Enhancing distributed optimization: A PID-Based approach for unbalanced graphs with bounded disturbances

While fractional calculus has shown appealing properties in optimization related fields, few results have been reported on distributed optimization with fractional dynamics over networks. In this letter, we propose a novel Nabla fractional distributed optimization algorithm over unbalanced graphs. Rigorous analysis demonstrates that the proposed algorithm converges at least with the Mittag—Leffler rate, achieving a globally optimal solution. The algorithm’s effectiveness is validated through numerical simulations. The research underscores the potential of utilizing Nabla fractional difference to improve the efficiency of distributed optimization algorithms.

Massive bipartite graphs are ubiquitous in real world and have important applications in social networks, biological mechanisms, etc. Consider one billion plus people on Facebook making trillions of connections with millions of organizations. Such big social bipartite graphs are often very skewed and unbalanced, on which traditional indexing algorithms do not perform optimally. In this paper, we propose Arowana, a data-driven algorithm for indexing large unbalanced bipartite graphs. Arowana achieves a high-performance efficiency by building an index tree that incorporates the semantic affinity among unbalanced graphs. Arowana uses probabilistic data structures to minimize space overhead and optimize search. In the experiments, we show that Arowana exhibits significant performance improvements and reduces space overhead over traditional indexing techniques.

In this note, a constrained consensus problem is studied for multi-agent systems in unbalanced networks in the presence of communication delays. Here each agent needs to lie in a closed convex constraint set while reaching a consensus. The communication graphs are directed, dynamically changing, and not necessarily balanced and only the union of the graphs is assumed to be strongly connected among each time interval of a certain bounded length. The analysis is performed based on an undelayed equivalent system that is composed of a linear main body and an error auxiliary. To tackle the loss of symmetry caused by unbalanced graphs and communication delays, a novel approach is proposed. The idea is to estimate the distance from each agent to the intersection set of all agents' constraint sets based on the properties of the projection on convex sets so as to show consensus convergence by contradiction. It is shown that the error auxiliary vanishes as time evolves and the linear main body converges to a vector with an exponential rate as a separate system. It is also shown that the communication delays do not affect the consensus stability and constrained consensus is reached even if the communication delays are arbitrarily bounded. Finally, a numerical example is included to illustrate the obtained theoretical results.

In this paper, we use nonlinear continuous opinion dynamics models to investigate how the opinions in social networks are affected by the existence of stubborn individuals who remain attached to their initial opinions. Specifically, we are interested in the case where trust-mistrust relations also exist in the network. We extend the existing models, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stubborn positives</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stubborn extremists</i> , by using signed graphs and allowing negative edges to model mistrust interactions. The effect of stubborn individuals is studied for both structurally balanced and unbalanced graphs. We show that, due to the stubborn individuals, a bipartite consensus (polarization) might not occur even if the associated graph is structurally balanced. Moreover, for structurally unbalanced graphs, the opinion vector does not always converge to 0 as compared to the well-known Altafini's model because of stubborn individuals. We present theorems that provide the above convergence guarantees.

Practical synchronization with diffusive couplings

In this paper, a constrained consensus problem is studied in unbalanced networks in the presence of communication delays. Here each agent needs to lie in a closed convex set while reaching a consensus. The communication graphs considered are directed, dynamically changing, and not necessarily balanced and only the union of the graphs is assumed to be strongly connected among each time interval of a certain bounded length. The analysis is performed based on an undelayed equivalent system that is composed of a linear main body and an error auxiliary. It is showed that the error auxiliary vanishes as time evolves and the linear main body converges to a vector with an exponential rate as a separate system. It is also showed that the communication delays do not affect the consensus stability and consensus is achieved even though the communication delays are arbitrary bounded.

We present a more powerful heuristic algorithm for the NP-complete problem of finding a minimum size subset of edges in an unbalanced signed graph G whose ‘+’/‘−’ labels can be flipped to balance G. Our algorithm finds a minimal flipping edge-set, starting with a given spanning tree T of G, by considering both the edges not in T and those in T because flipping a tree-edge can sometimes balance multiple fundamental unbalanced cycles at the same time. This can give a much smaller minimal flipping edge-set than the current algorithm where only the edges not in T are considered for flipping.

We introduce the Minimum-size bounded-capacity cut (MinSBCC) problem, in which we are given a graph with an identified source and seek to find a cut minimizing the number of nodes on the source side, subject to the constraint that its capacity not exceed a prescribed bound B. Besides being of interest in the study of graph cuts, this problem arises in many practical settings, such as in epidemiology, disaster control, military containment, as well as finding dense subgraphs and communities in graphs.In general, the MinSBCC problem is NP-complete. We present an efficient \((\frac{1}{{\rm \lambda}},\frac{1}{1-{\rm \lambda}})\)-bicriteria approximation algorithm for any 0 < λ < 1; that is, the algorithm finds a cut of capacity at most \(\frac{1}{{\rm \lambda}}B\), leaving at most \(\frac{1}{1-{\rm \lambda}}\) times more vertices on the source side than the optimal solution with capacity B. In fact, the algorithm’s solution either violates the budget constraint, or exceeds the optimal number of source-side nodes, but not both. For graphs of bounded treewidth, we show that the problem with unit weight nodes can be solved optimally in polynomial time, and when the nodes have weights, approximated arbitrarily well by a PTAS.KeywordsTree DecompositionNode WeightSource SideDense SubgraphEdge CapacityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This article addresses the optimization problem with its global objective function being composed of multiple convex functions under multiple nonidentical local constraints. The specific purpose is to resolve the studied optimization problem in a distributed manner in the presence of the time-varying unbalanced graph sequence. For this purpose, an efficient distributed discrete-time algorithm is developed over the time-varying unbalanced graph sequence by improving the classical push–pull algorithms. Moreover, for the developed distributed discrete-time algorithm, a rigorous analysis is made of its convergence property to the optimal solution as well as its convergence rate under some standard assumptions. Finally, numerical simulations are carried out to demonstrate the good performance of the designed algorithm.