Graph Algebra I: An Introduction Courtney Brown, Ph.D. Emory University

The Origin of Graph Algebra The language of graph algebra was derived from the engineering literature, and was developed by Fernando Cortés, Adam Przeworski, and John Sprague. See Systems Analysis for Social Scientists, by F. Cortés, A. Przeworski, and J. Sprague New York: John Wiley & Sons.

Rule #1: Things on the same path get multiplied. pC(t) = V(t), where p is a parameter of proportional transformation.

Rule #2: Things that meet at an intersection get added. C(t) + R(t) = V(t) C(t)C(t) R(t)R(t) + V(t)V(t)

Linear Regression Model Using Graph Algebra

Overview of Graph Algebra Rules #1 and #2

Positive Feedback Loop The Xs are states of the system. X 1 = Input + X 3 ; X 2 = pX 1 ; X 3 = mX 2. Since Output=X 2, substitution yields Output = p(Input + mOutput), or re-arranging Output = Input[p/(1 – pm)]

Mason’s Rule This derives Mason’s Rule. The function of a single feedback loop is Forward Path/[1 – (Forward Path)(Feedback Path)] This gets multiplied by the Input to equal the Output.

Negative Feedback Loop From Mason’s Rule, Output = Input[p/(1+pm)] Note the positive sign in the denominator. This is because of the sign of –m.

The Keynesian Multiplier Economic output t = Investment t [1/(1-c)]