The paper proposes a game of weighted network formation in which each agent has a limited resource to form links of possibly different intensities with other agents and to use for private purposes. We show that every equilibrium is either “reciprocal” or “non-reciprocal”. In a reciprocal equilibrium, any two agents invest equally in the link between them. In a non-reciprocal equilibrium, agents are partitioned into “concentrated” and “diversified” agents and a concentrated agent is only linked to diversified agents and vice versa. For every link, the concentrated agent invests more in the link than the diversified agent. The unweighted relationship graph of an equilibrium, in which two agents are linked if they both invest positively in each other, uniquely predicts the equilibrium values of each agent's network investment and utility level, as well as the ratio of any two agents' investments in each other. We show that equilibria are not pairwise stable and not efficient due to the positive externalities of investing in a link.