Learning network structure via Hawkes processes

Many of the most challenging problems in modern signal processing and machine learning involve the study of complex networks. Networks arise in a wide variety of applications, ranging from the analysis of traditional communication networks and social networks to problems in genomics and fMRI. In all of these settings, one of the most fundamental questions involves how to discover the relationships between different nodes in a network. In some applications, this information is (seemingly) readily available. For example, in many social networks, the relationships for a particular user can be immediately identified by looking at the user’s “friends.” However, in many cases the links are far less clear. How can we learn this information when it is not readily available to us? Moreover, even if we have some estimate of the network, how can we determine which relationships are superficial and which ones are truly meaningful? In many of the applications mentioned above, it can be difficult to directly observe the types of interactions between nodes that would lead to the most direct approach for estimating the network structure. For example, this might be because of technical constraints or privacy concerns in the context of a communication or social network, or because of limited measurement ability in many biological applications. However, in these applications it may still be possible to observe the activity of a particular node, even when the interaction is ambiguous. For example, we may be able to observe a particular user transmitting in a communication network without necessarily being able to determine with whom they are communicating. In this and many other applications, the only information that we can observe regarding the network structure is the timing of events at various nodes in the network. We aim to quantify when it is possible to accurately recover the structure of a network from this kind of simple information about the co-occurrence of events. We will see that, under natural assumptions on the network structure and the number of observations, reliable recovery is indeed possible.