Random Geometric Graphs Research Articles

Distributed Mechanism for Detecting Average Consensus with Maximum-Degree Weights in Bipartite Regular Graphs

In recent decades, distributed consensus-based algorithms for data aggregation have been gaining in importance in wireless sensor networks since their implementation as a complementary mechanism can ensure sensor-measured values with high reliability and optimized energy consumption in spite of imprecise sensor readings. In the presented article, we address the average consensus algorithm over bipartite regular graphs, where the application of the maximum-degree weights causes the divergence of the algorithm. We provide a spectral analysis of the algorithm, propose a distributed mechanism to detect whether a graph is bipartite regular, and identify how to reconfigure the algorithm so that the convergence of the average consensus algorithm is guaranteed over bipartite regular graphs. More specifically, we identify in the article that only the largest and the smallest eigenvalues of the weight matrix are located on the unit circle; the sum of all the inner states is preserved at each iteration despite the algorithm divergence; and the inner states oscillate between two values close to the arithmetic means determined by the initial inner states from each disjoint subset. The proposed mechanism utilizes the first-order forward and backward finite-difference of the inner states (more specifically, five conditions are proposed) to detect whether a graph is bipartite regular or not. Subsequently, the mixing parameter of the algorithm can be reconfigured the way it is identified in this study whereby the convergence of the algorithm is ensured in bipartite regular graphs. In the experimental part, we tested our mechanism over randomly generated bipartite regular graphs, random graphs, and random geometric graphs with various parameters, thereby identifying its very high detection rate and proving that the algorithm can estimate the arithmetic mean with high precision (like in error-free scenarios) after the suggested reconfiguration.

Minimum spanning trees of random geometric graphs with location dependent weights

Consider n nodes {Xi}1≤i≤n independently distributed in the unit square S, each according to a distribution f. Nodes Xi and Xj are joined by an edge if the Euclidean distance d(Xi,Xj) is less than rn, the adjacency distance and the resulting random graph Gn is called a random geometric graph (RGG). We now assign a location dependent weight to each edge of Gn and define MSTn to be the sum of the weights of the minimum spanning trees of all components of Gn. For values of rn above the connectivity regime, we obtain upper and lower bound deviation estimates for MSTn and L2-convergence of MSTn appropriately scaled and centred.

Exponential rate for the contact process extinction time

We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.

Seeking the Truth in a Decentralized Manner

In networks where massive sources make observations of same entities, we intend to seek the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">truth</i> – the most trustworthy value of each entity from conflicting information claimed by multiple sources. Various methods are proposed for accurately inferring both source reliability and truths, yet relying heavily on centralized settings that incur tremendous overhead to source side. In this paper, we offer a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">decentralized</i> design of truth discovery task that can fit favorably to the environments with limited resources. Considering that sources forming the connected network and making individual observations, we undertake the joint maximum likelihood estimation (MLE) of truth and source reliability. Our decentralization framework simply allows each source to maintain local information exchange at a time, and computes very basic functions of data observations. To this end, we facilitate the decentralization by simplifying the MLE problem into optimizing an objective function. Upon the proof of NP-hardness, two proposed decentralized algorithms (exact and approximation) are decentralized and randomized via a combination of algorithms from their centralized counterparts that ensure performance guarantee. The derived time complexity features explicit data/network dependent terms, which leads to further acceleration in truth finding. Remarkably, in two well connected networks like random geometric and preferential attachment graphs, the accelerated approximation method enjoys logarithmic time complexity while preserving comparable accuracy to the centralized counterparts. The effectiveness of the proposed decentralizations are further empirically confirmed.

Degree distributions in AB random geometric graphs

In this paper, we provide degree distributions for AB random geometric graphs, in which points of type A connect to the closest k points of type B. The motivating example to derive such degree distributions is in 5G wireless networks with multi-connectivity, where users connect to their closest k base stations. In this setting, it is important to know how many users a particular base station serves, which gives the degree of that base station. To obtain these degree distributions, we investigate the distribution of area sizes of the kth order Voronoi cells of B-points. Assuming that the A-points are Poisson distributed, we investigate the amount of users connected to a certain B-point, which is equal to the degree of this point. In the simple case where the B-points are placed in an hexagonal grid, we show that all kth order Voronoi areas are equal and thus all degrees follow a Poisson distribution. However, this observation does not hold for Poisson distributed B-points, for which we show that the degree distribution follows a compound Poisson–Erlang distribution in the 1-dimensional case. We then approximate the degree distribution in the 2-dimensional case with a compound Poisson-Gamma degree distribution and show that this one-parameter fit performs well for different values of k. Moreover, we show that for increasing k, these degree distributions become more concentrated around the mean. This means that k-connected AB random graphs balance the loads of B-type nodes more evenly as k increases. Finally, we provide a case study on real data of base stations. We show that with little shadowing in the distances between users and base stations, the Poisson distribution does not capture the degree distribution of these data, especially for k>1. However, under strong shadowing, our degree approximations perform quite good even for these non-Poissonian location data.

The role of spatial embedding in mouse brain networks constructed from diffusion tractography and tracer injections

Diffusion MRI tractography is the only noninvasive method to measure the structural connectome in humans. However, recent validation studies have revealed limitations of modern tractography approaches, which lead to significant mistracking caused in part by local uncertainties in fiber orientations that accumulate to produce larger errors for longer streamlines. Characterizing the role of this length bias in tractography is complicated by the true underlying contribution of spatial embedding to brain topology. In this work, we compare graphs constructed with ex vivo tractography data in mice and neural tracer data from the Allen Mouse Brain Connectivity Atlas to random geometric surrogate graphs which preserve the low-order distance effects from each modality in order to quantify the role of geometry in various network properties. We find that geometry plays a substantially larger role in determining the topology of graphs produced by tractography than graphs produced by tracers. Tractography underestimates weights at long distances compared to neural tracers, which leads tractography to place network hubs close to the geometric center of the brain, as do corresponding tractography-derived random geometric surrogates, while tracer graphs place hubs further into peripheral areas of the cortex. We also explore the role of spatial embedding in modular structure, network efficiency and other topological measures in both modalities. Throughout, we compare the use of two different tractography streamline node assignment strategies and find that the overall differences between tractography approaches are small relative to the differences between tractography- and tracer-derived graphs. These analyses help quantify geometric biases inherent to tractography and promote the use of geometric benchmarking in future tractography validation efforts.

Isotropic random geometric networks in two dimensions with a penetrable cavity

In this work, a novel model of the random geometric graph (RGG), namely the isotropic random geometric graphs (IRGG) has been developed and its topological properties in two dimensions have been studied in details. The defining characteristics of RGG and IRGG are the same — two nodes are connected by an edge if their distance is less than a fixed value, called the connection radius. However, IRGGs have two major differences from regular RGGs. Firstly, the shape of their boundaries — which is circular. It brings very little changes in final results but gives a significant advantage in analytical calculations of the network properties. Secondly, it opens up the possibility of an empty concentric region inside the network. The empty region contains no nodes but allows the communicating edges between the nodes to pass through it. This second difference causes significant alterations in physically relevant network properties such as average degree, connectivity, clustering coefficient and average shortest path. Analytical expressions for most of these features have been provided. These results agree well with those obtained from simulations. Apart from the applicability of the model due to its symmetry and simplicity, the scope of incorporating a penetrable cavity makes it suitable for potential applications in wireless communication networks that often have a node-free region.

Normalized Sombor Indices as Complexity Measures of Random Networks.

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

Hierarchical effects facilitate spreading processes on synthetic and empirical multilayer networks.

In this paper we consider the effects of corporate hierarchies on innovation spread across multilayer networks, modeled by an elaborated SIR framework. We show that the addition of management layers can significantly improve spreading processes on both random geometric graphs and empirical corporate networks. Additionally, we show that utilizing a more centralized working relationship network rather than a strict administrative network further increases overall innovation reach. In fact, this more centralized structure in conjunction with management layers is essential to both reaching a plurality of nodes and creating a stable adopted community in the long time horizon. Further, we show that the selection of seed nodes affects the final stability of the adopted community, and while the most influential nodes often produce the highest peak adoption, this is not always the case. In some circumstances, seeding nodes near but not in the highest positions in the graph produces larger peak adoption and more stable long-time adoption.

Increasing efficacy of contact-tracing applications by user referrals and stricter quarantining.

We study the effects of two mechanisms which increase the efficacy of contact-tracing applications (CTAs) such as the mobile phone contact-tracing applications that have been used during the COVID-19 epidemic. The first mechanism is the introduction of user referrals. We compare four scenarios for the uptake of CTAs—(1) the p% of individuals that use the CTA are chosen randomly, (2) a smaller initial set of randomly-chosen users each refer a contact to use the CTA, achieving p% in total, (3) a small initial set of randomly-chosen users each refer around half of their contacts to use the CTA, achieving p% in total, and (4) for comparison, an idealised scenario in which the p% of the population that uses the CTA is the p% with the most contacts. Using agent-based epidemiological models incorporating a geometric space, we find that, even when the uptake percentage p% is small, CTAs are an effective tool for mitigating the spread of the epidemic in all scenarios. Moreover, user referrals significantly improve efficacy. In addition, it turns out that user referrals reduce the quarantine load. The second mechanism for increasing the efficacy of CTAs is tuning the severity of quarantine measures. Our modelling shows that using CTAs with mild quarantine measures is effective in reducing the maximum hospital load and the number of people who become ill, but leads to a relatively high quarantine load, which may cause economic disruption. Fortunately, under stricter quarantine measures, the advantages are maintained but the quarantine load is reduced. Our models incorporate geometric inhomogeneous random graphs to study the effects of the presence of super-spreaders and of the absence of long-distant contacts (e.g., through travel restrictions) on our conclusions.

Comparative Study of Distributed Consensus Gossip Algorithms for Network Size Estimation in Multi-Agent Systems

Determining the network size is a critical process in numerous areas (e.g., computer science, logistic, epidemiology, social networking services, mathematical modeling, demography, etc.). However, many modern real-world systems are so extensive that measuring their size poses a serious challenge. Therefore, the algorithms for determining/estimating this parameter in an effective manner have been gaining popularity over the past decades. In the paper, we analyze five frequently applied distributed consensus gossip-based algorithms for network size estimation in multi-agent systems (namely, the Randomized gossip algorithm, the Geographic gossip algorithm, the Broadcast gossip algorithm, the Push-Sum protocol, and the Push-Pull protocol). We examine the performance of the mentioned algorithms with bounded execution over random geometric graphs by applying two metrics: the number of sent messages required for consensus achievement and the estimation precision quantified as the median deviation from the real value of the network size. The experimental part consists of two scenarios—the consensus achievement is conditioned by either the values of the inner states or the network size estimates—and, in both scenarios, either the best-connected or the worst-connected agent is chosen as the leader. The goal of this paper is to identify whether all the examined algorithms are applicable to estimating the network size, which algorithm provides the best performance, how the leader selection can affect the performance of the algorithms, and how to most effectively configure the applied stopping criterion.

Connectivity and Coverage in Hybrid Wireless Sensor Networks using Dynamic Random Geometric Graph Model

Connectivity and Coverage in Hybrid Wireless Sensor Networks using Dynamic Random Geometric Graph Model

Rare Events in Random Geometric Graphs

This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we use conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance.

Lower bounds for in-network computation of arbitrary functions

In this paper, we provide a family of bounds for the rate at which the functions of many inputs can be computed in-network on general topologies. Going beyond simple symmetric functions where the output is invariant to the permutation of the operands, e.g., average, parity, we describe an algorithm that is analyzed to provide throughput bounds (both lower and upper) for the general functions. In particular, we analyze our algorithm when the function to be computed is given as a binary tree schema. Our lower bounds depend on schema parameters like the number of operands and graph parameters like the second largest eigenvalue of the transition matrix of simple random walk on network graph, the maximum and minimum degree of any node in the network. The lower bounding technique that we have used is based on the network flows and can capture general multi-commodity flow settings. Our proposed algorithm uses the well-known simple random walk on a network as its basic primitive for routing. We show that the lower bound obtained on the rate of computation is tight for the complete network topology, the hypercube and the star topology. We also present an upper bound on the expected latency of any data operand in terms of the height of schema, well-studied random walk parameter like the hitting time, and the relative distance from the critical data rate. For the computation time of symmetric functions on the random geometric graph under the gossip model, our approach achieves an order-optimal \(\widetilde{O}(n)\) time despite enforcing a binary tree schema for function computation. In general, Big-O notation represents an upper bound and \(\widetilde{O}\) hides \(\text {poly}\log n\) factors.

Functional central limit theorems for persistent Betti numbers on cylindrical networks

AbstractWe study functional central limit theorems for persistent Betti numbers obtained from networks defined on a Poisson point process. The limit is formed in large volumes of cylindrical shape stretching only in one dimension. The results cover a directed sublevel‐filtration for stabilizing networks and the Čech and Vietoris–Rips complex on the random geometric graph. The presented functional central limit theorems open the door to a variety of statistical applications in topological data analysis and we consider goodness‐of‐fit tests in a simulation study.

Ollivier-Ricci curvature convergence in random geometric graphs

Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. This result establishes the first rigorous link between a definition of curvature applicable to networks and a traditional definition of curvature of smooth spaces.

Modeling Rydberg gases using random sequential adsorption on random graphs

The statistics of strongly interacting, ultracold Rydberg gases are governed by the interplay of two factors: geometrical restrictions induced by blockade effects, and quantum mechanical effects. To shed light on their relative roles in the statistics of Rydberg gases, we compare three models in this paper: a quantum mechanical model describing the excitation dynamics within a Rydberg gas, a Random Sequential Adsorption (RSA) process on a Random Geometric Graph (RGG), and a RSA process on a Decomposed Random Intersection Graph (DRIG). The latter model is new, and refers to choosing a particular subgraph of a mixture of two other random graphs. Contrary to the former two models, it lends itself for a rigorous mathematical analysis; and it is built specifically to have particular structural properties of a RGG. We establish for it a fluid limit describing the time-evolution of number of Rydberg atoms, and show numerically that the expression remains accurate across a wider range of particle densities than an earlier approach based on an RSA process on an Erdos-Renyi Random Graph (ERRG). Finally, we also come up with a new heuristic using random graphs that gives a recursion to describe a normalized pair-correlation function of a Rydberg gas. Our results suggest that even without dissipation, on long time scales the statistics are affected most by the geometrical restrictions induced by blockade effects, while on short time scales the statistics are affected most by quantum mechanical effects.

Step-isometries

Motivated by a problem in the theory of random geometric graphs a step-isometry on a dense subset A of a real Banach space X is defined as a map f:A→X satisfying ⌊‖x−y‖⌋=⌊‖f(x)−f(y)‖⌋, x,y∈A. Under the additional assumption that f maps A onto itself a complete description of such maps on finite-dimensional spaces has been obtained by Balister, Bollobás, Gunderson, Leader, and Walters. What happens in the absence of this additional assumption and in the case that X is infinite-dimensional? Studying step-isometries on classical sequence spaces we show that for certain finite-dimensional spaces we can get their result for general step-isometries satisfying no surjectivity assumption. And under the assumption of almost surjectivity, which is weaker than the assumption f(A)=A, this result can be extended to the infinite-dimensional case. By an example we show that the result of Balister, Bollobás, Gunderson, Leader, and Walters is optimal for Rn equipped with the ℓ∞ norm and extend also this result to the infinite-dimensional setting.

Public goods games on random hyperbolic graphs with mixing

Understanding the evolution of cooperation in structured populations remains one of the fundamental challenges of the 21st century, with far-reaching implications for the wellbeing of modern human societies. Studies over the past two decades showed that the structure of the network of contacts plays a crucial role in determining whether cooperation prevails or not. An important step to more realistic networks was made with the shift from regular grids and lattices to complex social networks at the turn of the century. Real networks exhibit a high mean local clustering coefficient, short average path lengths, and community structure. Recent studies have revealed that random geometric graphs in hyperbolic spaces exhibit properties that are frequently found in real networks. We here study the public goods game on random geometric graphs in hyperbolic spaces, and we consider assortative and disassortative mixing with different frequencies. We show that in hyperbolic spaces heterogeneous networks promote the evolution of public cooperation in comparison to the more homogeneous networks. We also confirm that assortative and disassortative mixing on random hyperbolic networks both impair the evolutionary success of cooperators, regardless of the network architecture. The differences between the two mixing protocols are most expressed at low mixing frequencies, whilst at high mixing frequencies the two almost converge.

Modeling Quantum Dot Systems as Random Geometric Graphs with Probability Amplitude-Based Weighted Links.

This paper focuses on modeling a disorder ensemble of quantum dots (QDs) as a special kind of Random Geometric Graphs (RGG) with weighted links. We compute any link weight as the overlap integral (or electron probability amplitude) between the QDs (=nodes) involved. This naturally leads to a weighted adjacency matrix, a Laplacian matrix, and a time evolution operator that have meaning in Quantum Mechanics. The model prohibits the existence of long-range links (shortcuts) between distant nodes because the electron cannot tunnel between two QDs that are too far away in the array. The spatial network generated by the proposed model captures inner properties of the QD system, which cannot be deduced from the simple interactions of their isolated components. It predicts the system quantum state, its time evolution, and the emergence of quantum transport when the network becomes connected.