The satellite will be held in presence. link

The Dynamics On and Of Complex Networks (DOOCN) workshop series,
aims at exploring statistical dynamics on and off complex networks.
*Dynamics on networks* refers to the different types of
processes that take place on networks, like spreading, diffusion, and synchronization.
Modeling such processes is strongly affected by the topology
and temporal variation of the network structure,
i.e., by the *dynamics of networks*.

The 14^{th} edition of the DOOCN workshop,
“DOOCN-XV: Dynamical Multiscale Engineering of Network Architecture; will be held on **July 22-23, 2024** in conjunction with the upcoming
Compeng2024 -2024 IEEE Workshop on Complexity in Engineering
which will take place during 22– 24 July 2024, in Florence, Italy.

Alessandro Torcini

CY Cergy Paris Université,CNRS, Cergy-Pontoise, France

Pietro De Lellis

University of Naples Federico II, Naples 80125, Italy

Alessandro Scaglione

Department of Physics and Astronomy, University of Florence, Italy, Italy

Eleonora Russo

The BioRobotics Institute, Sant’Anna School of Advanced Studies, Italy

Angelo di Garbo

CNR Institute of Biophysics, Pisa, Italy

Paolo Bonifazi

Computational Neuroimaging Lab, Biocruces-Bizkaia Health Research Institute, Barakaldo, Spain

Emanuele Crisostomi

University of Pisa, Italy

Marco Storace

University of Genoa, Italy

Diego Febbe

Department of Physics and Astronomy, University of Florence, Sesto Fiorentino, Italy

Pietro De Lellis

University of Naples Federico II, Naples, Italy.

**All times are in Central European Time (CET)****July 22**- 14:30–14:35: Opening
- 14:35–15:00:
Invited Talk by
**Alessandro Torcini**#### Title: Is the cortical dynamics ergodic?

Abstract: Cortical neurons in vivo display significant temporal variability in their spike trains even in response to the same stimulus. This variability is explained only in part by the intrinsic stochasticity of the spike-generation mechanism. In fact, to account for the levels of variability observed in the experiment, one needs to assume additional fluctuations in the level of activity over longer time scales [1]. But what is the origin of these fluctuations in the ”firing rates” ? Some theories explain them as a result of precise adjustments of the synaptic connectivity [2]. According to an alternative scenario slow fluctuations in the ”rates” are instead a signature of non-ergodicity [3], due to the partially-symmetric synaptic connectivity, consistently with anatomical observations [4]. It is unclear, however, whether such ergodicity breaking occurs also in spiking neural networks, due to the presence of fast temporal fluctuations in the synaptic inputs generated by the spiking variability [5]. To address this question we study the dynamics of sparsely-connected networks of inhibitory quadratic-integrate-and-fire (QIF) neurons [6], with arbitrary levels of symmetry, q, in the synaptic connectivity. The connectivity (i.e., adjacency) matrix is random for q = 0 and fully symmetric for q =1. The neurons also receive a spatially-homogeneous, time- constant excitatory drive, which is dynamically balanced by the recurrent synaptic inputs [6]. This results in low and heterogeneous average levels of activity across neurons, and temporally irregular spike trains, mimicking prominent features of the activity observed in the cortex [7]. To investigate the ergodicity of the network dynamics, we estimate over time intervals, T , of increasing duration, the single-neuron ”firing rates”, starting from different initial distribution of the membrane voltages (for the same network). If the dynamics is ergodic, the difference D between the estimates obtained from different initial settings should go to zero for suitably long time windows (i.e., as 1/T for sufficiently large T). This is, in fact, what happens in random networks (i.e., q = 0). In partially symmetric networks q > 0, the onset of the ”ergodic” regime occurs at longer and longer times. The situation becomes dramatic for the fully symmetric network (q=1), where D does not decay even for time windows that are 5 order of magnitudes longer than the membrane time constant as shown in Fig 1 (c) (red solid line); the network dynamics is non-ergodic, at least in a weak sense. In this regime, the network activity is sparse, with a large fraction of almost-silent neurons, and the auto-covariance function of the spike trains exhibits long time scales, as routinely observed in experimental recordings [8]. Our results support the idea that many features of cortical activity can be explained by the non-ergodicity of the network dynamics [3]. In particular, in this regime, the activity level of the single neurons can significantly change depending on the ”microscopic” initial conditions, providing a simple explanation for the large trial-to-trial fluctuations observed in the experiment.

[1] A. K. Churchland, et al., Neuron, 69(4):818– 831, 2011; R. L.T. Goris, J. A. Movshon, E. P. Simoncelli, Nature neuroscience, 17(6):858–865, 2014. [2] C. Huang, B. Doiron, Current opinion in neurobiology, 46:31–38, 2017. [3] K. Berlemont, G. Mongillo, bioRxiv, pages 2022–03, 2022. [4] L. Campagnola, et al., Science, 375(6585) 5861, 2022. [5] S. Rao, D. Hansel, C. van Vreeswijk, Sci. Rep., 9(1):3334, 2019. [6] M. Di Volo, A. Torcini, Phys. Rev. Lett., 121(12):128301, 2018. [7] A. Roxin, N. Brunel, D. Hansel, G. Mongillo, C. van Vreeswijk, Journal of Neuroscience, 31(45):16217–16226, 2011. [8] S. Shoham, D. H O’Connor, R. Segev, Journal of Comparative Physiology A, 192:777–784, 2006; J.D. Murray et al., Nature neuroscience, 17(12):1661– 1663, 2014 - 15:00–15:25:
Invited Talk by
**Pietro De Lellis**#### Title: Early warning signals for psychopathology

Abstract: Many real-world phenomena share a common feature: the potential for sudden and difficult to reverse transitions into less desirable states. This applies to a wide range of domains, from ecosystems to social structures [1]. The theory of dynamical systems has identified early warning signals (EWSs) that can precede these so called “critical transitions,” enabling intervention before the system becomes locked into an undesired state. Recently, mental health researchers have explored the utility of EWSs for predicting transitions into unhealthy states (i.e. psychiatric disorders), which can be perceived as alternative stable states opposed to the “healthy” ones. Notably, critical slowing down-based early warning signals have exhibited potential as a predictive tool for identifying transitions into depression or other mental diseases. In this work, we define \(x_i = [x_{i1} , . . . , x_{in} ]^T\) as the mental state of the individual i, with \(x_{ij}∈R\) being the variable associated to the j-th symptom, emotion or transdiagnostic factor pertaining to the i-th individual. The mental health dynamics for individual i are given by the following dynamical equation: $$\dot{x}_{ij}=f_j(x_{ij} , x_{ij1} , . . . , x_{ijm} , u_i , w_i ), j = 1, . . . , n; y_i = h(x_{i1}, . . . , x_{in}, ν_{ij} ), (1)$$ where \(j_1 , . . . , j_m\) are the neighbors of node j, i.e., the symptoms affecting the dynamics of symptom j; \(f_j\) describes how an individual’s mental state is influenced by their own symptom history and the contextual factors in their environment, represented by the state of neighboring symptoms, and by the “outside world” through the variable \(u_i∈R^m\). Note that \(f_j\) also depends on the random variable \(w_i∈R^q\), which represents the uncertainty inherent in the model. This uncertainty arises from various sources, such as the complexity of the underlying dynamics, and the inability to fully capture all the relevant factors affecting the symptoms within the mathematical formalization. The variable \(y_i∈R^p\) corresponds to what we are actually able to measure of the overall mental state of an individual, whereby we typically have p < n ; function h is a nonlinear function of the symptoms, and of a random variable \(ν_{ij}∈R\), which represents the uncertainty that may arise from recall biases, subjective interpretation, distortions or other influences that may affect the accuracy of the patient’s narration. In practical applications, we are not aware of the functional form of fj and h, and only have at our disposal a multivariate time-series \(y_i(1), . . . , y_i(T)\) of the noisy measured mental state of an individual. The objective is then that of an early identification of the onset of critical transitions in an individual mental state, so to make an early detection, for instance, of a crisis that is about to take place and change medications in a timely manner. In this work, we compare two viable approaches for early warning, the kernel change point on running statistics, kcpRS [3], and the time-varying change point autoregressive model of order 1, TVCP-AR(1) [4] on a case study of a 57 years-old man with a history of major depression [2].

[1] M. Scheffer, Critical Transitions in Nature and Society. Princeton: Princeton University Press, 2009. [2] M. Wichers, P. C. Groot, and E. G. Psychosystems, ESM Group, “Critical slowing down as a personalized early warning signal for depression,” Psy- chotherapy and Psychosomatics, vol. 85, no. 2, pp. 114–116, 2016. [3] J. Cabrieto, K. Meers, E. Schat et al., “An R package for performing kernel change point detection on the running statistics of multivariate time series,” Behavior Research Methods, vol. 54, no. 3, pp. 1092–1113, 2022. [4] C. J. Albers and L. F. Bringmann, “Inspecting gradual and abrupt changes in emotion dynamics with the time-varying change point autoregressive model,” European Journal of Psychological Assessment, vol. 36, no. 3, pp. 492–499, 2020 - 15:25–15:50:
Invited Talk by
**Alessandro Scaglione**#### Title: Group ICA of wide-field calcium imaging data reveals the retrosplenial cortex as a major contributor to cortical activity during anesthesia

Abstract: Large-scale cortical dynamics play a crucial role in many cognitive functions such as goal- directed behaviors, motor learning and sensory processing. It is well established that brain states including wakefulness, sleep, and anesthesia modulate neuronal firing and synchronization both within and across different brain regions. However, how the brain state affects cortical activity at the mesoscale level is less understood. This work aimed to identify the cortical regions engaged in different brain states. To this end, we employed group ICA (Independent Component Analysis) to wide-field imaging recordings of cortical activity in mice during different anesthesia levels and the awake state. Thanks to this approach we identified independent components (ICs) representing elements of the cortical networks that are common across subjects under decreasing levels of anesthesia toward the awake state. We found that ICs related to the retrosplenial cortices exhibited a pronounced dependence on brain state, being most prevalent in deeper anesthesia levels and diminishing during the transition to the awake state. Analyzing the occurrence of the ICs we found that activity in deeper anesthesia states was characterized by a strong correlation between the retrosplenial components and this correlation decreases when transitioning toward wakefulness. Overall, these results indicate that during deeper anesthesia states coactivation of the posterior-medial cortices is predominant over other connectivity patterns, whereas a richer repertoire of dynamics is expressed in lighter anesthesia levels and the awake state.

- 15:50–16:20: Coffee Break
- 16:20–16:45:
Invited Talk by
**Eleonora Russo**#### Title: Integration of rate and phase codes by hippocampal cell-assemblies supports flexible encoding of spatiotemporal context

Abstract: Spatial information is encoded by location-dependent hippocampal place cell firing rates and sub-second, rhythmic entrainment of spike times. These ‘rate’ and ‘temporal’ codes have primarily been characterized in low-dimensional environments under limited cognitive demands; but how is coding configured in complex environments when individual place cells signal several locations, individual locations contribute to multiple routes and functional demands vary? Quantifying rat CA1 population dynamics during a decision-making task, we show that the phase of individual place cells’ spikes relative to the local theta rhythm shifts to differentiate activity in different place fields. Theta phase coding also disambiguates repeated visits to the same location during different routes, particularly preceding spatial decisions. Using unsupervised detection of cell assemblies alongside theoretical simulation, we show that integrating rate and phase coding mechanisms dynamically recruits units to different assemblies, generating spiking sequences that disambiguate episodes of experience and multiplexing spatial information with cognitive context.

- 16:45–17:10:
Invited Talk by
**Paolo Bonifazi**#### Title: Linking hubness, embryonic neurogenesis, transcriptomics and diseases in human brain networks

Abstract: Understanding the architectural principles that shape human brain networks is a major challenge for systems neuroscience. We hypothesize that the centrality of the different brain circuits in the human connectome is a product of their embryogenic age, such that early-born nodes should become stronger hubs than those born later. Using a human brain segmentation based on embryogenic age, we observed that nodes’ structural centrality correlated with their embryogenic age, fully confirming our hypothesis. Distinct trends were found at different resolutions on a functional level. The difference in embryonic age between nodes inversely correlated with the probability of existence of links and their weights. Brain transcriptomic analysis revealed strong associations between embryonic age, structure-function centrality, and the expression of genes related to nervous system development, synapse regulation and human neurological diseases. Our results highlight two key principles regarding the wiring of the human brain, “preferential age attachment” and “the older gets richer”. In this talk, I will especially focus and link some evidences from this model to the genetics, transcriptomics and pharmacology relating to epilepsy and channelopathies. In particular, we will focus on the GABAergic circuits' lesion induced by epilepsy and the impact of the inhibition of the neuroinflammatory response via the JAK/STAT pathways.

- 17:10–17:35:
Invited Talk by
**Angelo Di Garbo**#### Title: Nonlinear Time-series Analysis of MEA Recordings in cultured neuronal networks

Abstract: The brain is a complex system, characterized by many interacting regions with distinct identities and functionalities. Understanding differences, in electrophysiological activity and connectivity among these regions, is crucial for gaining deeper insights into brain dynamics. In this work we studied neural cultures derived from mouse embryonic stem cells, focusing on their differentiation into three different brain regions: hippocampus, isocortex, and entorhinal cortex. Utilizing multi-electrode arrays (MEA) to record electrophysiological activity, we observed distinct behaviors among these networks. Specifically, entorhinal and isocortical networks exhibited critical behavior, leading to a synchronous activity, characterized by precise onsets within the network. Conversely, hippocampal networks did not exhibit such behavior. We analyzed these features by considering the approach of self-organized criticality theory, characterizing the formation of neuronal avalanches. The analysis of avalanche size and duration distributions revealed differences between cultures. In particular, we observed two different behaviors, i.e. critical, characterized by power-law distributions, and subcritical, characterized by power-law with exponential cut-off distributions. These behaviors depend on the activity synchronization level. Indeed, connectivity analysis highlights the disparity between critical networks displaying high-level synchronization, and sub-critical ones displaying lower-level synchronization.

**July 23**- 14:30–14:55:
Invited Talk by
**Emanuele Crisostomi**#### Title: Synchronization of interconnected microgrids

Abstract: Energy communities have been recently emerging as a concrete tool to accomplish the so-longed energy transition, in an attempt to achieve social energy awareness, generation from renewable sources and new energy markets in a unique framework [1]. This is not the first time that the idea of splitting current power grids into smaller building blocks has been discussed in the recent years, as energy communities may be seen as a special case of microgrids - or virtual power plants - which have already been largely investigated in the literature. A rather neglected aspect is however what will be the task of the transmission grid in a future power network where most of the energy exchanges (and thus, most of the power flows) will occur within microgrids, and less frequently there will be energy exchanges among different communities. Indeed, it is expected that different microgrids will be owned by different owners, and in addition to satisfying the internal energy requirements, they will be interested in maximizing their revenues by selling the surplus energy to other communities. This raises the problem of who should control the power flows, and who should regulate energy exchanges in order to maintain a desired level of reliability of the overall power grid. In this talk we shall show the impact of energy exchanges between different microgrids, and we shall explain a simple synchronization algorithm that can be utilized to orchestrate the functioning of the microgrids while maintaining a desired quality of service of the power network (e.g., in terms of grid frequency). The talk is largely based on references [2]–[4].

AIMD SYNCHRONIZATION The AIMD (Additive-Increase, Multiplicative-Decrease) algorithm has been widely utilized in the Internet congestion control problem to optimally and fairly share bandwidth among connected users [5]. Similarly to the original problem, here we assume that microgrids (MGs) gently increase in an additive fashion their probability to operate in a desired “market mode” where they can sell power to other MGs. This scenario is known to increase frequency oscillations that may be inconvenient for the power grid. Accordingly, when the frequency starts assuming values that may be dangerous for the safe operation of the power grid (denoted as “capacity event”), a multiplicative step occurs and the probability of operating in “market mode” drops to a lower value. Conversely, more frequently the MGs operate in “ancillary mode” where they start providing frequency regulation services. In this fashion, the overall mechanism allows the MGs to maintain their operational freedom as much as possible (e.g., to increase their revenues by operating in the energy market), while mitigating the effect on the power grid. The advantages of this solution are that it can be implemented in a fully decentralized fashion (each MG can measure the local frequency at the point of connection of the MG with the grid), and that it can be also implemented in an unsyncrhonized way to prioritize some MGs, if desired. In the last case, some MGs may have a larger or smaller probability to react to the capacity event, so that some MGs may be selected with a larger probability to provide the required frequency regulation services. CONCLUSIONS Appropriate synchronization strategies are required to orchestrate a large number of interconnected microgrids in a transmission grid. If microgrids operate in an uncoordinated fashion, frequency stability issues may occur in the power grid. Decentralized and flexible synchronization algorithms are attractive to guarantee autonomous managements and operations of microgrids. [1] I. Mariuzzo, B. Fina, S. Stroemer, M. Raugi, “Economic assessment of multiple energy community participation,” Applied Energy, vol. 353, pp. 1–13, 2024. [2] P. Ferraro, E. Crisostomi, M. Raugi, and F. Milano, “Analysis of the Impact of Microgrid Penetration on Power System Dynamics,” IEEE Transactions on Power Systems, vol. 32, no. 5, pp. 4101–4109, 2017. [3] P. Ferraro, E. Crisostomi, R. Shorten, and F. Milano, “Stochastic Frequency Control of Grid-Connected Microgrids,” IEEE Transactions on Power Systems, vol. 33, no. 5, pp. 5704–5713, 2018. [4] E. Dudkina, E. Crisostomi, P. Ferraro, and F. Milano, “Dynamic and Control of Grid-connected Microgrids,” Chapter in “Control and Optimisation of Microgrids”, Editors A. Parisio, J. Schiffer, and C. Hans, IET, 2024 (in Press). [5] D. Chiu, and R. Jain, “Analysis of the Increase and Decrease Algorithms for Congestion Avoidance in Computer Networks,” North Holland Computer Networks and ISDN Systems, vol. 17, pp. 1–14, 1989. - 14:55–15:20:
Invited Talk by
**Marco Storace**#### Title: A comparative analysis of different virtual inertia controllers in power grids with renewable energy sources: effect on the stability

Abstract: Maintaining a synchronous state of generators is of central importance to standard operation in a power grid; indeed, a loss of synchrony among power generators may cause the outages of power grids with cascading catastrophic failures, as in the Western American network in 1996 [1]. The so-called inertia of the grid [2] is a measure of the power system’s ability to counteract the frequency changes, i.e., to remain synchronized and stable when triggered by external disturbances. Renewable energy sources (RESs) are interfaced with the grid through power converters without the intrinsic inertia of synchronous machines, and their presence in power grids has recently increased [3], [4]. A strong presence of RESs reduces the global inertia of the system, thus compromising its stability and reliability [5], [6]. For this reason, it is fundamental to develop power converter controllers that provide synthetic inertia by mimicking the electro-mechanical characteristic of a synchronous machine. In this talk, we investigate the performances of three different power converter control techniques able to generate virtual inertia, namely grid-following with swing equation emulation [7], grid-forming with swing equation emulation [8], and synchronverter [9]. The reference framework is a 3-bus system that comprises (i) a synchronous generator, serving as a grid equivalent, with a rated power of 10 MW and a fixed inertia of 3s, (ii) a RES with a rated power of 1MW and variable inertia and (iii) a time-varying load. The analysis focuses on the grid reaction to a rapid load increase; this sudden change allows analyzing the grid stability and its capability to adapt to variable demand scenarios. In particular, we investigate how the virtual inertia values provided by the three compared techniques and the physical distance d between the grid and the RES affect the grid frequency and stability. The obtained results show that, by increasing d, the network can become unstable when the grid-following control method is used.

[1] V. Venkatasubramanian and Y. Li, “Analysis of 1996 western american electric blackouts,” Bulk Power System Dynamics and Control-VI, Cortina d’Ampezzo, Italy, pp. 22–27, 2004. [2] P. Kundur, Power System Stability and Control. McGraw- Hill, New York, 1994. [3] M. Rezkalla, M. Pertl, and M. Marinelli, “Electric power system inertia: requirements, challenges and solutions,” Electrical Engineering, vol. 100, pp. 2677–2693, 2018. [4] D. Linaro, F. Bizzarri, D. Del Giudice, C. Pisani, G. M. Giannuzzi, S. Grillo, and A. M. Brambilla, “Continuous estimation of power system inertia using convolutional neural networks,” Nature Communications, vol. 14, no. 1, p. 4440, 2023. [5] A. Fernández-Guillamón, E. Gómez-Lázaro, E. Muljadi, and Á. Molina-Garcia, “A review of virtual inertia techniques for renewable energy-based generators,” Renewable Energy–Technologies and Applications, 2021. [6] B. Kroposki and A. Hoke, “A path to 100 percent renewable energy: Grid-forming inverters will give us the grid we need now,” IEEE Spectrum, vol. 61, no. 5, pp. 50–57, 2024. [7] B. K. Poolla, D. Groß, and F. Dörfler, “Placement and implementation of grid-forming and grid-following virtual inertia and fast frequency response,” IEEE Transactions on Power Systems, vol. 34, no. 4, pp. 3035–3046, 2019. [8] S. D’Arco and J. A. Suul, “Equivalence of virtual synchronous machines and frequency-droops for converter-based microgrids,” IEEE Transactions on Smart Grid, vol. 5, no. 1, pp. 394–395, 2013. [9] Q.-C. Zhong and G. Weiss, “Synchronverters: Inverters that mimic synchronous generators,” IEEE Transactions on Industrial Electronics, vol. 58, no. 4, pp. 1259–1267, 2010. - 15:20–15:50: Coffee Break
- 15:50–16:15:
Invited Talk by
**Diego Febbe**#### Title: Chaos and Synchronization in the UJT relaxation oscillator

Abstract: Nonlinear electronic oscillators play an important role in a variety of electrical applications, ranging from individual components to the modelling of networks and the development of power grids. Additionally, the study of nonlinear oscillators can find many applications in other fields such as geology, chemistry, and biology. A preeminent example is the use of nonlinear relaxation oscillators for the modelling of neural spiking behaviour. Hence, understanding the collective interaction and synchronization phenomena among these oscillators is essential, as evidenced by important recent studies in this area [1-4]. In this work, we present the dynamical behaviour of a two-dimensional nonlinear relaxation oscillator described by a system of coupled nonlinear differential equations. In electronic circuits the presence of nonlinear active elements can enable the emergence of self- sustained oscillations and in this context, the UJT (Uni Junction Transistor) can be employed to construct relaxation oscillators where the junction acts as an electrically controlled switch that allows the current to flow only after a triggering potential point. In this circuit the UJT is connected to a capacitor and a load resistor, where the charge and the current are measured. The resulting waveforms present two distinct time scales corresponding to the charge and discharge of the capacitor and the impulsive current flowing through the resistor creating a two-stroke oscillator. In the work [5] a new mathematical model capable of satisfactory describe its behaviour was presented. This continuous model reproduces the dynamics of the free oscillator, and when subject to perturbation, it is capable of reproducing the observed complex dynamical behaviour, such as the transition to chaos through the torus breakdown that was not properly described in previous models of this system [6-7]. Finally, we explored synchronization phenomena by diffusively coupling a network of oscillators and by employing Turing pattern formation techniques we assessed the stability of the synchronized state in relation to its topological properties.

[1] Pikovsky A, Rosenblum M, Kurths J. Synchronization: a universal concept in nonlinear science. New York: Cambridge University Press; 2002. [2] Tattini L., Olmi S., and Alessandro Torcini. “Coherent periodic activity in excitatory Erdos-Renyi neural networks: the role of network connectivity”. In: Chaos: An Interdisciplinary Journal of Nonlinear Science 22.2 (2012). [3] Totz, C. H., Olmi, S., & Schöll, E. (2020). Control of synchronization in two-layer power grids. Physical Review E, 102(2), 022311. [4] Lucas M., Fanelli D., Carletti T., and Petit J., Desynchronization induced by time-varying network, Euro-physics Letters 121, 50008 (2018). [5] Febbe, D., Mannella, R., Meucci, R., & Di Garbo, A. (2024). Dynamical behaviour of a new model for the UJT relaxation oscillator. Chaos, Solitons & Fractals, 183, 114906. [6] Di Garbo A., Euzzor S., Ginoux J.-M., Arecchi F. T., and Meucci R., Delayed dynamics in an electronic relaxation oscillator, Physical Review E 100, 032224 (2019). [7] Ducci D, Meucci R, Euzzor S, Ginoux J.-M., Di Garbo A. Dynamical behaviour of two coupled two-stroke relaxation oscillators. Fluctuation Noise Lett 2024; 23(1):2440008–20. - 16:15–16:40:
Invited Talk by
**Pietro De Lellis**#### Title: Modeling and control of opinion dynamics in the presence of higher-order interactions

Abstract: We consider a group of N individuals that need to form an opinion on a two-option choice influenced by an opinion leader, the individual N + 1. The i-th individual has opinion \(x_i\) , where \(x_i = 0\) means that individual i has a neutral opinion; \(x_i>0\) and \(x_i < 0\) correspond, instead, to an individual leaning toward option 1 or 2, respectively, and higher values of \(|x_i|\) mean that the opinion of the i-th agent is more extreme. The individuals are coupled through the hypergraph H=(V,E), whereby social interactions are not only pairwise, but they can involve more than two agents at the same time. Namely, each hyperedge ε ∈ E is a pair of ordered, disjoint subsets of V. The first subset, T(ε), contains the tails, and the second, H(ε), the heads of ε, with the tail nodes trying to influence the opinion of the head nodes in the group interaction taking place on hyperedge ε. The opinion dynamics of the i-th coupled agent is described by $$ẋ_i = f (x_i , μ_i ) + ∑ σ_ε (x_ε^τ α_ε − x_ε^h β_ε ) + ∑ k_ε (x_{N+1} − x_ε^h β_ε ), i = 1, . . . , N,$$ whereas the opinion leader has dynamics \(ẋ_{N+1}=f(x_{N+1},μ_i)\), as it is not influenced by the other individuals; note that function \(f(z,μ_i)=−3z+μ_itanhz\) is so that, when \(μ_i>3\), an isolated agent has an unstable equilibrium at the neutral opinion \(z=0\), and two stable equilibria \(z=\pm\bar{x}_i\) leaning toward each of the two options [1]. The vectors αε and βε stack the weights of the tails and heads of hyperedge ε, respectively, and are such that \(α_ε^T 1_{|T (ε)|} = β_ε^T 1_{|H (ε)|} = 1\), and the set \(E_{uc}^{∗,i} (E_p^{∗,i})\) is the set of hyperedges not containing the opinion leader (containing the opinion leader) that have i as a head; \(x_ε^τ∈R^{|T (ε)|}\) and (x_ε^h ∈ R^{|H (ε)|}\) are row vectors whose elements are the state of the agents that are tails and heads of hyperedge ε, respectively. Finally, for a hyperedge \(ε∈E_{uc}\) , \(σ_ε\) is its coupling strength, and for \(ε ∈ E_p\) , \(k_ε\) is the control gain of the associated leader’s feedback control action. We consider the case in which the hypergraph describing social interactions, extracted from [2], is divided into 15 communities of 10 nodes each. The scope of the opinion leader is to bring the opinion of the followers as close as possible to its own. However, in the presence of limited resources the leader can only decide to influence one of the 15 communities through directed hyperedges. By linearizing equation (1) and using Lyapunov stability theory, we are able to identify the selection that yields the smallest difference in norm between the opinion of the leader and that of the followers.

[1] F. Lo Iudice, F. Garofalo, and P. De Lellis, “Bounded partial pinning control of network dynamical systems,” IEEE Transactions on Control of Network Systems, vol. 10, no. 1, pp. 238–248, 2022. [2] J. Stehlé et al., “High-resolution measurements of face-to-face contact patterns in a primary school,” PloS one, vol. 6, p. e23176, 2011. - 16:40–16:45: Conclusion

Simona Olmi

Institute for Complex Systems, National Research Council, Florence, Italy

Mehrnaz Anvari,

Fraunhofer Institute for Algorithms and Scientific Computing SCAI, Germany

Fakhteh Ghanbarnejad

Ghanbarnejad, Potsdam Institute for Climate Impact Research (PIK),Germany

The first Dynamics On and Of Complex Networks (DOOCN I) took place in Dresden, Germany, on 4th October 2007, as a satellite workshop of the European Conference on Complex Systems 07. The workshop received a large number of quality submissions from authors pursuing research in multiple disciplines, thus making the forum truly inter-disciplinary. There were around 20 speakers who spoke about the dynamics on and of different systems exhibiting a complex network structure, from biological systems, linguistic systems, and social systems to various technological systems like the Internet, WWW, and peer-to-peer systems. The organizing committee has published some of the very high quality original submissions as an edited volume from Birkhauser, Boston describing contemporary research position in complex networks.

After the success of DOOCN I, the organizers launched Dynamics On and Of Complex Networks – II (DOOCN II), a two days satellite workshop of the European Conference of Complex Systems 08. DOOCN II was held in Jerusalem, Israel, on the 18th and 19th September 2008.

DOOCN III was held as a satellite of ECCS 2009 in the University of Warwick, UK on 23rd and 24th of September. In continuation, DOOCN IV was held again as a satellite of ECCS 2010 in the University Institute Lisbon, Portugal on 16th September.

DOOCN V was held as a satellite of ECCS 2011 in the University of Vienna on 14th – 15th September 2011.

DOOCN VI took place in Barcelona, as a satellite to ECCS 2013, and focused on Semiotic Dynamics in time-varying social media. As DOOCN I, the other five DOOCN workshops counted with a large number participants and attracted prominent scientist in the field.

DOOCN VII, held in Lucca as a satellite to ECCS 2014, focused on Big Data aspects. DOOCN VIII was held in Zaragoza with focus also on BigData aspects.

The 9th edition of DOOCN was held in Amsterdam at Conference on Complex Systems (CCS) with the theme “Mining and learning for complex networks”.

The 2017 edition of DOOCN was held in Indianapolis USA in conjunction with NetSci 2017.

The 2018 edition of DOOCN XI was held in Thessaloniki, Greece at Conference on Complex Systems (CCS) with the theme “Machine learning for complex networks”.

The 2019 edition of DOOCN XII was held in Burlington, Vermont, USA in conjunction with NetSci 2019 with the theme “Network Representation Learning”.

The 2020 edition of DOOCN XIII was held online in conjunction with NetSci 2020 with the theme “Network Learning”.

The 2023 edition of DOOCN XIV was held online in conjunction with Statphys28 with the theme “Cascading Failures in Complex Networks”.

The organizing committees of the DOOCN workshop series have published three Birkhäuser book volumes, from selected talks from the series.