In matching problems with externalities, prudence measures the importance an agent gives to others' potential reactions when she considers deviating and social connectedness measures others' capacity to react to the agent's deviation. Assuming that externalities and preferences are random, the roles of prudence and social connectedness are studied. It is shown that asymptotic stability---a property that never holds in the absence of externalities---is achieved when the product of social connectedness and prudence grows at least exponentially with the population. Since social connectedness increases quickly in both the marriage and roommates markets, stability obtains even when prudence vanishes (sufficiently slowly).