Random Network Research Articles

A local–global principle for nonequilibrium steady states

The global steady state of a system in thermal equilibrium exponentially favors configurations with lesser energy. This principle is a powerful explanation of self-organization because energy is a local property of configurations. For nonequilibrium systems, there is no such property for which an analogous principle holds, hence no common explanation of the diverse forms of self-organization they exhibit. However, a flurry of recent empirical results has shown that a local property of configurations called "rattling" predicts the steady states of some nonequilibrium systems, leading to claims of a far-reaching principle of nonequilibrium self-organization. But for which nonequilibrium systems is rattling accurate, and why? We develop a theory of rattling in terms of Markov processes that gives simple and precise answers to these key questions. Our results show that rattling predicts a broader class of nonequilibrium steady states than has been claimed and for different reasons than have been suggested. Its predictions hold to an extent determined by the relative variance of, and correlation between, the local and global "parts" of a steady state. We show how these quantities characterize the local-global relationships of various random walks on random graphs, spin-glass dynamics, and models of animal collective behavior. Surprisingly, we find that the core idea of rattling is so general as to apply to equilibrium and nonequilibrium systems alike.

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.

Approximation of subgraph counts in the uniform attachment model

Abstract We use Stein’s method to obtain distributional approximations of subgraph counts in the uniform attachment model or random directed acyclic graph; we provide also estimates of rates of convergence. In particular, we give uni- and multi-variate Poisson approximations to the counts of cycles and normal approximations to the counts of unicyclic subgraphs; we also give a partial result for the counts of trees. We further find a class of multicyclic graphs whose subgraph counts are a.s. bounded as $n\to \infty$ .

Black holes, complex curves, and graph theory: Revising a conjecture by Kasner

The ratios 8/9=22/3≈0.9428 and 3/2≈0.866 appear in various contexts of black hole physics, as values of the charge-to-mass ratio Q/M or the rotation parameter a/M for Reissner-Nordström and Kerr black holes, respectively. In this work, in the Reissner-Nordström case, I relate these ratios with the quantization of the horizon area, or equivalently of the entropy. Furthermore, these ratios are related to a century-old work of Kasner, in which he conjectured that certain sequences arising from complex analysis may have a quantum interpretation. These numbers also appear in the case of Kerr black holes, but the explanation is not as straightforward. The Kasner ratio may also be relevant for understanding the random matrix and random graph approaches to black hole physics, such as fast scrambling of quantum information, via a bound related to Ramanujan graph. Intriguingly, some other pure mathematical problems in complex analysis, notably complex interpolation in the unit disk, appear to share some mathematical expressions with the black hole problem and thus also involve the Kasner ratio.

Partial recovery and weak consistency in the non-uniform hypergraph stochastic block model

Abstract We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma \in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati ((2017) Ann. Stat.45(1) 289–315.) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularisation of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

The k-core in percolated dense graph sequences

Abstract We determine the order of the k-core in a large class of dense graph sequences. Let $G_n$ be a sequence of undirected, n-vertex graphs with edge weights $\{a^n_{i,j}\}_{i,j \in [n]}$ that converges to a graphon $W\colon[0,1]^2 \to [0,+\infty)$ in the cut metric. Keeping an edge (i,j) of $G_n$ with probability ${a^n_{i,j}}/{n}$ independently, we obtain a sequence of random graphs $G_n({1}/{n})$ . Using a branching process and the theory of dense graph limits, under mild assumptions we obtain the order of the k-core of random graphs $G_n({1}/{n})$ . Our result can also be used to obtain the threshold of appearance of a k-core of order n.

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.

When does the mean network capture the topology of a sample of networks?

The notion of Fréchet mean (also known as “barycenter”) network is the workhorse of most machine learning algorithms that require the estimation of a “location” parameter to analyse network-valued data. In this context, it is critical that the network barycenter inherits the topological structure of the networks in the training dataset. The metric–which measures the proximity between networks–controls the structural properties of the barycenter. This work is significant because it provides for the first time analytical estimates of the sample Fréchet mean for the stochastic blockmodel, which is at the cutting edge of rigorous probabilistic analysis of random networks. We show that the mean network computed with the Hamming distance is unable to capture the topology of the networks in the training sample, whereas the mean network computed using the effective resistance distance recovers the correct partitions and associated edge density. From a practical standpoint, our work informs the choice of metrics in the context where the sample Fréchet mean network is used to characterize the topology of networks for network-valued machine learning.

Realized Random Graphs, with an Application to the Interbank Network

Abstract We introduce a new inferential methodology for dynamic network models driven by latent state variables. The main idea is to obtain a noisy representation of the state variables dynamics by computing a sequence of cross-sectional estimates of the network model at each point in time. The dynamic modeling of these cross-sectional estimates, that we name realized random graphs, transforms the original nonlinear state-space network model into a linear time-series model that can be easily estimated. Under the assumption of a mixed-membership blockmodel structure, the model parameters and the unobservable state variables can be consistently estimated when both the size of the network and the time-series length are large. By allowing for an extremely rich parameterization of the model in high dimensions, the proposed methodology describes the heterogeneous topology of real-world networks. We corroborate our findings by using this novel framework to estimate and forecast the dynamic common factors driving the evolution of the Italian electronic market of interbank deposits, and we show that the interbank lending rate is a key factor determining the network topology.

SRSGCN: A novel multi-sensor fault diagnosis method for hydraulic axial piston pump with limited data

Deep learning has immense potential in ensuring the safe operation of hydraulic axial piston pumps (HAPP). However, the complex operating environment and high cost of labeling have resulted in a scarcity of labeled fault samples. This paper proposes a novel method called Siamese Random Spatiotemporal Graph Convolutional Network (SRSGCN). Firstly, based on graph convolutional networks, a Random Spatiotemporal Graph (RSG) is designed to aggregate multi-sensor information at different time stamps, fully exploiting the spatiotemporal features of the original data. Secondly, the Siamese Neural Network (SNN) is improved by retaining the twin subnetwork structure and removing the similarity output part. While preserving feature extraction capabilities, it is endowed with classification ability. Based on its strong feature mining capability, SRSGCN can fully utilize the scarce sample information to improve diagnostic accuracy. Finally, a case study was conducted on our HAPP experimental platform. The experiments show that compared with other existing methods, this method has higher diagnostic accuracy and can effectively solve the problem of HAPP fault diagnosis under limited data conditions.

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.

The rank of sparse symmetric matrices over arbitrary fields

AbstractLet be an arbitrary field and be a sequence of sparse weighted Erdős–Rényi random graphs on vertices with edge probability , where weights from are assigned to the edges according to a matrix . We show that the normalized rank of the adjacency matrix of converges to a constant, and derive the limiting expression. Our result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalized rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established for adjacency matrices of sparse nonweighted Erdős–Rényi matrices over . Our proof, which is purely combinatorial in its nature, is based on an intricate extension of a novel perturbation approach to the symmetric setting.

Min–max group consensus of discrete-time multi-agent systems under directed random networks

This paper studies the min–max group consensus of discrete-time multi-agent systems under a directed random graph, where the presence of each directed edge is randomly determined by a probability and independent of the presence of other edges. Firstly, we propose a min–max consensus protocol without memory, and give the necessary and sufficient conditions to ensure that the multi-agent system can achieve the min–max group consensus in the sense of almost sure and mean square, respectively. Secondly, we design a novel consensus protocol with memory and a behavior mechanism. Using the stochastic analysis theory and the extremal algebra, some necessary and sufficient conditions are obtained for achieving the min–max group consensus in the sense of almost sure and mean square, respectively. It is shown that the protocol with memory can solve the loss problem of the maximum and minimum initial states. Finally, the effectiveness of the two group consensus protocols and the behavior mechanism is verified by four numerical simulations.

Estimating the impact of physician risky-prescribing on the network structure underlying physician shared-patient relationships

Social network analysis and shared-patient physician networks have become effective ways of studying physician collaborations. Assortative mixing or “homophily” is the network phenomenon whereby the propensity for similar individuals to form ties is greater than for dissimilar individuals. Motivated by the public health concern of risky-prescribing among older patients in the United States, we develop network models and tests involving novel network measures to study whether there is evidence of homophily in prescribing and deprescribing in the specific shared-patient network of physicians linked to the US state of Ohio in 2014. Evidence of homophily in risky-prescribing would imply that prescribing behaviors help shape physician networks and would suggest strategies for interventions seeking to reduce risky-prescribing (e.g., strategies to directly reduce risky prescribing might be most effective if applied as group interventions to risky prescribing physicians connected through the network and the connections between these physicians could be targeted by tie dissolution interventions as an indirect way of reducing risky prescribing). Furthermore, if such effects varied depending on the structural features of a physician’s position in the network (e.g., by whether or not they are involved in cliques—groups of actors that are fully connected to each other—such as closed triangles in the case of three actors), this would further strengthen the case for targeting groups of physicians involved in risky prescribing and the network connections between them for interventions. Using accompanying Medicare Part D data, we converted patient longitudinal prescription receipts into novel measures of the intensity of each physician’s risky-prescribing. Exponential random graph models were used to simultaneously estimate the importance of homophily in prescribing and deprescribing in the network beyond the characteristics of physician specialty (or other metadata) and network-derived features. In addition, novel network measures were introduced to allow homophily to be characterized in relation to specific triadic (three-actor) structural configurations in the network with associated non-parametric randomization tests to evaluate their statistical significance in the network against the null hypothesis of no such phenomena. We found physician homophily in prescribing and deprescribing. We also found that physicians exhibited within-triad homophily in risky-prescribing, with the prevalence of homophilic triads significantly higher than expected by chance absent homophily. These results may explain why communities of prescribers emerge and evolve, helping to justify group-level prescriber interventions. The methodology may be applied, adapted or generalized to study homophily and its generalizations on other network and attribute combinations involving analogous shared-patient networks and more generally using other kinds of network data underlying other kinds of social phenomena.

Microgrid control under uncertainty

Microgrids – decentralized electrical grids that can function both in conjunction with wide area macrogrids and without – are a powerful tool to address energy resiliency and climate change mitigation. Microgrid control, however, remains a challenge; their bespoke nature and the existence of multiple sources of uncertainty lead to a control problem that traditional grid modeling and control techniques are ill-suited to handle. We build a microgrid interface to simulate microgrids under uncertainty and devise off-policy reinforcement learning algorithms to control microgrids. Our algorithms, which incorporate domain randomization and random network distillation for exploration and computational efficiency, achieve performance better than model predictive control and rule based control benchmarks under battery model uncertainty on seven of ten tested scenarios. Our model code is available at https://github.com/ahalev/Microgrid-Control-Under-Uncertainty and our microgrid simulator is available at https://github.com/ahalev/python-microgrid.

New metrics for influential spreaders identification in complex networks based on D-spectra of nodes

Identifying important nodes is of key significance in network sciences as it is closely related to information and disease spreading, structural robustness, etc. Till today, no theoretically proved optimal metric for measuring the importance of nodes is known, while a lot of metrics, mostly heuristic, have been studied. In this paper, we propose several heuristic metrics that are constructed from D-spectra of vertices for discussion. The latter is a graph invariant which is induced by D-chain decompositions of graphs, a recently introduced framework for studying network structures by the first author, Bura and Reidys (SIAM J. Appl. Dyn. Syst. 18, 2019, pp. 2181–2201). Statistical analyses based on numerous data from running the SIR model on real-world and random networks show that some of our proposed metrics outperform a number of well-known metrics such as the H-index, the core values, the betweenness centrality and the closeness centrality.

Epidemics on critical random graphs with heavy-tailed degree distribution

We study the susceptible-infected-recovered (SIR) epidemic on a random graph chosen uniformly over all graphs with certain critical, heavy-tailed degree distributions. We prove process level scaling limits for the number of individuals infected on day h on the largest connected components of the graph. The scaling limits contain non-negative jumps corresponding to some vertices of large degree. These weak convergence techniques allow us to describe the height profile of the α-stable continuum random graph (Goldschmidt et al., 2022; Conchon–Kerjan and Goldschmidt, 2023), extending results known in the Brownian case (Miermont and Sen, 2022). We also prove model-independent results that can be used on other critical random graph models.

Climb Performance Prediction in High Drag Configuration Middle-Class Transportation Aircraft: An Ensemble Learning Approach

This study addresses the application of machine learning and artificial neural network models for predicting the climb speed of the C-130H military transport aircraft. Random Forest, Neural Network, and Ensemble models were developed to overcome limitations of traditional chart reading and interpolation methods. Models were trained on flight manual data, considering factors such as gross weight, pressure altitude, drag index, temperature deviation, and engine efficiency. Comparative analysis revealed the Ensemble approach, combining Random Forest and Neural Network techniques, provided the highest accuracy (R² ≈ 0.4532), followed by Random Forest (R² ≈ 0.4303) and Neural Network (R² ≈ 0.3765) models. All significantly outperformed the traditional Young Method (R² = -1.2673). Feature importance analysis identified pressure altitude, gross weight, and engine efficiency as critical factors influencing climb speed. The ensemble approach demonstrated more reliable and accurate results in predicting C-130H climb rates, reducing risks associated with single-model reliance. This research highlights the potential of machine learning in aircraft performance prediction, offering possibilities for improving pre-flight preparation, reducing workload, and enhancing flight safety. Implications for the aviation industry and future research directions are discussed, emphasizing the role of advanced predictive models in shaping future flight operations and aircraft performance management.

Centrifugal Inertia-Induced Directional Alignment of AgNW Network for Preparing Transparent Electromagnetic Interference Shielding Films with Joule Heating Ability.

Transparent electromagnetic interference (EMI) shielding is highly desired in specific visual scenes, but the challenge remains in balancing their EMI shielding effectiveness (SE) and optical transmittance. Herein, this study proposed a directionally aligned silver nanowire (AgNW) network construction strategy to address the requirement of high EMI SE and satisfactory light transmittance using a rotation spraying technique. The orientation distribution of AgNW is induced by centrifugal inertia force generated by a high-speed rotating roller, which overcomes the issue of high contact resistance in random networks and achieves high conductivity even at low AgNW network density. Thus, the obtained transparent conductive film achieved a high light transmittance of 72.9% combined with a low sheet resistance of 4.5 Ω sq-1 and a desirable EMI SE value of 35.2dB at X band, 38.9dB in the K-band, with the highest SE of 43.4dB at 20.4GHz. Simultaneously, the excellent conductivity endowed the film with outstanding Joule heating performance and defogging/deicing ability, ensuring the visual transparency of windows when shielding electromagnetic waves. Hence, this research presents a highly effective strategy for constructing an aligned AgNW network, offering a promising solution for enhancing the performance of optical-electronic devices.

Meeting, coalescence and consensus time on random directed graphs

Meeting, coalescence and consensus time on random directed graphs