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The Erdős-Rényi model is one of the most studied models of random graphs where a graph G consists of n vertices, and each pair of vertices is connected by an edge with probability p, independently from every other edge. Its significance lies in its simplicity and in providing a foundation for the study of properties such as connectivity, diameter, and distribution of degrees in random graphs.

The connectivity threshold in a random graph is a critical value for the probability p, above which a random graph is almost surely connected, and below which it almost surely is not. This concept is important in network design and understanding when a large-scale network can sustain comprehensive connectivity.

The small-world property refers to the phenomenon in which most nodes in a random graph can be reached from every other by a small number of steps. This concept is significant as it is commonly observed in many real-world networks, such as social networks and the World Wide Web, and impinges on the efficiency of network communication.

The clustering coefficient of a random graph is a measure of the degree to which vertices in the graph tend to cluster together. It is significant as it provides insight into the local structure of the network, and in many real-world networks, a high clustering coefficient is indicative of a tendency towards community formation.

In the study of random graphs, a giant component is a connected subgraph that contains a positive fraction of the total number of vertices in the graph, as the number of vertices goes to infinity. The emergence of a giant component marks a phase transition in the connectivity of the graph.