Peter L. Simon (ELTE MI)
Modeling propagation processes on
networks by using differential equations
The exact mathematical model
of a network process, like epidemic prop-
agation on a graph, can be
formulated as a large system of linear ordinary
differential equations. The mathematical model is given by a graph the
nodes of which can be in different states, in the case of epidemic dynamics
each node can be either
susceptible or infected. The state of the network
containing N nodes is given by
an N-tuple of S and I symbols, i.e. there
are 2 N states
altogether. The transition rules determine how the state of the
network evolves by infection
from one node to its neighbours and by recovery,
when an I node becomes S
again. The system of master equations is formu-
lated in terms of the
probabilities of the states, i.e. the system consists of 2 N
differential equations. Similar differential equations can be
derived for mod-
eling neural activity in a
neural network (when each neurone, a node of the
network, can be either active
or inactive) or for the voter model describing
the collective behaviour of
voters.
Despite of the fact, that the
mathematical model is relatively simple,
the analytical or numerical
study of the system can be carried out only for
small graphs or graphs with
many symmetries like the complete graph or
the star graph. For real-world
large graphs the master equations are beyond
tractability, hence the system
is approximated by simple non-linear differ-
ential equations, called
mean-field equations. These models give accurate
approximation for random
graphs, like the Erdős-Rényi graph, regular ran-
dom graphs,
configuration random graphs with a given degree distribution
and power-law random graphs.
The mean-field equations are derived for
these graphs and are studied
by the methods of dynamical system theory.
Simon, P.L., Kiss., I.Z.,
Super compact pairwise model for SIS epidemic
on heterogeneous networks, J.
Complex Networks , 4(2), 187-200, (2016).
M. Taylor, P. L. Simon, D. M.
Green, T. House, I. Z. Kiss, From Marko-
vian to pairwise epidemic
models and the performance of moment closure
approximations, J. Math. Biol.
64, 1021-1042, (2012)..
Date: Nov. 22, Tuesday 4:15pm
Place: BME, Building „Q”, Room QBF13