The　conformation　spaces　generated　by　short　hydrophobic-hydrophilic　(HP)　lattice　chains　are　mapped　to　conformation　space　networks　(CSNs).　The　vertices　(nodes)　of　the　network　are　the　conformations　and　the　links　are　the　transitions　between　them.　It　has　been　found　that　these　networks　have　＂small-world＂　properties　without　considering　the　interaction　energy　of　the　monomers　in　the　chain,　i.e.　the　hydrophobic　or　hydrophilic　amino　acids　inside　the　chain.　When　the　weight　based　on　the　interaction　energy　of　the　monomers　in　the　chain　is　added　to　the　CSNs,　it　is　found　that　the　weighted　networks　show　the　＂scale-free＂　characteristic.　In　addition,　it　reveals　that　there　is　a　connection　between　the　scale-free　property　of　the　weighted　CSN　and　the　folding　dynamics　of　the　chain　by　investigating　the　relationship　between　the　scale-free　structure　of　the　weighted　CSN　and　the　noted　parameter　Z　score.　Moreover,　the　modular　(community)　structure　of　weighted　CSNs　is　also　studied.　These　results　are　helpful　to　understand　the　topological　properties　of　the　CSN　and　the　underlying　free-energy　landscapes.