created with NetLogo
view/download model file: SmallWoldDiffusion.nlogo
This model is a mish-mash between the original SW model that comes with NetLogo (whose documentation this partly is) and a diffusion model: http://www.cs.northwestern.edu/~hrb281/SocialTies.html
The original social ties documentation is placed in quotes.
I've only really modified it to only optionally lay out the network, to have a probability 'p' of an infection spreading over an edge at each time step, and to plot the number of infected individuals over time.
"This model explores the formation of networks that result in the "small world" phenomenon -- the idea that a person is only a couple of connections away any other person in the world." It also lets you explore the role of random connections (vs. knowing people who are close-by), can affect the speed with which disease propagates in the network.
A popular example of the small world phenomenon is the network formed by actors appearing in the same movie (e.g. the "six degrees of Kevin Bacon" game), but small worlds are not limited to people-only networks. Other examples range from power grids to the neural networks of worms. This model illustrates some general, theoretical conditions under which small world networks between people or things might occur."
The disease model is the SI model - nodes are either susceptible or infected - and once they are infected, they remain so. If the network is connected, eventually all nodes will be infected - it is just a question of how long it will take.
"This model is an adaptation of a model proposed by Duncan Watts and Steve Strogatz (1998). It begins with a network where each person (or "node") is connected to his or her two neighbors on either side... The REWIRING-PROBABILITY slider determines the probability that an edge will get rewired" (so that one of its endpoints goes to a random node instead of a neighbor). Clicking the setup button will produce different network configurations, all with the same rewiring probability.
The p slider determines the probability that an infected individual will infect a susceptible contact at every time step.
"The NUM-NODES slider controls the size of the network. Choose a size and press SETUP.
Changing the REWIRING-PROBABILITY slider changes the fraction of nodes rewired." Press SETUP to generate the new network. SETUP will also infect one node.
Adjust the p value to determine the infectiousness of the spreading agent.
To re-infect one will infect a single individual while keeping the same topology - press "reinfect-one".
Now to allow the disease to spread, you can advance on time step at a time (each infected node will infect each of its neighbors with probability p) with the "spread once" button. To let the disease run its full course, you can click the "spread complete" button.
Try plotting the values for different rewiring probabilities observe how long it takes the infection to spread completely.
What is the shape of the infection curve when the rewiring probability is 0? Is there much difference in the speed of spread past a certain value of p?
Try to see if you can create the SIS model - nodes recover and return to the 'susceptible state' after either a fixed time period, or with some probability at each time step. In this case you are looking for the conditions under which you will observe epidemics - outbreaks that affect a significant fraction of the network, vs. conditions under which the outbreak remains small and contained.
See other models in the Networks section of the Models Library, such as Giant Component and Preferential Attachment. Also check out Lada's other NetLogo models:
http://www-personal.umich.edu/~ladamic/netlogo/
This model is adapted from:
Duncan J. Watts, Six Degrees: The Science of a Connected Age (W.W. Norton & Company, New York, 2003), pages 83-100.
The work described here was originally published in:
DJ Watts and SH Strogatz. Collective dynamics of 'small-world' networks, Nature,
393:440-442 (1998)
For more information please see Watts' website: http://smallworld.columbia.edu/index.html
The small worlds idea was first made popular by Stanley Milgram's famous experiment (1967) which found that two random US citizens where on average connected by six acquaintances (giving rise to the popular "six degrees of separation" expression):
Stanley Milgram. The Small World Problem, Psychology Today, 2: 60-67 (1967).
This experiment was popularized into a game called "six degrees of Kevin Bacon" which you can find more information about here: http://www.cs.virginia.edu/oracle/
To refer to this model in academic publications, please use: Wilensky, U. (2005). NetLogo Small Worlds model. http://ccl.northwestern.edu/netlogo/models/SmallWorlds. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.
In other publications, please use: Copyright 2005 Uri Wilensky. All rights reserved. See http://ccl.northwestern.edu/netlogo/models/SmallWorlds for terms of use.
This small world model was modified by Lada Adamic in 2007 to include a diffusion simulation.