- Population of N people that fall into one of the two groups
- Susceptible
- Infected
- People can move from being susceptible to being infected
- Initially,
people are infected, and the rest are susceptible to infection - In each time period, you encounter c other people at random
- For each infected person you encounter, you become infected with probability p
- Only one outcome: everyone is infected
- Three groups
- Susceptible
- Infected
- Recovered (Sometimes it means death)
- Susceptible -> Infected -> Recovered -> Susceptible (sometimes)
- People recover with probability r
- Several outcomes (It depends on the balance between r and p)
- Everyone becomes infected
- The disease dies out before infecting everyone
- Average degree
- Average distance
- The size of the largest connected component
- Disease will not spread as far when the network is broken into chunks
- Solomon Asch Experiment
- Perception of Neighbors (Positive vs. Negative)
- Self Fulfilling Prophecies
- People might think the party will be well-attended -> they will go -> the party is well-attended
- Tulips
- Network Effects
- Hydrogen-powered cars
- Early adopters -> Enough people -> Explosive growth -> Everyone is using it
- Threshold Models of Decision Making
- One person has different answer -> suspect him/her
- More people -> Doubt yourself
- Schelling Segregation Model
- Two groups of people: red and blue, distributed randomly in a grid
- Each step
- Check your eight neighborhood
- If more than p% are different from your color -> switch to a new random location
- You need more than one contact -- a threshold number of contacts.
- In simple contagions, you only need one contact to become infected.
- It relies on redundancy
- A few weak links can change a network, reducing siloing
- Things that can change the spreading speed
- Who you give the information to
- Who gets the information
- Our preferences and opinions depend on those around us
- We gather information
- We reduce cognitive load
- We bow to social pressure
- We value things differently depending on how our neighbors value them
- It’s convenient to coordinate
- Based on the paper The Diffusion of Microfinance by Abhijit Banerjee, Arun G. Chandrasekhar, Esther Duflo, Matthew O. Jackson
- Context: An organization called BSS provides access to a group-based microcredit program in rural southern Karnataka
- Who to spread the information
- Local leaders
- Teachers
- Reasons for the different adoption rates in villages
- Different social networks that the information is spreading on
- Results
- The best people to spread information are those with high eigenvector centrality (those who know other well-connected people)
- People are more likely to talk about the micro-finance program if they have entered the program themselves (The endorsement effect)
- People are not swayed by what their neighbors did
How economists understand strategic behavior
- Each individual chooses an action
- Their outcomes depend on the choices of everyone in the system
A set of choices where no individual wants to change their action unilaterally
- People communicate along links
- People only communicate 0 < p < 1 of the information due to the lost of some information
- Everyone gets 1 unit of information
- The total amount of information you get is
- With cost, the total payoff you will get is
- C gets
- p from B, D, E, F
from A-
- E gets
- p from C
- from B, F, D
from A
A network is pairwise stable if
- No two people would both like to add a link
- No single person would like to sever a link
It depends on the value of the information and the cost of the link
- If c > p then no new links
- If c < p and
, then keep the links for the middle node, but no new links between leaf nodes - If
, then everyone wants to link to everyone else
A node might be added or removed
- A power station goes down
- A new faculty member joins a department
- A person is incarcerated
- etc.
An edge might be added or removed
- Two former collaborators lose touch
- A road is shut down for repairs
- A business partnership is disrupted
- A physical wire is severed
- etc.
When we remove nodes/edges
- Path length ↑
- Average degree ↓
- Size of largest connected component ↓
Preventing epidemics by changing network structure
- Change how people interact
- Change who is susceptible
- Change who is infected
Goals
- Limit the number of links
- Break up the network
- Vaccination
- People who are vaccinated neither get the disease nor spread the disease
- Vaccination essentially works by removing nodes from the infection network
- Vaccinate a small number of people
- Vaccinating the right people can break up the infection network with few resources
- Good people to vaccinate are hubs (health care workers) and bridges (bus drivers)
- Vaccinate a large number of people
- Vaccinating enough people reduces the size of the largest component to 0
- The fact that the network broken up protects prople who are unable to be vaccinated (Herd immunity)
- Subpopulations that fail to vaccinate may lose herd immunity and have outbreaks, which may infect people in other communities who cannot be vaccinated
- Quarantine
- Quarantines remove links from the network
- It will break the network into much smaller chunks
- Degree distribution
- Higher average degree -> Less resilience to node deletion
- More skewed degree distribution -> Less resilience to node deletion
- Assortativity
- r -> The assortativity coefficient
- Assortativity: Nodes with high degree are connected to other nodes with high degree
- r > 0
- Assortative networks fail quickly, but the failure is not wide-spread
- Disassortativity: Nodes with high degree tend to be connected to nodes with low degree
- r < 0
- Disassortative networks fail in bursts, and usually comprehensively
- Random failure or deliberate attack
- Scale-free networks are robust to random failure, but susceptible to attack