Estimating and understanding exponential random graph models

Abstract

4 The social environment is a pervasive influence on the ecological and evolutionary dynamics 5 of animal populations. Recently, social network analysis has provided an increasingly 6 powerful and diverse toolset to enable animal behaviour researchers to quantify the social 7 environment of animals and the impact that it has on ecological and evolutionary processes. 8 However, there is considerable scope for improving these methods further. We outline an 9 approach specifically designed to model the formation of network links, exponential random 10 graph models (ERGMs), which have great potential for modelling animal social structure. 11 ERGMs are generative models that treat network topology as a response variable. This 12 makes them ideal for answering questions related directly to how and why social 13 associations or interactions occur, from the modelling of population-level transmission, 14 through within-group behavioural dynamics to social evolutionary processes. We discuss 15 how ERGMs have been used to study animal behaviour previously, and how recent 16 developments in the ERGM framework can increase the scope of their use further. We also 17 highlight the strengths and weaknesses of this approach relative to more conventional 18 methods, and provide some guidance on the situations and research areas in which they can 19 be used appropriately. ERGMs have the potential to be an important part of an animal 20 behaviour researcher's toolkit and fully integrating them into the field should enhance our 21 ability to understand what shapes animal social interactions, and identify the underlying 22 processes that lead to the social structure of animal populations. 23 2 assortative behaviour, transitivity 25 26 the study of the social and spatial components of dispersal behaviours (Blumstein et al., 47 2009; Fletcher et al., 2011). 48 The statistical analysis of social networks is complicated by the non-independence of 49 individuals within a population that results from linking individuals together within a 50 network (Croft et al., 2011; Farine & Whitehead, 2015). This confounds the use of the 51 conventional statistical approaches used in ecology such as the linear model and linear 52 mixed model, as these methods assume independence of the residuals, which is an invalid 53 assumption for individuals that are linked in a network. In light of this, numerous statistical 54 methodologies have been developed to analyse social network structure. Typically, the 55 analysis of animal social networks has revolved around randomisation-based approaches to 56 significance testing (Croft et al., 2011; Farine & Whitehead, 2015). The data used to 57 construct networks is permuted to generate uncertainty around the null hypothesis (e.g. 58 social interactions are assorted by a phenotype of interest), with permutations typically 59 constrained to produce biologically plausible null models. For example, if researchers are 60 studying how body size relates to social network connections in a population spread over 61 several ...

Econometrics of network models

“…Another issue that appears in working in coalitional games on the continuum, when coalitions themselves can be a continuum, is avoided in this framework by restricting individuals to form a finite number of links where only the characteristics on the potential connection (and not her identity) matter. 40 The corresponding approximation to dense graphs, known as graphons, is used, for example, by Chatterjee and Diaconis (2013) in their asymptotic study of ERGMs.…”

Reciprocity in directed networks

“…Many people have delved into this area. A particularly significant discovery was made by Chatterjee and Diaconis [6], who showed that the supremum in (5.16) is always attained and a random graph drawn from the model must lie close to the maximizing set with probability vanishing in n. When β 3 , β 4 ≥ 0, Yin [35] further showed that the 3-parameter space would consist of a single phase with first-order phase transition(s) across one (or more) surfaces, where all the first derivatives of χ exhibit (jump) discontinuities, and second-order phase transition(s) along one (or more) critical curves, where all the second derivatives of χ diverge. The second special situation is when β 3 = 0, Following similar arguments as in Kenyon et al [17], we conclude that d(x) can take only finitely many values.…”

“…Large deviations techniques are used throughout this paper. We refer the readers to the works of Chatterjee and Diaconis [6] and Chatterjee and Varadhan [7] for more details of this framework.…”

From Data to Assessments and Decisions: Epi-Spline Technology

“…. , β I ) being a vector of parameters and c β a normalization coefficient; see Chatterjee and Diaconis [8] and references therein. From observed data in the form of a collection of graphs, there are significant challenges to estimate the parameters β.…”

“…From observed data in the form of a collection of graphs, there are significant challenges to estimate the parameters β. In Chatterjee and Diaconis [8], we find an approach that relies on graph limits in the sense of Lovász and large-deviation results for random graphs, and leads to an infinite-dimensional problem whose optimal value is an estimate of the crucial normalization constant c β .…”