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.

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.

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.

Hegselmann-Krause (HK) models exhibit complex behaviors which are not easily tractable through mathematical analysis. In this paper, a characterization of the steady-state behaviors of homogeneous HK models and sensitivity to confidence thresholds is discussed by commenting on existing and new numerical results. The typical decreasing of number of clusters and convergence time by increasing the confidence thresholds are discussed and motivations for the behavior of some counterexamples are provided. A tighter upper bound for the dependence of the number of clusters with respect to the confidence thresholds is proposed. Differences and analogies between the opinions evolution for symmetric and asymmetric HK models are commented.

This article investigates a two-timescale opinion dynamics model, named the concatenated Friedkin-Johnsen (FJ) model, which describes the evolution of the opinions of a group of agents over a sequence of discussion events. The topology of the underlying graph changes with the event, in the sense that the agents can participate or less to an event, and the agents are stubborn, with stubbornness that can vary from one event to the other. Concatenation refers to the fact that the final opinions of an event become initial conditions of the next event. We show that a concatenated FJ model can be represented as a time-varying product of stochastic transition matrices having a special form. Conditions are investigated under which a concatenated FJ model can achieve consensus in spite of the stubbornness. Four different sufficient conditions are obtained, mainly based on the special topological structure of our stochastic matrices.

This paper investigates the impact of addition/removal/reweighting of edges in a complex networked linear control system. For networks of positive edge weights, we show that when adding edges leads to the creation of new cycles, these in turn may lead to instabilities. Dynamically, these cycles correspond to positive feedback loops. Conditions are provided under which the modified network is guaranteed to be stable. These conditions are related to the steady state value of the transfer function matrix of the newly created positive feedbacks. The tools we develop in the paper can be used to investigate the fragility of a network, i.e., its robustness to structured perturbations.(c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

BackgroundMultiple sclerosis (MS) is a neuroinflammatory disease in which pregnancy leads to a temporary amelioration in disease activity as indicated by the profound decrease in relapses rate during the 3rd trimester of pregnancy. CD4(+) and CD8(+) T cells are implicated in MS pathogenesis as being key regulators of inflammation and brain lesion formation. Although Tcells are prime candidates for the pregnancy-associated improvement of MS, the precise mechanisms are yet unclear, and in particular, a deep characterization of the epigenetic and transcriptomic events that occur in peripheral T cells during pregnancy in MS is lacking.MethodsWomen with MS and healthy controls were longitudinally sampled before, during (1st, 2nd and 3rd trimesters) and after pregnancy. DNA methylation array and RNA sequencing were performed on paired CD4(+) and CD8(+) T cells samples. Differential analysis and network-based approaches were used to analyze the global dynamics of epigenetic and transcriptomic changes.ResultsBoth DNA methylation and RNA sequencing revealed a prominent regulation, mostly peaking in the 3rd trimester and reversing post-partum, thus mirroring the clinical course with improvement followed by a worsening in disease activity. This rebound pattern was found to represent a general adaptation of the maternal immune system, with only minor differences between MS and controls. By using a network-based approach, we highlighted several genes at the core of this pregnancy-induced regulation, which were found to be enriched for genes and pathways previously reported to be involved in MS. Moreover, these pathways were enriched for in vitro stimulated genes and pregnancy hormones targets.ConclusionThis study represents, to our knowledge, the first in-depth investigation of the methylation and expression changes in peripheral CD4(+) and CD8(+) T cells during pregnancy in MS. Our findings indicate that pregnancy induces profound changes in peripheral T cells, in both MS and healthy controls, which are associated with the modulation of inflammation and MS activity.

This paper addresses the problem of propagation of opinions in a Signed Friedkin-Johnsen (SFJ) model, i.e., an opinion dynamics model in which the agents are stubborn and the interaction graph is signed. We provide sufficient conditions for the stability of the SFJ model and for convergence to consensus of a concatenation of such SFJ models.

Sensitive and reliable protein biomarkers are needed to predict disease trajectory and personalize treatment strategies for multiple sclerosis (MS). Here, we use the highly sensitive proximity-extension assay combined with next-generation sequencing (Olink Explore) to quantify 1463 proteins in cerebrospinal fluid (CSF) and plasma from 143 people with early-stage MS and 43 healthy controls. With longitudinally followed discovery and replication cohorts, we identify CSF proteins that consistently predicted both short- and long-term disease progression. Lower levels of neurofilament light chain (NfL) in CSF is superior in predicting the absence of disease activity two years after sampling (replication AUC = 0.77) compared to all other tested proteins. Importantly, we also identify a combination of 11 CSF proteins (CXCL13, LTA, FCN2, ICAM3, LY9, SLAMF7, TYMP, CHI3L1, FYB1, TNFRSF1B and NfL) that predict the severity of disability worsening according to the normalized age-related MS severity score (replication AUC = 0.90). The identification of these proteins may help elucidate pathogenetic processes and might aid decisions on treatment strategies for persons with MS.

Even for nonnegative graphs, the pseudoinverse of a Laplacian matrix is not an ``ordinary (i.e., unsigned) Laplacian matrix but rather a signed Laplacian. In this paper, we show that the property of eventual positivity provides a natural embedding class for both signed and unsigned Laplacians, class which is closed with respect to pseudoinversion as well as to stability. Such a class can deal with both undirected and directed graphs. In particular, for digraphs, when dealing with pseudoinverse-related quantities such as effective resistance, two possible solutions naturally emerge, differing in the order in which the operations of pseudoinversion and of symmetrization are performed. Both lead to an effective resistance which is a Euclidean metric on the graph.

We consider complex multistage multiagent negotiation processes such as those occurring at climate conferences and ask ourselves how can an agent maximize its social power, intended as influence over the outcome of the negotiation. This question can be framed as a strategic game played over an opinion dynamics model, in which the action of an agent consists in stubbornly defending its own opinion. We show that for consensus-seeking opinion dynamics models in which the interaction weights are uniform, the optimal action obeys to an early mover advantage principle, i.e. the agents behaving stubbornly in the early phases of the negotiations achieve the highest social power. When looking at data collected from the climate change negotiations going on at the United Nations Framework Convention on Climate Change, we find evidence of the use of the early mover strategy. Furthermore, we show that the social powers computed through our model correlate very well with the perceived leadership roles assessed through independent survey data, especially when non-uniform weights incorporating economical and demographic factors are considered.

If a concatenated Friedkin-Johnsen model is used to describe the evolution of the opinions of stubborn agents in a sequence of discussion events, then the social power achieved by the agents at the end of the discussions depends from the stubbornness coefficients adopted by the agents through the sequence of events. In this paper we assume that the agents are free to choose their stubbornness profiles, and ask ourselves what strategy should an agent follow in order to maximize its social power. Formulating the problem as a strategic game, we show that choosing the highest possible values of stubbornness in the early discussions leads to the highest possible social power.

Bounded confidence opinion dynamics are dynamic networks in which agents are connected if their opinions are similar, and each agent updates her opinion as the average of the neighbors’ opinions. In homogeneous asymmetric Heglselmann–Krause (HK) models, all agents have the same confidence thresholds which could be different for the selection of upper and lower neighbors. This paper provides conditions for the convergence of the opinions to consensus and to clustering for this class of HK models. A new tighter bound on the time interval for reaching the steady state is also provided.

Laplacian dynamics on signed graphs have a richer behavior than those on nonnegative graphs. In particular, their stability is not guaranteed a priori. Consequently, also the time-varying case must be treated with care. In particular, instabilities can occur also when switching in a family of systems each of which corresponds to a stable signed Laplacian. In the paper we obtain sufficient conditions for such a family of signed Laplacians to form a consensus set, i.e., to be stable and converging to consensus for any possible switching pattern. The conditions are that all signed Laplacian matrices are eventually exponentially positive (a Perron-Frobenius type of property) and normal.

In this article we use high-throughput epigenomics, transcriptomics, and proteomics data to construct fine-graded models of the "protein-coding units"gathering all transcript isoforms and chromatin accessibility peaks associated with more than 4000 genes in humans. Each protein-coding unit has the structure of a directed acyclic graph (DAG) and can be represented as a Bayesian network. The factorization of the joint probability distribution induced by the DAGs imposes a number of conditional independence relationships among the variables forming a protein-coding unit, corresponding to the missing edges in the DAGs. We show that a large fraction of these conditional independencies are indeed verified by the data. Factors driving this verification appear to be the structural and functional annotation of the transcript isoforms, as well as a notion of structural balance (or frustration-free) of the corresponding sample correlation graph, which naturally leads to reduction of correlation (and hence to independence) upon conditioning.

When data are available for all nodes of a Gaussian graphical model, then, it is possible to use sample correlations and partial correlations to test to what extent the conditional independencies that encode the structure of the model are indeed verified by the data. In this paper, we give a heuristic rule useful in such a validation process: When the correlation subgraph involved in a conditional independence is balanced (i.e., all its cycles have an even number of negative edges), then a partial correlation is usually a contraction of the corresponding correlation, which often leads to conditional independence. In particular, the contraction rule can be made rigorous if we look at concentration subgraphs rather than correlation subgraphs. The rule is applied to real data for elementary gene regulatory motifs.

Profiling of mRNA expression is an important method to identify biomarkers but complicated by limited correlations between mRNA expression and protein abundance. We hypothesised that these correlations could be improved by mathematical models based on measuring splice variants and time delay in protein translation. We characterised time-series of primary human naive CD4(+) T cells during early T helper type 1 differentiation with RNA-sequencing and mass-spectrometry proteomics. We performed computational time-series analysis in this system and in two other key human and murine immune cell types. Linear mathematical mixed time delayed splice variant models were used to predict protein abundances, and the models were validated using out-of-sample predictions. Lastly, we re-analysed RNA-seq datasets to evaluate biomarker discovery in five T-cell associated diseases, further validating the findings for multiple sclerosis (MS) and asthma. The new models significantly out-performing models not including the usage of multiple splice variants and time delays, as shown in cross-validation tests. Our mathematical models provided more differentially expressed proteins between patients and controls in all five diseases. Moreover, analysis of these proteins in asthma and MS supported their relevance. One marker, sCD27, was validated in MS using two independent cohorts for evaluating response to treatment and disease prognosis. In summary, our splice variant and time delay models substantially improved the prediction of protein abundance from mRNA expression in three different immune cell types. The models provided valuable biomarker candidates, which were further validated in MS and asthma.

A biased assimilation model of opinion dynamics is a nonlinear model, in which opinions exchanged in a social network are multiplied by a state-dependent term having the bias as exponent and expressing the bias of the agents toward their own opinions. The aim of this article is to extend the bias assimilation model to signed social networks. We show that while for structurally balanced networks, polarization to an extreme value of the opinion domain (the unit hypercube) always occurs regardless of the value of the bias, for structurally unbalanced networks, a stable state of indecision (corresponding to the centroid of the opinion domain) also appears, at least for small values of the bias. When the bias grows and passes a critical threshold, which depends on the amount of "disorder" encoded in the signed graph, then a bifurcation occurs and opinions become again polarized.

In this article, 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 exhibit 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.

Motivation: The simultaneous availability of ATAC-seq and RNA-seq experiments allows to obtain a more in-depth knowledge on the regulatory mechanisms occurring in gene regulatory networks. In this article, we highlight and analyze two novel aspects that leverage on the possibility of pairing RNA-seq and ATAC-seq data. Namely we investigate the causality of the relationships between transcription factors, chromatin and target genes and the internal consistency between the two omics, here measured in terms of structural balance in the sample correlations along elementary length-3 cycles. Results: We propose a framework that uses the a priori knowledge on the data to infer elementary causal regulatory motifs (namely chains and forks) in the network. It is based on the notions of conditional independence and partial correlation, and can be applied to both longitudinal and non-longitudinal data. Our analysis highlights a strong connection between the causal regulatory motifs that are selected by the data and the structural balance of the underlying sample correlation graphs: strikingly, >97% of the selected regulatory motifs belong to a balanced subgraph. This result shows that internal consistency, as measured by structural balance, is close to a necessary condition for 3-node regulatory motifs to satisfy causality rules.

In parliamentary democracies, government negotiations talks following a general election can sometimes be a long and laborious process. In order to explain this phenomenon, in this paper we use structural balance theory to represent a multiparty parliament as a signed network, with edge signs representing alliances and rivalries among parties. We show that the notion of frustration, which quantifies the amount of "disorder" encoded in the signed graph, correlates very well with the duration of the government negotiation talks. For the 29 European countries considered in this study, the average correlation between frustration and government negotiation talks ranges between 0.42 and 0.69, depending on what information is included in the edges of the signed network. Dynamical models of collective decision-making over signed networks with varying frustration are proposed to explain this correlation.

The purpose of this paper is to propose a dynamical model describing the achievement of the 2015 Paris Agreement on climate change. To represent the complex, decade-long, multiparty negotiation process that led to the accord, we use a two time scale dynamical model. The short time scale corresponds to the discussion process occurring at each meeting and is represented as a Friedkin-Johnsen model, a dynamical multiparty model in which the parties show stubbornness, i.e., tend to defend their positions during the discussion. The long time scale behavior is determined by concatenating multiple Friedkin-Johnsen models (one for each meeting). The proposed model, tuned on real data extracted from the Paris Agreement meetings, achieves consensus on a time horizon similar to that of the real negotiations. Remarkably, the model is also able to identify a series of parties that exerted a key leadership role in the Paris Agreement negotiation process.

A concatenated Friedkin-Johnsen (FJ) model is a two time-scale opinion dynamics model in which stubborn agents discuss a sequence of issues. For each issue, a FJ model is adopted, and concatenation refers to the fact that the final opinion of the agents at issue s becomes the initial condition at issue s + 1. In this paper we deal with the case in which the system is open, i.e., the group of interacting agents changes at each issue, and so does their stubbornness. A concatenated FJ model can in this case be represented as an infinite product of stochastic matrices. For such system, we obtain sufficient conditions under which the opinions of the agents converge to consensus.

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. We show that the concept is also related to the idea of balanced realization.

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.

This work introduces antagonistic interactions into the so-called biased assimilation model of opinion dynamics, a nonlinear model expressing the bias of the agents towards their own opinions. In this model, opinions exchanged in a signed network are multiplied by a state dependent term having the bias as exponent. For small values of the bias parameters, while for structurally balanced networks polarization always occurs, we show that for structurally unbalanced networks also a state of indecision (corresponding to the centroid of the opinion hypercube) is a local attractor. When instead the biases are all large, the opinions usually polarize. In particular, for large enough biases if all agents take the same initial stance (i.e., are all on the same side of the indecision state), then the opinions polarize all to the same extreme value for both structurally balanced network and structurally unbalanced network.

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. 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. It is shown that dynamical privacy is not compatible with the existence of equilibria. Nevertheless the masked system retains the same convergence properties of the original system: the equilibria of the original systems become attractors for the masked system but lose the stability property. Application of dynamical privacy to popular examples of multiagent dynamics, such as models of social opinions, average consensus and synchronization, is investigated in detail. (c) 2020 Elsevier Ltd. All rights reserved.

The constrained consensus problem considered in this paper, denoted interval consensus, is characterized by the fact that each agent can impose a lower and upper bound on the achievable consensus value. Such constraints can be encoded in the consensus dynamics by saturating the values that an agent transmits to its neighboring nodes. We show in the paper that when the intersection of the intervals imposed by the agents is nonempty, the resulting constrained consensus problem must converge to a common value inside that intersection. In our algorithm, convergence happens in a fully distributed manner, and without need of sharing any information on the individual constraining intervals. When the intersection of the intervals is an empty set, the intrinsic nonlinearity of the network dynamics raises new challenges in understanding the node state evolution. Using Brouwer fixed-point theorem we prove that in that case there exists at least one equilibrium, and in fact the possible equilibria are locally stable if the constraints are satisfied or dissatisfied at the same time among all nodes. For graphs with sufficient sparsity it is further proven that there is a unique equilibrium that is globally attractive if the constraint intervals are pairwise disjoint.

This paper investigates the impact of addition/removal of edges in complex networks. Growing a network by the addition of edges has for instance been suggested as a way to improve network robustness to external disturbances. Moreover, when network controllability is considered, designing edge additions is a promising alternative to add more actuation capabilities in order to improve different performance metrics. We quantify the impact of an edge modification with the H-infinity and H-2 norms. For networks with positive edge weights we show how the H-infinity norm can be computed exactly for each possible single edge modification, while for the H-2 norm we instead obtain a lower bound. This bound is linked to the trace of the controllability Gramian, hence it can be used for instance to reduce the energy needed for control. Copyright (C) 2020 The Authors.

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 its only attractor. However, in order to preserve privacy, it cannot share an equilibrium point with the unmasked system, meaning that in the masked system the attractor cannot be also stable.

This paper investigates the problem of controlling a complex network with reduced control energy. Two centrality measures are defined, one related to the energy that a control, placed on a node, can exert on the entire network, and the other related to the energy that the network exerts on a node. We show that by combining these two centrality measures 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, can be simultaneously taken into account. From an algebraic point of view, the node ranking that we obtain from the combination of our centrality measures is related to the non-normality of the adjacency matrix of the graph.

This paper investigates the problem of controlling a complex network with reduced control energy. Two centrality measures are defined, one related to the energy that a control, placed on a node, can exert on the entire network, and the other related to the energy that the network exerts on a node. We show that by combining these two centrality measures 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, can be simultaneously taken into account. From an algebraic point of view, the node ranking that we obtain from the combination of our centrality measures is related to the non-normality of the adjacency matrix of the graph.

A signed network is a network in which each link is associated with a positive or negative sign. Models for nodes interacting over such signed networks 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 for studying node state evolution over signed networks can be obtained utilizing generalized Perron-Frobenius theory, graph theory, and elementary algebraic recursions.

Signed Laplacian matrices generally fail to be diagonally dominant and may fail to be stable. For both undirected and directed graphs, in this paper we present conditions guaranteeing the stability of signed Laplacians based on the property of eventual positivity, a Perron-Frobenius type of property for signed matrices. Our conditions are necessary and sufficient for undirected graphs, but only sufficient for digraphs, the gap between necessity and sufficiency being filled by matrices who have this Perron-Frobenius property on the right but not on the left side (i.e., on the transpose). An exception is given by weight balanced signed digraphs, where eventual positivity corresponds to positive semidefinitness of the symmetric part of the Laplacian. Analogous conditions are obtained for signed stochastic matrices.

Complex eukaryotic promoters normally contain multiple cis-regulatory sequences for different transcription factors (TFs). The binding patterns of the TFs to these sites, as well as the way the TFs interact with each other and with the RNA polymerase (RNAp), lead to combinatorial problems rarely understood in detail, especially under varying epigenetic conditions. The aim of this paper is to build a model describing how the main regulatory cluster of the olfactory receptor Or59b drives transcription of this gene in Drosophila. The cluster-driven expression of this gene is represented as the equilibrium probability of RNAp being bound to the promoter region, using a statistical thermodynamic approach. The RNAp equilibrium probability is computed in terms of the occupancy probabilities of the single TFs of the cluster to the corresponding binding sites, and of the interaction rules among TFs and RNAp, using experimental data of Or59b expression to tune the model parameters. The model reproduces correctly the changes in RNAp binding probability induced by various mutation of specific sites and epigenetic modifications. Some of its predictions have also been validated in novel experiments.

In this work we consider a nonlinear interconnected system describing a decision-making process in a community of agents characterized by the coexistence of collaborative and antagonistic interactions. The resulting signed graph is in general not structurally balanced. It is shown in the paper that the decision-making process is affected by the frustration of the signed graph, in the sense that a nontrivial decision can be reached only if the social commitment of the agents is high enough to win the disorder introduced by the frustration in the network. The higher the frustration of the graph, the higher the commitment strength required from the agents.

The aim of this paper is to shed light on the problem of controlling a complex network with minimal control energy. We show first that the control energy depends on the time constant of the modes of the network, and that the closer the eigenvalues are to the imaginary axis of the complex plane, the less energy is required for complete controllability. In the limit case of networks having all purely imaginary eigenvalues (e.g. networks of coupled harmonic oscillators), several constructive algorithms for minimum control energy driver node selection are developed. A general heuristic principle valid for any directed network is also proposed: the overall cost of controlling a network is reduced when the controls are concentrated on the nodes with highest ratio of weighted outdegree vs indegree.

Wireless power transfer technology provides a possible sustainable and cost-effective way to prolong indefinitely the lifetime of networks of smart devices needed in future Internet-of-Things, while equipping them with batteries of limited capacity. In this paper we show that the theory of compartmental systems, positive interconnected systems exchanging mass (here power) and ruled by mass conservation laws, provides a suitable framework to describe wireless power transfer networks. In particular we show that sustainability of the network of smart devices corresponds to each of them being alimented, directly or indirectly, by nodes having an external source of power, condition known as inflow connectivity in the compartmental systems literature. The framework allows to compute the topology which is optimal in terms of maximizing the overall efficiency of the power transfer.

The models of collective decision-making considered in this paper are nonlinear interconnected cooperative systems with saturating interactions. These systems encode the possible outcomes of a decision process into different steady states of the dynamics. In particular, they are characterized by two main attractors in the positive and negative orthant, representing two choices of agreement among the agents, associated to the Perron-Frobenius eigenvector of the system. In this paper we give conditions for the appearance of other equilibria of mixed sign. The conditions are inspired by Perron-Frobenius theory and are related to the algebraic connectivity of the network. We also show how all these equilibria must be contained in a solid disk of radius given by the norm of the equilibrium point which is located in the positive orthant.

The aim of this paper is to modify continuous-time bounded confidence opinion dynamics models so that "changes of opinion" (intended as changes of the sign of the initial states) are never induced during the evolution. Such sign invariance can be achieved by letting opinions of different sign localized near the origin interact negatively, or neglect each other, or even repel each other. In all cases, it is possible to obtain sign-preserving bounded confidence models with state-dependent connectivity and with a clustering behavior similar to that of a standard bounded confidence model. (C) 2018 Elsevier Ltd. All rights reserved.

Nonlinear cooperative systems associated to vector fields that are concave or subhomogeneous describe well interconnected dynamics that are of key interest for communication, biological, economical, and neural network applications. For this class of positive systems, we provide conditions that guarantee existence, uniqueness and stability of strictly positive equilibria. These conditions can be formulated directly in terms of the spectral radius of the Jacobian of the system. If control inputs are available, then it is shown how to use state feedback to stabilize an equilibrium point in the interior of the positive orthant.

\This paper deals with the problem of controlling linear complex networks in an efficient way, i.e., with limited control energy. A general principle is provided, based on the eigenvalues of the network. It is shown numerically that the cost of controlling a network grows with the (absolute value of the) real part of the eigenvalues of the adjacency matrix. Constructive rules for driver node selection are also provided, based on the (weighted) topology of the network. In particular, we show that the key to have an energetically efficient driver node placement strategy is to use the skewness of the outdegree versus indegree distributions of the network, a topological property not associated before to controllability. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

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 types of interactions are defined along the positive and negative links, respectively, arise from various biological, social, political, and economical systems. Starting from standard consensus dynamics, there are two basic types of negative interactions along negative links, namely state flipping or relative-state flipping. In this paper, we provide an algebraic-graphical method serving as a systematic tool of studying these dynamics over signed networks. Utilizing generalized Perron-Frobenius theory, graph theory, and elementary algebraic recursions, we show this method is useful to establish a series of basic convergence results for dynamics over signed networks.

In this paper, we study the problem of controlling complex networks with unilateral controls, i.e., controls which can assume only positive or negative values, not both. Given a complex network represented by the adjacency matrix A, an algorithm is developed that constructs an input matrix B such that the resulting system (A, B) is controllable with a near minimal number of unilateral control inputs. This is made possible by a reformulation of classical conditions for controllability that casts the minimal unilateral input selection problem into well known optimization problems. We identify network properties that make unilateral controllability relatively easy to achieve as compared to unrestricted controllability. The analysis of the network topology for instance allows us to establish theoretical bounds on the minimal number of controls required. For various categories of random networks as well as for a number of real-world networks these lower bounds are often achieved by our heuristics.

In stable biological and ecological networks, the steady-state influence matrix gathers the signs of steady-state responses to step-like perturbations affecting the variables. Such signs are difficult to predict a priori, because they result from a combination of direct effects (deducible from the Jacobian of the network dynamics) and indirect effects. For stable monotone or cooperative networks, the sign pattern of the influence matrix can be qualitatively determined based exclusively on the sign pattern of the system Jacobian. For other classes of networks, we show that a semi-qualitative approach yields sufficient conditions for Jacobians with a given sign pattern to admit a fully positive influence matrix, and we also provide quantitative conditions for Jacobians that are translated eventually nonnegative matrices. We present a computational test to check whether the influence matrix has a constant sign pattern in spite of parameter variations, and we apply this algorithm to quasi-Metzler Jacobian matrices, to assess whether positivity of the influence matrix is preserved in spite of deviations from cooperativity. When the influence matrix is fully positive, we give a simple vertex algorithm to test robust stability. The devised criteria are applied to analyse the steady-state behaviour of ecological and biomolecular networks.

In the consensus problem considered in this paper, each agent can impose a lower and an upper bound on the achievable consensus values. We show that if such state constraints are implemented by saturating the value transmitted to the neighboring nodes, the resulting constrained consensus problem must converge to the intersection of the intervals imposed by the individual agents.

In this paper we provide necessary conditions for the existence of multiple equilibrium points for a class of non-linear cooperative networked systems with saturating interactions which describe models of collective decision-making. The multiple steady states of the dynamics represent the possible outcomes of a decision process, and, except for one positive and one negative, have all mixed signs. The conditions we obtain can be formulated in terms of the algebraic connectivity of the network and are inspired by Perron-Frobenius arguments. It is also shown that the mixed-sign equilibria are contained in a ball of radius given by the norm of the positive equilibrium point and centered in the origin. Numerical examples are given to illustrate the results.

In a continuous-time nonlinear driftless control system, an involutive flow is a composition of input profiles that does not excite any Lie bracket. Such flow composition is trivial, as it corresponds to a "forth and back" cyclic motion obtained rewinding the system along the same path. The aim of this paper is to show that, on the contrary, when a (nonexact) discretization of the nonlinear driftless control system is steered along the same trivial input path, it produces a net motion, which is related to the gap between the discretization used and the exact discretization given by a Taylor expansion. These violations of involutivity can be used to provide an estimate of the local truncation error of numerical integration schemes. In the special case in which the state of the driftless control system admits a splitting into shape and phase variables, our result corresponds to saying that the geometric phases of the discretization need not obey an area rule, i.e., even zero-area cycles in shape space can lead to nontrivial geometric phases. (C) 2017 Elsevier B.V. All rights reserved.