Mathematical Models of Leadership

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Mathematical Models of Leadership

• By
• Matthew Allinder

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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

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An Example of a Leadership Approach is the Style
Approach

• Task (concern for production)

• Relationship (concern for people)

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Blake and Moutons Managerial (Leadership) Grid
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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

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What does leadership have to do with math???
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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
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What is a signed digraph?

• A signed digraph is a digraph in which each arc
  is labeled with a sign or ?

?
u1
u2


u3
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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).

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w(u1, u2) 2 w(u2, u3) -1 w(u3, u2) -1 w(u3,
u1) 1
u1
u2
1
-1
u3
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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

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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 gt 0.

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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(t1) vi(t) ?w(uj, ui)pj(t)
• Since pi(t) vi(t) vi(t-1), then
• pi(t1) ?w(uj, ui)pj(t)

i
i
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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)
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


u
u

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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.
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The End