giving your opponent the upper hand. This position is known as ‘Nash Equilibrium’: a position whereby players do not have any incentive to deviate from their chosen strategy or where the dominant strategy for each player is to leave things as they are. This equilibrium occurs when Tom and Sam place their carts right next to each other at the center of the beach.

Scenario 3: Sam breaks the pacts and encroaches in Tom’s space.

Notice that in this situation, they both get half of the beach to sell their ice cream, the exact amount they had at the beginning of their pact. At the start, however, their initial positions were more advantageous to their customers as the maximum walking distance to get ice cream from anywhere on the beach was 0.5km as opposed to 1km in this new positioning. The reason however that they settled in this final position was that this final position is the only stable position whereby no competitor can deviate to a more advantageous position.

Scenario #5: Nash Equilibrium position

This line of reasoning is the same one that fast-food executives follow when choosing new locations for their chains. Intuitively, the average consumer would assume that spreading out their locations will allow them to capture a higher percentage of local markets, however, no company wants to be susceptible to a power play by any of their competitors, thereby forcing them to choose positions where Nash Equilibrium occurs, whether or not it is more or less advantageous to their customers.


‘Why Are Fast-Food Restaurants Always Found Next To Each Other?’ https://themakingofamillionaire.com/why-are-fast-food-restaurants-always-found-next-to-each-other-7ed836a46c9e

Big Brother and Alliance Networks

Big Brother is a show televised on CBS each summer. Players live in a house together and compete to win a half million dollar prize, and there are 16 contestants typically competing to win the prize. Typically, one person is the “head” of the house for a week, nominates 2 people for eviction, the contestants have a chance to compete to possibly ‘save’ themselves, and then there is a vote in which one person is “evicted” from the house each week.

The alliance networks of Big Brother are particularly interesting. It’s a game of back-stabbing, numbers, and strategy all into one. Who you make friends with matters, as your friends may turn you into a target. As the numbers dwindle and the game becomes smaller and smaller, people typically have an idea of where they stand, but there’s always a chance of betrayal, which leads to paranoia and conflict between the houseguests. In a game like Big Brother, cascading information also commonly occurs. Rumors can easily be conceived as truth, particularly given paranoia levels.

Here are the alliance networks from one game of Big Brother as it evolved throughout:

On the second day in the house, Brian, Dan and Ollie make an alliance. Brian then goes and makes an alliance with Jerry, which blows up in his face and results in a mega-alliance between the members below, with only Dan staying by Brian’s side. Brian is then evicted.

On Day 10, some members of the 8 person alliance realized that the 8 person alliance could not survive — and decided to create their own.

On Day 13, the other 4 members of the 8 person alliance decided to create their own alliance. As we can see here, there are now 2, 4 person alliances, and 3 “floaters” left in the game — players who have no alliances and are often picked off.

On Day 26, a key vote emerged — Memphis and Jessie were up for eviction, and the house was divided into two camps. With the vote clearly deadlocked at 3-3… the deciding vote was clearly on Dan. However, a twist put into the game earlier that week ensured his vote was up in the air and controlled by the viewers, and not him…

On day 35 in the house, Dan and Memphis make a “Final 2” deal. This is about the most serious deal you can make in the house, as you are promising to go to the end with one another… and something, that if betrayed, often leaves others very bitter.

On Day 39, the power dynamic had evolved as follows… Ollie/Michelle/April were aligned, Renny/Dan/Memphis/Keesha were aligned, and Jerry was a “floater”.

On Day 51, as the odd one out, Jerry grabs power. Memphis aligns himself with Jerry to help protect Dan and himself… which eventually leads to Dan and Memphis successfully controlling the vote and evicting Renny.

In Day 58, Dan and Memphis decide to flip on Keesha and evict her out of the house. On the left is how Keesha perceived the alliances, whereas on the right was how the alliances actually were aligned. Keesha is then evicted.

Dan and Memphis then initiate a big fight between themselves, to trick Jerry into thinking that he is going to win and to lull him into a sense of complacency. Their trap works and Dan wins the final “head” of household, leading him to evict Jerry and to make it to the final vote with him and Memphis.

The jury (a group of 9 houseguests) then vote for who they think should win — and it is unanimous — Dan wins.

Alliance Networks in Survivor

Survivor is a game show / social experiment where ~20 contestants are left on a remote island/region and forced to build a fire and shelter and find food (aside from one cup of rice allotted to each player daily). Initially, in the tribal phase, at the beginning of the game, is when contestants are divided into anywhere from two to four groups. In the second stage, the tribes merge. Approximately every three days (or one episode in the show), a contestant is voted out at the tribal council and eliminated from the game. Immunity from being cast off can be one for your tribe in part one and individually in part two of the game in challenges. In this blog post, I will explore the voting alliances in the show through the framework of networks.

In the voting in Survivor, the votes for who is going home are all written individually and read aloud after all votes are cast. The United States judicial system could learn a lesson from the show in this regard as doing the votes blindly prevents information cascades from occurring. They do not get to see who people are voting for and therefore don’t fall victim to the trap of the information cascade.

In many cases in the show, human subject behavior deviates from equilibrium predictions due to many factors. One may be socially inclined to vote someone off or keep them because of the nature of their personal relationship. Someone could not trust their alliance and due to this uncertainty, change their final decision. There are also hidden immunity idols that can be played after the votes are cast to protect oneself from being voted off. If you think someone in the other team’s alliance has one, this can heavily impact your decision and even can lead your alliance to split your votes between two members of the opposing alliance to ensure one of them goes home.

Interestingly, in the season I am watching at the moment (Season 29 San Juan Del Sur Blood vs. Water) there is an interesting aspect of having pairs of loved ones come on the show together. This feature creates weak bonds that are people in your alliance that you can’t fully trust and strong bonds between loved ones. In Survivor, you never know if someone in your alliance is going to turn on you and blindside you, but with your loved one, you can be sure they won’t be trying to vote you out. Below I modeled out how this network looks in episode 8 after the tribes merged. There are two alliances at this point, one with a 7 – 4 advantage over the other. The strong ties for family are represented by solid lines and the weak ties for normal alliances are represented by dashed lines. At this point, the people in the larger alliance are comfortable and think they are safe at tribal council from being voted off.

Things quickly change when they got too comfortable and didn’t feel the need to socialize or interact with Jaclyn. Instead, Wes farted in her face at the campsite and Alex treated her and the other women poorly, asking them to take out the trash for him. This pissed Jaclyn off and using her strong tie with her boyfriend Jon, she was able to blindside one of the strongest players in the game Josh to vote him out with a 6 – 5 advantage.

This huge play in the game shifted the alliances and was a true shocker. Jaclyn wouldn’t have been able to pull this off if it wasn’t for the unique feature of this season of having a loved one on the show and the numbers working perfectly in her favor.

The Prisoner’s Dilemma in the context of Sports Trades

I see lots of similarities between the Prisoner’s Dilemma of game theory that is discussed in chapter 6 of the textbook and sports trades where a player is trying to force themselves out of a team. There are many recent examples of high profile athletes attempting to leave their current team while still under contract with this team, this is known as “demanding a trade”. To do so the player will tell the team to find a trade partner for them and if the team does not find a trade partner the player will choose to sit out and not play that season. 

The Prisoner’s Dilemma  is when two people are arrested and are in different interrogation rooms. They are suspected to have committed a robbery, but there is not enough evidence to put them away for this, but there is enough evidence to put them both away for resisting arrest. The Dilemma comes into play when the police say that if they say what happened and their crime partner does not confess then the person who confesses will get off free and the person who does not confess will get 10 years in jail. If both people confess then the duo will both get 4 years. If neither confess they will both get 1 year for resisting arrest.

Although there is no interrogation room, the scenario of a player demanding a trade has some similarities I wish to highlight. 

First, It does not benefit the team or the player to sit out the entire season, the team is paying a player to not play and the player is likely getting fined everyday for not working and also does not get to play they game they love. Thus the situation where the player does not play and the team does not trade the player is much like the situation where both people confess to the crime. They are both not getting a good deal. The player is getting fined and is disgruntled by the team they are on and the team is not getting the services of a player they are paying a lot of money. 

Next, there is the situation that is the most beneficial for most parties, the scenario where the player is quickly traded for lots of assets. In this scenario the player gets what they want, the ability to play on a new team. The team is able to receive lots of assets such as players and draft picks for the player they trade away. This situation is like the scenario where neither robbers confess and they both get one year. It is not ideal because the team wishes the player would just play, but it is the best of a series of bad option. 

Then there are a series of situations that are alright for one party and dreadful for the other party involved. These include the player sitting out so long that no more teams want the player and also the team having to cut the player without recuperating assets in return. 

Rich get Richer Scenario: Alabama Football Program

For the most part, the best college football teams have the best players. Recruiting websites rank players across the country while they are in high school, using high school film and in person camps to determine who the best players in high school football are. The top programs recruit and land these top players, allowing them to continue their success. Although there are many players that were not rated properly in high school, there is a high correlation with getting the top recruiting classes and winning games. The recruits realize this too, knowing that the best programs will get even better if great high school recruits commit to them, thus this creates a rich get richer scenario. 

Chapters 16, 17 and 18 discussed scenarios where decisions are made based off of other people’s choices. A great example of this is Alabama and their ability to consistently get 5 star football recruits, the 5 star recruits base their decision off of the winning pedigree of the program and their desire to play with other 5 stars in order to continue the success of the program. Once the high school players see other prominent and good players commit to Alabama, they commit because they know that the team will be incredibly good if many of the good players go there. 

Going from 2022 to 2012, Alabama’s recruiting classes have been ranked: 2, 1, 2, 1, 5, 1, 1, 1, 1, 1, and 1. In eleven recruiting classes Alabama has never had a class ranked below 5 in the nation and has had 8 #1 ranked recruiting classes. In that same span they have won 4 national championships, meaning that every recruiting class that has come through Alabama has one a National Championship in their 4 years at the program. 

The recruits that commit to Alabama make a correlated decision to attend the school, knowing that if they go with other great players they will win games, this creates a power law. Just like someone is more likely to choose a web link to an article when the views are listed next to the article, a recruit is more likely to join Alabama’s team after seeing how many other good players are committed there. Alabama has a higher probability of getting more recruits because of their current probability (the current players committed to the University).


Using “Small World” Theory to better understand team dynamics in soccer

As professional sports continue to grow increasingly competitive, with every team searching for every advantage they can incur, increasingly advanced methods of analysis are being used to optimize performance. The 2011 movie Moneyball depicts just this, telling the true story of the resurrection of the baseball team the Oakland A’s. Due to budget limitations, the team is unable to acquire conventionally high-valued players. Instead, the team employs revolutionary statistical analysis to create a winning team from less desired players. Since the Oakland A’s did so, statistical analysis has become commonplace amongst professional sports programs. However, another form of objective analysis may prove to be of use to professional sports programs as well: the “small world” theory.

The “Small World” theory began with a short story but was born into theory in the famous six degrees of separation study by Stanley Milgram. By asking a group of people to attempt to get a letter from them to a specific target in Boston. The people were able to mail it to whomever they deemed most likely to be able to get the letter to the target themselves. Eventually, many of the letters made it to the target, and Milgram studied the chain of people these letters took. This led Milgram to conclude that, on average, any given person is six degrees of separation away from any other given person. More broadly, Milgram’s discovery proved that social networks with many short-distance links and just a few long-distance links can lead to very short avg paths between nodes.

In recent years, it has become evident that the “Small World” theory could be used to very precisely represent the complicated inter-player dynamics of soccer. In a 2015 study, researchers gathered data from 30 premier league soccer matches in which the championship-winning team played. In total, they recorded 7583 collective offensive actions and 22518 intra-team interactions. They used this information to construct a network representation of the team, in which each player was a node. Due to the nature of the sport, the researchers found that the network exhibited a very short average path length considering the number of nodes. Through the use of this network, researchers were able to derive many useful metrics about each player. By defining the scaled connectivity as the sum of the connection weights of a player i to those connected to him, where connection weight between players is a measure of how much the pair interacted in play, the researchers were able to objectively measure how cooperative each player was. Furthermore, clustering coefficients for each player were used to describe the extent to which the teammates a player interacted with the most cooperated. Combining these two metrics, the researchers assigned each player with a global rank, representative of each players total team productivity. By using a network to encode the actions of the team, the researchers were able to quanitifiably measure many aspects of performance that are usually left to subjective observation.

On top of assigning objective metrics to every player for different aspects of performance, the researchers also concluded that defenders and midfielders are the players who exhibit the highest levels of connectivity, whereas strikers tended to interact less with their teammates. They further suggested that similar analysis could be used within the sport to “detect under-performing players, fix weak spots, detect potential problems amongst teammates, as well as to detect weaknesses in the opposing team.” As professional sports teams continue to search for advantages wherever they can be found, network analysis could become a crucial component of objective evaluations of player performance and team dynamics.

How real is the friendship paradox?

I think the friendship paradox is an interesting concept. I am a little skeptical of this concept because it is hard to believe that each friend down the line has more friends than the previous. The cycle has to end somewhere right. An article I found shared the same skepticism about the friendship paradox. 

The article The ‘friendship paradox’ doesn’t always explain real friendships, mathematicians say by Yasemin Saplakoglu examines the phenomenon, the friendship paradox. The article also introduces a new concept that “takes the friendship paradox beyond averages”. This enabled the mathematicians who created this new theory to discover that their equations describe real-world popularity differences among friends.

The initial description of the friendship paradox “is that the number of friends of a person’s friends is, on average, greater than the number of friends of that individual person.” But this can not be true for all friends. Some people are more popular than their friends. The article describes an example:

“Think about a person with just two friends contrasted with a person who has hundreds of friends. Now imagine entering this social bubble: You are more likely to be friends with the social butterfly than the wallflower, simply because there are more “chances” that you are one of the hundreds of the social butterfly’s friends than one of the wallflower’s two best buds. But it’s still possible for you to become friends with the wallflower.”

This example explains the friendship paradox but also addresses what might happen if you befriend the “wallflower”. Even though there is a higher percentage of you becoming friends with the “social butterfly”, focusing on this percentage can prohibit you from noticing the percentage of people who befriend the “wallflower”. The mathematician, George Cantwell, developed new mathematical equations that will not overlook the friendships with the “wallflowers”. Their new equation has two assumptions from real-world studies.

  1. There’s a significant degree of variation in how many friends people have, depending on the social network analyzed
  2. Popular people are more likely to have popular friends, whereas unpopular people are more likely to have unpopular friends

Overall, the article concluded that the new equation can account for “95% of the variance in real-world situations”. The mathematician’s equation also shows that the friendship paradox has a tendency to be more accurate in social networks that consist of people with very different popularities. The takeaway from this article is that “our social circles are biased samples of the population.” Therefore, it is unclear to see how that bias will affect a specific case.



Game Theory in Football: Pass or Run vs. Pass, Run, or Blitz

One topic that interested me in this class was game theory. It was a concept I had learned about in a previously taken class, but I was excited to learn more about it. It was also interesting to see the connection between game theory and social networks. I started to wonder where else in my day-to-day life I can see game theory in action. I soon realized that it is present in a lot of things I do in life, one of them being football. 

More specifically, I found an article that supported my idea and talked about how game theory affected the run/pass balance in football. The article Game Theory in American Football Play Calling by Brian Burke introduced the idea that game theory influenced the “thought process behind the decisions made on each play call”. This chart explains all the hypothetical payoffs for the offense if they decide to run or pass and also the decision of the defense whether they choose pass defense, run defense, and blitz.   

The article continues with attempting to find a Nash equilibrium within this chart. Burke begins to look at all the payoffs and tries to find the dominant strategy. He explains that running would be the safest option for the offensive team because both of the payoffs are positive. However, the passing option when going up against a run defense has the highest positive payoff of 9. Therefore, there is no dominant strategy for the offense. But Burke states that this does not mean that there isn’t a Nash equilibrium. He states that “the real optimal path is through a mixed strategy in Nash equilibrium”. 

Game theory in football affects the decisions coaches make when they play calling. They have an opportunity to go for the highest payoff, but it is dependent on what the defense runs. Since teams are competing against each other it is impossible for the two sides to try and work together to reach the optimal strategy for both teams. Therefore, this process of play-calling is always each coach attempting to call a better play strategy than the other.


Cascading Behavior in College Football Transfer Portal


This offseason, the transfer portal, and coaching carrousel have changed the landscape of college football, creating new powerhouses and leaving perenially strong programs void of talent. For this case study, I will particularly look at the networks of players in the USC and Oklahoma football programs before and after Oklahoma superstar Head Coach Lincoln Riley accepted the head coaching job at USC. Just before the season finished, USC fired their head coach, Clay Helton, for a lack of success under his supervision. Tempting Lincoln Riley away from Oklahoma with a big-money contract and the benefits of leading a major sports team in Los Angeles, many of his players at Oklahoma (present and future) followed him by transferring from Oklahoma to USC. In response, many of USC’s players followed suit by transferring away in pursuit of an opportunity to play in a less competitive depth chart or play for a different coach. With a yard sale of players between these two programs, rippling out into the rest of college football, networks were changed, broken, and recreated. 

This example is very similar, yet much more complicated, to the example of the dance club from Pset 4. In football, a player that has a strong relationship with a coach that moves to another team is likely to follow due to built-up trust and promise of opportunity. This was the case for a myriad of players on Oklahoma such as star Quarterback Caleb Williams. Likewise, players from USC who were strongly in favor of Clay Helton are likely to leave to another school to find opportunities. With this first layer of movement, the remaining players are likely to follow their teammates if the majority are changing schools. Staying in a rebuilding program without your network of friends is not appealing to most. As a small example, I have analyzed a hypothetical network of each head coach and just 5 players before and after Helton’s firing and Riley’s move. The players with an x refer to those who started 2021 at USC and O refers to players who started at Oklahoma in 2021.

 As you can see, players that are directly connected to their coach followed their leader’s move, such as Caleb Williams. Next, players that are friends with more people to move than stay, will also follow. Sometimes, cross-team relationships can also play a factor, such as 4x and 4o’s friendship that is common in sports due to prior overlap or hometowns. Ultimately, This split provides a great model of cascading behavior and changing networks. As transferring becomes more and more common, network analysis like this will continue to play a role in College Football’s transfer portal and recruiting. 

U.S. Physician Shortage: A Product of Imperfect Matching

Since the start of the pandemic, there has been a higher demand for doctors and medical professionals. While much of this can be attributed to the global pandemic, research done by Association of American Medical Colleges (AAMC) suggests that the growing shortage of U.S. physicians is also a product of the matching program for medical school graduates to residency programs. This National Resident Matching Program is a computer algorithm that has been described by STAT News as a “joyous transition from student to doctor.” However, the AAMC outlines the significant number of applicants who do not ever get placed and predicts a possible shortage of 139,000 physicians by 2033.

I would like to examine how the principles of matching markets can be applied to this scenario. The medical graduates can be described as buyers and the residency spots can be seen as sellers. As stated in the STAT News article, applicants must commit to whatever program they are matched with by the algorithm. Therefore, each of their valuations will be the same for each residency. Additionally, because we know there is an imperfect matching rate that does not reach 100%, we can assume that there is no perfect matching. Lastly, because there are more applicants than actual spots in the residency programs, this ‘market’ can be seen as a constricted set.

60% of applicants are accepted, this is represented by 3 Residency options (A,B,C) and 5 Applicants (A,B,C,D,E) all with the same valuation

In order to have a perfect match, there needs to be two additional Residency options to match with the five Applicants– those can be seen as “Rejections.”

The article in STAT News goes on to look at the social implications of this matching system, ones in which I think are important to address when thinking about matching markets. It is stated that only 5% of the U.S. medical students are from the bottom quintile of socioeconomic status. As this is mainly represented by Black, Hispanic, and Native American communities, they are disproportionately represented in both medical fields and the matching system. Moreover, the lack of representation of such communities in the medical field has been shown to create worse outcomes for patients of color as well. It would be interesting to see how the principles of matching markets can be altered in order to create a more representative pairing, and a more socially equitable market.

College Basketball Recruiting: Game Theory

To an audience who is unfamiliar with the inter-workings of college basketball, the offseason may simply appear as a time for teams to focus on player development and the preparation for the year to come. However, this time is primarily focused on finding the next group of players. Coaching staffs use the offseason to determine which players are the best for their program and must determine the best recruiting strategy to get the desired player to commit. The most highly sought-after recruits are designated as “5 stars” of whom there are 30 in each recruiting class. Colleges are designated as being High Major (power 5 conference- PAC12, BIG10, BIG12, ACC, SEC) or Mid Major (not power 5) *No low major has ever landed a 5-star recruit. One of the most important components in a student-athlete decision is how well they feel connected with the school and its coaches, oftentimes making it advantageous for schools to recruit a student early; however, the level of school can ultimately dictate which strategy the school chooses to implement.

To analyze this from a game theory lens, the two “players” will be two colleges, and their strategies will be either recruiting an athlete early or late. The payoff will be the probability that the prospect commits to their school. A key assumption is that all High major schools are viewed equally, and all mid-major schools are equal to one another. (Specific preferences disregarded).

In this first example, the game is being played between Two high Major schools, UNC and Duke. This case is clear-cut, given that a dominant strategy exists for both players. Despite what the other player does, it is in each school’s best interest to recruit early, creating a pure nash equilibrium with payoff values of (5,5) in the upper left quadrant (neither party has an incentive to unilaterally deviate). Despite the same payoff being present in the lower right quadrant, both schools would have an incentive to deviate, and would eventually arrive at the aforementioned equilibrium. The practical application of this theory is clear, for two high major schools recruiting against each other, the key is to be early (absent of other factors). This is why such emphasis is placed on talent evaluation, particularly at a young age. There are risks of recruiting early and the player not developing as expected, which would reduce the payoff for all values corresponding to strategies “early”. With this in mind, it makes more sense for high major schools to recruit late, as the next example will show that they are truly only in competition against themselves (Not against Mid-Majors)

The next case to explore is a high major school recruiting against a Mid-major school. In This example, Duke is a High major, And Santa Clara is mid-major. Looking at the 2022 recruiting class statistics, of the 30 “5 stars”, only 3 have committed to mid-major schools. Further research into their recruiting timeline shows that this was only the case when the mid-major school recruited early (sophomore year) and the high major school came in after the junior year. (this is how the payoff 9,1 was determined; as there is a 9 to 1 ratio between high major to mid-major commitments in this specific case) The ensuing payoff matrix is shown above.

If the mid-major school does not recruit early, the high major school is indifferent to its strategy. The high major school will obtain the maximum payoff regardless of its strategy. However, if a mid-major school recruits early (their only chance at positive payoff), the high major school is inclined to recruit early and guarantee the maximum payoff. This once more creates a dominant strategy for both schools to recruit early. A more complex dive into this example would be to consider a potentially negative payoff if a school does not get its intended prospect. If this was implemented, the mid-major school would likely choose to not play the game with the high major school, as they would rather have a payoff of zero rather than negative. After all, it takes time to recruit, a finite resource. Using a significant amount of time recruiting and not getting the player is more realistically a negative payoff, as you lose out on other potential players that you never had the chance to recruit. it is for this reason that mid-major schools often steer clear of 5-star prospects, and focus their recruiting on prospects without high major offers.
Another complexity that is beside the topic of this post is the cascade effect in recruiting. As shown, it is essential to be early in recruiting to have the best chance to maximize payoff. Therefore, when one school begins recruiting an athlete, many others follow suit. At a certain point, schools ignore their own intuition and settle for the idea that if so many other schools are recruiting, there must be a reason. This can be potentially an explanation for why scholarship offers come in bunches, as schools are all racing to recruit as early as possible, and do not want to risk the consequences of being late.


Information sharing in global climate policy networks

As climate change becomes a more pressing issue in the global community, it is important to consider how collaboration occurs – what actors are involved, how they communicate, and how they work together. Networks can help generate insight about these questions.

Research from Goritz, Jörgens, and Kolleck shed light on these questions, with a specific focus on international public administrators (IPAs) in global climate governance networks. Although organizations themselves are important actors in international climate negotiations, another important actor are IPAs – the hierarchically organized administrative units within international organizations. IPAs are particularly important because they share information about climate and policy topics to other actors in the policymaking process. Information is seen as a particularly important mechanism for influence from IPAs as well. Although previous research had verified the importance of information through qualitative document analysis, interviews, and descriptive statistics, little research had been done on through network analysis.

Thus, the researchers conducted some pretty interesting analysis through a network view. The actors specifically use survey data to determine close relationships between different actors and actor popularity in the information network. They found that IPA are part of dyads (so paired with another organization/actor in the information network) significantly more often than other actors in the information network. They also showed that there is general homophily for a couple of other actors, including businesses, NGOs, and research organizations. This means that those organizations are more likely to pass information  amongst each other. However, they did not find the same thing for IPAs, which is an interesting observation. Because of the prominence of IPAs in the information network, they also conclude that IPAs play a particularly influential role in information sharing amongst different actors in the global climate policy network. This complex network, and the dominance of IPAs (in blue) like UNFCC, UNDP, and UNEP is shown in the figure.

I think it could be interesting to take this network analysis and run a modeling exercise. We learned in class about information cascades and network effects – at a high level, the phenomenon of initial actors sharing certain information affects the behavior, assumptions, and conclusions of other actors in a network. I wonder what would happen if one or more prominent actors in the global climate policy network shared some information that happened to be false – would this affect the conclusions of other actors in the network? Would we see a cascading effect across the network? If not, what factors are not being taken into account in this network model?



Galapagos Syndrome = (Incomplete Cascade) + (R_0 < 1)

I. Definitions and motivation

Galapagos Syndrome (GS) is a term that originated in Japan to describe the isolated development trajectory of institutional practices and consumer products despite global standards/development trajectory that the rest of the world follows. As its name suggests, it is a reference to the isolated development of various Galapagos finches that Darwin discovered. In application to Japan, Mizuho puts it eloquently, “development in isolation leads to deviation.”

Two examples help illustrate this syndrome with its very real social and economic impacts. The most well-known example of GS is for Japan’s mobile phone industry from 1999 until 2007 (iPhone’s release). In 1999, Japanese (flip) phones were the most advanced in the world, with myriad features including browsing, streaming, and gaming. However, as the rest of the world picked rapidly and slowly entered a smartphone era, Japan continued to develop intricate, complex flip phones that were packed with features that were too complicated for foreigners (or the global market to understand). It was only until iPhone penetrated the market did smartphones finally become increasingly mainstream.

Understanding the Galapagos Syndrome through Japan's Unique Quirks | by  Alexander Gee | Medium
Galapagos Syndrome in flip phones

A Japanese institutional practice that exhibits GS is Japan’s difficult transition to paperless banking. The Japanese practice of requiring a hanko (or personal passbook with personal stamp/steal) has remained a norm in Japanese banks despite the rest of the world moving onto paperless banking. This is not only becoming increasingly costly for Japanese banks (to record and store all transactions in paper for Japanese accounts but also to invest in a separate paperless system for international transactions and clients) but also dragging down these bank’s efficiency as they struggle to seamlessly integrate with the rest of international banking.

This concept of GS may be connected to two concepts in class: incomplete Cascade and the epidemic networks in relation to ideas. The existence of GS may be understood as the combined effects of an incomplete cascade of globalized standards/products as well as the epidemic-like spread of globalized standards/products with an R_0 < 1 in the context of Japan.

II. Incomplete cascades

The spread of globally available and marketed products may be understood as a diffusion effect within a network. Using the same logic of cascades, the individual adoption of global-standards smartphones is an even that occurs with probability p and threshold q. Given that international markets and globalization was already advanced enough to support the adoption of other things like western trends and other technologies, the lack of adoption of modernized phones in Japan should not be attributed to a infrastructural shortcomings in logistics but should be attributed to the existence of a dense cluster. Understanding Japan as a dense cluster is also not difficult. It is a largely homogenous country that speaks a non-Western language, with the average Japanese individual having a friend group with a q fraction that adopted non-Japanese phones. This means that Japan, or more precisely, the SCC (super connected component) in the Japanese network, does in effect represent a cluster of density 1-q, preventing the cascade of globalized technology to become mainstream in Japan.

In short, for the average individual, they just simply have no incentive to switch because their threshold of friends using the product has simply not (and will have a hard time to be met), and so they are stuck using a highly localized product.

III. The spread of “viral ideas” with R_0 < 1

Another angle to look at GS would be to look at why certain ideas have trouble taking hold in Japan (more applicable to GS in practices). Just like how products can diffuse through a market, “viral” ideas can spread, much like viruses in an epidemic contact network.

It may be argued that globalized ideas that spread rapidly in the ex-Japan world has an R_0 < 1, which is why even if they do affect some people they fail to spread throughout the population. (e.g. executives at companies that wanted to embrace globalization but could not due to rigid market preferences, a consumer that believes in the “superiority” smart-phone to flip phones but still chose the flip phone).

R_0, or the reproductive number is a key aspect of epidemic networks, where R_0 = p*k, with p being the probability that the idea influences a person, and k being the number of person a person can influence. A simple but important nuance that must be explained here is the extent to which being infected by a disease is similar to being infected by an idea. Though ideas are normally adopted via self-will, ideas influence people regardless of the adoption. The difference between adoption and influence is that the prior is a result of a rational decision making mechanism while the latter is the result of the intrinsic power of the idea. Even if people decide against adoption of a particular idea, the very act of “deciding against adoption” is the result of an influence, that prompted the formation of a new opinion by the person which culminated in a rejection. To this extent, the influence of ideas is similar to the infectiousness of diseases.

A simple argument may be made for Japan that both p and k are low, and hence R_0 is low, or below 1, meaning that the spread of new ideas will die out with the probability one.

p is low because of 1) natural barriers to access of international ideas by the average Japanese individual (language) and 2) cultural differences to the bulk of western-centric globalization of ideas.

Point 1) may be understood beyond just not having the language to understand popular western ideas that usually find primary footing among English-dominant social networking sites but it is also that Western infiltration of Japanese-native networking applications of the early 2000s was very low (LINE, Mixi, etc). The average Japanese now (2022) now spends the most amount of time on Facebook, Twitter, Instagram, and TikTok, all of which allow for quick access to globalized ideas, but back in the 2000s, local Japanese social media apps were mainstream.

Point 2) has a two nuanced implications. Firstly the importing of globalized ideas is usually in bulks (see a Westernized phone in an MTV music video of an American pop singer or a Hollywood movie) and rarely do ideas spread individually (unless via a marketing scheme purposely crafted for that product). Hence, given that ideas are often presented in bulk, even if a certain idea (smart phones are better than flip phones) is easily influential, the rest, and most, of the bulk of ideas would have a hard time finding cultural relevance and grounding in Japan. The realistic illustration of an idea spreading in Japan would be that because an individual has little interest in the majority of the bulk of globalized ideas, they will also miss out on “being influenced by” singular ideas threaded within this large tapestry of Westernization. Hence, the true expectation of individual buy-in of ideas is effectively buy-in of all ideas, or none, and because the all-in scenario is too rigorous of an expectation, p is low.

k’s small magnitude takes on a similar logic to the incomplete cascade: an influenced individual although has a network of friends to spread to, the effective “available” friends to spread to would be significantly smaller as the 1-q cluster is too dense.

IV. Conclusions

By applying the concepts of incomplete cascades and R_0 to Japan, we can come up with a network theory explanation of Japan’s Galapagos Syndrome. This will allow us to better understand Japan’s transition out of GS now. Effectively, the increasingly popularity of smart phones and the transition to paperless banking raises the question of what has changed in Japan: has the threshold q to adopt novelties decreased significantly? Did p and k decrease? Or is it a combination of both.

The Galapagos Syndrome | POP | TECH | JAM
Variety of complex flip phones
Insane lines for the iPhone 6 from around the world | SoraNews24 -Japan  News-
Line outside Apple in Tokyo





The Unregulated Matching Market of the Professional Ballet World

As a lifelong dancer, one thing I always dreamed about was dancing professionally. For a long time, I thought it might be fun to be an NBA dancer, but by the time I was old enough to begin seriously thinking about a professional dance career, my love for ballet had grown tremendously. I decided to take it upon myself to talk to a teammate I had danced with formerly – she was a few years older than me and dancing professionally in New York at a Ballet company and she told me a story eerily similar to the article I referenced in this blog post. 

She first told me about how much she loved dancing in New York and how all of her dreams were coming true, but not without a few setbacks. When first starting her career she signed a 1-year contract with a company in New Jersey after months of auditions. After dancing with them for eight months, she felt things going downhill. She didn’t like the teachers and wasn’t improving. Without warning her contract was cut short at 10 months and she was left to fend for herself. She decided to take the leap and move to New York where after another grueling round of auditions she was given a spot at the company she was currently at. 

While I was still excited by the idea of professionally dancing, I realized the system was less talent-based and instead mimicked a matching market. I pieced together that dancers do not simply choose which companies they want to dance at, the companies must also want to hire that dancer. In creating a match, there are a plethora of criteria by which the dancer chooses a company and a company chooses a dancer. An interesting fact I recently learned about this specific matching market: there is not very much overlap in terms of the dancers that companies want due to their criteria for preferred dancers just being that specific. 

While much has changed in the ballet world in terms of racial and body type criteria for dancers, there is still no efficient or “fair” way to go about carrying out this matching market – hence why I decided it wasn’t for me. The main reason is that the market is almost completely unregulated. This means that dancers enter, exit, or are forced out of the market at random times. This also means that the market almost never reaches a market-clearing state – as there are always dancers left “hanging in the balance.” 

Although there is some flexibility required in a creative industry like this, I hope to see some sort of standardization of this process, especially for the dancer’s sake. It is generally the case with Ballet companies and the market at large that demand – dancers looking to be hired – is never met by the supply – the number of spots available, but I see a standardization of this process beneficial in creating less movement in the market and more stability. 

Reference: https://www.dancemagazine.com/ballet-job-market/

Herding and the iPhone Wave


The majority of people that we know who have ‘smartphones’ have an iPhone. In fact, I’ve had friends who get made fun of for having a Google smartphone. What started this trend of Americans buying only iPhones to the point where Apple is a monopoly in possession of 62% of the smartphone industry? According to this article, only 28% of new iPhone users claimed that they bought the phone because they truly believed it was the ‘best’ phone. This shows how Apple used herding to their advantage. The majority of Apple customers bought iPhones because they saw so many other people around them make the choice to buy an iPhone. “Well, if this many people are making the switch to an iPhone, they must be making the best decision possible.” Once this trend caught fire, we have seen nothing but iPhones since.

The article also goes into detail about how much better Apple’s competitors’ smartphone products are. Google, Samsung, and Android all make smartphones that are cheaper and arguably better. Yet we still see everyone buying and staying loyal to their iPhones. This is a clear example of herding, as Americans demonstrate their inability to decide on their own and trust the decisions of earlier customers over their own.