Multistability of Self-Attention Dynamics in Transformers Claudio Altafini IEEE Transactions on Automatic Control, 2026 In machine learning, a self-attention dynamics is a continuous-time multiagent-like model of the attention mechanisms of transformers. In this paper we show that such dynamics is related to a multiagent version of the Oja flow, a dynamical system that computes the principal eigenvector of a matrix corresponding for transformers to the value matrix. We classify the equilibria of the “single head” self-attention system into four classes: consensus, bipartite consensus, clustering and polygonal equilibria. Multiple asymptotically stable equilibria from the first three classes often coexist in the self-attention dynamics. Interestingly, equilibria from the first two classes are always aligned with the eigenvectors of the value matrix, often but not exclusively with the principal eigenvector
Wisdom of crowds in signed opinion dynamics models Muhammad Ahsan Razaq, Claudio Altafini Proceedings of the IEEE Conference on Decision and Control, 2025 This paper investigates the wisdom of crowds of linear opinion dynamics models in presence of antagonistic interactions. Conditions are given under which models such as the DeGroot, Friedkin-Johnsen (FJ) and concatenated FJ models may or may not improve wisdom.
Signed Friedkin–Johnsen Models: Opinion Dynamics With Stubbornness and Antagonism Muhammad Ahsan Razaq, Claudio Altafini IEEE Transactions on Automatic Control, 2025 The aim of this article is to investigate how opinions propagate in presence of stubborn behavior (i.e., agents that tend to defend their stance) and of signed (i.e., antagonistic) interactions among the agents. The stability properties of the resulting system, denoted signed Friedkin–Johnsen (SFJ) model, are studied in detail, and also the convergence to consensus for a concatenation of such SFJ models. A noteworthy feature of the SFJ model is that the opinion dynamics need not be confined to the convex hull of the initial conditions, meaning that cleavage can increase because of the antagonistic interactions. In particular, we show that the concatenated SFJ model can behave as a nonnormal system, in which the opinions transiently distance themselves only to rejoin a common consensus value asymptotically.
Optimal Disturbance Decoupling over Networks via State Feedback Luca Claude Gino Lebon, Claudio Altafini Proceedings of the IEEE Conference on Decision and Control, 2025 This paper addresses the problem of optimal disturbance decoupling over networks through state feedback control. We propose an algorithm that isolates and eliminates the impact of disturbance nodes on specific target nodes to be protected, leveraging well-known properties of controlled invariance and reformulating them for systems over networks. The optimal solution, intended as a minimal-cardinality set of control inputs needed to achieve the decoupling, can be characterized as a function of the maximal controlled invariant set contained in the null space of the target nodes, and can be computed by solving a modified min-cut/max-flow problem. A closed-form solution for the state feedback matrix that achieves disturbance decoupling is also provided, and its graphical meaning is explained.
Social Power Games for Parallel Friedkin-Johnsen Models Lingfei Wang, Yu Xing, Shijie Huang, Claudio Altafini, Karl Henrik Johansson IEEE Transactions on Automatic Control, 2025 In this paper we consider a strategic game played by a group of agents on a set of opinion dynamics models. The models are all Friedkin-Johnsen (FJ) models, which are independent of each other (we call them “parallel FJ models”). The task of an agent is to maximize her overall social power by allocating a given budget of stubbornness across the parallel FJ models. For this game, the cost function is shown to be convex in the action profile set, but discontinuous at some boundary points when for some FJ model only one agent is stubborn (i.e., assigning non-zero stubbornness in the FJ model). Despite the discontinuity, an Nash equilibrium is shown to exist, but is not necessarily unique. Some sufficient conditions that can guarantee the uniqueness are proposed, relying on the strictly monotone pseudo-gradient mappings associated to the game. The conditions are applied to complete graphs with rank-1 weight matrices, for which the link weights are unequal for different agents and on different FJ models. Moreover, for the complete graph case, given the actions of the other agents, the best response of each agent is analytically characterized.
On controllability of temporal networks Luca Claude Gino Lebon, Francesco Lo Iudice, Claudio Altafini European Journal of Control, 2024 Temporality has been recently identified as a useful feature to exploit when controlling a complex network. Empirical evidence has in fact shown that, with respect to their static counterparts, temporal networks (i) are often endowed with larger reachable sets and (ii) require less control energy when steered towards an arbitrary target state. However, to date, we lack conditions guaranteeing that the dimension of the controllable subspace of a temporal network is larger than that of its static counterpart. In this work, we consider the case in which a static network is input connected but not controllable. We show that when the structure of the graph underlying the temporal network remains the same throughout each temporal snapshot while the (nonvanishing) edge weights vary, then the temporal network will be completely controllable almost always, even when its static counterpart is not. An upper bound on the number of snapshots needed to achieve controllability is also provided.
From populations to networks: Relating diversity indices and frustration in signed graphs Angela Fontan, Marco Ratta, Claudio Altafini Pnas Nexus, 2024 Diversity indices of quadratic type, such as fractionalization and Simpson index, are measures of heterogeneity in a population. Even though they are univariate, they have an intrinsic bivariate interpretation as encounters among the elements of the population. In the paper, it is shown that this leads naturally to associate populations to weakly balanced signed networks. In particular, the frustration of such signed networks is shown to be related to fractionalization by a closed-form expression. This expression allows to simplify drastically the calculation of frustration for weakly balanced signed graphs.
Multidimensional signed Friedkin-Johnsen model Muhammad Ahsan Razaq, Claudio Altafini Proceedings of the IEEE Conference on Decision and Control, 2024 The multidimensional signed Friedkin-Johnsen (SFJ) model introduced in this paper describes opinion dynamics on a signed network in which the agents hold opinions on multiple interconnected topics and are allowed to be stubborn. In the paper, we establish sufficient conditions for the stability of the multidimensional SFJ model, and analyze convergence to consensus in concatenated instances of this model.
Connectivity and Synchronization in Bounded Confidence Kuramoto Oscillators Trisha Srivastava, Carmela Bernardo, Claudio Altafini, Francesco Vasca IEEE Control Systems Letters, 2024 Frequency synchronization of bounded confidence Kuramoto oscillators is analyzed. The dynamics of each oscillator is defined by the average of the phase differences with its neighbors, where any two oscillators are considered neighbors if their geodesic distance is less than a certain confidence threshold. A phase-dependent graph is defined whose nodes and edges represent the oscillators and their connections, respectively. It is studied how the connectivity of the graph influences steady-state behaviors of the oscillators. It is proved that the oscillators synchronize asymptotically if the subgraph of each partition, possibly not complete, eventually remains constant over time. Simulation results show the application of the theoretical findings also in the presence of oscillators having different natural frequencies.
Bounded confidence opinion dynamics: A survey Carmela Bernardo, Claudio Altafini, Anton Proskurnikov, Francesco Vasca Automatica, 2024 At the beginning of this century, Hegselmann and Krause proposed a dynamical model for opinion formation that is referred to as the Bounded Confidence Opinion Dynamics (BCOD) model and that has since attracted a wide interest from different research communities. The model can be viewed as a dynamic network, in which each agent is endowed with a state variable representing an opinion and two agents interact if the distance between their opinions does not exceed a constant confidence bound. This relation of instantaneous proximity between the opinions naturally induces a dynamic interaction graph. At each stage of the opinion iteration, all agents synchronously update their opinion to the average of all opinions that belong to the neighbors in the interaction graph. BCOD models exhibit a broad variety of phenomena that cannot be studied by traditional methods, and their analysis has enriched the systems and control field with a number of novel mathematical tools. This fact, together with the existence of an extensive literature on the topic scattered across different fields, calls for a systematic presentation of the existing results on this class of dynamic networks. The aim of this survey is to provide an overview of BCOD models with time-synchronous interactions, with possibly asymmetric and heterogeneous confidence bounds. Conditions on the different classes of BCOD which ensure the convergence (in finite time or asymptotically) of the opinions are discussed, and the possible structures of the terminal opinions are described. The numerous phenomena highlighted in the literature from numerical studies, e.g., the characterization of steady state behaviors and the sensitivity to confidence thresholds, are also reviewed. Finally, some recent modifications and applications of BCOD models are discussed, and suggestions of directions for future research are provided.
Feedback stabilization of spin systems C. Altafini Proceedings of the 44th IEEE Conference on Decision and Control and the European Control Conference Cdc Ecc 05, 2005
Tensor of coherences parametrization of multiqubit density operators for entanglement characterization Physical Review A Atomic Molecular and Optical Physics, 2004
Quantum Markovian master equation driven by coherent controls: A controllability analysis Proceedings of the IEEE Conference on Decision and Control, 2003
On the generation of discrete unitary gates from continuous time forced Schrödinger equations Proceedings of the IEEE Conference on Decision and Control, 2002
Explicit Wei-Norman formulæ for matrix Lie groups Proceedings of the IEEE Conference on Decision and Control, 2002
Redundant robotic chains on Riemannian manifolds Proceedings of the IEEE Conference on Decision and Control, 2002
Hybrid control of a truck and trailer vehicle Claudio Altafini, Alberto Speranzon, Karl Henrik Johansson Lecture Notes in Computer Science Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics, 2002
Inverse kinematics along a geometric spline for a holonomic mobile manipulator Proceedings IEEE International Conference on Robotics and Automation, 2001
Backward line tracking control of a radio-controlled truck and trailer Proceedings IEEE International Conference on Robotics and Automation, 2001
General n-trailer, differential flatness and equivalence Proceedings of the IEEE Conference on Decision and Control, 1999
Zero dynamics and off-tracking bounds for the path following problem of wheeled vehicles Proceedings of the IEEE Conference on Decision and Control, 1999
Why to use an articulated vehicle in underground mining operations? Proceedings IEEE International Conference on Robotics and Automation, 1999
General n-trailer problem: Conversion into chained form Proceedings of the IEEE Conference on Decision and Control, 1998
Path following with reduced off-tracking for the n-trailer system Proceedings of the IEEE Conference on Decision and Control, 1998
Robust control of a Flash Dryer plant IEEE Conference on Control Applications Proceedings, 1997
RECENT SCHOLAR PUBLICATIONS
Gradient flow equations for deep linear neural networks: A survey from a network perspective J Wendin, C Altafini SIAM Review 68 (2), 293-345 , 2026 2026 Citations: 2
Geometric Control Theory Over Networks: Minimal Node Cardinality Disturbance Decoupling Problems LCG Lebon, C Altafini IEEE Transactions on Automatic Control , 2026 2026 Citations: 1
Quantifying opinion homophily in online social networks: A bounded confidence perspective Y Luan, C Ancona, C Bernardo, V Pansanella, FL Iudice, G Rossetti, ... arXiv preprint arXiv:2604.23157 , 2026 2026
Multistability of self-attention dynamics in transformers C Altafini IEEE Transactions on Automatic Control , 2026 2026 Citations: 3
Minimal input cardinality disturbance decoupling of coupled oscillators via output feedback with application to power networks LCG Lebon, J Lindberg, C Altafini arXiv preprint arXiv:2604.14977 , 2026 2026
Signed DeGroot-Friedkin Dynamics with Interdependent Topics Y Luan, MA Razaq, X Wu, C Altafini arXiv preprint arXiv:2604.12685 , 2026 2026
Optimal Disturbance Decoupling over Networks via State Feedback LCG Lebon, C Altafini 2025 IEEE 64th Conference on Decision and Control (CDC), 5258-5263 , 2025 2025 Citations: 2
Wisdom of crowds in signed opinion dynamics models MA Razaq, C Altafini 2025 IEEE 64th Conference on Decision and Control (CDC), 3800-3805 , 2025 2025 Citations: 1
Social Power Games for Parallel Friedkin-Johnsen Models L Wang, Y Xing, S Huang, C Altafini, KH Johansson IEEE Transactions on Automatic Control , 2025 2025
Wisdom of Crowds Effects under Antagonistic Interactions and Correlated Opinions MA Razaq, C Altafini arXiv preprint arXiv:2510.19123 , 2025 2025
Computing frustration and near-monotonicity in deep neural networks J Wendin, EG Larsson, C Altafini arXiv preprint arXiv:2510.05286 , 2025 2025 Citations: 1
Combining homophily in interaction and opinion spaces: A bounded confidence approach Y Luan, C Ancona, C Bernardo, V Pansanella, FL Iudice, G Rossetti, ... 2025
Signed Friedkin–Johnsen Models: Opinion Dynamics With Stubbornness and Antagonism MA Razaq, C Altafini IEEE Transactions on Automatic Control 70 (8), 5037-5051 , 2025 2025 Citations: 8
Multidimensional signed friedkin-johnsen model MA Razaq, C Altafini 2024 IEEE 63rd Conference on Decision and Control (CDC), 5304-5309 , 2024 2024 Citations: 3
On controllability of temporal networks LCG Lebon, FL Iudice, C Altafini European Journal of Control 80, 101046 , 2024 2024 Citations: 7
Maximizing social power in multiple independent Friedkin-Johnsen models L Wang, Y Xing, C Altafini, KH Johansson 2024 European Control Conference (ECC), 3422-3427 , 2024 2024 Citations: 3
Connectivity and synchronization in bounded confidence Kuramoto oscillators T Srivastava, C Bernardo, C Altafini, F Vasca IEEE Control Systems Letters 8, 874-879 , 2024 2024 Citations: 1
Social power games in concatenated opinion dynamics L Wang, G Chen, C Bernardo, Y Hong, G Shi, C Altafini IEEE Transactions on Automatic Control 69 (11), 7614-7629 , 2024 2024 Citations: 12
From populations to networks: Relating diversity indices and frustration in signed graphs A Fontan, M Ratta, C Altafini PNAS nexus 3 (2), pgae046 , 2024 2024 Citations: 2
Multidimensional signed Friedkin-Johnsen model C Altafini, R Muhammad Ahsan 63rd IEEE Conf. on Decision and Control, Milan. , 2024 2024
MOST CITED SCHOLAR PUBLICATIONS
Consensus problems on networks with antagonistic interactions C Altafini Automatic Control, IEEE Transactions on 58 (4), 935-946 , 2013 2013 Citations: 2305
Computing global structural balance in large-scale signed social networks G Facchetti, G Iacono, C Altafini Proceedings of the National Academy of Sciences 108 (52), 20953-20958 , 2011 2011 Citations: 517
Modeling and control of quantum systems: An introduction C Altafini, F Ticozzi IEEE Transactions on Automatic Control 57 (8), 1898-1917 , 2012 2012 Citations: 353
Dynamics of opinion forming in structurally balanced social networks. C Altafini PloS ONE, 7 (6), e38135 , 2012 2012 Citations: 343
Controllability of quantum mechanical systems by root space decomposition of su (N) C Altafini J. Mathematical Physics 43 (5), 2051-2062 , 2002 2002 Citations: 253
A feedback control scheme for reversing a truck and trailer vehicle C Altafini, A Speranzon, B Wahlberg IEEE Transactions on robotics and automation 17 (6), 915-922 , 2001 2001 Citations: 245
Predictable dynamics of opinion forming for networks with antagonistic interactions C Altafini, G Lini IEEE Transactions on Automatic Control 60 (2), 342-357 , 2015 2015 Citations: 241
Controllability properties for finite dimensional quantum Markovian master equations C Altafini J. of Mathematical Physics 44 (6), 2357-2372 , 2003 2003 Citations: 203
Dynamics over signed networks G Shi, C Altafini, JS Baras Siam review 61 (2), 229-257 , 2019 2019 Citations: 186
Some properties of the general n-trailer C Altafini International Journal of Control 74 (4), 409-424 , 2001 2001 Citations: 185
Comparing association network algorithms for reverse engineering of large-scale gene regulatory networks: synthetic versus real data N Soranzo, G Bianconi, C Altafini Bioinformatics 23 (13), 1640-1647 , 2007 2007 Citations: 172
Bounded confidence opinion dynamics: A survey C Bernardo, C Altafini, A Proskurnikov, F Vasca Automatica 159, 111302 , 2024 2024 Citations: 166
A system-theoretic framework for privacy preservation in continuous-time multiagent dynamics C Altafini Automatica 122, 109253 , 2020 2020 Citations: 166
Hybrid control of a truck and trailer vehicle C Altafini, A Speranzon, KH Johansson International Workshop on Hybrid Systems: Computation and Control, 21-34 , 2002 2002 Citations: 154
Coherent control of open quantum dynamical systems C Altafini Phys. Rev. A 70, 062321 , 2004 2004 Citations: 139
Minimum energy control for complex networks G Lindmark, C Altafini Scientific reports 8 (1), 3188 , 2018 2018 Citations: 129
Proteomics reveal biomarkers for diagnosis, disease activity and long-term disability outcomes in multiple sclerosis J Åkesson, S Hojjati, S Hellberg, J Raffetseder, M Khademi, R Rynkowski, ... Nature Communications 14 (1), 6903 , 2023 2023 Citations: 108
Path following with reduced off-tracking for multibody wheeled vehicles C Altafini IEEE transactions on control systems technology 11 (4), 598-605 , 2003 2003 Citations: 96
A path-tracking criterion for an LHD articulated vehicle C Altafini The International Journal of Robotics Research 18 (5), 435-441 , 1999 1999 Citations: 87
Feedback stabilization of isospectral control systems on complex flag manifolds: application to quantum ensembles C Altafini IEEE Transactions on Automatic Control 52 (11), 2019-2028 , 2007 2007 Citations: 86
