A Topological Approach Mark D. Flood, Jonathan Simon, and Mathew Timm We present a general definition of complexity appropriate for financial counterparty networks, and derive several topologically based implementations. These range from simple and obvious metrics to others that are more mathematically subtle. It is important to tailor a complexity measure to the specific context in which it is used. This paper introduces measures of the complexity of search and netting in dealer markets. We define measures of line graph homology and collateral line graph homology that are sensitive to network interactions, such as collateral commingling and interdependent chains of obligations, that can be difficult or intractable to unwind.