arXiv subjects
Publications and source records attributed to Guodong Shi.
Firstly, we introduce a class of new algebraic systems which generalize Hopf quasigroups and Hopf $π-$algebras called $Q$-graded Hopf quasigroups, and research some properties of them. Secondly, we define the representations of $Q$-graded Hopf quasigroups, i.e $Q$-graded Hopf quasimodules, research the construction method and fundamental theorem of them. Thirdly, we research the smash products of $Q$-graded Hopf quasigroups.
In the first part of the paper, we have studied the computational privacy risks in distributed computing protocols against local or global dynamics eavesdroppers, and proposed a Privacy-Preserving-Summation-Consistent (PPSC) mechanism as a generic privacy encryption subroutine for consensus-based distributed computations. In this part of this paper, we show that the conventional deterministic and random gossip algorithms can be used to realize the PPSC mechanism over a given network. At each time step, a node is selected to interact with one of its neighbors via deterministic or random gossiping. Such node generates a random number as its new state, and sends the subtraction between its current state and that random number to the neighbor; then the neighbor updates its state by adding the received value to its current state. We establish concrete privacy-preservation conditions by proving the impossibility for the reconstruction of the network input from the output of the gossip-based PPSC mechanism against eavesdroppers with full network knowledge, and by showing that the PPSC mechanism can achieve differential privacy at arbitrary privacy levels. The convergence is characterized explicitly and analytically for both deterministic and randomized gossiping, which is essentially achieved in a finite number of steps. Additionally, we illustrate that the proposed algorithms can be easily made adaptive in real-world applications by making realtime trade-offs between resilience against node dropout or communication failure and privacy preservation capabilities.
A distributed computing protocol consists of three components: (i) Data Localization: a network-wide dataset is decomposed into local datasets separately preserved at a network of nodes; (ii) Node Communication: the nodes hold individual dynamical states and communicate with the neighbors about these dynamical states; (iii) Local Computation: state recursions are computed at each individual node. Information about the local datasets enters the computation process through the node-to-node communication and the local computations, which may be leaked to dynamics eavesdroppers having access to global or local node states. In this paper, we systematically investigate this potential computational privacy risks in distributed computing protocols in the form of structured system identification, and then propose and thoroughly analyze a Privacy-Preserving-Summation-Consistent (PPSC) mechanism as a generic privacy encryption subroutine for consensus-based distributed computations. The central idea is that the consensus manifold is where we can both hide node privacy and achieve computational accuracy. In this first part of the paper, we demonstrate the computational privacy risks in distributed algorithms against dynamics eavesdroppers and particularly in distributed linear equation solvers, and then propose the PPSC mechanism and illustrate its usefulness.
We study online linear regression problems in a distributed setting, where the data is spread over a network. In each round, each network node proposes a linear predictor, with the objective of fitting the \emph{network-wide} data. It then updates its predictor for the next round according to the received local feedback and information received from neighboring nodes. The predictions made at a given node are assessed through the notion of regret, defined as the difference between their cumulative network-wide square errors and those of the best off-line network-wide linear predictor. Various scenarios are investigated, depending on the nature of the local feedback (full information or bandit feedback), on the set of available predictors (the decision set), and the way data is generated (by an oblivious or adaptive adversary). We propose simple and natural distributed regression algorithms, involving, at each node and in each round, a local gradient descent step and a communication and averaging step where nodes aim at aligning their predictors to those of their neighbors. We establish regret upper bounds typically in ${\cal O}(T^{3/4})$ when the decision set is unbounded and in ${\cal O}(\sqrt{T})$ in case of bounded decision set.
A signed network is a network with each link associated with a positive or negative sign. Models for nodes interacting over such signed networks, where two different types of interactions take place along the positive and negative links, respectively, arise from various biological, social, political, and economic systems. As modifications to the conventional DeGroot dynamics for positive links, two basic types of negative interactions along negative links, namely the opposing rule and the repelling rule, have been proposed and studied in the literature. This paper reviews a few fundamental convergence results for such dynamics over deterministic or random signed networks under a unified algebraic-graphical method. We show that a systematic tool of studying node state evolution over signed networks can be obtained utilizing generalized Perron-Frobenius theory, graph theory, and elementary algebraic recursions.
This paper presents a first-order {distributed continuous-time algorithm} for computing the least-squares solution to a linear equation over networks. Given the uniqueness of the solution, with nonintegrable and diminishing step size, convergence results are provided for fixed graphs. The exact rate of convergence is also established for various types of step size choices falling into that category. For the case where non-unique solutions exist, convergence to one such solution is proved for constantly connected switching graphs with square integrable step size, and for uniformly jointly connected switching graphs under the boundedness assumption on system states. Validation of the results and illustration of the impact of step size on the convergence speed are made using a few numerical examples.
In this paper, we study network linear equations subject to digital communications with a finite data rate, where each node is associated with one equation from a system of linear equations. Each node holds a dynamic state and interacts with its neighbors through an undirected connected graph, where along each link the pair of nodes share information. Due to the data-rate constraint, each node builds an encoder-decoder pair, with which it produces transmitted message with a zooming-in finite-level uniform quantizer and also generates estimates of its neighbors' states from the received signals. We then propose a distributed quantized algorithm and show that when the network linear equations admit a unique solution, each node's state is driven to that solution exponentially. We further establish the asymptotic rate of convergence, which shows that a larger number of quantization levels leads to a faster convergence rate but is still fundamentally bounded by the inherent network structure and the linear equations. When a unique least-squares solution exists, we show that the algorithm can compute such a solution with a suitably selected time-varying step size inherited from the encoder and zooming-in quantizer dynamics. In both cases, a minimal data rate is shown to be enough for guaranteeing the desired convergence when the step sizes are properly chosen. These results assure the applicability of various network linear equation solvers in the literature when peer-to-peer communication is digital.
This paper aims to provide a systemic analysis to social opinion dynamics subject to individual biases. As a generalization of the classical DeGroot social interactions, defined by linearly coupled dynamics of peer opinions that evolve over time, biases add to state-dependent edge weights and therefore lead to highly nonlinear network dynamics. Previous studies have dealt with convergence and stability analysis of such systems for a few specific initial node opinions and network structures, and here we focus on how individual biases affect social equilibria and their stabilities. First of all, we prove that when the initial network opinions are polarized towards one side of the state space, node biases will drive the opinion evolution to the corresponding interval boundaries. Such polarization attraction effect continues to hold under even directed and switching network structures. Next, for a few fundamental network structures, some important interior network equilibria are presented explicitly for a wide range of system parameters, which are shown to be locally unstable in general. Particularly, the interval centroid is proven to be unstable regardless of the bias level and the network topologies.
The constrained consensus problem considered in this paper, denoted interval consensus, is characterized by the fact that each agent can impose a lower and upper bound on the achievable consensus value. Such constraints can be encoded in the consensus dynamics by saturating the values that an agent transmits to its neighboring nodes. We show in the paper that when the intersection of the intervals imposed by the agents is nonempty, the resulting constrained consensus problem must converge to a common value inside that intersection. In our algorithm, convergence happens in a fully distributed manner, and without need of sharing any information on the individual constraining intervals. When the intersection of the intervals is an empty set, the intrinsic nonlinearity of the network dynamics raises new challenges in understanding the node state evolution. Using Brouwer fixed-point theorem we prove that in that case there exists at least one equilibrium, and in fact the possible equilibria are locally stable if the constraints are satisfied or dissatisfied at the same time among all nodes. For graphs with sufficient sparsity it is further proven that there is a unique equilibrium that is globally attractive if the constraint intervals are pairwise disjoint.
In this paper, we investigate the fundamental limitations of feedback mechanism in dealing with uncertainties for network systems. The study of maximum capability of feedback control was pioneered in Xie and Guo (2000) for scalar systems with nonparametric nonlinear uncertainty. In a network setting, nodes with unknown and nonlinear dynamics are interconnected through a directed interaction graph. Nodes can design feedback controls based on all available information, where the objective is to stabilize the network state. Using information structure and decision pattern as criteria, we specify three categories of network feedback laws, namely the global-knowledge/global-decision, network-flow/local-decision, and local-flow/local-decision feedback. We establish a series of network capacity characterizations for these three fundamental types of network control laws. First of all, we prove that for global-knowledge/global-decision and network-flow/local-decision control where nodes know the information flow across the entire network, there exists a critical number $\big(3/2+\sqrt{2}\big)/\|A_{\mathrm{G}}\|_\infty$, where $3/2+\sqrt{2}$ is as known as the Xie-Guo constant and $A_{\mathrm{G}}$ is the network adjacency matrix, defining exactly how much uncertainty in the node dynamics can be overcome by feedback. Interestingly enough, the same feedback capacity can be achieved under max-consensus enhanced local flows where nodes only observe information flows from neighbors as well as extreme (max and min) states in the network. Next, for local-flow/local-decision control, we prove that there exists a structure-determined value being a lower bound of the network feedback capacity. These results reveal the important connection between network structure and fundamental capabilities of in-network feedback control.
In this paper, we investigate probabilistic stability of Kalman filtering over fading channels modeled by $\ast$-mixing random processes, where channel fading is allowed to generate non-stationary packet dropouts with temporal and/or spatial correlations. Upper/lower almost sure (a.s.) stabilities and absolutely upper/lower a.s. stabilities are defined for characterizing the sample-path behaviors of the Kalman filtering. We prove that both upper and lower a.s. stabilities follow a zero-one law, i.e., these stabilities must happen with a probability either zero or one, and when the filtering system is one-step observable, the absolutely upper and lower a.s. stabilities can also be interpreted using a zero-one law. We establish general stability conditions for (absolutely) upper and lower a.s. stabilities. In particular, with one-step observability, we show the equivalence between absolutely a.s. stabilities and a.s. ones, and necessary and sufficient conditions in terms of packet arrival rate are derived; for the so-called non-degenerate systems, we also manage to give a necessary and sufficient condition for upper a.s. stability.
This paper proposes and investigates a framework for clique gossip protocols. As complete subnetworks, the existence of cliques is ubiquitous in various social, computer, and engineering networks. By clique gossiping, nodes interact with each other along a sequence of cliques. Clique-gossip protocols are defined as arbitrary linear node interactions where node states are vectors evolving as linear dynamical systems. Such protocols become clique-gossip averaging algorithms when node states are scalars under averaging rules. We generalize the classical notion of line graph to capture the essential node interaction structure induced by both the underlying network and the specific clique sequence. We prove a fundamental eigenvalue invariance principle for periodic clique-gossip protocols, which implies that any permutation of the clique sequence leads to the same spectrum for the overall state transition when the generalized line graph contains no cycle. We also prove that for a network with $n$ nodes, cliques with smaller sizes determined by factors of $n$ can always be constructed leading to finite-time convergent clique-gossip averaging algorithms, provided $n$ is not a prime number. Particularly, such finite-time convergence can be achieved with cliques of equal size $m$ if and only if $n$ is divisible by $m$ and they have exactly the same prime factors. A proven fastest finite-time convergent clique-gossip algorithm is constructed for clique-gossiping using size-$m$ cliques. Additionally, the acceleration effects of clique-gossiping are illustrated via numerical examples.
This paper studies deterministic consensus networks with discrete-time dynamics under persistent flows and non-reciprocal agent interactions. An arc describing the interaction strength between two agents is said to be persistent if its weight function has an infinite $l_1$ norm. We discuss two balance conditions on the interactions between agents which generalize the arc-balance and cut-balance conditions in the literature respectively. The proposed conditions require that such a balance should be satisfied over each time window of a fixed length instead of at each time instant. We prove that in both cases global consensus is reached if and only if the persistent graph, which consists of all the persistent arcs, contains a directed spanning tree. The convergence rates of the system to consensus are also provided in terms of the interactions between agents having taken place. The results are obtained under a weak condition without assuming the existence of a positive lower bound of all the nonzero weights of arcs and are compared with the existing results. Illustrative examples are provided to show the critical importance of the nontrivial lower boundedness of the self-confidence of the agents.
This paper proposes and investigates a Boolean gossip model as a simplified but non-trivial probabilistic Boolean network. With positive node interactions, in view of standard theories from Markov chains, we prove that the node states asymptotically converge to an agreement at a binary random variable, whose distribution is characterized for large-scale networks by mean-field approximation. Using combinatorial analysis, we also successfully count the number of communication classes of the positive Boolean network explicitly in terms of the topology of the underlying interaction graph, where remarkably minor variation in local structures can drastically change the number of network communication classes. With general Boolean interaction rules, emergence of absorbing network Boolean dynamics is shown to be determined by the network structure with necessary and sufficient conditions established regarding when the Boolean gossip process defines absorbing Markov chains. Particularly, it is shown that for the majority of the Boolean interaction rules, except for nine out of the total $2^{16}-1$ possible nonempty sets of binary Boolean functions, whether the induced chain is absorbing has nothing to do with the topology of the underlying interaction graph, as long as connectivity is assumed. These results illustrate possibilities of {relating dynamical} properties of Boolean networks to graphical properties of the underlying interactions.
We study the approach to obtaining least squares solutions to systems of linear algebraic equations over networks by using distributed algorithms. Each node has access to one of the linear equations and holds a dynamic state. The aim for the node states is to reach a consensus as a least squares solution of the linear equations by exchanging their states with neighbors over an underlying interaction graph. A continuous-time distributed least squares solver over networks is developed in the form of the famous Arrow-Hurwicz-Uzawa flow. A necessary and sufficient condition is established on the graph Laplacian for the continuous-time distributed algorithm to give the least squares solution in the limit, with an exponentially fast convergence rate. The feasibility of different fundamental graphs is discussed including path graph, star graph, etc. Moreover, a discrete-time distributed algorithm is developed by Euler's method, converging exponentially to the least squares solution at the node states with suitable step size and graph conditions. The exponential convergence rate for both the continuous-time and discrete-time algorithms under the established conditions is confirmed by numerical examples. Finally, we investigate the performance of the proposed flow under switching networks, and surprisingly, switching networks at high switching frequencies can lead to approximate least square solvers even if all graphs in the switching signal fail to do so in the absence of structure switching.
We study distributed network flows as solvers in continuous time for the linear algebraic equation $\mathbf{z}=\mathbf{H}\mathbf{y}$. Each node $i$ has access to a row $\mathbf{h}_i^{\rm T}$ of the matrix $\mathbf{H}$ and the corresponding entry $z_i$ in the vector $\mathbf{z}$. The first "consensus + projection" flow under investigation consists of two terms, one from standard consensus dynamics and the other contributing to projection onto each affine subspace specified by the $\mathbf{h}_i$ and $z_i$. The second "projection consensus" flow on the other hand simply replaces the relative state feedback in consensus dynamics with projected relative state feedback. Without dwell-time assumption on switching graphs as well as without positively lower bounded assumption on arc weights, we prove that all node states converge to a common solution of the linear algebraic equation, if there is any. The convergence is global for the "consensus + projection" flow while local for the "projection consensus" flow in the sense that the initial values must lie on the affine subspaces. If the linear equation has no exact solutions, we show that the node states can converge to a ball around the least squares solution whose radius can be made arbitrarily small through selecting a sufficiently large gain for the "consensus + projection" flow under fixed bidirectional graphs. Semi-global convergence to approximate least squares solutions is demonstrated for general switching directed graphs under suitable conditions. It is also shown that the "projection consensus" flow drives the average of the node states to the least squares solution with complete graph. Numerical examples are provided as illustrations of the established results.
We study continuous-time consensus dynamics for multi-agent systems with undirected switching interaction graphs. We establish a necessary and sufficient condition for exponential asymptotic consensus based on the classical theory of complete observability. The proof is remarkably simple compared to similar results in the literature and the conditions for consensus are mild. This observability-based method can also be applied to the case where negatively weighted edges are present. Additionally, as a by-product of the observability based arguments, we show that the nodes' initial value can be recovered from the signals on the edges up to a shift of the network average.
We consider a basic quantum hybrid network model consisting of a number of nodes each holding a qubit, for which the aim is to drive the network to a consensus in the sense that all qubits reach a common state. Projective measurements are applied serving as control means, and the measurement results are exchanged among the nodes via classical communication channels. We show how to carry out centralized optimal path planning for this network with all-to-all classical communications, in which case the problem becomes a stochastic optimal control problem with a continuous action space. To overcome the computation and communication obstacles facing the centralized solutions, we also develop a distributed Pairwise Qubit Projection (PQP) algorithm, where pairs of nodes meet at a given time and respectively perform measurements at their geometric average. We show that the qubit states are driven to a consensus almost surely along the proposed PQP algorithm, and that the expected qubit density operators converge to the average of the network's initial values.