Model for Heterogeneous Random Networks Using Continuous Latent Variables and an Application to a Tree–Fungus Network
Article first published online: 22 JAN 2010
© 2010, The International Biometric Society
Volume 66, Issue 4, pages 1043–1051, December 2010
How to Cite
Daudin, J.-J., Pierre, L. and Vacher, C. (2010), Model for Heterogeneous Random Networks Using Continuous Latent Variables and an Application to a Tree–Fungus Network. Biometrics, 66: 1043–1051. doi: 10.1111/j.1541-0420.2009.01378.x
- Issue published online: 22 JAN 2010
- Article first published online: 22 JAN 2010
- Received May 2009. Revised October 2009. Accepted October 2009.
- Ecological network;
- Grade of membership model;
- Heterogeneous random graph;
- Maximum likelihood;
- Mixture model
Summary The mixture model is a method of choice for modeling heterogeneous random graphs, because it contains most of the known structures of heterogeneity: hubs, hierarchical structures, or community structure. One of the weaknesses of mixture models on random graphs is that, at the present time, there is no computationally feasible estimation method that is completely satisfying from a theoretical point of view. Moreover, mixture models assume that each vertex pertains to one group, so there is no place for vertices being at intermediate positions. The model proposed in this article is a grade of membership model for heterogeneous random graphs, which assumes that each vertex is a mixture of extremal hypothetical vertices. The connectivity properties of each vertex are deduced from those of the extreme vertices. In this new model, the vector of weights of each vertex are fixed continuous parameters. A model with a vector of parameters for each vertex is tractable because the number of observations is proportional to the square of the number of vertices of the network. The estimation of the parameters is given by the maximum likelihood procedure. The model is used to elucidate some of the processes shaping the heterogeneous structure of a well-resolved network of host/parasite interactions.