A survey on exponential random graph models: an application perspective

A form of the likelihood function

We aim to find the best values of the θ vector in Eq. (3) which maximize the probability over the observed data. In a more formal expression, we want to solve the following equation: (10)${\displaystyle {\theta }_{ML}:=argma{x}_{\theta \in R^k}P\left(X|\theta \right)}$ where, P is the same probability function as the Eq. (3) and, R^k represents all possible real values over a k-dimentional space. Note that the θ is a vector of coefficients rather than a single value; thus, its space value should be a vector space. Different methods exist for solving such equations. Here, we are going to name a few of them which are mostly used in the ERGM related works. Also, we intend to present a number of state-of-the-art methods that have been presented after 2016.

Sampling methods

There are two important applications of sampling methods in the ERGMs parameter estimation model:

• In all methods, there is a need to simulate graphs from the fitted model or simulate some graphs to gain more insight into the distribution of the graphs and their configurations (Lusher, Koskinen & Robins, 2012). This distribution is also used to test whether the distribution of the fitted model is close to the observed data or not.

• Predicting the prior distribution of the graphs for Bayesian learning models

So, there is a need for sampling methods to draw a sample from the given graph distribution. In this section, we present some of the sampling methods that have been used extensively in the literature.

Monte Carlo Markov Chain sampling method which is abbreviated to MCMC (Metropolis et al., 1953) is a well-known sampling method which has been used in many works. Here, we only discuss it in the context of graph generation. In this method, we start with an initial graph which can also be an empty graph. Then, in each iteration, a new graph is generated by making a small change to the graph from the last step. The form of this “change” is different from work to work. The most straightforward change is adding or removing a tie. The procedure is as follows: two nodes are chosen randomly. After which the state of their connection is altered (if they are already connected, they become disconnected while if they are not connected, they become connected.). In the next step, the probability of the generated graph is computed according to Eq. (3). This probability is compared to that of the graph generated in the previous step. Then, we accept or reject the new graph based on the comparison of these two probabilities. If the new graph is more probable, it is more likely to substitute the old graph in the next iteration. The probability of whether the new graph is chosen for the next iteration or the graph from the last step is re-chosen depends on which one of them has a higher probability score in Eq. (3). Note that only having a higher probability score is not a guarantee that the graph gets chosen. It only increases the chance of selection. All these outlined the scheme of all MCMC methods. However, the details including how many of ties are altered in each iteration or the probabilistic selection between the old graph and the new one are different in literature. We intend to present a quick introduction to the Metropolis-Hasting sampling methods which is mostly used in ERGM related literature. Figure 1 displays the overall procedure of an MCMC method.

Metropolis-Hasting Metropolis et al. (1953) is the most widely used MCMC derivation in ERGM studies. Metropolis Hasting in the context of graph generation is as follows. Initially, as we explained in the general MCMC scheme, we start with an empty or random graph. Our goal is to generate N samples from the distribution of graphs, implying that we want to generate a sequence of x₁, x₂, …, x_N graphs. We choose two random nodes at each step and change the tie situation between them. The probability of the newly generated graph and the graph from the last step is then computed using the following formula: (11)${\displaystyle min\left\{1,\frac{P\left(X=x_{new}|\theta \right)}{P\left(X=x_{N-1}|\theta \right)}\right\}.}$

This formula computes the probability of whether to accept the new move or substitute the last step graph as the new one.

Classic methods

So far, we have reviewed the necessary preliminaries. Now, we can review the most widely used methods in the literature for estimating the value for statistical parameters (θ in Eq. (4)) best representing the observed data. In other words, our aim is to solve Eq. (10). Most of the methods use the following steps: initially, they start with an initial value for the parameter vector. Then, the distribution of the graphs is generated by one of the sampling methods. Next, the difference between the distribution and the observed data is computed ($E_{\theta }\left(C\left(X\right)\right)-C\left(X_{observed}\right)$. If the difference is satisfactory, the learning process is halted and the current vector of the parameter is considered as the final answer which best fits the observed data; Otherwise, based on the learning method the algorithm moves to the subsequent values of θ and goes back to step 2. Figure 2 demonstrates the finite automata of this method.

The ultimate goal of all learning methods is to find a vector of θ values in Eq. (3) that can also generate graphs which are similar to the observed graphs. To this end, different learning methods exist and this section describes the most important of them namely importance sampling and stochastic approximation. We used the description presented in Lusher, Koskinen & Robins (2012).

Importance sampling

The goal as we said is to minimize the expected value of θ which minimizes the expected value between observed statistics and the ones generated by the ERGM model. The aim is to use Maximum Likelihood that we discussed before to find the best value of vector θ which maximize the right hand side of Eq. (10). One possible approach is to search over all possible θ values in the search space and try them one by one. But since the search space is very large and θ values are continuous this approach is not practical. Instead of such brute force algorithm, one of the methods for ERGMs parameter estimation is the one inspired by the general framework ML estimation method for dependent data introduced by (Geyer & Thompson, 1992). The main idea is to instead of generating the whole possible graphs of a particular θ vector, we can draw a large sample of the graphs and consider it as a representation of the whole possible graphs at each iteration. This sample is generated from the current value of the θ vector using the Eq. (3) and is used in the formula to compute $E_{\theta }\left(C\left(X\right)\right)$ and then compute how much the value $E_{\theta }\left(C\left(X\right)\right)-C\left(X_{observed}\right)$ is close to zero. At each iteration an average over the generated graphs statistics is computed to measure $E_{\theta }\left(C\left(X\right)\right)$ and decide whether to continue the estimation or not based on how much the $E_{\theta }\left(C\left(X\right)\right)-C\left(X_{observed}\right)$ is close to zero. Other than the mentioned halting situation we need an algorithm to move from each θ vector to a new one (if the halting is not satisfied). A Newton-Raphson formula is used to move from one statistic to another. For more detail on the mathematical details of the sampling and the used Newton-Raphson based method see (Lusher, Koskinen & Robins, 2012).

Stochastic approximation

The This model presented in Snijders (2002) can handle both bimodal and multimodal and enhance the speed of convergence. They also used the Newton-Raphson method for the learning step of the algorithm. As mentioned in Lusher, Koskinen & Robins (2012), they used a three-step method. At phase one, a limited number of iterations is performed to determine initial values of the algorithm. In the second step, the Newton-Raphson algorithm is employed to optimize the answer. Finally, the convergence criteria are checked.

Newly presented methods for ERGM estimation

In Byshkin et al. (2016), the authors improved the MCMC sampling part of the ERGM estimation by adding an auxiliary parameter to the model. In their method, which they called Improved Fixed Density (IFD) MCMC sampler, they tried to decrease the state space of the network to reduce the time complexity of the algorithm. This new auxiliary variable which was based on the number of ties helped the model to converge faster without the need of making the MCMC overall model more complicated.

In some works like (Stivala et al., 2016), snowball sampling (Coleman, 1958; Goodman, 1961) was used to overcome the computational complexity of the MCMC method over large network datasets.

As mentioned earlier, the Bayesian estimation of the parameters requires prior knowledge about the network posterior distribution. However, this posterior probability distribution is not always easily available. To overcome this issue, (Bouranis, Friel & Maire, 2017) introduced a pseudo-likelihood estimation approach by replacing the posterior distribution with a more achievable pseudo-distribution. Although this method resulted in faster computation of the likelihood function, as mentioned in the (Schmid & Desmarais, 2017), its results are not still as precise as they should. To handle this problem, the same mentioned article introduced another pseudo-likelihood estimator based on the bootstrapping parameters which culminated in more accurate convergence.

In a recent work (Bouranis, Friel & Maire, 2018), the authors proposed yet another heuristic model based on pseudo-likelihood estimation. They did so by performing three adjustments to the pseudo-likelihood function: (1) mode corrections to overcome the bias of the pseudo-likelihood function; (2) curvature adjustment, which is a modification in the selection of the transformation matrix and the corresponding Hessian matrix; and (3) magnitude adjustment, which is a linear transformation to scale the curvature-adjusted pseudo likelihood to the right values.

Despite all the progress in the ERGMs parameter estimation and modeling, it is still a hard task in large graphs. Thiemichen & Kauermann (2017) addressed two of the main challenges of ERGMs, including the instability of the model especially in the models with more straightforward statistics like triangles and the time-consuming nature of the ERGM parameter estimation procedure due to large number of numerical simulations. For solving the first problem, they proposed a technique to produce smooth stable statistics. Further, to overcome the second issue, they employed a novel subsampling model which instead of fitting the model to the whole network it only fit the model to subgraphs from the network and then aggregated these sample estimates. The two mentioned ideas yielded a significant improvement for modeling large graphs.

ERGMs variations

Apart from the basic definition of ERGMs there are also some other variations of ERGMs. Each year a number of new extensions of the original ERGM definition are introduced. In this chapter we introduce three of the most widely used ERGMs variations.

Evolution of networks in dynamic environment like social networks has attracted scientist to make an extension of the ERGMs called Temporal ERGMs a.k.a. TERGMs which is capable of capturing the information underlying dynamics of such networks (Hanneke et al., 2010). A Markov assumption between snapshots of the network at each timestep is taken. Then the model is created based upon the relation between each two consecutive snapshots S_t and S_t−1. (12)${\displaystyle P\left(X=S_t|S_{t-1},\theta \right)=\frac{1}{N\left(\theta ,S_{t-1}\right)}exp\left\{{\theta }^T,\psi \left(S_t,S_{t-1}\right)\right\}.}$

As it can be seen in Eq. (12) most parts of the formula for TERGM is similar to normal ERGM. However, the time snapshots are now considered and each new time snapshot S_t is dependent to its previous one S_t−1. Also, the normal count of the networks statistics has been substituted with temporal potential count ψ over two consecutive snapshots. For more information see Hanneke et al. (2010) and for the information about the btergm which is a library for temporal ERGMs see Leifeld, Cranmer & Desmarais (2017).

Most of the real-world network are associated with a value on their edges which are referred to as weighted graphs in graph theory. A plethora of researches have been done to consider these types of networks into the ERGM general schema. GERGM (Desmarais & Cranmer, 2012) and the model proposed by Krivitsky (2012) are the two most well-known models which have incorporated the networks’ edges’ weights into the model. The normalizing factor in Eq. (3) which is the denominator of Eq. (4) is not assured to be convergent when the network statistics (C(x) in the Eq. (4)) are infinite set like continuous valued edges. GERGM is a model aimed to overcome this issue by using a probability model for such continuous values. They build a transformed version of the original ERGM formula that no longer suffers from the mentioned problem. The Krivitsky (2012) have also extended the previous binary version of ERGM which only models edges existence rather than their value into a model which is capable of capturing the information of weighted graphs. However, his method is restricted to natural valued weights on the edges.

In network science there is a special kind of networks called multiplex or multilayer networks. These are networks which their nodes are connected in the context of more than one attribute. For example, in a social relation network, actors might have several relations between them like friendship network or co-working network. Each of these relations can be abstracted as a layer in a network model. Also, in some situations, there is a hierarchical structure in the data like modeling the relations inside a university. There are schools, which are divided into groups and lecturers and students. An extension of ERGM which is applicable to model such scenarios in multilevel networks is proposed for these networks (Wang et al., 2013). They considered relation between the nodes in each level and also the inter-level relations into the model. For example, consider a two layer network with layers A, B and an imaginary layer between them called x which is for the purpose of modelling inter-level relations between A and B. Then the Eq. (1) is re-written as: (13)${\displaystyle P\left(A=a,X=x,B=b|\theta \right)=\frac{1}{N}exp\left\{{\theta }_a^T{C}_a\left(a\right)+{\theta }_b^T{C}_b\left(b\right)+{\theta }_x^T{C}_x\left(x\right)+{\theta }_{a,x}^T{C}_{a,x}\left(a,x\right)+{\theta }_{b,x}^T{C}_{b,x}\left(b,x\right)+{\theta }_{a,b,x}^T{C}_{a,b,x}\left(a,b,x\right)\right\}}$

Which the ${\theta }_a^T,{\theta }_b^T,{\theta }_x^T,{\theta }_{b,x}^T,{\theta }_{a,x}^T,{\theta }_{a,b,x}^T$ are the parameters for statistics which are extracted from layers a, b and the inter-level relations a, x and b, x and the inter-level relation of layers a, b. The same is true for the count functions of the statistics. θ is the set of all types of statistics.

Medical imaging

In order to take care of the limitations of the descriptive analysis of brain neural networks, the author of Sinke et al. (2016) used ERGMs to be able to model the observed network using the joint contribution of network structure. They also compared the changes in brain networks statistics across different ages. This study was conducted to examine the effects of aging during lifetime in the brain global and local structures. Graphs where extracted from brain images obtained from diffusion tensor imaging (DTI). Four network statistics were used to model these networks:

• The number of edges

• The geometrically weighted edgewise shared partner (Hunter, 2007)

• The geometrically weighted non-edgewise shared partner (Hunter, 2007)

• The hemispheric node match: a binary indicator which shows whether two nodes are in the same hemisphere of the brain.

The Bayesian learning schema from Caimo & Friel (2011) was used to fit the model.

In a recent work, (Dellitalia et al., 2018) employed ERGMs to study the structure of neural networks of the brain. They aimed to increase the chance of unconscious and injured patients to recover by analyzing brain functional data. In their work, they overcame four shortcomings of previous methods by incorporating ERGMs into their study. For example, one of them was the ability to assess the dynamics of the network over time.They used the Separable Temporal ERGMs (TERGM) (Krivitsky & Handcock, 2014) for their modeling. One of the aspects of their work that successfully handled with ERGMs was that the network structures they chose should have not been necessarily independent. This restriction was one of the main drawbacks of previous methods.

Functional Magnetic Resonance Imaging or fMRI is a method for observing brain activities and their changes over time. There are components in the fMRI images which can be explained using network analysis methods. Nodal signals, network architecture, and network function are the three essential network properties in building fMRI-based networks (Solo et al., 2018). ERGMs are one the main important network analysis methods which have been used to explain such networks. The authors of a review paper (Solo et al., 2018) introduced the most critical efforts with the aim to explain these brain networks. Note that there are plenty of works which used ERGM as their method (Simpson, Hayasaka & Laurienti, 2011; Simpson, Moussa & Laurienti, 2012).

Healthcare applications

Having a healthy life is one the central concerns of human life. If we look at this issue from a macro perspective, we can see that many health-related problems can be alleviated by analyzing their corresponding inter-related actors. For example, in epidemiology, there is a direct connection between the patient relationships and the extent that the disease can spread. In most cases, these relations between the actors will result in the formation of a network. This network can be analyzed using ERGMs to answer different questions underpinning its formation and dynamics. This kind of analysis is something that has already been done extensively by researchers in the healthcare community.

Analyzing inter-hospital patient referral network is a significant problem which (Caimo, Pallotti & Lomi, 2017) has recently investigated using ERMGs. They used a combination of the edges and nodes of the network and utilized the Bayesian approach introduced in Caimo & Friel (2011) to fit their model. This task was done using BERGM (Caimo & Friel, 2014) R language package for their implementation.

Another work (Baggio, Luisier & Vladescu, 2017) shed light on the relationship between social isolation and mental health. The connection between these two subjects was investigated by analyzing the network of Romanian adolescents using ERGM modeling. They concluded that there is a strong link between the two mentioned concepts.

Application of statistical network in epidemiology and disease spreading is another interesting topic which has attracted from the attention of the biological science community. (Silk et al., 2017) provided an important opportunity to advance the understanding of the pattern and evolution of infections in static and dynamic environments. They also used ERGMs for their models. In their ERGM model, they employed a fair number of both structural and node-based attributes. The ERGM (Hunter et al., 2008) R language package was used for the tests.

Social ties can reveal a wide range of aspects of human life. The networks formed by such ties and edges among individuals can transfer life habits and behaviors in a society. For example, in many kinds of literature, the relationship between social tie and analysis of obesity has been investigated. Zhang et al. (2018b) thoroughly studied the articles related to applications of social network analysis to obesity. In another work related to eating disorders, (Becker et al., 2018) have presented some findings using ERGM network analysis about the relationship between the eating disorders and human relationships. They conducted their study on members of a sorority at Southeastern University.

Economics and management

Marketing organizations that are responsible for promoting tourist destination have also been analyzed using ERGMs. (Williams & Hristov, 2018) intended to study the networks underpinning Destination Marketing Organizations (DMOs). They developed four models with the most complex one consisting of the following statistics:

• Number of edges

• The geometrically weighted edgewise shared partner (Hunter, 2007)

• Properties of membership and industry background.

Global migration and different attributes of immigrants can be considered as a network. There are many theories on how these networks shape and evolve and how they depend on immigrants and country backgrounds. ethnicity, wealth, religion). (Windzio, 2018) applied ERGM in order to examine theories and hypotheses about creation and evolution of these networks. He used both the graph structure and node attributes in a large number of statistics.

Global tourism and its corresponding network, Global Tourism Network (GTN), is yet another field of study, given the tremendous financial importance of tourism market. As mentioned in Lozano & Gutiérrez (2018), it is essential to gain insight into the connections between its components. In the same article, an ERGM approach was used to find the critical local substructures of the GTNs.

Handling the budget and resources during crises is always a challenging task for humanitarian organizations. There is a need for a tradeoff between the use of asset supplies for the current crises and the usual ongoing projects. This problem has been formulated in the form of asset supply networks. Stauffer et al. (2018) used ERGMs as an empirical model to understand the asset flows during a crisis.

The applications of ERGMs have even been extended to the analysis of online drug distribution networks. In a recent work, (Duxbury & Haynie, 2018) conducted the mentioned research on a dataset of an online drugstore on the dark web. They studied such networks concerning their topological dynamics, suppliers, and customer demand as well as the resistance of such networks to disruptions.

Does economic partnership between professionals will result in further trust and solidarity? This is the central question of Bianchi, Casnici & Squazzoni (2018). They developed an ERGM multiplex network model collaboration network and a number of other attributes and then analyzed it using multivariate ERGMs to examine social support and trust for each of the network statistics.

Political science

A large number of articles in the political science community have used ERGMs for their modeling. This enthusiasm toward ERGMs among political science scholars well suggests that it is among the most famous mathematical modeling in the field. Here we introduce a handful of these articles.

Sustainable development policy is a major concern both for the government and the private sector. It is only achievable by interaction among individuals. In particular, the role of the connection between funding sectors and those in need of money is important for carrying out their projects. This is the central problem of Gallemore & Jespersen (2016). The dataset consisted of 91 donor organizations. The role of ERGM in this work was modeling the donor agent relationship networks.

Another major issue that has been addressed through ERGMs is collaborative governance between different sectors and individuals of multiple organizations. In Ulibarri & Scott (2016), the authors used ERGM to test their hypothesis about what should be observed in low-collaboration vs. high-collaboration networks. Four simple ERGMs’ configurations were used, including:

• The number of networks ties

• The number of nonzero ties

• The number of reciprocity relations in the network

• The number of transitivity relations in the network.

In a more recent work, (Scott & Thomas, 2017) addressed the same problem. However, they used different datasets and network statistics. Hamilton & Lubell (2018) also took the same ERGM modeling approach in discussing the collaborative governance, in the special domain of climate change adoption.

In an exciting work, Li et al. (2017) investigated the effectiveness of military alliances in making peace between states. They used temporal random graph models for longitudinal network data of alliance. They employed two different sets of network statistics and developed two models upon them.

Communications via internet social networks have helped the human to take a huge step further. People from multiple backgrounds and societies are engaged in conversations that have never been possible before the widespread popularity of online social networks. In the case of political conversations in social networks, there is always the dilemma whether this freedom has resulted in more communications between people with different ideologies or adversely it will cause people with same viewpoints tend to dominate most of the conversation thereby self-reinforcing the same way of thinking. (Song, Cho & Benefield, 2018) addressed this issue by studying the network of message selection of users during a presidential election and then analyzed the mentioned network by a Temporal ERGM (TERMG) to answer the questions above. The world trade network has also been investigated via TERGMs. Pan (2018) studied these networks to answer the underlying questions about them and their effects on related subjects.

Even further, some works such as Chen (2019) took the use of ERGM networks in modeling political networks a step further by incorporating multilayer networks properties into their models. He proved with experimental results that this multilayer approach toward ERGMs could better fit the model to the observed data.

The analysis and challenges of power transition in a personalized authoritarian system is a problem that has been discussed in Osei (2018) using ERGM modeling. In addition to qualitative methods, the author employed ERGM as a quantitative method to answer questions about the regime survival of the regime under the mentioned situations. The network in this context consisted of elite interactions network in authoritarian countries. They found that many of the important people in the past ruler administration still play a crucial role in the current government.

Environmental treaties among governments play a vital rule in solving environmental issues. However, coming to an agreement in such commitments is not straightforward. The aim of Campbell et al. (2019) is to study the model of ratification in such treaties among different parties or states. The main contribution of this research is to find out how the influence network between countries can affect the interdependency of countries decisions on environmental politics. To this end, they have used Bipartite Longitudinal Influence Network (BLIN) model to extract two latent influence network using which show negative and positive influence among different countries. Later these two networks have been analyzed using ERGMs to find the effective contextual and structural network statistics on the shaping of influence (negative or positive) networks (Marrsetal, 2018).

Network of international arm trade is yet another subject that has been studied using ERGM simulations (Thurner et al., 2019). The structure of weapon exchanges network between countries and alias is very complicated. A plethora of effective factors are effective in the formation of the network. Economic enhancement of the seller and the desire to strengthen they allay in different regions of the words are two important considerations from the dealers. Their datasets are extracted from available data of arm deals after world war II. Temporal ERGMs were used for during the analysis. They have used a number of statistics based on their hypothesis about importer and exporter effects, size of the countries’ domestic economic markets, national material capabilities, conflict involvement joint membership in defense agreement, geographic distance between two countries. Most of the statistics used in this work were exogenous statistics.

Missing data and link prediction

Link prediction is the problem of finding missing links in a network. As we explained, ERGM deals with estimating graph distribution and generating a new graph based on them. Graph generation part is the exact process of finding missing links. However, in link prediction, we do not want to estimate the whole graph distribution, and we desire to find the probability of link formation between two nodes based on the current structure of the graph.

Smith (2012) used ERGMs to create a global view of networks with missing data based on sampled data. In their approach, they took sampled ego networks and tried to estimate features of the whole network. The interesting fact about this work was that not both the structure of the network and its size were unknown. A three-step algorithm was used, and in the last step, the aim was to predict the global structure of the network from the fitted model. Two real-world network data were used in the tests including addition of health network and sociology co-authorship network.

Koskinen et al. (2013) used the same approach of leveraging ERGM for data augmentation in graphs with missing tie variable. In an empirical test, they were able to estimate the missing tie variable of a network with 74% missing tie with fair precision. As the article name suggests, they used a Bayesian estimation method for fitting the parameters of their model.

Zhang, Zhai & Wu (2013) applied ERGMs for predicting links in microblogs. They used five kinds of graph statistics with four of them (2–5) introduced in Hunter (2007):

• Number of edges

• Gwidegree (Geometrically weighted indegree): the weighting indegree of the network.

• Gwidegree (Geometrically weighted outdegree): the weighting outdegree of the network.

• Gwodegree (Geometrically weighted dyadwise shared partner): the number of shared nodes of all node pairs in the network.

• Gwesp (Geometrically weighted edgewise shared partner): similar with Gwdsp, it is the number of shared nodes for linked node pairs in the network.

The link prediction based on the ERGM method introduced in this article is an iterative approach. At each step, they compute the conditional probability of adding an edge between two arbitrary nodes having the observed part of the network. This process is performed several times through an MCMC simulation, and at last the average of all these steps is computed: (14)${\displaystyle P\left(X_{ij}=1|X^c=x^c\right)=\frac{1}{N}exp\left({\theta }^{T}C\left(x_{ij}=1,x^c\right)\right)}$

In Eq. (14), X_ij is the probability of presence of an edge between nodes i and j. X^c is also the state of all other edges in the time predicting X_ij.

Five datasets have been used:

• Sina Microblog dataset community of “Beijing badminton community.”

• Sina Microblog dataset community of “Beijing bicycle community.”

• Sina Microblog dataset community of “Data mining community.”

• Scientist co-authorship dataset GR (General Category).

The authors of Krause & Caimo (2019) have presented a new estimation algorithm for Bayesian Exponential Random Multi-graphs model which is an imputation model applicable to such multi-layer networks. This work is an extension of the Koskinen, Robins & Pattison (2010) to multi-layer networks.

An interested reader can refer to a recent survey on different imputation method on network missing data (Krause et al., 2018). One of the advantages of this methods is that it is solely about missing data in the context of networks. Different missing data treatment methods have been tested on different missing data in a complete benchmarking framework.

Scientific collaboration

Finding the best colleagues or best-related research papers and topics is always a significant issue for anyone in the scientific community. Co-author and citation networks analyses are two important topics that have been extensively studied in research related to analysis of networks addressing these issues.

The researchers in Zhang et al. (2018a) addressed the effect of three major network properties in scientific collaboration networks including Homophily, Transitivity, and Preferential attachment. Performing an ERGM study on these networks, they argued that incorporating the mentioned properties we can provide more insight into how collaborations form. The data for this study were collected using the metadata of papers’ citations from the Web of Science from 1956 to 2014.

As we approach more complex scientific phenomena, we more feel the need for collaboration between different scientific communities. Fagan et al. (2018) also studied a co-authorship network to evaluate the changes in inter-disciplinary scientific articles. More precisely, they applied a special form of ERGMs called the Separable Temporal ERGMs (STERGM) Krivitsky & Handcock (2014) to evaluate the co-authorship network over time and make prediction ties in the network. They employed some structural and nodal attributes. Structural attributes refer to a number of edges, degree, and triadic closure, while some nodal attributes capture whether two individuals have the same professor rank, gender, and college.

ERGMs are widely used for citation networks analysis. An & Ding (2018) performed the same study in the special case of publications on causal inference. They argued that some technical and social processes are underpinning citation networks. Their ultimate goal was to explain the essential factors in forming a citation network and predicting the citation patterns.

An in-depth study of polarization among researchers of the field of social science was performed in a recent work (Leifeld, 2018). He used both qualitative and quantitative methods to address the most compelling reasons and strategies causing the polarization. He applied ERGM as his qualitative method over two co-authorship networks in the field of social science in two separate countries.

Other than the studies on co-authorships and citation networks there are other aspects of scientific collaboration that have been widely studied. One of such studies is the study on how the recruitment of new members of scientific collaborators in scientific organizations takes place. In a study (Leifeld & Fisher, 2017) the dynamics underlying the membership procedure of new scientist in international scientific assessments has been evaluated. The authors have used a dataset extracted from an international well-known research program on world’s ecosystem called Millennium Ecosystem Assessment (MA). Their method is based on analyzing the pattern of the network formation by ERGM using a number of exogenous and endogenous network’s statistics. The analysis approved the authors hypothesis which suggests that factors like having the same nationality to the previous researcher in the research group or being in the same institution with them have a high impact on the recruitment of new researchers. This could result in lack of diversity of opinions in the final outcomes of the assessments conducted by the research group.

Wireless networks modelling

Random Geometric Graphs also known as RGGs are defined as the group of graphs which are obtained by placing a number of nodes randomly in a geometrical space and draw vertices between those nodes which their distance is less than a threshold d in a given norm (Penrose et al., 2003). One issue in the wireless sensor networks is that there is not a fixed placement for the nodes in most of times. The nodes are randomly distributed and therefore the shape of connecting graph tend to be very volatile (Raghavendra, Sivalingam & Znati, 2006). Studying these graphs formation and the statistical dynamic behind their formation has been extensively investigated in the literature related to RGGs (Iyer & Manjunath, 2006). For example, exponential RGGs which are the RGGs that the distribution of their nodes is also exponential (Gupta, Iyer & Manjunath, 2008). These graphs have been used for modeling wireless sensor networks Shang (2009) abd Kenniche & Ravelomananana (2010). In the Shang (2009) the wireless sensors are assumed to be on a line and to evolve over time with respect to a dynamic RGG process. The effect of statistical properties for a particular time snapshot has also been considered in this paper. Such analysis with using one-dimensional RGG has also been done in the past in Karamchandani et al. (2006). Vehicular Ad Hoc Network are yet another use case for RGGs (Zhang et al., 2014). Due to the movement of the vehicles and the rapid changes in the graph they have many similarities to previous applications of RGGs.

Other applications

The applications of ERGMs are so extensive that some works cannot be organized in a particular category. In this section, we introduce some of them.

The concept of social networks is not limited only to human relationships. There are some complex interactions in animal behaviors which can be modeled as graphs. ERGMs are also a useful tool for analyzing these kinds of networks. In a recent work, (Silk & Fisher, 2017) reviewed the use of ERGMs in such studies. Also, more specifically, other recent works have leveraged ERGMs capabilities in their specific context. Hellmann & Hamilton (2018) is a work in which the authors investigated the effect of neighbors’ mediation in cooperative fish breeding by analyzing their interactions with an ERGM model. In another work (Silk et al., 2018), the same approach was used to investigate sex-related disease spreading through animal contact networks in three sorts of animal networks.

In a novel work, Müller, Grund & Koskinen (2018) studied the social inequalities in Sweden by analyzing an immigrant movement flow network on both the micro and macro levels. Their network was a directed binary graph with Stockholm’s neighborhoods as the nodes and ties as the representative of the movement flow across neighborhoods. Only structural features (statistics) were used in ERGMs.

How do networks respond to a sudden change? Which sort of disruptions is most influential in the network upcoming status? How the network will react to a change or what is the best reaction? These are all questions that can be summarized as “forecasting social network reaction to disruption.” In a recent article, Mellon & Evans (2018) reviewed state-of-the-art research articles concerning these topics in various fields. According to them and by mentioning one of their previous works (Mellon, Yoder & Evans, 2016), ERGMs can play a crucial role in this issue. In the mentioned work, they used ERGMs to examine the network formation mechanism before and after the intervention. According to their findings, networks tend to preserve these mechanisms following the disruptions.