The emergence of Software-Defined Networking (SDN) has revolutionised network management, offering improved flexibility, programmability, and scalability through the introduction of centralised controllers. Such a controller, e.g. an SDN controller, typically uses an OpenFlow discovery protocol to establish and maintain a global view of the topology in the underlying network. An accurate global view of the network topology is essential for effective routing, load balancing, and deployment of mobilitybased applications. However, in a hybrid multi-controller SDN network, the OpenFlow discovery protocol introduces repetitive operations, degrading effectiveness, efficiency, and scalability. This paper addresses this issue by presenting a novel link discovery framework for establishing and maintaining topology information in a hybrid multi-controller SDN network. The framework, named EESLD, uses an event-driven approach utilising the Bidirectional Forwarding Detection (BFD) protocol to detect direct and indirect SDN links in intra-domain and inter-domain networks. Additionally, EESLD uses the sFlow protocol to discover and monitor legacy links and employs a distributed messaging system to maintain a consistent network view across controllers. The EESLD framework has been implemented and evaluated on a Mininet emulator with an RYU controller and sFlow server. Performance evaluation results show that the EESLD framework can discover direct SDN links 10.3 times faster than OFDPv2, and indirect SDN links 12.9 times faster than BDDP in a network with 85 switches. Evaluation results also indicate that the sFlow-based link discovery outperforms the OSPF-based link discovery in legacy link discovery and removal times. These results show that the EESLD framework is a more effective, efficient and scalable solution for dynamic and large-scale hybrid multi-controller SDN networks.INDEX TERMS Software-Defined Networking (SDN), topology discovery, link discovery protocols, OpenFlow protocol, hybrid SDN, Multi-controller SDN.