# Characterization of Extremal Epidemic Networks with Diffusion Characters

### Daniel Ashlock and Colin Lee.

Submitted to the CIBCB 2008

Epidemic models often incorporate contact net-
works along which the disease can be passed. The connectivity
of the network can have a substantial impact on the course
of the epidemic. In this study an evolutionary computation
system is used to optimizes networks with a fixed distribution
of contacts to yield either long-lasting epidemics or epidemics
in which a maximal number of individuals are infected in a
given time step. These networks represent extremal cases of
network behavior. A novel network analysis tool called the
diffusion character matrix, derived from the Leontief inverse
of a modified adjacency matrix, is used to demonstrate that the
networks located for the two optimizations are substantially
different. The diffusion character matrix analysis allows us to
place several metric-like dissimilarity measure on the space
of graphs with a fixed number of nodes. The evolutionary
algorithm used searches the space of networks with a specified
degree sequence, with degrees representing the number of
contacts for each member of the population. The representation
used to evolve networks is a linear chromosome specifying a
series of degree-preserving editing moves applied to an initial
network that specifies the degree sequence of the searched
networks. The evolutionary algorithm uses a non-standard type
of restart called recentering in which the currently best network
in the population replaces the initial network at intervals.
The recentering operator moves the evolving population to
successively higher fitness regions of the search space. In this
study the algorithm is applied to networks with constant degrees
from 3 to 7. The diffusion character matrix analysis also
demonstrates that the volume of the search space occupied by
networks maximizing the number of individuals that fall sick in
one time step is much smaller than that occupied by networks
that maximize epidemic length.