arXiv subjects

Saber Jafarizadeh

Publications and source records attributed to Saber Jafarizadeh.

At least 19 recordsLinked to original sources

arXiv2025

Fastest Mixing Reversible Markov Chain: Clique Lifted Graphs and Subgraphs

Markov chains are one of the well-known tools for modeling and analyzing stochastic systems. At the same time, they are used for constructing random walks that can achieve a given stationary distribution. This paper is concerned with determining the transition probabilities that optimize the mixing time of the reversible Markov chains towards a given equilibrium distribution. This problem is referred to as the Fastest Mixing Reversible Markov Chain (FMRMC) problem. It is shown that for a given base graph and its clique lifted graph, the FMRMC problem over the clique lifted graph is reducible to the FMRMC problem over the base graph, while the optimal mixing times on both graphs are identical. Based on this result and the solution of the semidefinite programming formulation of the FMRMC problem, the problem has been addressed over a wide variety of topologies with the same base graph. Second, the general form of the FMRMC problem is addressed on stand-alone topologies as well as subgraphs of an arbitrary graph. For subgraphs, it is shown that the optimal transition probabilities over edges of the subgraph can be determined independent of rest of the topology.

cs.IT↗

arXiv2025

Fastest mixing reversible Markov chain on friendship graph: Trade-off between transition probabilities among friends and convergence rate

A long-standing goal of social network research has been to alter the properties of network to achieve the desired outcome. In doing so, DeGroot's consensus model has served as the popular choice for modeling the information diffusion and opinion formation in social networks. Achieving a trade-off between the cost associated with modifications made to the network and the speed of convergence to the desired state has shown to be a critical factor. This has been treated as the Fastest Mixing Markov Chain (FMMC) problem over a graph with given transition probabilities over a subset of edges. Addressing this multi-objective optimization problem over the friendship graph, this paper has provided the corresponding Pareto optimal points or the Pareto frontier. In the case of friendship graph with at least three blades, it is shown that the Pareto frontier is reduced to a global minimum point which is same as the optimal point corresponding to the minimum spanning tree of the friendship graph, i.e., the star topology. Furthermore, a lower limit for transition probabilities among friends has been provided, where values higher than this limit do not have any impact on the convergence rate.

cs.IT↗

arXiv2024

Optimal Diffusion Processes

Of stochastic differential equations, diffusion processes have been adopted in numerous applications, as more relevant and flexible models. This paper studies diffusion processes in a different setting, where for a given stationary distribution and average variance, it seeks the diffusion process with optimal convergence rate. It is shown that the optimal drift function is a linear function and the convergence rate of the stochastic process is bounded by the ratio of the average variance to the variance of the stationary distribution. Furthermore, the concavity of the optimal relaxation time as a function of the stationary distribution has been proven, and it is shown that all Pearson diffusion processes of the Hypergeometric type with polynomial functions of at most degree two as the variance functions are optimal.

math.PR↗

arXiv2016

Continuous Time Quantum Consensus & Quantum Synchronisation

Distributed consensus algorithm over networks of quantum systems has been the focus of recent studies in the context of quantum computing and distributed control. Most of the progress in this category have been on the convergence conditions and optimizing the convergence rate of the algorithm, for quantum networks with undirected underlying topology. This paper aims to address the extension of this problem over quantum networks with directed underlying graphs. In doing so, the convergence to two different stable states namely, consensus and synchronous states have been studied. Based on the intertwining relation between the eigenvalues, it is shown that for determining the convergence rate to the consensus state, all induced graphs should be considered while for the synchronous state only the underlying graph suffices. Furthermore, it is illustrated that for the range of weights that the Aldous' conjecture holds true, the convergence rate to both states are equal. Using the Pareto region for convergence rates of the algorithm, the global and Pareto optimal points for several topologies have been provided.

eess.SY↗

arXiv2016

Optimizing the Diffusion System Based on Continuous-Time Consensus Algorithm

Traditionally, systems governed by linear Partial Differential Equations (PDEs) are spatially discretized to exploit their algebraic structure and reduce the computational effort for controlling them. Due to beneficial insights of the PDEs, recently, the reverse of this approach is implemented where a spatially-discrete system is approximated by a spatially-continuous one, governed by linear PDEs forming diffusion equations. In the case of distributed consensus algorithms, this approach is adapted to enhance its convergence rate to the equilibrium. In previous studies within this context, constant diffusion parameter is considered for obtaining the diffusion equations. This is equivalent to assigning a constant weight to all edges of the underlying graph in the consensus algorithm. Here, by relaxing this restricting assumption, a spatially-variable diffusion parameter is considered and by optimizing the obtained system, it is shown that significant improvements are achievable in terms of the convergence rate of the obtained spatially-continuous system. As a result of approximation, the system is divided into two sections, namely, the spatially-continuous path branches and the lattice core which connects these branches at one end. The optimized weights and diffusion parameter for each of these sections are optimal individually but considering the whole system, they are suboptimal. It is shown that the symmetric star topology is an exception and the obtained results for this topology are globally optimal. Furthermore, through variational method, the results obtained for the symmetric star topology are validated and it is shown that the variable diffusion parameter improves the robustness of the system too.

eess.SY↗

arXiv2015

Optimizing the Gossip Algorithm with Non-Uniform Clock Distribution over Classical & Quantum Networks

Distributed gossip algorithm has been studied in literature for practical implementation of the distributed consensus algorithm as a fundamental algorithm for the purpose of in-network collaborative processing. This paper focuses on optimizing the convergence rate of the gossip algorithm for both classical and quantum networks. A novel model of the gossip algorithm with non-uniform clock distribution is proposed which can reach the optimal convergence rate of the continuous-time consensus algorithm. It is described that how the non-uniform clock distribution is achievable by modifying the rate of the Poisson process modeling the clock of the gossip algorithm. The minimization problem for optimizing the asymptotic convergence rate of the proposed gossip algorithm and its corresponding semidefinite programming formulation is addressed analytically. It is shown that the optimal results obtained for uniform clock distribution are suboptimal compared to those of the non-uniform one and for non-uniform distribution the optimal answer is not unique i.e. there is more than one set of probabilities that can achieve the optimal convergence rate. Based on the optimal continuous-time consensus algorithm and the detailed balance property, an effective method of obtaining one of these optimal answers is proposed. Regarding quantum gossip algorithm, by expanding the density matrix in terms of the generalized Gell-Mann matrices, the evolution equation of the quantum gossip algorithm is transformed to the state update equation of the classical gossip algorithm. By defining the quantum gossip operator, the original optimization problem is formulated as a convex optimization problem, which can be addressed by semidefinite programming.

eess.SY↗

arXiv2015

Optimizing the Convergence Rate of the Quantum Consensus: A Discrete Time Model

Motivated by the recent advances in the field of quantum computing, quantum systems are modelled and analyzed as networks of decentralized quantum nodes which employ distributed quantum consensus algorithms for coordination. In the literature, both continuous and discrete time models have been proposed for analyzing these algorithms. This paper aims at optimizing the convergence rate of the discrete time quantum consensus algorithm over a quantum network with $N$ qudits. The induced graphs are categorized in terms of the partitions of integer $N$ by arranging them as the Schreier graphs. It is shown that the original optimization problem reduces to optimizing the Second Largest Eigenvalue Modulus (SLEM) of the weight matrix. Exploiting the Specht module representation of partitions of $N$, the Aldous' conjecture is generalized to all partitions (except ($N$)) in the Hasse diagram of integer $N$. Based on this result, it is shown that the spectral gap of Laplacian of all induced graphs corresponding to partitions (other than ($N$)) of $N$ are the same, while the spectral radius of the Laplacian is obtained from the feasible least dominant partition in the Hasse diagram of integer $N$. The semidefinite programming formulation of the problem is addressed analytically for $N \leq d^2 + 1$ and a wide range of topologies where closed-form expressions for the optimal results are provided. For a quantum network with complete graph topology, solution of the optimization problem based on group association schemes is provided for all values of $N$.

eess.SY↗

arXiv2015

Optimizing the Convergence Rate of the Continuous Time Quantum Consensus

Inspired by the recent developments in the fields of quantum distributed computing, quantum systems are analyzed as networks of quantum nodes to reduce the complexity of the analysis. This gives rise to the distributed quantum consensus algorithms. Focus of this paper is on optimizing the convergence rate of the continuous time quantum consensus algorithm over a quantum network with $N$ qudits. It is shown that the optimal convergence rate is independent of the value of $d$ in qudits. First by classifying the induced graphs as the Schreier graphs, they are categorized in terms of the partitions of integer $N$. Then establishing the intertwining relation between one level dominant partitions in the Hasse Diagram of integer $N$, it is proved that the spectrum of the induced graph corresponding to the dominant partition is included in that of the less dominant partition. Based on this result, the proof of the Aldous' conjecture is extended to all possible induced graphs and the original optimization problem is reduced to optimizing spectral gap of the smallest induced graph. By providing the analytical solution to semidefinite programming formulation of the obtained problem, closed-form expressions for the optimal results are provided for a wide range of topologies.

eess.SY↗

arXiv2012

Distributed Data Storage in Large-Scale Sensor Networks Based on LT Codes

This paper proposes an algorithm for increasing data persistency in large-scale sensor networks. In the scenario considered here, k out of n nodes sense the phenomenon and produced ? information packets. Due to usually hazardous environment and limited resources, e.g. energy, sensors in the network are vulnerable. Also due to the large size of the network, gathering information from a few central hopes is not feasible. Flooding is not a desired option either due to limited memory of each node. Therefore the best approach to increase data persistency is propagating data throughout the network by random walks. The algorithm proposed here is based on distributed LT (Luby Transform) codes and it benefits from the low complexity of encoding and decoding of LT codes. In previous algorithms the essential global information (e.g., n and k) are estimated based on graph statistics, which requires excessive transmissions. In our proposed algorithm, these values are obtained without additional transmissions. Also the mixing time of random walk is enhanced by proposing a new scheme for generating the probabilistic forwarding table of random walk. The proposed method uses only local information and it is scalable to any network topology. By simulations the improved performance of developed algorithm compared to previous ones has been verified.

cs.IT↗

arXiv2012

A Solution to Fastest Distributed Consensus Problem for Generic Star & K-cored Star Networks

Distributed average consensus is the main mechanism in algorithms for decentralized computation. In distributed average consensus algorithm each node has an initial state, and the goal is to compute the average of these initial states in every node. To accomplish this task, each node updates its state by a weighted average of its own and neighbors' states, by using local communication between neighboring nodes. In the networks with fixed topology, convergence rate of distributed average consensus algorithm depends on the choice of weights. This paper studies the weight optimization problem in distributed average consensus algorithm. The network topology considered here is a star network where the branches have different lengths. Closed-form formulas of optimal weights and convergence rate of algorithm are determined in terms of the network's topological parameters. Furthermore generic K-cored star topology has been introduced as an alternative to star topology. The introduced topology benefits from faster convergence rate compared to star topology. By simulation better performance of optimal weights compared to other common weighting methods has been proved.

cs.IT↗

arXiv2011

Weight Optimization for Distributed Average Consensus Algorithm in Symmetric, CCS & KCS Star Networks

This paper addresses weight optimization problem in distributed consensus averaging algorithm over networks with symmetric star topology. We have determined optimal weights and convergence rate of the network in terms of its topological parameters. In addition, two alternative topologies with more rapid convergence rates have been introduced. The new topologies are Complete-Cored Symmetric (CCS) star and K-Cored Symmetric (KCS) star topologies. It has been shown that the optimal weights for the edges of central part in symmetric and CCS star configurations are independent of their branches. By simulation optimality of obtained weights under quantization constraints have been verified.

cs.IT↗

arXiv2011

Fastest Distributed Consensus Problem on Branches of an Arbitrary Connected Sensor Network

This paper studies the fastest distributed consensus averaging problem on branches of an arbitrary connected sensor network. In the previous works full knowledge about the sensor network's connectivity topology was required for determining the optimal weights and convergence rate of distributed consensus averaging algorithm over the network. Here in this work for the first time, the optimal weights are determined analytically for the edges of certain types of branches, independent of the rest of network. The solution procedure consists of stratification of associated connectivity graph of the branches and Semidefinite Programming (SDP), particularly solving the slackness conditions, where the optimal weights are obtained by inductive comparing of the characteristic polynomials initiated by slackness conditions. Several examples and numerical results are provided to confirm the optimality of the obtained weights.

cs.IT↗

arXiv2010

Fastest Distributed Consensus on Petal Networks

Providing an analytical solution for the problem of finding Fastest Distributed Consensus (FDC) is one of the challenging problems in the field of sensor networks. Here in this work we present analytical solution for the problem of fastest distributed consensus averaging algorithm by means of stratification and semi-definite programming, for two particular types of Petal networks, namely symmetric and Complete Cored Symmetric (CCS) Petal networks. Our method in this paper is based on convexity of fastest distributed consensus averaging problem, and inductive comparing of the characteristic polynomials initiated by slackness conditions in order to find the optimal weights. Also certain types of leaves are introduced along with their optimal weights which are not achievable by the method used in this work if these leaves are considered individually.

cs.IT↗

arXiv2010

Fastest Distributed Consensus on Star-Mesh Hybrid Sensor Networks

Solving Fastest Distributed Consensus (FDC) averaging problem over sensor networks with different topologies has received some attention recently and one of the well known topologies in this issue is star-mesh hybrid topology. Here in this work we present analytical solution for the problem of FDC algorithm by means of stratification and semidefinite programming, for the Star-Mesh Hybrid network with K-partite core (SMHK) which has rich symmetric properties. Also the variations of asymptotic and per step convergence rate of SMHK network versus its topological parameters have been studied numerically.

cs.IT↗

arXiv2010

Fastest Mixing Markov Chain on Symmetric K-Partite Network

Solving fastest mixing Markov chain problem (i.e. finding transition probabilities on the edges to minimize the second largest eigenvalue modulus of the transition probability matrix) over networks with different topologies is one of the primary areas of research in the context of computer science and one of the well known networks in this issue is K-partite network. Here in this work we present analytical solution for the problem of fastest mixing Markov chain by means of stratification and semidefinite programming, for four particular types of K-partite networks, namely Symmetric K-PPDR, Semi Symmetric K-PPDR, Cycle K-PPDR and Semi Cycle K-PPDR networks. Our method in this paper is based on convexity of fastest mixing Markov chain problem, and inductive comparing of the characteristic polynomials initiated by slackness conditions in order to find the optimal transition probabilities. The presented results shows that a Symmetric K-PPDR network and its equivalent Semi Symmetric K-PPDR network have the same SLEM despite the fact that Semi symmetric K-PPDR network has less edges than its equivalent symmetric K-PPDR network and at the same time symmetric K-PPDR network has better mixing rate per step than its equivalent semi symmetric K-PPDR network at first few iterations. The same results are true for Cycle K-PPDR and Semi Cycle K-PPDR networks. Also the obtained optimal transition probabilities have been compared with the transition probabilities obtained from Metropolis-Hasting method by comparing mixing time improvements numerically.

cs.DM↗

arXiv2010

Fastest Distributed Consensus Averaging Problem on Chain of Rhombus Networks

Distributed consensus has appeared as one of the most important and primary problems in the context of distributed computation and it has received renewed interest in the field of sensor networks (due to recent advances in wireless communications), where solving fastest distributed consensus averaging problem over networks with different topologies is one of the primary problems in this issue. Here in this work analytical solution for the problem of fastest distributed consensus averaging algorithm over Chain of Rhombus networks is provided, where the solution procedure consists of stratification of associated connectivity graph of the network and semidefinite programming, particularly solving the slackness conditions, where the optimal weights are obtained by inductive comparing of the characteristic polynomials initiated by slackness conditions. Also characteristic polynomial together with its roots corresponding to eigenvalues of weight matrix including SLEM of network is determined inductively. Moreover to see the importance of rhombus graphs it is indicated that convergence rate of path network increases by replacing a single node by a rhombus sub graph within the path network.

cs.IT↗

arXiv2010

Fastest Distributed Consensus Averaging Problem on Perfect and Complete n-ary Tree networks

Solving fastest distributed consensus averaging problem (i.e., finding weights on the edges to minimize the second-largest eigenvalue modulus of the weight matrix) over networks with different topologies is one of the primary areas of research in the field of sensor networks and one of the well known networks in this issue is tree network. Here in this work we present analytical solution for the problem of fastest distributed consensus averaging algorithm by means of stratification and semidefinite programming, for two particular types of tree networks, namely perfect and complete n-ary tree networks. Our method in this paper is based on convexity of fastest distributed consensus averaging problem, and inductive comparing of the characteristic polynomials initiated by slackness conditions in order to find the optimal weights. Also the optimal weights for the edges of certain types of branches such as perfect and complete n-ary tree branches are determined independently of rest of the network.

cs.IT↗

arXiv2010

Fastest Distributed Consensus Problem on Fusion of Two Star Networks

Finding optimal weights for the problem of Fastest Distributed Consensus on networks with different topologies has been an active area of research for a number of years. Here in this work we present an analytical solution for the problem of Fastest Distributed Consensus for a network formed from fusion of two different symmetric star networks or in other words a network consists of two different symmetric star networks which share the same central node. The solution procedure consists of stratification of associated connectivity graph of network and Semidefinite Programming (SDP), particularly solving the slackness conditions, where the optimal weights are obtained by inductive comparing of the characteristic polynomials initiated by slackness conditions. Some numerical simulations are carried out to investigate the trade-off between the parameters of two fused star networks, namely the length and number of branches.

cs.IT↗