Joke Propagation

Ethan Fast, Elais Jackson, Abbie Jacobs, Eric Scott

Date: 21 June, 2009

Model

We use an artificial neural network model to simulate joke propagation in a heterogeneous social network. Nodes $n_i$ represent persons. Each node is assigned a "personality" value, $\mu_i \in U(-2,2)$, which is the mean of a Gaussian distribution, all of which have the same variance $\sigma^2$. Jokes are represented by integers $J_k \in U(-2,2)$. The probability that a person $n_i$ likes a joke $J_k$ and tells it to his or her friends depends on where in the distribution $N(\mu_i,\sigma^2)$ the joke falls. Specifically, we define $P_{like}(J_k, \mu_i) = G(-\vert J_k-\mu_i\vert)$, where $G(x)=\frac{1}{\sqrt{2{\pi}{\sigma^2}}}\int_{-\infty}^{x}{e^{-\frac{y^2}{2\sigma^2}}dy}$ is the cumulative distribution of $N(0,\sigma^2)$, so that jokes falling closest to the mean of the distribution highest probability of being retold.

The network is initialized to be fully connected with random weights.¹ Each node has an edge direct to itself, simulating memory. The random weights are initially adjusted depending on the similarity between the two nodes personalities, i.e.

$$w_{ij} = w_{0ij}(1-\frac{1}{4}\vert\mu_i - \mu_j\vert),$$ (1)

where $w_{0ij}$ is drawn from $U(0,1)$, and the division by four is to prevent negative values. This adjustment represents and decreased probability of a node telling a joke to other nodes who have very different senses of humor.

Instaces of the the jokes are randomly assigned as inputs to nodes ("told" to them) at time $t_0$, and the propogation loop begins. For each joke $J_k$, every node has a probability $P_{activation}(J_k, n_i)$ of sending an activation signal for $J_k$, which depends on the incoming signals $S(n_{ji})$ for that joke weighted by the probability $P_{like}(J_k, \mu_i)$ of liking the joke, divided by crowding effects for all jokes: taking inspiration from Malthusian population capacity, we model a limit to the number of jokes a person can remember and tell by dividing the probability by the total input signal of all jokes raised to an explonent $\alpha$. Note that we made a mistake in the equation by putting a normalizing factor in the denomonator (since $P_{activation}$ must be less than or equal to one) which actually cancels out the dependence on incoming signals for $J_k$.

$$P_{activation}(J_k, n_i) = \frac{\sum_j{S_{k}(n_{ji})P_{like}(J_k,\mu_{i})}}{\sum_j{S_{k}(n_{ji})}}\frac{1}{(\sum_{j,k}{S_k(n_{ji})})^\alpha}$$ (2)

The signal is binary, and is outputted to child nodes with strength $S(n_{ij}) = w_{ij}$.

Results

We developed two visualizations of the simulation. The first displays a circle for each node which shrinks over time, but grows when it tells a joke. When a joke is told, the node flashes a color for a moment. Each color corresponds to a unique joke. The movement of the circles is meaningless.

The second visualization is a bar graph of the cumulative number of jokes told:

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Joke Propagation

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Footnotes

... weights.¹

Preferably this would be replaced with a small-world/scale-free initialization, to more accurately represent realy social networks.