  We consider the problem of testing small set expansion for general graphs. A
graph $G$ is a $(k,\phi)$-expander if every subset of volume at most $k$ has
conductance at least $\phi$. Small set expansion has recently received
significant attention due to its close connection to the unique games
conjecture, the local graph partitioning algorithms and locally testable codes.
  We give testers with two-sided error and one-sided error in the adjacency
list model that allows degree and neighbor queries to the oracle of the input
graph. The testers take as input an $n$-vertex graph $G$, a volume bound $k$,
an expansion bound $\phi$ and a distance parameter $\varepsilon>0$. For the
two-sided error tester, with probability at least $2/3$, it accepts the graph
if it is a $(k,\phi)$-expander and rejects the graph if it is $\varepsilon$-far
from any $(k^*,\phi^*)$-expander, where $k^*=\Theta(k\varepsilon)$ and
$\phi^*=\Theta(\frac{\phi^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)})$. The
query complexity and running time of the tester are
$\widetilde{O}(\sqrt{m}\phi^{-4}\varepsilon^{-2})$, where $m$ is the number of
edges of the graph. For the one-sided error tester, it accepts every
$(k,\phi)$-expander, and with probability at least $2/3$, rejects every graph
that is $\varepsilon$-far from $(k^*,\phi^*)$-expander, where
$k^*=O(k^{1-\xi})$ and $\phi^*=O(\xi\phi^2)$ for any $0<\xi<1$. The query
complexity and running time of this tester are
$\widetilde{O}(\sqrt{\frac{n}{\varepsilon^3}}+\frac{k}{\varepsilon \phi^4})$.
We also give a two-sided error tester with smaller gap between $\phi^*$ and
$\phi$ in the rotation map model that allows (neighbor, index) queries and
degree queries.
