Random Graph Models Research Articles

Stochastic step-wise feature selection for Exponential Random Graph Models (ERGMs).

This study introduces a novel methodology for endogenous variable selection in Exponential Random Graph Models (ERGMs) to enhance the analysis of social networks across various scientific disciplines. Addressing critical challenges such as ERGM degeneracy and computational complexity, our method integrates a systematic step-wise feature selection process. This approach effectively manages the intractable normalizing constants characteristic of ERGMs, ensuring the generation of accurate and non-degenerate network models. An empirical application to nine real-life binary networks demonstrates the method's effectiveness in accommodating network dependencies and providing meaningful insights into complex network interactions. Particularly notable is the adaptability of this methodology to both directed and undirected networks, overcoming the limitations of traditional ERGMs in capturing realistic network structures. The findings contribute to network analysis, offering a robust framework for modeling and interpreting social networks and laying a foundation for future advancements in statistical network analysis techniques.

Research on Interprovincial Embodied Carbon Transfer Network in China and Its Endogenous Dynamic Evolutionary Mechanism

This paper uses the multi-regional input–output model to measure China’s interprovincial embodied carbon transfer and constructs an interprovincial network; then, the temporal exponential random graph model is applied to analyze the spatial correlation characteristics and endogenous evolutionary mechanism of the network. The results show that interprovincial embodied carbon transfer relationships are increasingly close in China, but the weak symmetric accessibility between the eastern and central regions leads to less reciprocity in the embodied carbon network, and carbon emission inequality still exists. Based on the identification of networks, it is shown that the global network structures are stable, with obvious small-world characteristics and a core–periphery structure. And a structure-dependent effect and time-dependent effect also exist in the formation and evolution of the interprovincial embodied carbon transfer network in China. The popularity, multi-connectivity, and path-dependent effects among the provinces are significant, but the imperfection of interprovincial communication and the cooperation mechanism leads to the failure to form stable structures of ternary closed loops. Interprovincial embodied carbon transfer relationships tend to occur between provinces, with large differences in energy consumption structures, while geographical distance can hinder the formation of embodied carbon transfer relationships. Consequently, considering the spatial network correlation and its endogenous dynamic evolutionary mechanism, it is important to implement policies to guide coordinated carbon reduction among the provinces and to improve the fairness in embodied carbon transferring, in order to promote the fine governance of all links in the transferring process of embodied carbon.

Average distance in a general class of scale-free networks

Abstract In Chung–Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution, and two vertices form an edge independently with probability proportional to the product of their weights. Chung–Lu graphs have average distance $O(\log\log n)$ and thus reproduce the small-world phenomenon, a key property of real-world networks. Modern, more realistic variants of this model also equip each vertex with a random position in a specific underlying geometry. The edge probability of two vertices then depends, say, inversely polynomially on their distance. In this paper we study a generic augmented version of Chung–Lu random graphs. We analyze a model where the edge probability of two vertices can depend arbitrarily on their positions, as long as the marginal probability of forming an edge (for two vertices with fixed weights, one fixed position, and one random position) is as in a Chung–Lu random graph. The resulting class contains Chung–Lu random graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs as special cases. Our main result is that every random graph model in this general class has the same average distance as a Chung–Lu random graph, up to a factor of $1+o(1)$ . This shows in particular that specific choices, such as taking the underlying geometry to be Euclidean, do not significantly influence the average distance. The proof also shows that every random graph model in our class has a giant component and polylogarithmic diameter with high probability.

ALAAMEE: Open-source software for fitting autologistic actor attribute models

The autologistic actor attribute model (ALAAM) is a model for social influence, derived from the more widely known exponential-family random graph model (ERGM). ALAAMs can be used to estimate parameters corresponding to multiple forms of social contagion associated with network structure and actor covariates. This work introduces ALAAMEE, open-source Python software for estimation, simulation, and goodness-of-fit testing for ALAAM models. ALAAMEE implements both the stochastic approximation and equilibrium expectation (EE) algorithms for ALAAM parameter estimation, including estimation from snowball sampled network data. It implements data structures and statistics for undirected, directed, and bipartite networks. We use a simulation study to assess the accuracy of the EE algorithm for ALAAM parameter estimation and statistical inference, and demonstrate the use of ALAAMEE with empirical examples using both small (fewer than 100 nodes) and large (more than 10 000 nodes) networks.

Study on the pattern and driving factors of water scarcity risk transfer networks in China from the perspective of transfer value—Based on complex network methods

The transfer value of risk, measured by the asymmetric impact of virtual water flows, is vital for regions addressing the global water crisis, achieving Sustainable Development Goal 6 (SDG6), and implementing sustainable water management. This study uses multi-regional input-output models, complex network analysis, and the Exponential Random Graph Model (ERGM) to construct and analyze China's water scarcity risk transfer network. Findings show frequent but weakly connected risk transfers between provinces, with significant spatial clustering. The network displays both random and hierarchical features, with clustered development being prominent. Reciprocal cooperation and extensive connections are crucial to avoiding the “prisoner's dilemma” in water conservation. High levels of “dual circulation” and water resource utilization enhance risk transfer value, while lower industrialization and digitization levels facilitate value reception. The efficiency of water resource use is key in shaping high-risk value pathways, underscoring the need for risk management at both source and sink ends. This study offers a scientific basis for improving water resource management, ensuring national water security, and redefining risk flow patterns.

Self-similarity of communities of the ABCD model

The Artificial Benchmark for Community Detection (ABCD) graph is a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs similar to the well-known LFR model but it is faster and can be investigated analytically. In this paper, we show that the ABCD model exhibits some interesting self-similar behaviour, namely, the degree distribution of ground-truth communities is asymptotically the same as the degree distribution of the whole graph (appropriately normalized based on their sizes). As a result, we can not only estimate the number of edges induced by each community but also the number of self-loops and multi-edges generated during the process. Understanding these quantities is important as (a) rewiring self-loops and multi-edges to keep the graph simple is an expensive part of the algorithm, and (b) every rewiring causes the underlying configuration models to deviate slightly from uniform simple graphs on their corresponding degree sequences.

Unveiling the dynamics and determinants of world city network: Insights from global airline data

Existing research on world city networks (WCNs) has predominantly focused on macro-structure analysis, leaving a gap in understanding the dynamic changes from a multi-scale perspective and underlying factors shaping WCNs. Therefore, this study utilizes airline data to explore WCN dynamics at both macroscale and mesoscale, and employ a weighted exponential random graph model to investigate determinants of WCN formation. The research findings reveal that WCN displays hierarchical and regional tendencies. At the macroscale, the evolution of urban hierarchy demonstrates both stability and change, and at the mesoscale, WCN morphology follows a core-periphery spatial arrangement with strong regional patterns. ERGM results highlight the influence of both endogenous network structures and exogenous city characteristics on WCN formation. Preferential attachment emerges as a crucial driving force, with disassortativity facilitating connectivity between peripheral and core cities. Additionally, GDP, language, institutional proximity and path dependence positively contribute to WCN formation, while population and distance have negative effects.

Structural characteristics and mechanism of collaborative environmental governance network of urban agglomerations: perspective of multilevel network

Collaborative environmental governance (CEG) is increasingly advocated to address the environmental risk issues in the integrated development of urban agglomerations. Constructing an effective CEG network from the perspective of interdependent multilevel network plays a vital role in promoting the environmental governance of urban agglomerations. To investigate the structure characteristics and formation mechanism of CEG network, this paper takes the Yangtze River Delta urban agglomeration as the research area, and employes the social network analysis and Exponential Random Graph Model (ERGM) methods to analyze the CEG network, which consists of the collaborative network of cities, relationship network of topics, and affiliation network connecting cities to topics. Research results show that the CEG level in the Yangtze River Delta urban agglomeration continues to improve, while the CEG network is still not in a tightly connected state. For the collaborative network of cities, it presents the small world characteristics and forms a cooperative trend of “central-subcentral-peripheral city.“For the relationship network of topics, the evolution of environmental governance topics is characterized by “from aspect to point.” For the affiliation network connecting cities to topics, as the diversity of environmental governance topics increases among cities, cities within the Yangtze River Delta urban agglomeration tend to share the similar topics. In addition, the interactive triangular structures, star structures, open triangular structures and closed triangular structures in the network can promote the formation of new cooperative relationships in CEG network.

Scalable Estimation and Two-Sample Testing for Large Networks via Subsampling

In recent years, large networks are routinely used to represent data from many scientific fields. Statistical analysis of these networks, such as estimation and hypothesis testing, has received considerable attention. However, most of the methods proposed in the literature are computationally expensive for large networks. In this paper, we propose a subsampling-based method to reduce the computational cost of estimation and two-sample hypothesis testing. The idea is to divide the network into smaller subgraphs with an overlap region, then draw inference based on each subgraph, and finally combine the results together. We first develop the subsampling method for random dot product graph models, and establish theoretical consistency of the proposed method. Then we extend the subsampling method to a more general setup and establish similar theoretical properties. We demonstrate the performance of our methods through simulation experiments and real data analysis. Supplemental materials for the article are available online. The code is available in the following GitHub repository: https://github.com/kchak19/SubsampleTestingNetwork.

Characterizing the spatial correlation network structure and impact mechanism of carbon emission efficiency: Evidence from China's transportation sector

Numerous countries and regions are actively seeking to reduce carbon emissions through policy guidance and technological innovation. In this process, balancing economic development with environmental protection and achieving synergistic carbon reduction between regions pose challenges for policymakers and the academic community alike. This study analyzes data from 30 provinces in China over the period from 2005 to 2020, employing the SBM-DEA, block model, and the Exponential Random Graph Models (ERGM) to explore the spatial association network structure characteristics of carbon emission efficiency and its driving factors. The findings indicate that: the carbon emission efficiency of the transportation industry is generally on an upward trend, with the eastern region having the highest carbon emission efficiency; the spatial association network exhibits a “dense in the east, sparse in the west” pattern; the block model demonstrates clear inter-regional carbon emission transfer behaviors; the result of ERGM shows that factors such as the level of economic development and population density significantly affect the network structure. The macro-micro individual analysis framework for the carbon emission efficiency network fills the theoretical gap in the context of the digital economy, providing a scientific basis and decision-making reference for policymakers when formulating and optimizing carbon reduction policies, which holds significant theoretical and practical value.

Evolutionary characteristics and structural dependence determinants of global lithium trade network: An industry chain perspective

As a globally emerging critical mineral, lithium is increasingly prominent in international trade, emphasizing the importance for energy transition and industrial structure upgrading to understand the evolution and formation of the global lithium trade networks (GLTN). From the perspective of industry chain, this paper uses complex network theory to construct GLTN for 2000–2021, investigates the determinants of network structure dependence through an exponential random graph model (ERGM), for analyzing the evolutionary characteristics and formation mechanisms of trade networks in both the holistic and local dimensions. The results show that, firstly, GLTN exhibits the overall characteristics of a "sparse upstream and tight midstream and downstream" network. The downstream exhibits a distinctive small-world network feature. China, the United States and Europe are the most active countries and regions in GLTN, covering the whole industry chains. Second, endogenous structure, node attributes and exogenous network effects all exert an influence on network structure. Regarding the impact of network structural dependence, lithium trade has transitivity effect, connectivity effect, and popularity effect. Third, the shocks of COVID-19 cut off the transitivity effect formed by indirect dependencies to the downstream industry chain first, resulting in a high network volatility. There is a strong heterogeneity in the network structural dependency for the industry chain.

On the edge: a multilevel perspective on innovation complexities and dynamic attractors in the supply network

PurposeIn innovation management, the complexity inherited in the supply network may be necessary for success. This study aims to holistically examine innovation complexities and system attractors within a hierarchically nested supply network and explore how they dynamically interact and influence adaptive innovation processes.Design/methodology/approachTaking a complexity theory perspective, we employed a methodological bricolage approach using a single case study with multiple embedded units of analysis – namely, a supply network encompassing 36 firms. We drew upon primary data obtained from 42 interviewees and rich secondary data, and we employed a temporal exponential random graph model to examine the micro-foundations of the evolution of the sampled supply network over a decade.FindingsThis study presents a comprehensive overview of the innovation complexities—relational, temporal, dynamic, operational and structural – and how they manifest within a supply network. It also identifies three systemic attractors – point, periodic and strange – and elucidates their relationships with the complexities and their impact on innovative supply network dynamics. The resulting conceptual framework and working propositions provide a detailed perspective on the complex interplay between balanced order and chaos and the potentially unbalanced innovation states within a supply network.Originality/valueThis research offers an in-depth perspective on the innovation complexities and dynamic attractors within a supply network from a holistic, multilevel perspective. It advances complexity theory and deepens the understanding of supply networks as complex adaptive systems.

The scaling limit of random cubic planar graphs

AbstractWe study the random cubic planar graph with an even number of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of as tends to infinity, after rescaling distances by for a specific constant . This is the first time a model of random graphs that are not embedded into the plane is shown to converge to the Brownian map. Our approach features a new method that allows us to relate distances on random graphs to first‐passage percolation distances on their 3‐connected core.

Score-driven exponential random graphs: A new class of time-varying parameter models for temporal networks.

Motivated by the increasing abundance of data describing real-world networks that exhibit dynamical features, we propose an extension of the exponential random graph models (ERGMs) that accommodates the time variation of its parameters. Inspired by the fast-growing literature on dynamic conditional score models, each parameter evolves according to an updating rule driven by the score of the ERGM distribution. We demonstrate the flexibility of score-driven ERGMs (SD-ERGMs) as data-generating processes and filters and show the advantages of the dynamic version over the static one. We discuss two applications to temporal networks from financial and political systems. First, we consider the prediction of future links in the Italian interbank credit network. Second, we show that the SD-ERGM allows discriminating between static or time-varying parameters when used to model the U.S. Congress co-voting network dynamics.

Asymptotic Analysis of k-Hop Connectivity in the 1D Unit Disk Random Graph Model

Asymptotic Analysis of k-Hop Connectivity in the 1D Unit Disk Random Graph Model

Improved MRF rail surface defect segmentation method based on clustering features

Aiming at the characteristics of small number and many types of rail surface defect samples, as well as the problems of unstable transfer learning effect and threshold segmentation being easily affected by environmental factors in real scenes, an improved Markov defect segmentation method with zero samples is proposed. Firstly, the collected data is processed by Gabor function to highlight the defect features and reduce the data dimension to obtain the reduced dimension feature map; Kmeans clustering is performed on the processed feature map to reduce the distribution of data and reduce the influence of reflection and shadow, and the clustering result is used as the pre-classification matrix; an improved Markov random field two-layer graph model is constructed and inferred through the reduced dimension feature map and the pre-classification matrix; the local geometric structure of the defect part is analyzed according to the eigenvalues of the classification matrix inferred by the model; finally, the defect area is marked and the defect segmentation is completed. The experimental part uses a self-sampling data set, and the final conclusion is drawn based on the comparative experiment and ablation experiment. The experimental results show that the pixel accuracy, average pixel accuracy, weighted intersection-over-union ratio, and average intersection-over-union ratio of this method on the self-sampling data set are respectively 93.6%、80.7%、89.4%、68.2% , which exceeds the accuracy of other comparative detection algorithms.

Crisis management from a relational perspective: an analysis of interorganizational transboundary crisis networks

Abstract Although transboundary crises have gained relevance in an increasingly interdependent world, our understanding of the relational dynamics governing these phenomena remains limited. This paper addresses this knowledge gap by identifying common characteristics across interorganizational transboundary crisis networks and drivers of tie formation in successful structures. For this purpose, it applies descriptive Social Network Analysis and Exponential Random Graph Models to an original dataset of three networks. Results show that these structures combine elements of issue networks and policy communities. Common features include moderately high centralization, reciprocated ties, core-periphery structures, and the popularity of international organizations. Additionally, successful networks display smooth communication between NGOs and international organizations, whereas unsuccessful networks have fewer heterophilous interactions. Transitivity seems to play a role in network success too. These findings suggest that crisis networks are robust structures that reconcile bridging and bonding dynamics, thereby highlighting how evidence from relational studies could guide transboundary crisis management.

Exponential Random Graph Model Perspective: Formation and Evolution of a Collaborative Innovation Network in China’s New Energy Vehicle Industry

In light of the crucial role of collaborative R&D in advancing technology within the new energy vehicle industry, this study seeks to explore ways to overcome the barriers to technological innovation by establishing an effective collaborative innovation network. Utilizing joint patent-authorized data from China’s new energy vehicles between 2005 and 2019, the collaborative innovation network was developed, and the Exponential Random Graph Model (ERGM) was employed to analyze its formation and evolution mechanisms. The results indicate that the network has undergone significant expansion, closely linked to strong national policy support and the active involvement of innovation participants. The network exhibits effects of expansion, transfer, and closure. External attribute analysis revealed the Matthew effect and geographical compatibility effect and found that organizational compatibility tends to foster complementary cooperation. The findings offer insights into optimizing collaborative innovation networks in the NEVs industry and suggest strategies for policymakers and industry players to promote collaborative innovation.

Switchover phenomenon for general graphs

AbstractWe study SIR‐type epidemics (susceptible‐infected‐resistant) on graphs in two scenarios: (i) when the initial infections start from a well‐connected central region and (ii) when initial infections are distributed uniformly. Previously, Ódor et al. demonstrated on a few random graph models that the expectation of the total number of infections undergoes a switchover phenomenon; the central region is more dangerous for small infection rates, while for large rates, the uniform seeding is expected to infect more nodes. We rigorously prove this claim under mild, deterministic assumptions on the underlying graph. If we further assume that the central region has a large enough expansion, the second moment of the degree distribution is bounded and the number of initial infections is comparable to the number of vertices, the difference between the two scenarios is shown to be macroscopic.

From Protectionist to Regulator: Policy-Driven Transformation of Digital Urban Networks in China’s Online Gaming Industry

In the digital era, data-driven production organizes digital urban networks. This study explores the critical role of government policies in shaping these networks, focusing on China’s evolving policy contexts. While existing research has mainly emphasized qualitative analyses, this paper quantitatively assesses the impact of policy changes on digital urban networks, specifically through the lens of China’s online gaming industry. The study aimed to elucidate the relationship between the policy environment and digital urban networks. By examining China’s transition from protectionist to regulatory policies, this research employed a social network analysis and valued exponential random graph models (ERGMs) across two key phases: the competitive protection phase (2014–2017) and the systematic regulatory phase (2018–2022). The findings revealed a significant transformation in urban network structure, shifting from a centralized model dominated by a few core cities to a decentralized, multi-centered network. The key factors influencing this evolution include the institutional proximity and cross-regional collaborations. This study offers valuable insights into how policy shifts affect urban networks in the digital economy, contributing both theoretically and practically to future policy design.