We consider the ferromagnetic Ising model on a highly inhomogeneous network
created by a growth process. We find that the phase transition in this system
is characterised by the Berezinskii--Kosterlitz--Thouless singularity, although
critical fluctuations are absent, and the mean-field description is exact.
Below this infinite order transition, the magnetization behaves as
$exp(-const/\sqrt{T_c-T})$. We show that the critical point separates the phase
with the power-law distribution of the linear response to a local field and the
phase where this distribution rapidly decreases. We suggest that this phase
transition occurs in a wide range of cooperative models with a strong
infinite-range inhomogeneity. {\em Note added}.--After this paper had been
published, we have learnt that the infinite order phase transition in the
effective model we arrived at was discovered by O. Costin, R.D. Costin and C.P.
Grunfeld in 1990. This phase transition was considered in the papers: [1] O.
Costin, R.D. Costin and C.P. Grunfeld, J. Stat. Phys. 59, 1531 (1990); [2] O.
Costin and R.D. Costin, J. Stat. Phys. 64, 193 (1991); [3] M. Bundaru and C.P.
Grunfeld, J. Phys. A 32, 875 (1999); [4] S. Romano, Mod. Phys. Lett. B 9, 1447
(1995). We would like to note that Costin, Costin and Grunfeld treated this
model as a one-dimensional inhomogeneous system. We have arrived at the same
model as a one-replica ansatz for a random growing network where expected to
find a phase transition of this sort based on earlier results for random
networks (see the text). We have also obtained the distribution of the linear
response to a local field, which characterises correlations in this system. We
thank O. Costin and S. Romano for indicating these publications of 90s.