This　paper　considers　the　mean-square　consensus　problem　for　a　nonlinear　complex　dynamical　network　system　with　Markovian　switching　topologies,　topology　uncertainties　and　time-varying　delays.　For　the　system,　each　node　is　a　Lure　system　whose　nonlinearity　satisfies　a　sector　condition,　and　more　than　one　interconnections　are　considered　in　view　of　the　fact　that　the　subsystems　of　networks　may　be　multiple　interconnected.　By　using　the　linear　matrix　inequality　method　and　the　partial　stability　theory,　sufficient　conditions　of　the　exponentially　mean-square　consensus　are　obtained　for　the　nonlinear　complex　dynamical　network　Lure　system　in　the　Markovian　switching　topologies.　Moreover,　a　novel　static　output　controller　has　been　designed　to　make　the　system　achieve　exponentially　mean-square　consensus　successfully.　Finally,　a　numerical　example　is　presented　to　illustrate　the　efficiency　and　applicability　of　the　proposed　methods.