Abstract
AbstractAutoregressive and moving average models for temporally dynamic networks treat time as a series of discrete steps which assumes even intervals between data measurements and can introduce bias if this assumption is not met. Using real and simulated data from the London Underground network, this paper illustrates the use of continuous time multilevel models to capture temporal trajectories of edge properties without the need for simultaneous measurements, along with two methods for producing interpretable summaries of model results. These including extracting ‘features’ of temporal patterns (e.g. maxima, time of maxima) which have utility in understanding the network properties of each connection and summarising whole-network properties as a continuous function of time which allows estimation of network properties at any time without temporal aggregation of non-simultaneous measurements. Results for temporal pattern features in the response variable were captured with reasonable accuracy. Variation in the temporal pattern features for the exposure variable was underestimated by the models. The models showed some lack of precision. Both model summaries provided clear ‘real-world’ interpretations and could be applied to data from a range of spatio-temporal network structures (e.g. rivers, social networks). These models should be tested more extensively in a range of scenarios, with potential improvements such as random effects in the exposure variable dimension.
Highlights
Coefficients from models like those in this example can be difficult to interpret, so the results were simplified in two ways: through extracting pattern features and estimating continuous temporal network properties
The Monte-Carlo Markov Chain (MCMC) estimation procedure used to fit models in this paper provided an ideal solution to derive estimates of uncertainty for pattern features and continuous temporal edge density
This paper illustrates the use of multilevel models to capture continuous temporal patterns of delays in a transport network
Summary
Like that shown, are commonly used to represent connections or relationships between objects, places or individuals They are typically cast such that the objects are represented by vertices (or nodes) and connected by edges (or arcs). Several structures have been developed for such analyses including time-aggregated networks which represent temporally dynamic networks as a series of static graphs capturing temporal snapshots or windows (Blonder et al 2012). These approaches summarise the temporal dynamics, but there can be aggregation problems when discretising events, for example when measurements of properties for different edges are not simultaneous. Machine learning methods are often easier to implement than statistical methods and impose fewer restrictions (for example, on error distributions), this paper examines statistical methods as they more allow us to use prior knowledge of a process to inform modelling, which can be important when making inferences (Comber and Wulder 2019)
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