A survey of statistical network models (0912.5410v1)

Abstract: Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.

• The paper presents a comprehensive review categorizing static and dynamic network models, from ERGMs to preferential attachment techniques.
• It evaluates methodologies like blockmodels and Markov models, highlighting strengths and limitations in capturing network structures.
• The survey emphasizes challenges such as scalability and model fit, while proposing integrative approaches for improved network analysis.

A Survey of Statistical Network Models

The paper, "A Survey of Statistical Network Models," authored by Stephen E. Fienberg and Edoardo M. Airoldi, provides a thorough overview of statistical methodologies for network data analysis, spanning from foundational models to recent developments. The paper systematically categorizes and evaluates the diverse approaches within this rapidly evolving field.

Static Network Models

The authors delve into static network models, which attempt to explain observed link patterns in a fixed snapshot of a network. These models encapsulate:

• Erdős-Rényi-Gilbert Models: This foundational model treats networks probabilistically, assigning a uniform probability to the existence of each edge independently. Notably, it leads to a binomial degree distribution, which is often inadequate for capturing the heterogeneous nature of real-world networks.
• Exponential Random Graph Models (ERGMs): Extending beyond the simple binomial models, ERGMs incorporate dependencies between edges. They allow for modeling subgraph configurations like triangles, k-stars, and more complex motifs. Despite their expressive power, ERGMs can suffer from degeneracy issues, which are actively being addressed by recent advancements.
• Stochastic Blockmodels: These models partition a network into blocks or communities, which can either be pre-specified or inferred from the data. They are particularly useful for uncovering community structures and have been extended into mixed membership stochastic blockmodels (MMSB) for nuanced role assignment within networks.

Dynamic Network Models

Recognizing the temporal aspect of networks, the authors review models designed to capture network dynamics. These include:

• Preferential Attachment Models: A widely cited mechanism, where nodes preferentially attach to existing nodes with higher degrees, is used to explain the emergence of scale-free degree distributions. The model is pivotal in understanding hubs and influential nodes within growing networks.
• Continuous Time Markov Chain Models (CMPMs): CMPMs model network evolution as a continuous-time stochastic process where changes (additions or removals of edges) occur according to certain rates. These methods are particularly suitable for systems where interactions are not confined to discrete time steps.
• Discrete Time Markov Models: Representing network evolution across discrete time points, these models are akin to their static counterparts but include temporal dependencies. Hanneke and Xing's discrete Markov ERGM is a prominent example in this category.

Theoretical and Practical Implications

The survey highlights several implications for both theory and practice. It underscores the necessity for robust computational methods in handling large-scale network data and calls for a better integration of latent space models with network data to uncover hidden patterns and relationships. From a theoretical perspective, addressing identifiability and degeneracy in complex models remains a critical challenge.

Future Directions

The paper discusses challenges and future directions in network modeling. Key areas for development include:

1. Scalability: As networks grow in size, scalable algorithms that maintain computational efficiency without sacrificing model accuracy are essential.
2. Inference and Model Fit: Developing reliable methods for model evaluation and comparison, including robust asymptotic properties and goodness-of-fit metrics, is crucial.
3. Integrative Models: There is a growing interest in models that integrate network data with node attributes and edge covariates to provide richer insights.
4. Dynamic and Longitudinal Networks: Innovative approaches for capturing and predicting temporal changes in networks will be increasingly important.

Conclusion

Overall, "A Survey of Statistical Network Models" is a comprehensive resource that charts the progress and current state of network modeling. It serves as both a primer for new researchers in the field and a reference for experienced modelers looking to deepen their understanding of advanced techniques and their applications. The authors' emphasis on bridging theoretical foundations with practical computational strategies provides a balanced perspective valuable to the ongoing evolution of statistical network analysis.