# Longitudinal Models

Tom Snijders and Johan Koskinen

## Network Dynamics

Like all things in the world, networks are subject to change. Sometimes researchers observe networks at one point in time - this is often called a “cross-sectional observation of a network.” At other times, researchers make repeated observations of a network where the node set remains the same but ties may change - this can be called “longitudinal or dynamic network data.” Although the data collection requires more effort, there can be an important payoff because dynamics often tell us more about what governs social behavior than a cross-sectional view. This chapter presents a natural extension of exponential random graph models (ERGMs) to the longitudinal case.

## Data Structure

A finite number of observations on a network of the same relation on the same node set (e.g., collaboration among the employees in a firm) is called a “network panel data set.” Such a data set may be denoted by X(to), X(t_{1}),..., X(t_{M-1}), where *X(t _{m})* is the adjacency matrix representing the network at observation moment

*t*and

_{m},*M >*2 is the number of observation moments. On all observation moments, we have the same set of

*n*actors, but the ties between them may be different. The question treated in this chapter is not how to explain, or model, each network

*X(t*by itself, but how to model the changes from

_{m})*X(t*to X(t

_{m})_{m+1}), for

*m =*0,..., M.

Throughout, we take the initial network as given, and we only intend to model the subsequent changes to the network. The basic premise is that we observe a network at different points in time, and for every new observation, there is change that we want to explain. It is also assumed that the changes in the network from one observation to the next have arisen gradually.