Statistical inference for valued-edge networks: the generalized exponential random graph model

Review of weighted exponential random graph models frameworks applied to neuroimaging

Neuro-imaging data can often be represented as statistical networks, especially for functional magnetic resonance imaging (fMRI) data, where brain regions are defined as nodes and the functional interactions between those regions are taken as edges. Such networks are commonly divided into classes depending on the type of edges, namely binary or weighted. A binary network means edges can either be present or absent. Whereas the edges of a weighted network are associated with weight values, and fMRI networks belong to weighted networks. Statistical methods are often adopted to analyse such networks, among which, the exponential random graph model (ERGM) is an important network analysis approach. Typically ERGMs are applied to binary networks, and weighted networks often need to be binarised by arbitrarily selecting a threshold value to define the presence of the edges, which can lead to non-robustness and loss of valuable edge weight information representing the strength of fMRI interaction in fMRI networks. While it is therefore important to gain deeper insight in adopting ERGM on weighted networks, there only exists a few different ERGM frameworks for weighted networks; some of these are not directly implementable on fMRI networks based on their original proposal. We systematically review, implement, analyse and compare five such frameworks via a simulation study and provide guidelines on each modelling framework as well as conclude the suitability of them on fMRI networks based on a range of criteria. We concluded that Multi-Layered ERGM is currently the most suitable framework.

A Bayesian multilevel model for populations of networks using exponential-family random graphs

The collection of data on populations of networks is becoming increasingly common, where each data point can be seen as a realisation of a network-valued random variable. Moreover, each data point may be accompanied by some additional covariate information and one may be interested in assessing the effect of these covariates on network structure within the population. A canonical example is that of brain networks: a typical neuroimaging study collects one or more brain scans across multiple individuals, each of which can be modelled as a network with nodes corresponding to distinct brain regions and edges corresponding to structural or functional connections between these regions. Most statistical network models, however, were originally proposed to describe a single underlying relational structure, although recent years have seen a drive to extend these models to populations of networks. Here, we describe a model for when the outcome of interest is a network-valued random variable whose distribution is given by an exponential random graph model. To perform inference, we implement an exchange-within-Gibbs MCMC algorithm that generates samples from the doubly-intractable posterior. To illustrate this approach, we use it to assess population-level variations in networks derived from fMRI scans, enabling the inference of age- and intelligence-related differences in the topological structure of the brain's functional connectivity.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11222-024-10446-0.

Statistical models of complex brain networks: a maximum entropy approach

The brain is a highly complex system. Most of such complexity stems from the intermingled connections between its parts, which give rise to rich dynamics and to the emergence of high-level cognitive functions. Disentangling the underlying network structure is crucial to understand the brain functioning under both healthy and pathological conditions. Yet, analyzing brain networks is challenging, in part because their structure represents only one possible realization of a generative stochastic process which is in general unknown. Having a formal way to cope with such intrinsic variability is therefore central for the characterization of brain network properties. Addressing this issue entails the development of appropriate tools mostly adapted from network science and statistics. Here, we focus on a particular class of maximum entropy models for networks, i.e. exponential random graph models, as a parsimonious approach to identify the local connection mechanisms behind observed global network structure. Efforts are reviewed on the quest for basic organizational properties of human brain networks, as well as on the identification of predictive biomarkers of neurological diseases such as stroke. We conclude with a discussion on how emerging results and tools from statistical graph modeling, associated with forthcoming improvements in experimental data acquisition, could lead to a finer probabilistic description of complex systems in network neuroscience.

Simplifying the Process of Going From Cells to Tissues Using Statistical Mechanics

The value of digital twins for prototyping controllers or interventions in a sandbox environment are well-established in engineering and physics. However, this is challenging for biophysics trying to seamlessly compose models of multiple spatial and temporal scale behavior into the digital twin. Two challenges stand out as constraining progress: (i) ensuring physical consistency of conservation laws across composite models and (ii) drawing useful and timely clinical and scientific information from conceptually and computationally complex models. Challenge (i) can be robustly addressed with bondgraphs. However, challenge (ii) is exacerbated using this approach. The complexity question can be looked at from multiple angles. First from the perspective of discretizations that reflect underlying biophysics (functional tissue units) and secondly by exploring maximum entropy as the principle guiding multicellular biophysics. Statistical mechanics, long applied to understanding emergent phenomena from atomic physics, coupled with the observation that cellular architecture in tissue is orchestrated by biophysical constraints on metabolism and communication, shows conceptual promise. This architecture along with cell specific properties can be used to define tissue specific network motifs associated with energetic contributions. Complexity can be addressed based on energy considerations and finding mean measures of dependent variables. A probability distribution of the tissue's network motif can be approximated with exponential random graph models. A prototype problem shows how these approaches could be implemented in practice and the type of information that could be extracted.

Worth the Weight: Conceptualizing and Measuring Strong Versus Weak Tie Homophily

Homophily, or the tendency for social contact to occur among those who are similar, plays a crucial role in structuring our social networks. Most previous work considers whether homophily shapes the patterns of all social ties, regardless of their frequency of interaction or level of intimacy. As complex network data become increasingly available, however, researchers need to evaluate whether homophily operates differently for ties defined by strong versus weak measures of strength. Here, I take this approach by first defining two variants of homophily: (1) strong tie homophily, or the tendency for ties with high measures of strength to cluster together similar peers, and (2) weak tie homophily, or the tendency for ties with low edge weights to connect same-attribute actors. Then, I apply valued ERGMs to demonstrate the utility of differentiating between the two variants across simulated and observed networks. In most networks, I find that there are observable differences in the magnitude of strong versus weak tie homophily. Additionally, when there are low levels of clustering on the attribute of interest, distinguishing between strong and weak tie homophily can reveal that these processes operate in opposite directions. Since strong and weak ties carry substantively different implications, I argue that differentiating between the two homophily variants has the potential to uncover novel insights on a variety of social phenomena.

Testing biological network motif significance with exponential random graph models

Analysis of the structure of biological networks often uses statistical tests to establish the over-representation of motifs, which are thought to be important building blocks of such networks, related to their biological functions. However, there is disagreement as to the statistical significance of these motifs, and there are potential problems with standard methods for estimating this significance. Exponential random graph models (ERGMs) are a class of statistical model that can overcome some of the shortcomings of commonly used methods for testing the statistical significance of motifs. ERGMs were first introduced into the bioinformatics literature over 10 years ago but have had limited application to biological networks, possibly due to the practical difficulty of estimating model parameters. Advances in estimation algorithms now afford analysis of much larger networks in practical time. We illustrate the application of ERGM to both an undirected protein-protein interaction (PPI) network and directed gene regulatory networks. ERGM models indicate over-representation of triangles in the PPI network, and confirm results from previous research as to over-representation of transitive triangles (feed-forward loop) in an E. coli and a yeast regulatory network. We also confirm, using ERGMs, previous research showing that under-representation of the cyclic triangle (feedback loop) can be explained as a consequence of other topological features.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s41109-021-00434-y.

A network analysis of patient referrals in two district health systems in Tanzania

Patient referral systems are fragile and overlooked components of the health system in Tanzania. Our study aims at exploring patient referral networks in two rural districts in Tanzania, Kilolo and Msalala. Firstly, we ask whether secondary-level facilities act as gatekeepers, mediating referrals from primary- to tertiary-level facilities. Secondly, we explore the facility and network-level determinants of patient referrals focusing on treatment of childhood illnesses and non-communicable diseases. We use data collected across all public health facilities in the districts in 2018. To study gatekeeping, we employ descriptive network analysis tools. To explore the determinants of referrals, we use exponential random graph models. In Kilolo, we find a disproportionate share of patients referred directly to the largest hospital due to geographical proximity. In Msalala, small and specialized secondary-level facilities seem to attract more patients. Overall, the results call for policies to increase referrals to secondary facilities avoiding expensive referrals to hospitals, improving timeliness of care and reducing travel-related financial burden for households.

Characterising group-level brain connectivity: A framework using Bayesian exponential random graph models

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.

The performance of permutations and exponential random graph models when analyzing animal networks

Social network analysis is a suite of approaches for exploring relational data. Two approaches commonly used to analyze animal social network data are permutation-based tests of significance and exponential random graph models. However, the performance of these approaches when analyzing different types of network data has not been simultaneously evaluated. Here we test both approaches to determine their performance when analyzing a range of biologically realistic simulated animal social networks. We examined the false positive and false negative error rate of an effect of a two-level explanatory variable (e.g., sex) on the number and combined strength of an individual’s network connections. We measured error rates for two types of simulated data collection methods in a range of network structures, and with/without a confounding effect and missing observations. Both methods performed consistently well in networks of dyadic interactions, and worse on networks constructed using observations of individuals in groups. Exponential random graph models had a marginally lower rate of false positives than permutations in most cases. Phenotypic assortativity had a large influence on the false positive rate, and a smaller effect on the false negative rate for both methods in all network types. Aspects of within- and between-group network structure influenced error rates, but not to the same extent. In "grouping event-based" networks, increased sampling effort marginally decreased rates of false negatives, but increased rates of false positives for both analysis methods. These results provide guidelines for biologists analyzing and interpreting their own network data using these methods.

A survey on exponential random graph models: an application perspective

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.

Model-Based Clustering of Nonparametric Weighted Networks with Application to Water Pollution Analysis

Water pollution is a major global environmental problem, and it poses a great environmental risk to public health and biological diversity. This work is motivated by assessing the potential environmental threat of coal mining through increased sulfate concentrations in river networks, which do not belong to any simple parametric distribution. However, existing network models mainly focus on binary or discrete networks and weighted networks with known parametric weight distributions. We propose a principled nonparametric weighted network model based on exponential-family random graph models and local likelihood estimation, and study its model-based clustering with application to large-scale water pollution network analysis. We do not require any parametric distribution assumption on network weights. The proposed method greatly extends the methodology and applicability of statistical network models. Furthermore, it is scalable to large and complex networks in large-scale environmental studies. The power of our proposed methods is demonstrated in simulation studies and a real application to sulfate pollution network analysis in Ohio watershed located in Pennsylvania, United States.

A theoretical and empirical comparison of the temporal exponential random graph model and the stochastic actor-oriented model

The temporal exponential random graph model (TERGM) and the stochastic actor-oriented model (SAOM, e.g., SIENA) are popular models for longitudinal network analysis. We compare these models theoretically, via simulation, and through a real-data example in order to assess their relative strengths and weaknesses. Though we do not aim to make a general claim about either being superior to the other across all specifications, we highlight several theoretical differences the analyst might consider and find that with some specifications, the two models behave very similarly, while each model out-predicts the other one the more the specific assumptions of the respective model are met.

Grand canonical ensemble of weighted networks

The cornerstone of statistical mechanics of complex networks is the idea that the links, and not the nodes, are the effective particles of the system. Here, we formulate a mapping between weighted networks and lattice gases, making the conceptual step forward of interpreting weighted links as particles with a generalized coordinate. This leads to the definition of the grand canonical ensemble of weighted complex networks. We derive exact expressions for the partition function and thermodynamic quantities, both in the cases of global and local (i.e., node-specific) constraints on the density and mean energy of particles. We further show that, when modeling real cases of networks, the binary and weighted statistics of the ensemble can be disentangled, leading to a simplified framework for a range of practical applications.

Exploring dependence structures in the international arms trade network: A network autocorrelation approach

In this article, we analyse dependence structures among international trade flows of major conventional weapons from 1952 to 2016. We employ a Network Disturbance Model commonly used in inferential network analysis and spatial econometrics. The dependence structure is represented by pre-defined weight matrices that allow for correlating flows from the network of international arms exchange. Three dependence structures are proposed, representing sender-, receiver- and sender–receiver-related dependencies. The appropriateness of the presumed structures is comparatively assessed using the Akaike Information Criterion (AIC). It turns out that the dependence structure among the arms trade flows is complex and can be represented best by a specification that relates each arms trade flow to all exports and imports of the sending and the receiving state. Controlling for exogenous variables, we find that the trade volume increases with the GDP of the sending and the receiving state while the impact of geographical distance, regime dissimilarity and formal alliance membership is rather small.

Connectivity in fMRI: Blind Spots and Breakthroughs

In recent years, driven by scientific and clinical concerns, there has been an increased interest in the analysis of functional brain networks. The goal of these analyses is to better understand how brain regions interact, how this depends upon experimental conditions and behavioral measures and how anomalies (disease) can be recognized. In this paper, we provide, first, a brief review of some of the main existing methods of functional brain network analysis. But rather than compare them, as a traditional review would do, instead, we draw attention to their significant limitations and blind spots. Then, second, relevant experts, sketch a number of emerging methods, which can break through these limitations. In particular we discuss five such methods. The first two, stochastic block models and exponential random graph models, provide an inferential basis for network analysis lacking in the exploratory graph analysis methods. The other three addresses: network comparison via persistent homology, time-varying connectivity that distinguishes sample fluctuations from neural fluctuations, and network system identification that draws inferential strength from temporal autocorrelation.

Statistical Modeling of the Default Mode Brain Network Reveals a Segregated Highway Structure

We investigate the functional organization of the Default Mode Network (DMN) - an important subnetwork within the brain associated with a wide range of higher-order cognitive functions. While past work has shown the whole-brain network of functional connectivity follows small-world organizational principles, subnetwork structure is less well understood. Current statistical tools, however, are not suited to quantifying the operating characteristics of functional networks as they often require threshold censoring of information and do not allow for inferential testing of the role that local processes play in determining network structure. Here, we develop the correlation Generalized Exponential Random Graph Model (cGERGM) - a statistical network model that uses local processes to capture the emergent structural properties of correlation networks without loss of information. Examining the DMN with the cGERGM, we show that, rather than demonstrating small-world properties, the DMN appears to be organized according to principles of a segregated highway - suggesting it is optimized for function-specific coordination between brain regions as opposed to information integration across the DMN. We further validate our findings through assessing the power and accuracy of the cGERGM on a testbed of simulated networks representing various commonly observed brain architectures.

Analyzing Policy Networks Using Valued Exponential Random Graph Models: Do Government-Sponsored Collaborative Groups Enhance Organizational Networks?

This paper examines collaborative management groups from the perspective of policymakers seeking to increase coordination within a policy network. While governments often support collaborative groups as a tool to address perceived network failures such as a lack of coordination, the net impact groups have is unclear. I use valued exponential random graph models (ERGMs) to model relationships of varying strength among a regional network of organizations involved in 57 collaborative groups. This provides a unique opportunity to study the interplay between numerous groups and organizations within a large-scale network. Valued ERGMs are a recently developed extension of standard ERGMs that model valued instead of binary ties; thus, this paper also makes a methodological contribution to the policy literature. Findings suggest that participation in collaborative groups does motivate coordination and cooperation amongst individual network organizations; however, this effect is strongest for: (i) organizations that are not already members of another group and (ii) organizations that do not have a preexisting tie. These results support a transaction-cost-based perspective of how government-sponsored collaborative groups can influence network coordination; further, they also provide an empirical example of the Ecology of Games, in which multiple collaborative institutions have interactive effects on one another within a policy network.

A two-part mixed-effects modeling framework for analyzing whole-brain network data

Whole-brain network analyses remain the vanguard in neuroimaging research, coming to prominence within the last decade. Network science approaches have facilitated these analyses and allowed examining the brain as an integrated system. However, statistical methods for modeling and comparing groups of networks have lagged behind. Fusing multivariate statistical approaches with network science presents the best path to develop these methods. Toward this end, we propose a two-part mixed-effects modeling framework that allows modeling both the probability of a connection (presence/absence of an edge) and the strength of a connection if it exists. Models within this framework enable quantifying the relationship between an outcome (e.g., disease status) and connectivity patterns in the brain while reducing spurious correlations through inclusion of confounding covariates. They also enable prediction about an outcome based on connectivity structure and vice versa, simulating networks to gain a better understanding of normal ranges of topological variability, and thresholding networks leveraging group information. Thus, they provide a comprehensive approach to studying system level brain properties to further our understanding of normal and abnormal brain function.

Reciprocity of weighted networks

In directed networks, reciprocal links have dramatic effects on dynamical processes, network growth, and higher-order structures such as motifs and communities. While the reciprocity of binary networks has been extensively studied, that of weighted networks is still poorly understood, implying an ever-increasing gap between the availability of weighted network data and our understanding of their dyadic properties. Here we introduce a general approach to the reciprocity of weighted networks, and define quantities and null models that consistently capture empirical reciprocity patterns at different structural levels. We show that, counter-intuitively, previous reciprocity measures based on the similarity of mutual weights are uninformative. By contrast, our measures allow to consistently classify different weighted networks according to their reciprocity, track the evolution of a network's reciprocity over time, identify patterns at the level of dyads and vertices, and distinguish the effects of flux (im)balances or other (a)symmetries from a true tendency towards (anti-)reciprocation.

Gravity's Rainbow: A dynamic latent space model for the world trade network

The gravity model, long the empirical workhorse for modeling international trade, ignores network dependencies in bilateral trade data, instead assuming that dyadic trade is independent, conditional on a hierarchy of covariates over country, time, and dyad. We argue that there are theoretical as well as empirical reasons to expect network dependencies in international trade. Consequently, standard gravity models are empirically inadequate. We combine a gravity model specification with “latent space” networks to develop a dynamic mixture model for real-valued directed graphs. The model simultaneously incorporates network dependencies in both trade incidence and trade volumes. We estimate this model using bilateral trade data from 1990 to 2008. The model substantially outperforms standard accounts in terms of both in- and out-of-sample predictive heuristics. We illustrate the model's usefulness by tracking trading propensities between the USA and China.

Analyzing complex functional brain networks: Fusing statistics and network science to understand the brain*†

Complex functional brain network analyses have exploded over the last decade, gaining traction due to their profound clinical implications. The application of network science (an interdisciplinary offshoot of graph theory) has facilitated these analyses and enabled examining the brain as an integrated system that produces complex behaviors. While the field of statistics has been integral in advancing activation analyses and some connectivity analyses in functional neuroimaging research, it has yet to play a commensurate role in complex network analyses. Fusing novel statistical methods with network-based functional neuroimage analysis will engender powerful analytical tools that will aid in our understanding of normal brain function as well as alterations due to various brain disorders. Here we survey widely used statistical and network science tools for analyzing fMRI network data and discuss the challenges faced in filling some of the remaining methodological gaps. When applied and interpreted correctly, the fusion of network scientific and statistical methods has a chance to revolutionize the understanding of brain function.

Clarifying the use of aggregated exposures in multilevel models: self-included vs. self-excluded measures

BACKGROUND

Multilevel analyses are ideally suited to assess the effects of ecological (higher level) and individual (lower level) exposure variables simultaneously. In applying such analyses to measures of ecologies in epidemiological studies, individual variables are usually aggregated into the higher level unit. Typically, the aggregated measure includes responses of every individual belonging to that group (i.e. it constitutes a self-included measure). More recently, researchers have developed an aggregate measure which excludes the response of the individual to whom the aggregate measure is linked (i.e. a self-excluded measure). In this study, we clarify the substantive and technical properties of these two measures when they are used as exposures in multilevel models.

METHODS

Although the differences between the two aggregated measures are mathematically subtle, distinguishing between them is important in terms of the specific scientific questions to be addressed. We then show how these measures can be used in two distinct types of multilevel models-self-included model and self-excluded model-and interpret the parameters in each model by imposing hypothetical interventions. The concept is tested on empirical data of workplace social capital and employees' systolic blood pressure.

RESULTS

Researchers assume group-level interventions when using a self-included model, and individual-level interventions when using a self-excluded model. Analytical re-parameterizations of these two models highlight their differences in parameter interpretation. Cluster-mean centered self-included models enable researchers to decompose the collective effect into its within- and between-group components. The benefit of cluster-mean centering procedure is further discussed in terms of hypothetical interventions.

CONCLUSIONS

When investigating the potential roles of aggregated variables, researchers should carefully explore which type of model-self-included or self-excluded-is suitable for a given situation, particularly when group sizes are relatively small.