Abstract
In any given suitably ordered collection of networks $\G\$ major topological
changes can be detected systematically by analyzing the variation of
homogeneity $Łambda$ with the parameter affecting the ordering, say $\xi$,
where
\
Łambda(G) = 1+\barz+121-P(z_\max)1+z_\max,
\
where the network $G$ has $P(z_\max)$ fraction of hubs of degree $z_\max$
and average degree in the network is $z$.
These changes, detectable as phase transitions, are signaled by singular
behavior of the second and higher derivatives of homogeneity with respect to
the ordering parameter or another parameter that is smoothly isomorphic to it,
i.e.,
\
d^nŁambdad\xi^n = \pmınfty
\
where $n 2$. The case $n = 2$ corresponds to first-order and $n > 2$
corresponds to continuous phase transitions. We show that irrespective of
whether the phase transitions are first-order or continuous, the networks
corresponding to the point of transitions, or the flat regions surrounded by
such points, have truncated power-law tails with the cumulative distributions
behaving like $a x^-\alpha \exp(-b x^\beta)$,
where $x = (z+1)/(z_\max+1)$. Furthermore, for all finite
sized networks the truncation function is an stretched exponential, i.e.,
$1$.
We also determine the order of the phase transitions from the observed
behavior of the tails.
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