Statics and dynamics of a sparsely connected oscillator network
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
0.3cmWe study interacting systems where the network of interactions
between the different units is of a sparse, random and often complex
nature 1. Among the many and varied
models of this type, we have a particular interest in systems where
elements have their own dynamics, i.e. the nodes of the network are
dynamic objects themselves.
0.3cmAs an example, we have been studying a dynamical system model of the
immune network . In this model, each element or node is composed of
a number of B-cells and antibodies with the same idiotype (i.e. they
have the same antigen 'detector') - this subunit is called a clone.
The dynamics of a given clone is described by a pair of ordinary
differential equations.
0.3cmOne particular feature of the immune system is that each clone
interacts with only a finite number of other clones irrespective of
the number of clones (i.e. number of idiotypes in the system), $N$.
This property allows protection against a wide range of antigen,
with antigen of a specific type only having a 'local' effect on the
entire immune network. We have derived a partial differential
equation to describe the population of B-cells and antibodies by
using the dynamical replica
theory2. However, this model is
rather complicated and is difficult to analyze theoretically. It is
desirable to study a model in which each element has a dynamical
nature is simple enough to be able to analyze theoretically.
0.3cmIn this talk, as an example of such a tractable system, we discuss $N$ coupled phase oscillators as introduced by
Kuramoto3. In this model, each oscillator has a
definite amplitude, and the state of a given oscillator is described
by its phase $R$. The evolution equation for phase $\phi_i$
of $i$-th oscillator is given by
eqnarray
ddt \phi_i &=& _i + \sum_j i J_ij\sin(\phi_j
- \phi_i) + _i, eq:phi
eqnarray
where $\eta_i(t)$ is Gaussian white noise with variance $2T$.
0.3cmThe system simplifies in the case where $ømega_i=ømega$ for any
$i$ and where $J_ij=J_ji$. Then eq. (1) can be
rewritten (in the frame of reference moving with angular velocity
$ømega$) as
eqnarray
ddt \phi_i &=& - \partial\phi_i
H + _i,\\
H & = & - \sum_i<j J_ij\cos(\phi_i - \phi_j).
eqnarray
These assumptions allow us to investigate a Hamiltonian system.
0.3cmFirst, we focus on the following result of Ichimomiya4: the sparse random network with finite
connectivity behaves similarly to a fully connected model with
disordered bonds. The quenched Gaussian disordered bonds can be seen
to have a similar effect on the system to the random number of
connections in a sparse system.
0.3cmBy restricting ourselves to this Hamiltonian subcase,
we are able to
examine Ichinomiya's result analytically in the regime of finite $c$
comparing phase diagrams and the order parameters in these
models. Further, we perform numerical simulations and compare our
theoretical and numerical results.
0.3cmWe then tackle the more challenging dynamical relaxation behaviour
of this system. By using a maximum entropy assumption we are able
to gain some analytic control of the overall system behaviour and we
investigate the distribution of the phases as the system relaxes.\\
1) S. Dorgovtsev and J.F.F Mendes
Evolution of networks: From Biological Nets to the internet and
WWW Oxford University Press, 2003\\
2) T. Uezu, C. Kadono, J.P.L. Hatchett and A.C.C. Coolen Prog. Theo. Phys. Supp. 161
pp. 385-388, 2006\\
3) Y. Kuramoto Chemical waves,
Oscillations and Turbulence Springer-Verlag, 1984\\
4) T. Ichinomiya Phys. Rev. E 72 016109, 2005