Problem 14 Solution - "Community Events"

Suppose there are N attributes - number the attributes from 1 to N. Then define an N-by-N matrix T as follows: The value in the ith row and jth column of T is equal to the intensity of the link going from attribute j to attribute i, or zero if there is no such link.
Now consider the N-dimensional column vector of attribute values. Suppose the initial event adds the vector J to the vector of attribute values (e.g. if only the i'th attribute was affected, J would have zeros in all positions but the i'th position, and the i'th position would ahve how much was added). Then the amount that is added in addition to that due to links is TJ. But these changes also affect other attributes through links, for an additional effect of (T^2)J. And then (T^3)J, and so on.

So the total change is (I + T + (T^2) + (T^3)...)J. We want to see if (I + T + T^2 + T^3 ...) converges. Since this is a power series of matrices, we can use the eigenvalue decomposition of T - see the section titled "functional calculus" in the linked article for information on how to use the eigenvalue decomposition to find power series of matrices. Since we have all the a_i = 1 (using the notation in the article), it just becomes the regular geometric series, so it converges provided all the eigenvalues of T have absolute value less than 1.