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{\bf Critical behaviour on complex networks: inhomogeneous
mean-field vs Lee-Yang-Fisher formalism}

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{\bf M. Krasnytska$^{1,4}$}, B. Berche$^{2,4}$, Yu. Holovatch$^{1,4}$, R. Kenna$^{3,4}$ \\

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$^1$\textit{Institute for Condensed Matter Physics, National Acad.
Sci. of Ukraine, UA--79011 Lviv, Ukraine}

$^2$\textit{ Institut Jean Lamour, CNRS/UMR 7198, Groupe de Physique
Statistique, Universit\'e de Lorraine, BP 70239, F-54506 Vand\oe
uvre-les-Nancy Cedex, France }

$^3$\textit{Applied Mathematics Research Centre, Coventry
University, Coventry CV1 5FB, United Kingdom}

$^4$\textit{${\mathbb L}^4$ Collaboration \& Doctoral College for
the Statistical Physics of Complex Systems,
Leipzig-Lorraine-Lviv-Coventry, D-04009 Leipzig, Germany}


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We study the critical behavior of spin models on a scale-free
network with a power-law node-degree probability distribution decay
$P(k)\sim k^{-\lambda}$, $k>>1$ and on a complete graph. To this
end, we apply traditional inhomogeneous mean filed approach as well
as we use the method of partition function zeros analysis  in the
complex temperature and magnetic field plane (Lee-Yang-Fisher
formalism). The last method, to our knowledge, has not been used so
far for scale-free networks.

For the Potts model  on a scale-free network in terms of an
inhomogeneous mean-field approach we find the set of critical
amplitude ratios and scaling functions, which depend on the
probability distribution decay exponent $\lambda$ in the region $3<
\lambda <5$. Moreover, we observe the non-typical behaviour for the
heat capacity jump $\delta c_H$ [1]: for the Ising model on a
scale-free network $\delta c_H$ is $\lambda$-dependent even at
$\lambda>5$, while all critical exponents are $\lambda$-independent
and correspond on the values predicted by mean field theory.

We  show that angles that characterize partition function zeros
location in the complex plane attain $\lambda$-dependency for $3<
\lambda <5$ too. In particular, we derive the angle at which the
Fisher zeros impact onto the real temperature axis.  Our analysis of
the Lee-Yang zeros reveals a difference in their behaviour for the
Ising model on a complete graph and on an annealed scale-free
network. Whereas in the former case the zeros are purely imaginary,
they have a non zero real part in the latter case, so that the
celebrated Lee-Yang circle theorem is violated [2,3].

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\par \noindent [1]\quad  M. Krasnytska, B. Berche, Yu. Holovatch, R. Kenna, {\em Condens. Matter Phys.}  {\bf 18}, 44601 (2015).

\par \noindent [2]\quad M. Krasnytska, B. Berche, Yu. Holovatch, R. Kenna, {\em J. Phys. A:
Math. Theor.} {\bf 49}, 135001 (2016).

\par \noindent [3]\quad  M. Krasnytska, B. Berche, Yu. Holovatch, R. Kenna, {\em EPL} {\bf 111}, 60009 (2015).




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