Cascading in Social Networks

Abstract

Cascades are described as periods during which individuals in a population exhibit herd-like behaviour because they are making decisions based on the actions of other individuals rather than relying on their own information about the problem. We will look at the two models of cascade: decision based models and probabilistic models. In the decision based model, through a coordination game, we will look at how a few individual’s behaviours can cascade through the network to decide the norm. We will learn what the optimal strategies are when there is a playoff between two incompatible competing systems, and also when bilinguality is allowed. We will also see some studies which observes cascading in real-world networks.

While decision models looks at situations where cascade propagates due to the adoption of behaviour, probabilistic models do not require the consent of an individual and instead looks at the susceptibility of the individual to be part of the cascade. This model mainly looks at the spread of diseases. Here, we will look at various concepts related to outbreak transmission. The focus will be on the SIR, SIS and the SIRS epidemic models. Finally, the chapter looks at hashtag cascades in Twitter, cascading of recommendations and the popularity of blogs in the Blogspace.

Problems

Generate the following two graphs with random seed of 10:

50

An Erdös-Rényi undirected random graph with 10000 nodes and 100000 edges.

51

A preferential attachment graph with 10000 nodes with out-degree 10.

Let the nodes in each of these graphs have IDs ranging from 0 to 9999.

Assume that the graphs represent the political climate of an upcoming election between yourself and a rival with a total of 10000 voters. If most of the voters have already made up their mind: $40\%$ will vote for you, $40\%$ for your rival, and the remaining $20\%$ are undecided. Let us say that each voter’s support is determined by the last digit of their node ID. If the last digit is 0, 2, 4 or 6, the node supports you. If the last digit is 1, 3, 5 or 7, the node supports your rival. And if the last digit is 8 or 9, the node is undecided.

Assume that the loyalties of the ones that have already made up their minds are steadfast. There are 10 days to the election and the undecided voters will choose their candidate in the following manner:

• 1. In each iteration, every undecided voter decides on a candidate. Voters are processed in increasing order of node ID. For every undecided voter, if the majority of their friends support you, they now support you. If the majority of their friends support your rival, they now support your rival.

• 2. If a voter has equal number of friends supporting you and your rival, support is assigned in an alternating fashion, starting from yourself. In other words, the first tie leads to support for you, the second tie leads to support for your rival, the third for you, the fourth for your rival, and so on.

• 3. When processing the updates, the values from the current iteration are used.

• 4. There are 10 iterations of the process described above. One happening on each day.

• 5. The 11th day is the election day, and the votes are counted.

52

Perform these configurations and iterations, and compute who wins in the first graph, and by how much? Similarly, compute the votes for the second graph.

Let us say that you have a total funding of Rs. 9000, and you have decided to spend this money by hosting a live stream. Unfortunately, only the voters with IDs 3000–3099. However, your stream is so persuasive that any voter who sees it will immediately decide to vote for you, regardless of whether they had decided to vote for yourself, your rival, or where undecided. If it costs Rs. 1000 to reach 10 voters in sequential order, i.e, the first Rs. 1000 reaches voters 3000–3009, the second Rs. 1000 reaches voters 3010–3019, and so on. In other words, the total of Rs. k reaches voters with IDs from 3000 to $3000 + k/100 - 1$. The live stream happens before the 10 day period, and the persuaded voters never change their mind.

53

Simulate the effect of spending on the two graphs. First, read in the two graphs again and assign the initial configurations as before. Now, before the decision process, you purchase Rs. k of ads and go through the decision process of counting votes.

For each of the two social graphs, plot Rs. k (the amount you spend) on the x-axis (for values k = $1000, 2000, \ldots , 9000$) and the number of votes you win by on the y-axis (that is, the number of votes for youself less the number of votes for your rival). Put these on the same plot. What is the minimum amount you can spend to win the election in each of these graphs?

Instead of general campaigning, you decide to target your campaign. Let’s say you have a posh Rs. 1000 per plate event for the high rollers among your voters (the people with the highest degree). You invite high rollers in decreasing order of their degree, and your event is so spectacular that any one who comes to your event is instantly persuaded to vote for you regardless of their previous decision. This event happens before the decision period. When there are ties between voters of the same degree, the high roller with lowest node ID gets chosen first.

54

Simulate the effect of the high roller dinner on the two graphs. First, read in the graphs and assign the initial configuration as before. Now, before the decision process, you spend Rs. k on the fancy dinner and then go through the decision process of counting votes.

For each of the two social graphs, plot Rs. k (the amount you spend) on the x-axis (for values $k = 1000, 2000, \ldots , 9000$) and the number of votes you win by on the y-axis (that is, the number of votes for yourself less the number of votes for your rival). What is the minimum amount you can spend to win the election in each of the two social graphs?

Assume that a mob has to choose between two behaviours, riot or not. However, this behaviour depends on a threshold which varies from one individual to another, i.e, an individual i has a threshold $t_{i}$ that determines whether or not to participate. If there are atleast $t_{i}$ individuals rioting, then i will also participate, otherwise i will refrain from the behaviour.

Assuming that each individual has full knowledge of the behaviour of all the other nodes in the network. In order to explore the impact of thresholds on the final number of rioters, for a mob of n individuals, the histogram of thresholds $N = (N_{0}, \dots , N_{n-1})$ is defined, where $N_{l}$ expresses the number of individuals that have threshold $l \in [n]$. For example, $N_{0}$ is the number of people who riot no matter what, $N_{1}$ is the number of people who will riot if one person is rioting, and so on.

Let T = [1, 1, 1, 1, 1, 4, 1, 0, 4, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 4, 0, 1, 4, 0, 1, 1, 1, 4, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 4, 1, 1, 4. 1, 4, 0, 1, 0, 1, 1, 1, 0, 4, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 4, 0, 4, 0, 0, 1, 1, 1, 4, 0, 4, 0] be the vector of thresholds of 101 rioters.

55

For this threshold vector, compute the histogram N.

56

Using the N calculated in Problem 55, compute its cumulative histogram $[N] = (N_{[1]}, \dots , N_{[n-1]})$, where $N_{[k]} = \sum _{l=0}^{k}N_{l}$. Plot the cumulative histogram and report the final number of rioters.