arXiv subjects

Claudio Altafini

Publications and source records attributed to Claudio Altafini.

At least 19 recordsLinked to original sources

arXiv2026

Analogies between Transformer Layers and Power Method

In the paper we show that there is an analogy between the operations occurring in a layer of a transformer (projections and layer normalizations, disregarding the feedforward neural network) and a step in the power method. Coherently with this analogy, we show that passing through a layer the tokens tend to be tilted towards the principal eigenvector of a matrix which is the product of the output and value weight matrices of that layer. In the special case of a transformer with shared weights (i.e., in which all layers have identical weights) then the alignment with this principal eigenvector is particularly evident empirically, and can also be shown analytically. The analogy also suggests a method to steer the output of the transformer towards an arbitrary desired direction in token space.

cs.LG↗

arXiv2026

Quantifying opinion homophily in online social networks: A bounded confidence perspective

The concept of homophily is pervasive in online social media. While many empirical studies have relied on external sociodemographic traits to investigate it, significantly less is known about homophily at the cognitive level, that is, at the level of shared opinions or values. For such "value homophily", in this paper we study interval-based patterns of opinion homophily from a bounded confidence perspective. We consider three heterogeneous datasets from Reddit and Twitter covering polarizing issues, with user opinions quantified via sentiment analysis and fact-checking, and analyze the interaction networks formed by weaker (reply-based) and stronger (follow-based) social ties. Our findings show that users' interaction neighborhoods are significantly more concentrated in opinion space than expected by chance, with tie strength and issue polarization further amplifying this effect. Moreover, users often exhibit asymmetric tolerance ranges, with asymmetry typically directed toward locally mainstream positions rather than more radical or opposing ones. These findings support a bounded confidence interpretation of online value homophily.

cs.SI↗

arXiv2026

Minimal Input Cardinality Disturbance Decoupling of Coupled Oscillators via Output Feedback with Application to Power Networks

In this paper, we identify the smallest set of control input nodes and an associated output feedback law that achieves complete disturbance decoupling for a class of coupled oscillator networks. The focus is specifically on systems linearized around a stable phase-locked synchronized state. The proposed theoretical framework is applied to the linearized swing dynamics of power grids operating near synchronization. In this context, the disturbance decoupling problem corresponds to isolating subsets of nodes from exogenous disturbances by means of batteries that can both add or withdraw active power. Numerical simulations carried out on the IEEE New England 39-bus system show that the proposed methodology not only yields a minimal actuator placement ensuring effective disturbance rejection, but also preserves the internal stability of the closed-loop system.

eess.SY↗

arXiv2026

Signed DeGroot-Friedkin Dynamics with Interdependent Topics

This paper investigates DeGroot-Friedkin (DF) dynamics over signed influence networks with interdependent topics. We propose a multi-topic signed framework that combines repelling interpersonal interactions with cross-issue self-appraisal, examining how antagonism and topic interdependence shape the evolution of agent-level social power. When the logic matrices (for topic interdependence) of all agents share a common dominant left eigenvector, we identify structural conditions under which the original dynamics admit an exact reduction to an explicit scalar DF map. This yields a complete classification of limiting social power configurations into pluralistic, mixed, and vertex-dominant types. In all three cases, the dynamics are globally convergent, and in the first two the ordering induced by the interaction centrality is preserved. We further show local robustness under small heterogeneous perturbations of the logic matrices. We also clarify what changes when this common-eigenvector structure is lost. These results extend signed social power dynamics beyond the standard nonnegative scalar setting and shed light on the robustness and scope of centrality-based social power formation in multi-topic signed influence systems.

physics.soc-ph↗

arXiv2025

Multistability of Self-Attention Dynamics in Transformers

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.

cs.LG↗

arXiv2025

Gradient Flow Equations for Deep Linear Neural Networks: A Survey from a Network Perspective

The paper surveys recent progresses in understanding the dynamics and loss landscape of the gradient flow equations associated to deep linear neural networks, i.e., the gradient descent training dynamics (in the limit when the step size goes to 0) of deep neural networks missing the activation functions and subject to quadratic loss functions. When formulated in terms of the adjacency matrix of the neural network, as we do in the paper, these gradient flow equations form a class of converging matrix ODEs which is nilpotent, polynomial, isospectral, and with conservation laws. The loss landscape is described in detail. It is characterized by infinitely many global minima and saddle points, both strict and nonstrict, but lacks local minima and maxima. The loss function itself is a positive semidefinite Lyapunov function for the gradient flow, and its level sets are unbounded invariant sets of critical points, with critical values that correspond to the amount of singular values of the input-output data learnt by the gradient along a certain trajectory. The adjacency matrix representation we use in the paper allows to highlight the existence of a quotient space structure in which each critical value of the loss function is represented only once, while all other critical points with the same critical value belong to the fiber associated to the quotient space. It also allows to easily determine stable and unstable submanifolds at the saddle points, even when the Hessian fails to obtain them.

cs.LG↗

arXiv2025

Wisdom of Crowds Effects under Antagonistic Interactions and Correlated Opinions

This paper investigates the wisdom of crowds of linear opinion dynamics models evolving on signed networks. Conditions are given under which models such as the DeGroot, Friedkin-Johnsen (FJ) and concatenated FJ models improve or undermine collective wisdom. The extension to dependent initial opinions is also presented, highlighting how the correlation structure influences the feasibility and geometry of the wisdom-improving regions.

eess.SY↗

arXiv2025

Geometric Control Theory Over Networks: Minimal Node Cardinality Disturbance Decoupling Problems

In this paper we show how to formulate and solve disturbance decoupling problems over networks while choosing a minimal number of input and output nodes. Feedback laws that isolate and eliminate the impact of disturbance nodes on specific target nodes to be protected are provided using state, output, and dynamical feedback. For that, we leverage the fact that when reformulated in terms of sets of nodes rather than subspaces, the controlled and conditional invariance properties admit a simple graphical interpretation. For state and dynamical feedback, the minimal input and output cardinality solutions can be computed exactly in polynomial time, via min-cut/max-flow algorithms.

math.OC↗

arXiv2025

Computing frustration and near-monotonicity in deep neural networks

For the signed graph associated to a deep neural network, one can compute the frustration level, i.e., test how close or distant the graph is to structural balance. For all the pretrained deep convolutional neural networks we consider, we find that the frustration is always less than expected from null models. From a statistical physics point of view, and in particular in reference to an Ising spin glass model, the reduced frustration indicates that the amount of disorder encoded in the network is less than in the null models. From a functional point of view, low frustration (i.e., proximity to structural balance) means that the function representing the network behaves near-monotonically, i.e., more similarly to a monotone function than in the null models. Evidence of near-monotonic behavior along the partial order determined by frustration is observed for all networks we consider. This confirms that the class of deep convolutional neural networks tends to have a more ordered behavior than expected from null models, and suggests a novel form of implicit regularization.

cs.LG↗

arXiv2022

Multi-agent consensus over time-invariant and time-varying signed digraphs via eventual positivity

Laplacian dynamics on signed digraphs have a richer behavior than those on nonnegative digraphs. In particular, for the so-called "repelling" signed Laplacians, the marginal stability property (needed to achieve consensus) is not guaranteed a priori and, even when it holds, it does not automatically lead to consensus, as these signed Laplacians may loose rank even in strongly connected digraphs. Furthermore, in the time-varying case, instability can occur even when switching in a family of systems each of which corresponds to a marginally stable signed Laplacian with the correct corank. In this paper we present conditions guaranteeing consensus of these signed Laplacians based on the property of eventual positivity, a Perron-Frobenius type of property for signed matrices. The conditions cover both time-invariant and time-varying cases. A particularly simple sufficient condition valid in both cases is that the Laplacians are normal matrices. Such condition can be relaxed in several ways. For instance in the time-invariant case it is enough that the Laplacian has this Perron-Frobenius property on the right but not on the left side (i.e., on the transpose). For the time-varying case, convergence to consensus can be guaranteed by the existence of a common Lyapunov function for all the signed Laplacians. All conditions can be easily extended to bipartite consensus.

eess.SY↗

arXiv2021

On the properties of Laplacian pseudoinverses

The pseudoinverse of a graph Laplacian is used in many applications and fields, such as for instance in the computation of the effective resistance in electrical networks, in the calculation of the hitting/commuting times for a Markov chain and in continuous-time distributed averaging problems. In this paper we show that the Laplacian pseudoinverse is in general not a Laplacian matrix but rather a signed Laplacian with the property of being an eventually exponentially positive matrix, i.e., of obeying a strong Perron-Frobenius property. We show further that the set of signed Laplacians with this structure (i.e., eventual exponential positivity) is closed with respect to matrix pseudoinversion. This is true even for signed digraphs, and provided that we restrict to Laplacians that are weight balanced also stability is guaranteed.

eess.SY↗

arXiv2021

Dynamics of Opinions with Bounded Confidence in Social Cliques: Emergence of Fluctuations

In this paper, we study the evolution of opinions over social networks with bounded confidence in social cliques. Node initial opinions are independently and identically distributed; at each time step, nodes review the average opinions of a randomly selected local clique. The clique averages may represent local group pressures on peers. Then nodes update their opinions under bounded confidence: only when the difference between an agent individual opinion and the corresponding local clique pressure is below a threshold, this agent opinion is updated according to the DeGroot rule as a weighted average of the two values. As a result, this opinion dynamics is a generalization of the classical Deffuant-Weisbuch model in which only pairwise interactions take place. First of all, we prove conditions under which all node opinions converge to finite limits. We show that in the limits the event that all nodes achieve a consensus, and the event that all nodes achieve pairwise distinct limits, i.e., social disagreements, are both nontrivial events. Next, we show that opinion fluctuations may take place in the sense that at least one agent in the network fails to hold a converging opinion trajectory. In fact, we prove that this fluctuation event happens with a strictly positive probability, and also constructively present an initial value event under which the fluctuation event arises with probability one. These results add to the understanding of the role of bounded confidence in social opinion dynamics, and the possibility of fluctuation reveals that bringing in cliques in Deffuant-Weisbuch models have fundamentally changed the behavior of such opinion dynamical processes.

cs.SI↗

arXiv2021

The role of frustration in collective decision-making dynamical processes on multiagent signed networks

In this work we consider a collective decision-making process in a network of agents described by a nonlinear interconnected dynamical model with sigmoidal nonlinearities and signed interaction graph. The decisions are encoded in the equilibria of the system. The aim is to investigate this multiagent system when the signed graph representing the community is not structurally balanced and in particular as we vary its frustration, i.e., its distance to structural balance. The model exhibits bifurcations, and a ``social effort'' parameter, added to the model to represent the strength of the interactions between the agents, plays the role of bifurcation parameter in our analysis. We show that, as the social effort increases, the decision-making dynamics exhibits a pitchfork bifurcation behavior where, from a deadlock situation of ``no decision'' (i.e., the origin is the only globally stable equilibrium point), two possible (alternative) decision states for the community are achieved (corresponding to two nonzero locally stable equilibria). The value of social effort for which the bifurcation is crossed (and a decision is reached) increases with the frustration of the signed network.

math.OC↗

arXiv2020

Investigating the effect of edge modifications on networked control systems

This paper investigates the impact of addition/removal of edges in a complex networked control system, for the purposes of improving its controllability, system performances or robustness to external disturbances. The transfer function formulation we obtain allows to quantify the impact of an edge modification with the $\hinf$ and $\htwo$ norms. For stable networks with positive edge weights, we show that the $\hinf$ norm can be computed exactly for each possible single edge modification, as well as the associated stability margin. For the $\htwo$ norm we instead obtain a lower bound. Since this bound is linked to the trace of the controllability Gramian, it can be used for instance to reduce the energy needed for control. When instead the dynamics is of Laplacian type, then the norms become unbounded. However, the associated displacement systems are stable and for them the effect of edge modifications can be quantified. In particular, in this case we provide an upper bound on the $ \hinf$ norm and compute the exact value of the $ \htwo $ norm for arbitrary edge additions.

eess.SY↗

arXiv2019

A system-theoretic framework for privacy preservation in continuous-time multiagent dynamics

In multiagent dynamical systems, privacy protection corresponds to avoid disclosing the initial states of the agents while accomplishing a distributed task. The system-theoretic framework described in this paper for this scope, denoted dynamical privacy, relies on introducing output maps which act as masks, rendering the internal states of an agent indiscernible by the other agents as well as by external agents monitoring all communications. Our output masks are local (i.e., decided independently by each agent), time-varying functions asymptotically converging to the true states. The resulting masked system is also time-varying, and has the original unmasked system as its limit system. When the unmasked system has a globally exponentially stable equilibrium point, it is shown in the paper that the masked system has the same point as a global attractor. It is also shown that existence of equilibrium points in the masked system is not compatible with dynamical privacy. Application of dynamical privacy to popular examples of multiagent dynamics, such as models of social opinions, average consensus and synchronization, is investigated in detail.

eess.SY↗

arXiv2018

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.

cs.SI↗

arXiv2018

A dynamical approach to privacy preserving average consensus

In this paper we propose a novel method for achieving average consensus in a continuous-time multiagent network while avoiding to disclose the initial states of the individual agents. In order to achieve privacy protection of the state variables, we introduce maps, called output masks, which alter the value of the states before transmitting them. These output masks are local (i.e., implemented independently by each agent), deterministic, time-varying and converging asymptotically to the true state. The resulting masked system is also time-varying and has the original (unmasked) system as its limit system. It is shown in the paper that the masked system has the original average consensus value as a global attractor. However, in order to preserve privacy, it cannot share an equilibrium point with the unmasked system, meaning that in the masked system the global attractor cannot be also stable.

eess.SY↗

arXiv2018

Centrality measures and the role of non-normality for network control energy reduction

Combinations of Gramian-based centrality measures are used for driver node selection in complex networks in order to simultaneously take into account conflicting control energy requirements, like minimizing the average energy needed to steer the state in any direction and the energy needed for the worst direction. The selection strategies that we propose are based on a characterization of the network non-normality, a concept we show is related to the idea of balanced realization.

eess.SY↗