Generating Synthetic Power Grids Using Exponential Random Graph Models

Advances in exponential random graph ( p*) models applied to a large social network

Since the 1970s, communication researchers have utilized social network analysis to understand mass, health, organizational, and interpersonal communication. This article introduces communication researchers to a new class of social network analysis methods, exponential random graph (p*) models. This new method represents the latest advancement in social network methodology and will enhance the trajectory of social network research in the communication discipline. The benefits of this class of models include allowing for the simultaneous estimation of attribute and structural parameters, accounting for the interdependent nature of network data, and retaining the complexity of network observations throughout the analysis. An example analysis using data from Shumate, Fulk, and Monge (2005) is provided to illustrate the potentials of exponential random graph modeling. Five different social network software programs capable of the analysis discussed in this article are introduced with regard to their respective benefits. Finally, a brief tutorial based on data from Palazzolo (2005) is given on how to conduct an ERGM analysis using the PNET software program.

This paper presents a methodology for the systematic generation of synthetic power grid test examples. The synthetic power grids have the same characteristics with actual power systems and do not correspond to any real grid. thus, the generated synthetic power grids are free from confidentiality requirements and available for public use. The examples constructed by this method have small world effect and their characteristics are consistent with that of China's power grid. Meanwhile, the evaluation method of the electrical characteristic parameters of the generated network is studied, and finally the test examples that can be used for various researches are generated. The statistical characteristics of the generated examples are basically consistent with the existing typical examples. The generated examples can provide help for the study of various kinds of power grid operation control methods and power flow calculation methods.

Dependent phenomena, such as relational, spatial and temporal phenomena, tend to be characterized by local dependence in the sense that units which are close in a well-defined sense are dependent. In contrast with spatial and temporal phenomena, though, relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent. Owing to the challenge of characterizing local dependence and constructing random graph models with local dependence, many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference. We take first steps to characterize local dependence in random graph models, inspired by the notion of finite neighbourhoods in spatial statistics and M-dependence in time series, and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference. We show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy, but conventional exponential family random graph models do not satisfy. In addition, we establish a central limit theorem for random graph models with local dependence, which suggests that random graph models with local dependence are amenable to statistical inference. We discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure. In the absence of observed neighbourhood structure, we take a Bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures. We present simulation results and applications to two real world networks with 'ground truth'.

The frequency constitutes a key state variable of electrical power grids. However, as the frequency is subject to several sources of fluctuations, ranging from renewable volatility to demand fluctuations and dispatch, it is strongly dynamic. Yet, the statistical and stochastic properties of the frequency fluctuation dynamics are far from fully understood. Here, we analyse properties of power grid frequency trajectories recorded from different synchronous regions. We highlight the non-Gaussian and still approximately Markovian nature of the frequency statistics. Further, we find that the frequency displays significant fluctuations exactly at the time intervals of regulation and trading, confirming the need of having a regulatory and market design that respects the technical and dynamical constraints in future highly renewable power grids. Finally, employing a recently proposed synthetic model for the frequency dynamics, we combine our statistical and stochastic analysis and analyse in how far dynamically modelled frequency properties match the ones of real trajectories.

How spacetime emerges from featureless nothingness is one of the most intriguing questions in fundamental physics. In this thesis, we take on the random geometry approach to study discretized spacetime and follow the assumption that, in the sim- plest form, geometric structures may arise from random connections between dots under certain rules. We study a family of random graph models called Exponential Random Graph Models (ERGMs). Although this family was extensively investigated in the network science community as a proxy to study real-world social networks, its strength is in its formulation as a Gibbs-Boltzmann distribution in equilibrium sta- tistical mechanics. Thus, one can modify, analyze, and simulate the ensemble using familiar tools from statistical mechanics. We are interested in modifying the basic ERGMs to arrive at a random graph model that possesses emergent geometric properties. This would serve as a proof-of-principle that geometric spacetime may emerge from randomly connected dots. Our study leads to novel classes of random graphs whose edges can self-assemble themselves into both simple geometric primitives (e.g. triangles) and more complex structures (e.g. hexagons). The number of such structures is relatively large compared to the amount of dots and connections available in the graph. Lastly, but interestingly, our model is free from the graph collapse problem that is often observed in the traditional ERGMs.

Exponential random graph models (ERGMs) provide conceptually elegant recipes for generating soft-constrained random graphs. This chapter begins by explaining the theory and describing how to properly specify an ERGM, including demonstrating Lagrange’s method to derive the values of the model parameters that correspond to the desired constraints. Three ERGMs, all with constraints depending linearly on the adjacency matrix, are solved exactly: the targeted total number of links, targeted individual node degrees and targeted number of two-way links in a directed graph. However, when the controlled features become more complicated, ERGMs have a tendency to produce graphs in extreme phases (very dense or very sparse). The two-star model and the Strauss model are worked through in detail using advanced techniques from statistical mechanics in order to analyze the phase transitions. The chapter closes with a discussion of the strengths and weaknesses of ERGMs as null models.

Exponential random graph models (ERGMs) are increasingly applied to observed network data and are central to understanding social structure and network processes. The chapters in this edited volume provide a self-contained, exhaustive account of the theoretical and methodological underpinnings of ERGMs, including models for univariate, multivariate, bipartite, longitudinal and social-influence type ERGMs. Each method is applied in individual case studies illustrating how social science theories may be examined empirically using ERGMs. The authors supply the reader with sufficient detail to specify ERGMs, fit them to data with any of the available software packages and interpret the results.

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.

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.

Man‐in‐The‐Middle (MiTM) attacks present numerous threats to a smart grid. In a MiTM attack, an intruder embeds itself within a conversation between two devices to either eavesdrop or impersonate one of the devices, making it appear to be a normal exchange of information. Thus, the intruder can perform false data injection (FDI) and false command injection (FCI) attacks that can compromise power system operations, such as state estimation, economic dispatch, and automatic generation control (AGC). Very few researchers have focused on MiTM methods that are difficult to detect within a smart grid. To address this, we are designing and implementing multi‐stage MiTM intrusions in an emulation‐based cyber‐physical power system testbed against a large‐scale synthetic grid model to demonstrate how such attacks can cause physical contingencies such as misguided operation and false measurements. MiTM intrusions create FCI, FDI, and replay attacks in this synthetic power grid. This work enables stakeholders to defend against these stealthy attacks, and we present detection mechanisms that are developed using multiple alerts from intrusion detection systems and network monitoring tools. Our contribution will enable other smart grid security researchers and industry to develop further detection mechanisms for inconspicuous MiTM attacks.

Network data often contain sensitive relational information. One approach to protecting sensitive information while offering flexibility for network analysis is to share synthesized networks based on the information in originally observed networks. We employ differential privacy (DP) and exponential random graph models (ERGMs) and propose the DP-ERGM method to synthesize network data. We apply DP-ERGM to two real-world networks. We then compare the utility of synthesized networks generated by DP-ERGM, the DyadWise Randomized Response (DWRR) approach, and the Synthesis through Conditional distribution of Edge given nodal Attribute (SCEA) approach. In general, the results suggest that DP-ERGM preserves the original information significantly better than two other approaches in network structural statistics and inference for ERGMs and latent space models. Furthermore, DP-ERGM satisfies node DP through modeling the global network structure with ERGM, a stronger notion of privacy than the edge DP under which DWRR and SCEA operate.

By introducing the statistical network method of Exponential Random Graph Modelling (ERGM) we adhere to the call of Lucarelli and Brorström (2013) in The Marketing Review to think of new ways to analyse and make inferences about participatory processes of place branding. The ERGM is more suited for studying the interactive dynamics in participatory processes of place branding than traditional methods like case study research or regression analysis. ERGM is a relatively new methodology. Only a few scholars in the neurosciences, international relations, and political sciences have worked with the inferential social network method. We show that the new ERGM methodology helps to gain more insights into the social processes and tie-formation mechanisms which generate emergent interaction patterns in participatory processes of place branding. Hence, this article brings a novel methodology to the area of place branding.

Learning and reasoning over graphs is increasingly done by means of probabilistic models, e.g. exponential random graph models, graph embedding models, and graph neural networks. When graphs are modeling relations between people, however, they will inevitably reflect biases, prejudices, and other forms of inequity and inequality. An important challenge is thus to design accurate graph modeling approaches while guaranteeing fairness according to the specific notion of fairness that the problem requires. Yet, past work on the topic remains scarce, is limited to debiasing specific graph modeling methods, and often aims to ensure fairness in an indirect manner.

A key challenge in the social network community is the problem of network generation -- that is, how can we create synthetic networks that match characteristics traditionally found in most real world networks? Important characteristics that are present in social networks include a power law degree distribution, small diameter, and large amounts of clustering. However, most current network generators, such as the Chung Lu and Kronecker models, largely ignore the clustering present in a graph and focus on preserving other network statistics, such as the power law distribution. Models such as the exponential random graph model have a transitivity parameter that can capture clustering, but they are computationally difficult to learn, making scaling to large real world networks intractable. In this work, we propose an extension to the Chung Lu random graph model, the Transitive Chung Lu (TCL) model, which incorporates the notion transitive edges. Specifically, it combines the standard Chung Lu model with edges that are formed through transitive closure (e.g., by connecting a 'friend of a friend'). We prove TCL's expected degree distribution is equal to the degree distribution of the original input graph, while still providing the ability to capture the clustering in the network. The single parameter required by our model can be learned in seconds on graphs with millions of edges, networks can be generated in time that is linear in the number of edges. We demonstrate the performance of TCL on four real-world social networks, including an email dataset with hundreds of thousands of nodes and millions of edges, showing TCL generates graphs that match the degree distribution, clustering coefficients and hop plots of the original networks.