1. Application is like a “spider web” of connections between objects
If that is what you see when you examine your application data model then a property graph database will be a great choice since it is intended for this type of applications with lots of connections (in graph terms called edges).
2. Applications requires search through a network of connections
Maybe your application is trying to discover how people are connected to each other via different types of connections like: family relationships, email exchanges, address, phone calls, organization, race, religion, friendship and so on. You can represent all these relationships as edges between vertices (nodes) in a graph and apply search/traverse through the connections to find out how people are connected.
3. Application is using nested SQL joins and is slow
Performance of a graph database does not suffer the cost associated with nested SQL joins. Following relationships can be done without an equivalent of a join, it is instead done by direct lookups or a lookup within a smaller indexed collection/array. A typical relational database does not directly support arrays/collections, each such lookup must be done using a join involving a search of all objects within a table which is slower than a direct traverse within a graph/object database.
PowerPoint presentation of VelocityGraph online here or download it. Feel free to reuse parts of it for your own presentations!
Anyone considering using Cassandra as database should read this great article about Behance at Adobe. It's showing the great advantages of a graph database like VelocityGraph over Cassandra.