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Given two discrete Morse functions on a simplicial complex, we introduce the {\em connectedness homomorphism} between the corresponding discrete Morse complexes. This concept leads to a novel framework for studying the connectedness in discrete Morse theory at the chain complex level. In particular, we apply it to describe a discrete analogy to `cusp-degeneration' of Morse complexes. A precise comparison between smooth case and our discrete cases is also given., Comment: 20 pages

In this paper, we aim to maximize the SSR for heterogeneous service demands in the cooperative MEC-assisted RAN slicing system by jointly considering the multi-node computing resources cooperation and allocation, the transmission resource blocks (RBs) allocation, and the time-varying dynamicity of the system. To this end, we abstract the system into a weighted undirected topology graph and, then propose a recurrent graph reinforcement learning (RGRL) algorithm to intelligently learn the optimal hybrid RA policy. Therein, the graph neural network (GCN) and the deep deterministic policy gradient (DDPG) is combined to effectively extract spatial features from the equivalent topology graph. Furthermore, a novel time recurrent reinforcement learning framework is designed in the proposed RGRL algorithm by incorporating the action output of the policy network at the previous moment into the state input of the policy network at the subsequent moment, so as to cope with the time-varying and contextual network environment. In addition, we explore two use case scenarios to discuss the universal superiority of the proposed RGRL algorithm. Simulation results demonstrate the superiority of the proposed algorithm in terms of the average SSR, the performance stability, and the network complexity.

In this paper, we study some useful properties of persistent pairs in a discrete Morse function on a simplicial complex $K$. In case of $\dim K=1$ (i.e., a graph), by using the properties, we characterize strongly connectedness of critical simplices between two distinct discrete Morse functions, and relate the number of such pairs to the Euler characteristic of $K$.

Lithosphere is an outer rigid part of the terrestrial body, usually consisting of the crust and part of the mantle. Characterizing the physical properties of the lithosphere is critical in investigating its evolution. By modeling mass-related loads within the lithosphere, physical parameters such as the elastic thickness of the lithosphere can be inferred from gravity and topography data. In the impact basin region, however, the low topography-gravity correlation and the sharp change in admittance from negative to positive within a narrow spheric harmonic degree make this model inapplicable. In this work, we incorporated mantle uplift structures commonly formed in impact basin regions into the lithospheric loading model. The crustal-mantle boundary of this mantle uplift structure is inferred from the global crustal thickness model. The gravity anomaly of the deflected lithosphere is calculated at the surface and crustal-mantle boundary, then the theoretical gravity admittance and correlation can be compared with the observed data. We sampled parameters using this mantle loading model at Argyre and Isidis basin on Mars with a novel crustal thickness model from the InSight mission. Our work suggests that proper modelling of the impact-induced load is critical to understanding the physical properties of the planetary lithosphere in the basin region.

Nowadays wireless communication is rapidly reshaping entire industry sectors. In particular, mobile edge computing (MEC) as an enabling technology for industrial Internet of things (IIoT) brings powerful computing/storage infrastructure closer to the mobile terminals and, thereby, significant lowers the response latency. To reap the benefit of proactive caching at the network edge, precise knowledge on the popularity pattern among the end devices is essential. However, the complex and dynamic nature of the content popularity over space and time as well as the data-privacy requirements in many IIoT scenarios pose tough challenges to its acquisition. In this article, we propose an unsupervised and privacy-preserving popularity prediction framework for MEC-enabled IIoT. The concepts of local and global popularities are introduced and the time-varying popularity of each user is modelled as a model-free Markov chain. On this basis, a novel unsupervised recurrent federated learning (URFL) algorithm is proposed to predict the distributed popularity while achieve privacy preservation and unsupervised training. Simulations indicate that the proposed framework can enhance the prediction accuracy in terms of a reduced root-mean-squared error by up to $60.5\%-68.7\%$. Additionally, manual labeling and violation of users' data privacy are both avoided., Comment: 17 pages, 15 figures, accepted for publication in IEEE INTERNET OF THINGS JOURNAL

In this paper, we introduce the concept of the weighted (harmonic) chromatic polynomials of graphs and discuss some of its properties. We also present the notion of the weighted (harmonic) Tutte--Grothendieck polynomials of graphs and give a generalization of the recipe theorem between the harmonic Tutte--Grothendieck polynomials graphs and the harmonic Tutte polynomials of matroids. Moreover, we give some constructions of the weighted (harmonic) Tutte--Grothendieck invariants for graphs and the weighted (harmonic) Tutte invariants for matroids. Finally, we give a remark on the categorification of the harmonic chromatic polynomials of graphs and harmonic Tutte polynomials of matroids., Comment: 26 pages

Mobile edge computing (MEC) is a prominent computing paradigm which expands the application fields of wireless communication. Due to the limitation of the capacities of user equipments and MEC servers, edge caching (EC) optimization is crucial to the effective utilization of the caching resources in MEC-enabled wireless networks. However, the dynamics and complexities of content popularities over space and time as well as the privacy preservation of users pose significant challenges to EC optimization. In this paper, a privacy-preserving distributed deep deterministic policy gradient (P2D3PG) algorithm is proposed to maximize the cache hit rates of devices in the MEC networks. Specifically, we consider the fact that content popularities are dynamic, complicated and unobservable, and formulate the maximization of cache hit rates on devices as distributed problems under the constraints of privacy preservation. In particular, we convert the distributed optimizations into distributed model-free Markov decision process problems and then introduce a privacy-preserving federated learning method for popularity prediction. Subsequently, a P2D3PG algorithm is developed based on distributed reinforcement learning to solve the distributed problems. Simulation results demonstrate the superiority of the proposed approach in improving EC hit rate over the baseline methods while preserving user privacy., Comment: 15 pages, 9 figures, under review with the IEEE Journal on Selected Areas in Communications

Virtual reality (VR) is promising to fundamentally transform a broad spectrum of industry sectors and the way humans interact with virtual content. However, despite unprecedented progress, current networking and computing infrastructures are incompetent to unlock VR's full potential. In this paper, we consider delivering the wireless multi-tile VR video service over a mobile edge computing (MEC) network. The primary goal is to minimize the system latency/energy consumption and to arrive at a tradeoff thereof. To this end, we first cast the time-varying view popularity as a model-free Markov chain to effectively capture its dynamic characteristics. After jointly assessing the caching and computing capacities on both the MEC server and the VR playback device, a hybrid policy is then implemented to coordinate the dynamic caching replacement and the deterministic offloading, so as to fully utilize the system resources. The underlying multi-objective problem is reformulated as a partially observable Markov decision process, and a deep deterministic policy gradient algorithm is proposed to iteratively learn its solution, where a long short-term memory neural network is embedded to continuously predict the dynamics of the unobservable popularity. Simulation results demonstrate the superiority of the proposed scheme in achieving a trade-off between the energy efficiency and the latency reduction over the baseline methods.