Mathematical Models of Leadership

By

Matthew Allinder

What is Leadership?

• Leadership is the ability to influence a group to achieve a common goal

• There are different approaches and theories on how to be an effective leader

• One approach may not necessarily be better than another

• Application of a certain approach or theory depends on variables of the situation

An Example of a Leadership Approach is the Style Approach

Task (concern for production) Relationship (concern for people)

Blake and Mouton’s Managerial (Leadership) Grid

Is one better than another?

It all depends on the situation, the leader, and the subordinates

Using the Country Club Style may not be as effective in a military setting as the Authority-Compliance Style.

On the contrary, using the Authority-Compliance Style as a Director of Activities for an organization may not be as effective as the Country Club Style

What does leadership have to do with math???

What is a digraph?

• A digraph (directed graph) D is a pair (V, A) where V is a set whose elements are vertices and A is a set whose elements are ordered pairs of vertices called arcs

V = {A, B, C, D}

A = {(A, B), (A, C), (B, D), (C, D)}D:

What is a signed digraph?

• A signed digraph is a digraph in which each

arc is labeled with a sign: + or ‒

u1 u2

u3

‒

+ +

What is a weighted digraph?

• A weighted digraph is a digraph in which a weight (value) w(u, v) is assigned to each arc (u, v).

1

2

-1

u1 u2

u3

w(u1, u2) = 2

w(u2, u3) = -1

w(u3, u2) = -1

w(u3, u1) = 1

Pulse Process

• Developed by our very own Dr. Fred Roberts

• Described in two books of his,

Discrete Mathematical Models, with Applications to Social, Biological, and Environmental Problems

and

Graph Theory and Its Applications to Problems of Society

How does the pulse process work?

• Let D be a weighted digraph with vertices u1, u2,…, un. Assume that each vertex ui attains a value vi(t) at each time t, and that time takes on discrete values, t = 0, 1, 2,…

Let pi(t) be the pulse (change of value) at ui at time t and let it be obtained by

pi(t) = vi(t) – vi(t-1) if t > 0.

pulse process continued…

• For t = 0, pi(t) and vi(t) must be given as initial conditions. Then for a given weight w(uj, ui) on a given arc (uj, ui),

vi(t+1) = vi(t) + ∑w(uj, ui)pj(t)

Since pi(t) = vi(t) – vi(t-1), then

pi(t+1) = ∑w(uj, ui)pj(t)

i

i

Example of an autonomous pulse process

+

+

–

+

Start with initial conditions of V(start) = (0, 0, 0) and P(0) = (1, 0, 0) so at time t = 0, V(0) = (1, 0, 0)

u

uu

At time t = 1, V(1) = (2, 1, -1) and so P(1) = (1, 1, -1)At time t = 2, V(2) = (4, 3, -2) and so P(2) = (2, 2, -1), and so on.

Possible Applications?

Take a signed digraph representing a relationship in society, apply the parameters of a certain leadership approach or theory and, using the pulse process, see how effective it is.

Not necessarily looking at values that are produced after a certain time t but rather at the general trend that occurs; whether or not the digraph is pulse and value stable. Looking at ways to make the digraph pulse and value stable as well as observing which leadership approach is optimal.

The End