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The uncertainty underlying real-world phenomena has attracted attention toward statistical analysis approaches. In this regard, many problems can be modeled as networks. Thus, the statistical analysis of networked problems has received special attention from many researchers in recent years. Exponential Random Graph Models, known as ERGMs, are one of the popular statistical methods for analyzing the graphs of networked data. ERGM is a generative statistical network model whose ultimate goal is to present a subset of networks with particular characteristics as a statistical distribution. In the context of ERGMs, these graph’s characteristics are called statistics or configurations. Most of the time they are the number of repeated subgraphs across the graphs. Some examples include the number of triangles or the number of cycle of an arbitrary length. Also, any other census of the graph, as with the edge density, can be considered as one of the graph’s statistics. In this review paper, after explaining the building blocks and classic methods of ERGMs, we have reviewed their newly presented approaches and research papers. Further, we have conducted a comprehensive study on the applications of ERGMs in many research areas which to the best of our knowledge has not been done before. This review paper can be used as an introduction for scientists from various disciplines whose aim is to use ERGMs in some networked data in their field of expertise.

While we know that adolescents tend to befriend peers who share their race and gender, it is unclear whether patterns of homophily vary according to the strength, intimacy, or connectedness of these relationships. By applying valued exponential random graph models to a sample of 153 adolescent friendship networks, I test whether tendencies towards same-race and same-gender friendships differ for strong versus weak relational ties. In nondiverse, primarily white networks, weak ties are more likely to connect same-race peers, while racial homophily is not associated with the formation of stronger friendships. As racial diversity increases, however, strong ties become more likely to connect same-race peers, while weaker bonds are less apt to be defined by racial homophily. Gender homophily defines the patterns of all friendship ties, but these tendencies are more pronounced for weaker connections. My results highlight the empirical value of considering tie strength when examining social processes in adolescent networks.

•Exponential random graph models (ERGMs) are a flexible class of network models.•We develop a novel ERGM-based Bayesian multilevel model for groups of brain networks.•Fitting networks simultaneously yields more precise estimates for model parameters.•Our method can also assess differences in network properties between multiple groups.•We demonstrate our method on resting-state fMRI data from Cam-CAN, an ageing study. The brain can be modelled as a network with nodes and edges derived from a range of imaging modalities: the nodes correspond to spatially distinct regions and the edges to the interactions between them. Whole-brain connectivity studies typically seek to determine how network properties change with a given categorical phenotype such as age-group, disease condition or mental state. To do so reliably, it is necessary to determine the features of the connectivity structure that are common across a group of brain scans. Given the complex interdependencies inherent in network data, this is not a straightforward task. Some studies construct a group-representative network (GRN), ignoring individual differences, while other studies analyse networks for each individual independently, ignoring information that is shared across individuals. We propose a Bayesian framework based on exponential random graph models (ERGM) extended to multiple networks to characterise the distribution of an entire population of networks. Using resting-state fMRI data from the Cam-CAN project, a study on healthy ageing, we demonstrate how our method can be used to characterise and compare the brain’s functional connectivity structure across a group of young individuals and a group of old individuals.

In today’s academic landscape, it is essential to continuously evaluate research centers, including assessing scientific output and inter-researcher collaboration. This paper delves into the intricate dynamics of collaboration among researchers at the Italian Institute of Technology, one of Italy’s leading research organizations, employing an Exponential Random Graph Model (ERGM) approach. Our analysis enabled the identification of specific network statistics and the social and demographic characteristics of authors that significantly influence the structure of the scientific collaboration network. The analysis encompasses 1,469 researchers based at the Institute’s headquarters in 2020, examining collaborative connections established through joint publications. Specifically, through the lens of an ERGM, this study examines the interplay between sex and research areas - Computational Sciences, Life Technologies, Nanomaterials, and Robotics - shedding light on how these factors shape collaborative relationships. By bridging the gap between network and non-network characteristics, this research contributes to the literature on network formation, one of the most promising areas within knowledge and innovation networks. The findings hold potential implications for the formulation of research policies aimed at promoting equity and fostering a more inclusive and synergistic research community.

Online Health Communities (OHCs) are a type of self-organizing platform that provide users with access to social support, information, and knowledge transfer opportunities. The medical expertise of registered physicians in OHCs plays a crucial role in maintaining the quality of online medical services. However, few studies have examined the effectiveness of OHCs in transferring knowledge between physicians and most do not distinguish between the explicit and tacit knowledge transferred between physicians. This study aims to demonstrate the cross-regional transfer characteristics of medical knowledge, especially tacit and explicit knowledge. Based on data collected from 4716 registered physicians on Lilac Garden (DXY.cn), a leading Chinese OHC, Exponential Random Graph Models are used to (1) examine the overall network and two subnets of tacit and explicit knowledge (i.e., clinical skills and medical information), and (2) identify patterns in the knowledge transferred between physicians, based on regional variations. Analysis of the network shows that physicians located in economically developed regions or regions with sufficient workforces are more likely to transfer medical knowledge to those from poorer regions. Analysis of the subnets demonstrate that only Gross Domestic Product (GDP) flows are supported in the clinical skill network since discussions around tacit knowledge are a direct manifestation of physicians’ professional abilities. These findings extend current understanding about social value creation in OHCs by examining the medical knowledge flows generated by physicians between regions with different health resources. Moreover, this study demonstrates the cross-regional transfer characteristics of explicit and tacit knowledge to complement the literature on the effectiveness of OHCs to transfer different types of knowledge.

Interaction with peers is an important contributor to young children’s social and cognitive development. Yet, little is known about the nature of social networks within preschool inclusive classrooms. The current study applied a social network analysis to characterize children’s peer interactions in inclusive classrooms and their relations with children’s disability status. The participants were 485 preschoolers from 64 early childhood special education (ECSE) inclusive classrooms. Results from teachers’ report of children’s social networks showed that children with disabilities formed smaller play networks compared to their typically developing peers in the classroom, but no evidence indicated that children with disabilities engaged in more conflict networks than their counterparts. Children’s play and conflict networks were segregated by children’s disability status.

Exponential-family Random Graph Models (ERGMs) constitute a large statistical framework for modeling dense and sparse random graphs with short- or long-tailed degree distributions, covariate effects and a wide range of complex dependencies. Special cases of ERGMs include network equivalents of generalized linear models (GLMs), Bernoulli random graphs, 𝛽-models, 𝑝1-models and models related to Markov random fields in spatial statistics and image processing. While ERGMs are widely used in practice, questions have been raised about their theoretical properties. These include concerns that some ERGMs are near-degenerate and that many ERGMs are non-projective. To address such questions, careful attention must be paid to model specifications and their underlying assumptions, and to the inferential settings in which models are employed. As we discuss, near-degeneracy can affect simplistic ERGMs lacking structure, but well-posed ERGMs with additional structure can be well-behaved. Likewise, lack of projectivity can affect non-likelihood-based inference, but likelihood-based inference does not require projectivity. Here, we review well-posed ERGMs along with likelihood-based inference. We first clarify the core statistical notions of \"sample\" and \"population\" in the ERGM framework, separating the process that generates the population graph from the observation process. We then review likelihood-based inference in finite, super and infinite population scenarios. We conclude with consistency results, and an application to human brain networks.

Maintaining ecological connectivity is crucial for biodiversity, yet effectively managing interconnected areas through actor collaboration is challenging. This study examines collaboration through social–ecological fit in interconnected aquatic “blue” and terrestrial “green” areas, encompassing natural and semi‐natural elements, in human‐dominated landscapes. Combining species distribution models and connectivity analyses focused on declining amphibians and survey data on actors’ area management and collaboration within interconnected areas, we create a spatially explicit social–ecological network that we analyze using network models. Results highlight diverse ecological dependencies shaping actor interactions. Strong collaboration is observed in interconnected blue‐rural‐green areas, whereas blue‐urban‐green areas lack collaboration, with minor rivers and urban‐green spaces at the network's core plagued by social–ecological misfit. Strengthening collaboration in these areas is essential to prevent further ecological network degradation. Incorporating a spatially explicit social–ecological perspective covering diverse blue and green areas guides targeted interventions and fosters effective conservation policy and practice.

Sustainable fisheries require strong management and effective governance. However, small-scale fisheries (SSF) often lack formal institutions, leaving management in the hands of local users in the form of various governance approaches (e.g. local, traditional, or co-management). The effectiveness of these approaches inherently relies upon some level of cohesion among resource users to facilitate agreement on common policies and practices regarding common pool fishery resources. Understanding the factors driving the formation and maintenance of community cohesion in SSF is therefore critical if we are to devise more effective participatory governance approaches and encourage and empower decentralized, localized, and community-based resource management approaches. Here, we adopt a social relational network perspective to propose a suite of hypothesized drivers that lead to the establishment of social ties among fishers that build the foundation for community cohesion. We then draw on detailed data from Jamaica’s small-scale fishery to empirically test these drivers by employing a set of nested exponential random graph models (ERGMs) based on specific structural building blocks (i.e. network configurations) theorized to influence the establishment of social ties. Our results demonstrate that multiple drivers are at play, but that collectively, gear-based homophily, geographic proximity, and leadership play particularly important roles. We discuss the extent to which these drivers help explain previous experiences, as well as their implications for future and sustained collective action in SSF in Jamaica and elsewhere.

When environmental processes cut across socioeconomic boundaries, traditional top-down government approaches struggle to effectively manage and conserve ecosystems. In such cases, governance arrangements that foster multiactor collaboration are needed. The effectiveness of such arrangements, however, depends on how well any ecological interdependencies across governed ecosystems are aligned with patterns of collaboration. This inherent interdisciplinary and complex problem has impeded progress in developing a better understanding of how to govern ecosystems for conservation in an increasingly interconnected world. We argue for the development of empirically informed theories, which are not only able to transcend disciplinary boundaries, but are also explicit in taking these complex social-ecological interdependences into account. We show how this emerging research frontier can be significantly improved by incorporating recent advances in stochastic modeling of multilevel social networks. An empirical case study from an agricultural landscape in Madagascar is reanalyzed to demonstrate these improvements.