# Chapter Notes [2]

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Eco and Soc Networks Note on Chapter Notes [2], created by cheekymonky52 on 12/04/2013.
Note by cheekymonky52, updated more than 1 year ago
 Created by cheekymonky52 about 11 years ago
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Ch 20 - Small World Phenomenon Watts-Strogatz Model - Assuming that all your friends are connected to new friends then the phenomenon that everyone is connected by a short path is quite understandable. However, assuming that triadic closure exists within a network this can be seen to limit the number of people you can reach by following short paths. Model shows how homophily creates a highly clustered network through triadic closure and how weak ties can create short paths. Two dimensional grid and each node can have two types of links; short links to neighbours within a distance of r and long range links to other random nodes. Clustering exponent can be used to determine whether two nodes v,w are connected by edge. If exponent is small then it is more likely to be a long range link i.e. more random but if exponent is small then it is more likely to be a short range link. The most optimal choice for exponent is 2 for decentralized search (how to find the short paths). Model can be adapted to not be placed in a grid i.e. using rank between two nodes which is the number of nodes closer to v then w, rank(w)-1 or social distance between two nodes which is the smallest size foci that both nodes share, s(v,w)-1.

Ch 21 - Epidemics 1. Branching Process - Disease spreads in waves. So the first person to enter into the network with the disease can pass on the disease to each person he meets with some probability, let’s say he meets k people. These k people then have a probability of infected a set of different people they meet, lets say they each meet k people so the maximum number of people infected become k2. The disease can die out after a finite number of steps if none of the k new people infected pass on the disease or it can continue infinitely. Basic Reproductive Number is the number of new cases of the disease caused by a single individual. If R < 1 then disease will die out with probability of 1 i.e. it is certain but if R > 1, the disease will permit with some probability greater than 0 i.e. it is not certain unless probability of infecting a new person is 1. 2. SIR Epidemic - Stages on the disease is susceptible, infected and removed. Progress of the epidemic is controlled by the probability of infection ‘p’ and the length of time an individual is infected ‘t’. 3. SIS Epidemic - There is no removed state meaning that there is not a bounded set of nodes. The epidemic can run for a very long time or if at any point all nodes are in the susceptible state then the disease will have died out. 4. SIRS Epidemic - Individuals can be temporarily immune but can become susceptible again after some time length. Transient Contacts - contact between two nodes can be between some time interval. Concurrency - a node is involved in two or more active partnerships that overlap in time. So disease can spread from both directions.

Voting involves opinions or preferences not numerical values.Individual Preferences - preference relation is a preference between TWO alternatives. Each individual has a set of preference relations for all pairs of alternatives. From this you can develop a ranking for all the alternatives. CONDORCET WINNER is the alternative that beats are other alternatives in a pairwise majority vote. Voting Systems -       1. Majority Rule: For each pair of alternatives count how many prefer X > Y or Y > X and keep the one that has the larger number. From the final set of preference relations you can produce a group ranking. This method can be subjected to the CONDORCET PARADOX which is when even if individual preferences are transitive the group ranking may not be. This voting systems can be used as part of an elimination tournament when the first pair of alternatives are compared and the winner is compared to the next alternative etc until there is one winner left. These tournaments can be arranged in a number of ways so therefore can be manipulated using strategic agenda setting.     2. Positional Voting: Each alternative is given a 'weight' depending on its position in each individual ranking, the alternatives are then ordered depending on their total weight to define the group ranking. This method can be used with the Borda Count where there are k alternatives and the top ranked alternative is given a weight equal to k - 1 and this is decreased by one for each lower ranked alternative with the last ranked alternative receiving 0. However, this method can be manipulated as the final group ranking depends on how the alternatives are ranked further down the list not including the highest ranked alternative in an individuals ranking, known as strategic misrepresenting of preferences. Arrow's Impossibility Theorem states that there is no voting system that can be free of pathologies i.e. has the following two principles, pareto principle and independence of irrelevant alternatives, that does not use dictatorship. A special circumstance where majority voting method can be used as the undesirable properties of pathologies and codercet paradox do not occur, which seems to meet both of the principles and does not use dictatorship is SINGLE PEAKED PREFERENCES which is used when voting on alternatives that correspond to numerical quantities or a linear ordering. In this voting method, each individual has a top favourite known as the 'peak' and all other alternatives 'fall off' on BOTH sides of this peak. In this circumstance no individual would benefit from lying about their ranking due to the medium individual favourite and the group ranking produced is complete and transitive.

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Ch 23

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