Figure 2

Schematic of maximum-likelihood estimation of κ and \(N_{\mathrm{p}}\). The top panel shows contact data that gives a combination \((N_{t},M_{t})\). The sequence \(\{(N_{t},M_{t})\}_{t=1}^{T}\) is plotted in the N-M space, in which a particular combination of \((11,8)\) is highlighted in red (bottom left). The joint distributions of \((N,M)\), or likelihood functions, are generated using the hidden-variable model for different combinations of \((\kappa ,N_{\mathrm{p}})=\boldsymbol{\theta}\), with each indexed by \(\ell ^{\prime}\) and \(\ell ^{\ast}\) (bottom right). A likelihood function gives the probability of observing a network with N nodes and M edges, for a given combination of \((\kappa ,N_{\mathrm{p}})\). The maximum-likelihood estimators, denoted by \(\widehat{\kappa}_{t}\) and \(\widehat{N}_{{\mathrm{p}},t}\), are given by a combination of κ and \(N_{\mathrm{p}}\) associated with the maximum-likelihood function \(f_{\ell ^{\ast}}=f(N_{t},M_{t}|{\boldsymbol{\theta}}^{\ell ^{\ast}})\)