Modeling Social Influence

Night Ride - April 18, 1775

Paul Revere and William Dawes.

Revere alerted influential militia leaders on his ride to Lexington.

Led to American forces winning the first battle of the Revolutionary War.

Medici Family in Renaissance Florence

15 prominent families

Medici had 50% higher degree than Strozzi and Guadagni.

But the Medicis influential power was much greater than explained by the 50%.

Graphs, \(G = (V, E) \)	\(N \times N \) Adjacency Matrix	\(N \times L \) Incidence Matrix
	\(A = \begin{bmatrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix} \)	\(\hat{A} = \begin{bmatrix}1 & 0 & -1 & 0 \\ -1 & 1 & 0 & 0 \\ 0 & -1 & 1 & -1 \\ 0 & 0 & 0 & 1\end{bmatrix} \)
\( V \) = set of \( N \) nodes \( E \) = set of \( L \) links	\( A_{ij} = 1 \) iff \( (i, j) \in E \) Note that \(A\) need not be symmetric for directed graphs.	For a directed graph, \( A_{ij} \) is \(1\) if \( i \) starts at \( j \); \(-1\) if \(i\) ends at \(j\), and \(0\) otherwise. For an undirected graph, \(A_{ij}\) is \(1\) if \(i\) on \(j\), and \(0\) otherwise.

Measurements of Node Importance

Degree Centrality of Node \(i\)

For undirected graphs, this is the number of nodes connecting to node \(i\).

For any node to possibly flip, there must be some nodes that have the disease at \(t = 0\). These nodes are called seeders.

Brings topology into the picture: flip iff \(p\) of your neighbors have flipped.

• Assumes that each node has the same flipping threshold \(p\)

Will the Whole Network Flip?

For a cluster of nodes, let \(d_i\) be the fraction of the neighbors of \(i\) that are in the cluster. The density \(d\) of the cluster is \( min_i d_i \), i.e. the node with the most outside influence. For example:

Models state transitions through differential equations over continuous time \(t\). Commonly used to model the spread of diseases.

Possible states that proportions of the population could be in at time \(t\) are: susceptible \(S(t)\), infected \(I(t)\), and recovered \(R(t)\)

The SI Model

Assumes that one can only move from state \(S\) to state \(I\), with probability \(\beta\).

$$ S’(t) = - \beta S(t) I(t);\ \ \ \ I’(t) = \beta S(t) I(t) $$