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 AuthorityCompliance 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
D:
V = {A, B, C, D}
A = {(A, B), (A, C), (B, D), (C, 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).
2
u1
u2
w(u1, u2) = 2
w(u2, u3) = -1
1
-1
u3
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)
i
Since pi(t) = vi(t) – vi(t-1), then
pi(t+1) = ∑w(uj, ui)pj(t)
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
+
–
+
u
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.
+
u
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