  A theory of relative species abundance on sparsely-connected networks is
presented by investigating the replicator dynamics with symmetric interactions.
Sparseness of a network involves difficulty in analyzing the fixed points of
the equation, and we avoid this problem by treating large self interaction $u$,
which allows us to construct a perturbative expansion. Based on this
perturbation, we find that the nature of the interactions is directly connected
to the abundance distribution, and some characteristic behaviors, such as
multiple peaks in the abundance distribution and all species coexistence at
moderate values of $u$, are discovered in a wide class of the distribution of
the interactions. The all species coexistence collapses at a critical value of
$u$, $u_c$, and this collapsing is regarded as a phase transition. To get more
quantitative information, we also construct a non-perturbative theory on random
graphs based on techniques of statistical mechanics. The result shows those
characteristic behaviors are sustained well even for not large $u$. For even
smaller values of $u$, extinct species start to appear and the abundance
distribution becomes rounded and closer to a standard functional form. Another
interesting finding is the non-monotonic behavior of diversity, which
quantifies the number of coexisting species, when changing the ratio of
mutualistic relations $\Delta$. These results are examined by numerical
simulations, and the multiple peaks in the abundance distribution are confirmed
to be robust against a certain level of modifications of the problem. The
numerical results also show that our theory is exact for the case without
extinct species, but becomes less and less precise as the proportion of extinct
species grows.
