In this paper, we study the entropy of a hard random geometric graph (RGG), a commonly used model for spatial networks, where the connectivity is governed by the distances between the nodes. Formally, given a connection range $r$, a hard RGG $G_m$ on $m$ vertices is formed by drawing $m$ random points from a spatial domain, and then connecting any two points with an edge when they are within a distance $r$ from each other. The two domains we consider are the $d$-dimensional unit cube $[0,1]^d$ and the $d$-dimensional unit torus $\mathbb{T}^d$. We derive upper bounds on the entropy $H(G_m)$ for both these domains and for all possible values of $r$. In a few cases, we obtain an exact asymptotic characterization of the entropy by proving a tight lower bound. Our main results are that $H(G_m) \sim dm \log_2m$ for $0 < r \leq 1/4$ in the case of $\mathbb{T}^d$ and that the entropy of a one-dimensional RGG on $[0,1]$ behaves like $m\log m$ for all $0

In this paper, we consider the rate-distortion-perception (RDP) trade-off for the lossy compression of a Bernoulli vector source, which is a finite collection of independent binary random variables. The RDP function quantifies in a way the efficient compression of a source when we impose a distortion constraint that limits the dissimilarity between the source and the reconstruction and a perception constraint that restricts the distributional discrepancy of the source and the reconstruction. In this work, we obtain an exact characterization of the RDP function of a Bernoulli vector source with the Hamming distortion function and a single-letter perception function that measures the closeness of the distributions of the components of the source using the total variation distance. The solution can be described by partitioning the set of distortion and perception levels$(D, P)$into three regions, where in each region the optimal distortion and perception levels we allot to the components have a similar nature. Finally, we introduce the RDP function for graph sources and apply our result to the Erdős-Rényi graph model.

Internet of Things (IoT) is an emerging next-generation technology in the fourth industrial revolution. The Industrial IoT is required to transmit the collected data in a timely manner to support real-time monitoring, control and automation. In such systems, the timeliness of information is very important, and meanwhile, different physical processes have different requirements on the accuracy of timeliness. However, existing performance metrics, such as the Age of Information (AoI), are unable to fully evaluate the timeliness of information with heterogeneous physical processes. Recently, we proposed an information-theoretic metric named the “Value of Information” (VoI) to measure the usefulness of information in the context of a heterogeneous and noisy environment. In this work, we study a joint path planning of the mobile robot and user scheduling optimization problem in Industrial IoT networks with the aim of maximizing the minimum VoI among all users under mobility and communication constraints. We formulate this optimization problem as a Markov decision process, and propose a reinforcement learning-based algorithm to find the VoI-aware mobility and communication strategy efficiently. Through numerical results, we show that the proposed method can capture the impact of data freshness, inherent correlation characteristics of underlying data sources and noise on the usefulness of information. Compared with the existing AoI-aware strategy, the proposed VoI-aware strategy achieves better performance by exploiting the heterogeneity of data sources especially when the wireless resource is limited.

We consider the line spectral estimation (LSE) from general linear/nonlinear measurements obtained through a generalized linear model (GLM). This paper develops expectation propagation (EP) based LSE (EPLSE) method. The proposed method automatically estimates the model order, noise variance, and can deal with the nonlinear measurements. Numerical experiments show the excellent performance of EPLSE.

In this paper, we address the lossy compression of spatial networks, namely random geometric graphs, where two nodes are connected by an edge with a probability that depends on the distance between the nodes. We carry out this study by considering the$n\mathbf{th}$order information-distortion function, which quantifies the complexity of a random graph under a distortion criterion. Our main result is a partial characterization of the information-distortion function for a random geometric graph with the Hamming distortion measure.

We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by$n$nodes randomly scattered in [0, 1] that connect if they are within the connection range$r\in [{0,1}]$. We provide bounds on the number of possible structures which give universal upper bounds on the structural entropy that hold for any$n$,$r$and distribution of the node locations. For fixed$r$, the number of structures is$\Theta (a^{2n})$with$a=a(r)=2 \cos {\left ({\frac {\pi }{\lceil 1/r \rceil +2}}\right)}$, and therefore the structural entropy is upper bounded by$2n\log _{2} a(r) + O(1)$. For large$n$, we derive bounds on the structural entropy normalized by$n$, and evaluate them for independent and uniformly distributed node locations. When the connection range$r_{n}$is$O(1/n)$, the obtained upper bound is given in terms of a function that increases with$n r_{n}$and asymptotically attains 2 bits per node. If the connection range is bounded away from zero and one, the upper and lower bounds decrease linearly with$r$, as$2(1-r)$and$(1-r)\log _{2} e$, respectively. When$r_{n}$is vanishing but dominates$1/n$(e.g.,$r_{n} \propto \ln n / n$), the normalized entropy is between$\log _{2} e \approx 1.44$and 2 bits per node. We also give a simple encoding scheme for random structures that requires 2 bits per node. The upper bounds in this paper easily extend to the entropy of the labeled random graph model, since this is given by the structural entropy plus a term that accounts for all the permutations of node labels that are possible for a given structure, which is no larger than$\log _{2}(n!) = n \log _{2} n {-} n + O(\log _{2} n)$.

In this paper, we investigate the performance of a reconfigurable intelligent surface (RIS) assisted large-scale network by characterizing the coverage probability and the average achievable rate using stochastic geometry. Considering the spatial correlation between transmitters (TXs) and RISs, their locations are jointly modelled by a Gauss-Poisson process (GPP). Two association strategies, i.e., nearest association and fixed association, are both discussed. For the RIS-aided transmission, the signal power distribution with a direct link is approximated by a gamma random variable using a moment matching method, and the Laplace transform of the aggregate interference power is derived in closed form. Based on these expressions, we analyze the channel hardening effect in the RIS-assisted transmission, the coverage probability, and the average achievable rate of the typical user. We derive the coverage probability expressions for the fixed association strategy and the nearest association strategy in an interference-limited scenario in closed form. Numerical results are provided to validate the analysis and illustrate the effectiveness of RIS-assisted transmission with passive beamforming in improving the system performance. Furthermore, it is also unveiled that the system performance is independent of the density of TXs with the nearest association strategy in the interference-limited scenario.

Routing tables in ad hoc and wireless routing protocols can be represented using rooted trees. The constant need for communication and storage of these trees in routing protocols demands an efficient rooted tree coding algorithm. This efficiency is defined in terms of the average code length, and the optimality of the algorithm is measured by comparing the average code length with the entropy of the source. In this work, TreeExplorer is introduced as an easy-to-implement and nearly optimal algorithm for coding rooted tree structures. This method utilizes the number of leaves of the tree as an indicator for choosing the best method of coding. We show how TreeExplorer can improve existing routing protocols for ad hoc and wireless systems, which normally entails a significant communication overhead.

The freshness and usefulness of information play an important role in offering ubiquitous connectivity for time-critical control applications. A concept named value of information (VoI) is proposed based on the field of information theory to quantity the usefulness of data for sensor-assisted Internet of Things (IoT) networks in the presence of transmission noise. In this work, we focus on general Gaussian random process models and study the rate of change of the VoI when generating more data samples and increasing the signal-to-noise ratio (SNR). We further look at Gauss-Markov random process models, and investigate the impact of the number of observations and the SNR on the VoI performance. It is interesting to find that using more data samples is effective to improve the VoI only in the low SNR regime, while it yields zero rate of change of the VoI in the high SNR regime. Moreover, the VoI can be improved by increasing the SNR in both high and low SNR regimes regardless of how many samples are used. We also find a trade-off between the SNR and the number of observations, and scale back SNR to achieve the same VoI improvement by adding one extra observation. The results illustrated in this work can be used in the design of practical real-time IoT networks.

In this paper, we analyze the coverage probability of a reconfigurable intelligent surface (RIS) aided cellular network with the theory of stochastic geometry. A Poisson cluster process (PCP) is applied to model the positions of transmitters (TXs) and RISs, capturing their spatial correlations. Considering the general Nakagami-m fading channel model, we derive the approximate distributions of the composite channel gains with RIS-assisted transmission, representing the desired signal channel and the interference channel, respectively. The coverage probability of the typical user is then obtained. The derived coverage probability is in a closed form, which can be evaluated efficiently. Simulation results are presented to show that the presented analysis is effective, demonstrate the significant performance gains brought by the passive beamforming of a RIS with a large number of elements, and show the impact of TX density on the performance of the proposed system.

In this paper, a general framework is formalised to characterise the value of information (VoI) in hidden Markov models. Specifically, the VoI is defined as the mutual information between the current, unobserved status at the source and a sequence of observed measurements at the receiver, which can be interpreted as the reduction in the uncertainty of the current status given that we have noisy past observations of a hidden Markov process. We explore the VoI in the context of the noisy Ornstein-Uhlenbeck process and derive its closed-form expressions. Moreover, we investigate the effect of different sampling policies on VoI, deriving simplified expressions in different noise regimes and analysing statistical properties of the VoI in the worst case. We also study the optimal sampling policy to maximise the average information value under the sampling rate constraint. In simulations, the validity of theoretical results is verified, and the performance of VoI in Markov and hidden Markov models is also analysed. Numerical results further illustrate that the proposed VoI framework can support timely transmission in status update systems, and it can also capture the correlation properties of the underlying random process and the noise in the transmission environment.

In this paper, we derive closed-form expressions for significant statistical properties of the link signal-to-noise ratio (SNR) and the separation distance in mobile ad hoc networks subject to Ornstein-Uhlenbeck (OU) mobility and Rayleigh fading. In these systems, the SNR is a critical parameter as it directly influences link performance. In the absence of signal fading, the distribution of the link SNR depends exclusively on the squared distance between nodes, which is governed by the mobility model. In our analysis, nodes move randomly according to an Ornstein-Uhlenbeck process, using one tuning parameter to control the temporal dependency in the mobility pattern. We derive a complete statistical description of the squared distance and show that it forms a stationary Markov process. Then, we compute closed-form expressions for the probability density function (pdf), the cumulative distribution function (cdf), the bivariate pdf, and the bivariate cdf of the link SNR. Next, we introduce small-scale fading, modeled by a Rayleigh random variable, and evaluate the pdf of the link SNR for rational path loss exponents. The validity of our theoretical analysis is verified by extensive simulation studies. The results presented in this work can be used to quantify link uncertainty and evaluate stability in mobile ad hoc wireless systems.

In this paper, a general value of information (VoI) framework is formalised for latent variable models. In particular, the mutual information between the current status at the source node and the observed noisy measurements at the destination node is used to evaluate the information value, which gives the theoretical interpretation of the reduction in uncertainty in the current status given that we have measurements of the latent process. Moreover, the VoI expression for a hidden Markov model is obtained in this setting. Numerical results are provided to show the relationship between the VoI and the traditional age of information (AoI) metric, and the VoI of Markov and hidden Markov models are analysed for the particular case when the latent process is an Ornstein-Uhlenbeck process. While the contributions of this work are theoretical, the proposed VoI framework is general and useful in designing wireless systems that support timely, but noisy, status updates in the physical world.

We analyse the performance of a communication link assisted by an intelligent reflective surface (IRS) positioned in the far field of both the source and the destination. A direct link between the transmitting and receiving devices is assumed to exist. Perfect and imperfect phase adjustments at the IRS are considered. For the perfect phase configuration, we derive an approximate expression for the outage probability in closed form. For the imperfect phase configuration, we assume that each element of the IRS has a one-bit phase shifter (0°,180°) and an expression for the outage probability is obtained in the form of an integral. Our formulation admits an exact asymptotic (high SNR) analysis, from which we obtain the diversity orders for systems with and without phase errors. We show these are N+1 and 1/2 (N+3), respectively. Numerical results confirm the theoretical analysis and verify that the reported results are more accurate than methods based on the central limit theorem (CLT).

The performance of mobile ad hoc networks in general and that of the routing algorithm, in particular, can be heavily affected by the intrinsic dynamic nature of the underlying topology. In this paper, we build a new analytical/numerical framework that characterizes nodes' mobility and the evolution of links between them. This formulation is based on a stationary Markov chain representation of link connectivity. The existence of a link between two nodes depends on their distance, which is governed by the mobility model. In our analysis, nodes move randomly according to an Ornstein-Uhlenbeck process using one tuning parameter to obtain different levels of randomness in the mobility pattern. Finally, we propose an entropy-rate-based metric that quantifies link uncertainty and evaluates its stability. Numerical results show that the proposed approach can accurately reflect the random mobility in the network and fully captures the link dynamics. It may thus be considered a valuable performance metric for the evaluation of the link stability and connectivity in these networks.

In this paper, we present a detailed framework to analyze the evolution of the random topology of a time-varying wireless network via the information theoretic notion of entropy rate. We consider a propagation channel varying over time with random node positions in a closed space and Rayleigh fading affecting the connections between nodes. The existence of an edge between two nodes at given locations is modeled by a Markov chain, enabling memory effects in network dynamics. We then derive a lower and an upper bound on the entropy rate of the spatiotemporal network. The entropy rate measures the shortest per-step description of the stationary stochastic process defining the state of the wireless system and depends both on the maximum Doppler shift and the path loss exponent. It characterizes the topological uncertainty of the wireless network and quantifies how quickly the underlying topology is varying with time.

Random geometric graphs (RGGs) are commonly used to model networked systems that depend on the underlying spatial embedding. We concern ourselves with the probability distribution of an RGG, which is crucial for studying its random topology, properties (e.g., connectedness), or Shannon entropy as a measure of the graph's topological uncertainty (or information content). Moreover, the distribution is also relevant for determining average network performance or designing protocols. However, a major impediment in deducing the graph distribution is that it requires the joint probability distribution of the n (n -1)/2 distances between n nodes randomly distributed in a bounded domain. As no such result exists in the literature, we make progress by obtaining the joint distribution of the distances between three nodes confined in a disk in \mathbbR2. This enables the calculation of the probability distribution and entropy of a three-node graph. For arbitrary n, we derive a series of upper bounds on the graph entropy; in particular, the bound involving the entropy of a three-node graph is tighter than the existing bound which assumes distances are independent. Finally, we provide numerical results on graph connectedness and the tightness of the derived entropy bounds.

The characterization of topological uncertainty in wireless networks using the formalism of graph entropy has received interest in the spatial networks community. In this paper, we develop lower bounds on the entropy of a wireless network by conditioning on potential network observables. Two approaches are considered: 1) conditioning on subgraphs, and 2) conditioning on node positions. The first approach is shown to yield a relatively tight bound on the network entropy. The second yields a loose bound, in general, but it provides insight into the dependence between node positions (modelled using a homogenous binomial point process in this work) and the network topology.

We design an algorithm for OFDM receivers operating in co-channel interference conditions, where the serving and interfering transmitters are synchronized in time. The channel estimation problem is formulated as one of sparse signal reconstruction using multiple measurement vectors. The proposed design harnesses the sparse common support of, respectively, the desired and interfering MIMO sub-channels by adopting a Bernoulli-Gaussian prior for the weights of the impulse responses of these sub-channels. Then, applying a variational Bayesian inference method we derive an algorithm that performs joint channel estimation, interference cancellation and decoding. Simulation results show how the performance of the proposed receiver depends on its knowledge of the interfering signals' modulation and code. When these are known, our receiver approaches the performance of a genie-aided receiver with perfect interference cancellation. Even with mismatched assumptions on the modulation, our proposed implementation still outperforms receivers which neglect the interference.

We study the problem of interference alignment in the multiple-input multiple-output interference channel. Aiming at minimizing the interference leakage power relative to the receiver noise level, we use the deterministic annealing approach to solve the optimization problem. In the corresponding probabilistic formulation, the precoders and the orthonormal bases of the desired signal subspaces are variables distributed on the complex Stiefel manifold. To enable analytically tractable computations, we resort to the variational mean field approximation and thus obtain a novel iterative algorithm for interference alignment. We also show that the iterative leakage minimization algorithm by Gomadam et al. and the alternating minimization algorithm by Peters and Heath, Jr. are instances of our method. Finally, we assess the performance of the proposed algorithm through computer simulations.

We design iterative receiver schemes for a generic communication system by treating channel estimation and information decoding as an inference problem in graphical models. We introduce a recently proposed inference framework that combines belief propagation (BP) and the mean field (MF) approximation and includes these algorithms as special cases. We also show that the expectation propagation and expectation maximization (EM) algorithms can be embedded in the BP-MF framework with slight modifications. By applying the considered inference algorithms to our probabilistic model, we derive four different message-passing receiver schemes. Our numerical evaluation in a wireless scenario demonstrates that the receiver based on the BP-MF framework and its variant based on BP-EM yield the best compromise between performance, computational complexity and numerical stability among all candidate algorithms.