# Discrete Mathematics

At PNNL, mathematicians study a diverse set of discrete systems using various techniques. One of the most prevalent is a graph (or network) which, in its simplest form, is nothing more than a set of objects and relationships between some of those objects. Graphs can be found in cyber systems, where computers are related to each other through the sending and receiving of information; in social networks, where users form relationships through following, friending, or retweeting; and in the power grid, which can be considered a graph where generators, buses, and substations are related through power lines. Other discrete systems studied include game theory, finite topology, and machine learning.

# Key Capabilities

**Clustering**: partitioning discrete elements into groups of similar objects that can be used to discover socioeconomic groups within social networks or different types of computers within a heterogeneous cyber network.**Label Prediction**: similar to clustering, label prediction breaks a set of objects into groups and learns labels on new objects based on the clusters to which they may be most similar. Discovering image labels is a well-known example.**Modeling and Simulation**: modeling a known system so it can be reproduced algorithmically or simulated forward in time. The power grid is a good example where models are built for both purposes.**Anomaly Detection**: determining when some of the discrete pieces are significantly different than expected. This is especially important in cyber security when trying to determine when computers have been compromised or when sources are trying to exfiltrate data.