On Stability and Consensus of Signed Networks: A Self-loop Compensation Perspective

Positive semidefinite is not an inherent property of signed Laplacians, which renders the stability and consensus of multi-agent system on undirected signed networks intricate. Inspired by the correlation between diagonal dominance and spectrum of signed Laplacians, this paper proposes a self-loop compensation mechanism in the design of interaction protocol amongst agents and examines the stability/consensus of the compensated signed networks. It turns out that self-loop compensation acts as exerting a virtual leader on these agents that are incident to negative edges, steering whom towards origin. Analytical connections between self-loop compensation and the collective behavior of the compensated signed network are established. Necessary and/or sufficient conditions for predictable cluster consensus of signed networks via self-loop compensation are provided. The optimality of self-loop compensation is discussed. Furthermore, we extend our results to directed signed networks where the symmetry of signed Laplacian is not free. Simulation examples are provided to demonstrate the theoretical results

Dynamics over Signed Networks

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.Comment: In press, SIAM Revie

Bures-Wasserstein Means of Graphs

Finding the mean of sampled data is a fundamental task in machine learning and statistics. However, in cases where the data samples are graph objects, defining a mean is an inherently difficult task. We propose a novel framework for defining a graph mean via embeddings in the space of smooth graph signal distributions, where graph similarity can be measured using the Wasserstein metric. By finding a mean in this embedding space, we can recover a mean graph that preserves structural information. We establish the existence and uniqueness of the novel graph mean, and provide an iterative algorithm for computing it. To highlight the potential of our framework as a valuable tool for practical applications in machine learning, it is evaluated on various tasks, including k-means clustering of structured aligned graphs, classification of functional brain networks, and semi-supervised node classification in multi-layer graphs. Our experimental results demonstrate that our approach achieves consistent performance, outperforms existing baseline approaches, and improves the performance of state-of-the-art methods

On the local metric property in multivariate extremes

Many multivariate data sets exhibit a form of positive dependence, which can either appear globally between all variables or only locally within particular subgroups. A popular notion of positive dependence that allows for localized positivity is positive association. In this work we introduce the notion of extremal positive association for multivariate extremes from threshold exceedances. Via a sufficient condition for extremal association, we show that extremal association generalizes extremal tree models. For H\"usler--Reiss distributions the sufficient condition permits a parametric description that we call the metric property. As the parameter of a H\"usler--Reiss distribution is a Euclidean distance matrix, the metric property relates to research in electrical network theory and Euclidean geometry. We show that the metric property can be localized with respect to a graph and study surrogate likelihood inference. This gives rise to a two-step estimation procedure for locally metrical H\"usler--Reiss graphical models. The second step allows for a simple dual problem, which is implemented via a gradient descent algorithm. Finally, we demonstrate our results on simulated and real data.Comment: 22 pages, 4 figure

Stability Bounds of Droop-Controlled Inverters in Power Grid Networks

The energy mix of future power systems will include high shares of electricity generation by wind turbines and solar photovoltaics. These generation facilities are generally connected via power-electronic inverters. While conventional generation responds dynamically to the state of the electric power system, inverters are power-electronic hardware and need to be programmed to react to the state of the system. Choosing an appropriate control scheme and the corresponding parameters is necessary to guarantee that the system operates safely. A prominent control scheme for inverters is droop control, which mimics the response of conventional generation. In this work, we investigate the stability of coupled systems of droop-controlled inverters in arbitrary network topologies. Employing linear stability analysis, we derive effective local stability criteria that consider both the overall network topology as well as its interplay with the inverters’ intrinsic parameters. First, we explore the stability of an inverter coupled to an infinite grid and uncover stability and instability regions. Second, we extend the analysis to a generic topology of inverters and provide mathematical criteria for the stability and instability of the system. Last, we showcase the usefulness of the criteria by examining two model systems using numerical simulations. The developed criteria show which parameters might lead to an unstable operating state.Stability Bounds of Droop-Controlled Inverters in Power Grid NetworkspublishedVersio

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum