Understanding financial contagion and instability, especially during financial crises, is an important issue in risk management. The emergence of alternative high-risk and speculative asset classes such as cryptocurrencies, make it imperative to effectively monitor the financial connectivity between heterogeneous asset classes across time, in conjunction with the associated risk, to avoid a substantial breakdown of financial systems during turmoil periods. To address this problem, this paper investigates the predictive capacity of time-varying graph connectivity measures on tail and systemic risk for heterogeneous asset classes. To this end, proper statistical and geometric rules are defined first, to infer the dynamic graph topology of asset returns. Then, a novel predictive signal is proposed to quantify and rank the predictive power of dynamic nodal and global graph measures. Finally, a minimum dominating set detection method is used to track the community structure of our asset classes over time and study its consistency with the time evolution of the top predictive measures. Our empirical findings show a remarkable variability of the predictive potential for the distinct connectivity measures, and reveal its importance in designing alerting mechanisms for risk management.