Many social networks have the following properties: (i) a short average distance between any two individuals; (ii) a high clustering coefficient; (iii) segregation patterns; the presence of (iv) brokers and (v) hubs. (i) and (ii) define a small world network. This paper develops a strategic network formation model where agents have heterogeneous knowledge of the network: cognizant agents know the whole network, while ignorant ones are less knowledgeable. For a broad range of parameters, all pairwise Nash (PN) networks have properties (i)-(iv). There are some PN networks with one hub. Cognizant agents have higher betweenness centrality: they are the brokers who connect different parts of the network. Ignorant agents cause the emergence of segregation patterns. The results are robust to varying the number of cognizant agents and to increasing the knowledge level of ignorant ones. An application shows the relevance of the results to assessing the welfare impact of an increase in network knowledge due to, e.g., improved access to social networking tools.