Bio-Inspired Computing and Networking

Ant-Inspired Allocation

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K sum to 0. Furthermore, the vector 1 exists in the nullspace of K

T

. From here we

can conclude that the system Kx = 0 subject to (10.6) has a unique equilibrium

point. Next, to show that the equilibrium point is in fact a stable one, consider the

matrix given by S = (1/s)(sI + K

T

),wheres > 0andI is the M × M identity

matrix. We note that for large enough s, S is a Markov matrix with nonnegative

entries which then allows us to prove that K

T

is negative semi-definite. We refer the

interested reader to Halász et al.

33

for more detailed exposition. 

10.4.2 Time-Delayed Model

We can arrive at a similar conclusion for the time-delayed ...