@unpublished{Asllani_polarization_2024,
  author = {Qian, H. and Asllani M.},
  title = {Explosive Opinion Polarization and Depolarization with Asymmetric Perception},
  year = {2024},
  note = {Under review in Physical Review Letters}
}

@unpublished{Asllani_chimera_2024,
  author = {Asllani M. and Arenas, A.},
  title = {Towards a Theory for the Formation of Chimera Patterns in Complex Networks},
  year = {2024},
  note = {Under review in Physical Review E}
}

@unpublished{Asllani_Synchronization_2024,
  author = {Muolo, R. and Carletti, T. and Gleeson, J. P. and Asllani M.},
  title = {Reply to Comment on ''Synchronization dynamics in non-normal networks: the trade-off for optimality''},
  note = {arXiv:2112.08700},
  year = {2022},
  doi = {2112.08700}
}

We reply to the recent note "Comment on Synchronization dynamics in non-normal networks: the trade-off for optimality", showing that the authors base their claims mainly on general theoretical arguments that do not necessarily invalidate the adequacy of our previous study. In particular, they do not specifically tackle the correctness of our analysis but instead limit their discussion on the interpretation of our results and conclusions, particularly related to the concept of optimality of network structure related to synchronization dynamics. Nevertheless, their idea of optimal networks is strongly biased towards their previous work and does not necessarily correspond to our framework, making their interpretation subjective and not consistent. We bring here further evidence from the existing and more recent literature, omitted in the Comment note, that the synchronized state of oscillators coupled through optimal networks, as intended by the authors, can indeed be highly fragile to small but finite perturbations, confirming our original results.

@article{Asllani_Broken_Balance_2024,
  title = {Broken detailed balance and entropy production in directed networks},
  author = {Nartallo-Kaluarachchi, R. and Asllani M. and Deco, G. and Kringelbach, M. L. and Goriely, A. and Lambiotte, R.},
  journal = {(To appear) Physical Review E},
  volume = {},
  pages = {},
  year = {2024},
  doi = {2402.19157}
}

The structure of a complex network plays a crucial role in determining its dynamical properties. In this work, we show that the degree to which a network is directed and hierarchically organised is closely associated with the degree to which its dynamics break detailed balance and produce entropy. We consider a range of dynamical processes and show how different directed network features affect their entropy production rate. We begin with an analytical treatment of a 2-node network followed by numerical simulations of synthetic networks using the preferential attachment and Erdös-Renyi algorithms. Next, we analyse a collection of 97 empirical networks to determine the effect of complex real-world topologies. Finally, we present a simple method for inferring broken detailed balance and directed network structure from multivariate time-series and apply our method to identify non-equilibrium dynamics and hierarchical organisation in both human neuroimaging and financial time-series. Overall, our results shed light on the consequences of directed network structure on non-equilibrium dynamics and highlight the importance and ubiquity of hierarchical organisation and non-equilibrium dynamics in real-world systems.

@article{Asllani_SOC_2024,
  title = {Persistence of Chimera States and the Challenge for Synchronization in Real-World Networks},
  author = {Muolo, R. and O’Brien, J. D. and Carletti, T. and Asllani M.},
  journal = {The European Physical Journal B},
  volume = {97},
  pages = {6},
  year = {2024},
  doi = {10.1140/epjb/s10051-023-00630-y}
}

The emergence of order in nature manifests in various phenomena, with synchronization being one of the most prominent examples. Understanding the role of interactions among components in a complex system’s synchronization is a key research question that bridges network science and dynamical systems. Particular attention has been given to the emergence of chimera states, where subsets of synchronized oscillators coexist with asynchronous ones. This coexistence of coherence and incoherence exemplifies the persistence of both order and disorder in a long-lasting regime. Despite significant progress in recent years, the manifestation of these states in real-world networks remains underexplored. In this paper, we investigate how non-normality, a ubiquitous structural property of real networks, contributes to the emergence of diverse dynamical phenomena, such as amplitude chimeras and oscillon patterns. Specifically, we demonstrate that networks with prevalent source or leader nodes are prone to the manifestation of phase chimera states. Throughout the study, we highlight that non-normality presents ongoing challenges to global synchronization and is crucial in the emergence of chimera states.

@article{Asllani_SOC_2025,
  title = {Emergence of power-law distributions in self-segregation reaction-diffusion processes},
  author = {de Kemmeter, J.-F. and Byrne, A. and Dunne, A. and Carletti, T. and Asllani M.},
  journal = {Physical Review E},
  volume = {110},
  number = {1},
  pages = {L012201},
  year = {2024},
  doi = {10.1103/PhysRevE.110.L012201}
}

Vegetation patterns in semi-arid areas manifest either through regular or irregular vegetation patches separated by bare ground. Of particular interest are the latter structures, which exhibit a distinctive power-law distribution of patch sizes. While a Turing-like instability mechanism can explain the formation of regular patterns, the emergence of irregular ones still lacks a clear understanding. To fill this gap, we present a novel self-organizing criticality mechanism driving the emergence of irregular vegetation patterns in semi-arid landscapes. The model integrates essential ecological principles, emphasizing positive interactions and limited resources. It consists of a single-species evolution equation with an Allee-logistic reaction term and a nonlinear diffusion term accounting for self-segregation. The model captures an initial mass decrease due to resource scarcity, reaching a predictable threshold. Beyond this threshold, and due to local positive interactions that promote cooperation, vegetation self-segregates into distinct clusters. Numerical investigations show that the distribution of cluster sizes obeys a power-law with an exponential cutoff, in accordance with empirical observations found in the literature. The study aims to establish a foundation for understanding self-organizing criticality in vegetation patterns, advancing the understanding of ecological pattern formation.

@article{Asllani_Self_Segregation_2022,
  title = {Self-segregation in heterogeneous metapopulation landscapes},
  author = {de Kemmeter, J.-F. and Carletti, T. and Asllani M.},
  journal = {Journal of Theoretical Biology},
  volume = {554},
  pages = {111271},
  year = {2022},
  doi = {10.1016/j.jtbi.2022.111271}
}

Complex interactions are at the root of the population dynamics of many natural systems, particularly for being responsible for the allocation of species and individuals across apposite niches of the ecological landscapes. On the other side, the randomness that unavoidably characterises complex systems has increasingly challenged the niche paradigm, providing alternative neutral theoretical models. We introduce a network-inspired metapopulation individual-based model (IBM), hereby named self-segregation, where the density of individuals in the hosting patches (local habitats) drives the individuals’ spatial assembling while still constrained by nodes’ saturation. In particular, we prove that the core–periphery structure of the networked landscape triggers the spontaneous emergence of vacant habitat patches, which segregate the population in multistable patterns of isolated (sub)communities separated by empty patches. Furthermore, a quantization effect in the number of vacant patches is observed once the total system mass varies continuously, emphasizing a striking feature of the robustness of population stationary distributions. Notably, our model reproduces the patch vacancy found in the fragmented habitat of the Glanville fritillary butterfly (Melitaea cinxia), an endemic species of the Åland islands. We argue that such spontaneous breaking of the natural habitat supports the concept of the highly contentious (Grinnellian) niche vacancy and also suggests a new mechanism for endogenous habitat fragmentation and consequently the peripatric speciation.

@article{Asllani_ESIR_Model_2022,
  title = {Nonlinear random walks optimize the trade-off between cost and prevention in epidemics lockdown measures: The ESIR model},
  author = {},
  journal = {Chaos, Solitons \& Fractals},
  volume = {161},
  pages = {112322},
  year = {2022},
  doi = {10.1016/j.chaos.2022.112322}
}

Contagious diseases can spread quickly in human populations, either through airborne transmission or if some other spreading vectors are abundantly accessible. They can be particularly devastating if the impact on individuals’ health has severe consequences on the number of hospitalizations or even deaths. Common countermeasures to contain the epidemic spread include introducing restrictions on human interactions or their mobility in general, which are often associated with an economic and social cost. In this paper, we present a targeted model of optimal social distancing on metapopulation networks, named the ESIR model, which can effectively reduce the disease spreading and at the same time minimize the impact on human mobility and related costs. The proposed model is grounded in a nonlinear random walk process that considers the finite carrying capacity of the network’s metanodes, the physical patches where individuals interact within mobility networks. This latter constraint is modeled as a slack compartment E for the classic SIR model and quantifies the density of vacant spaces to accommodate the diffusing individuals. Formulating the problem as a multi-objective optimization problem shows that when the walkers avoid crowded nodes, the system can rapidly approach Pareto optimality, thus reducing the spreading considerably while minimizing the impact on human mobility, as also validated in empirical transport networks. These results envisage ad hoc mobility protocols that can potentially enhance policymaking for pandemic control.

@article{Asllani_Cluster_Chimera_2022,
  title = {Symmetry-breaking mechanism for the formation of cluster chimera patterns},
  author = {Asllani M. and Siebert, B. A. and Arenas, A. and Gleeson, J. P.},
  journal = {Chaos},
  volume = {32},
  pages = {013107},
  year = {2022},
  doi = {10.1063/5.0060466}
}

The emergence of order in collective dynamics is a fascinating phenomenon that characterizes many natural systems consisting of coupled entities. Synchronization is such an example where individuals, usually represented by either linear or nonlinear oscillators, can spontaneously act coherently with each other when the interactions’ configuration fulfills certain conditions. However, synchronization is not always perfect, and the coexistence of coherent and incoherent oscillators, broadly known in the literature as chimera states, is also possible. Although several attempts have been made to explain how chimera states are created, their emergence, stability, and robustness remain a long-debated question. We propose an approach that aims to establish a robust mechanism through which cluster synchronization and chimera patterns originate. We first introduce a stability-breaking method where clusters of synchronized oscillators can emerge. At variance with the standard approach where synchronization arises as a collective behavior of coupled oscillators, in our model, the system initially sets on a homogeneous fixed-point regime, and, only due to a global instability principle, collective oscillations emerge. Following a combination of the network modularity and the model’s parameters, one or more clusters of oscillators become incoherent within yielding a particular class of patterns that we here name cluster chimera states. 

@article{Asllani_Leader_Nodes_2021,
  title = {Hierarchical route to the emergence of leader nodes in real-world networks},
  author = {O’Brien, J. D. and Oliveira, K. A. and Gleeson, J. P. and Asllani M.},
  journal = {Physical Review Research},
  volume = {3},
  pages = {023117},
  year = {2021},
  doi = {10.1103/PhysRevResearch.3.023117}
}

A large number of complex systems, naturally emerging in various domains, are well described by directed networks, resulting in numerous interesting features that are absent from their undirected counterparts. Among these properties is a strong non-normality, inherited by a strong asymmetry that characterizes such systems and guides their underlying hierarchy. In this work, we consider an extensive collection of empirical networks and analyze their structural properties using information-theoretic tools. A ubiquitous feature is observed amongst such systems as the level of non-normality increases. When the non-normality reaches a given threshold, highly directed substructures aiming towards terminal (sink or source) nodes, denoted here as leaders, spontaneously emerge. Furthermore, the relative number of leader nodes describe the level of anarchy that characterizes the networked systems. Based on the structural analysis, we develop a null model to capture features such as the aforementioned transition in the networks’ ensemble. We also demonstrate that the role of leader nodes at the pinnacle of the hierarchy is crucial in driving dynamical processes in these systems. This work paves the way for a deeper understanding of the architecture of empirical complex systems and the processes taking place on them.

@article{Asllani_Synchronization_2021,
  title = {Synchronization dynamics in non-normal networks: the trade-off for optimality},
  author = {Muolo, R. and Carletti, T. and Gleeson, J. P. and Asllani M.},
  journal = {Entropy},
  volume = {23},
  pages = {36},
  year = {2021},
  doi = {10.3390/e23010036}
}

Synchronization is an important behavior that characterizes many natural and human-made systems that are composed of several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, with the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework for tackling synchronization problems. Using this method, it has been shown that, for a class of models, synchronization in strongly directed networks is robust to external perturbations. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Moreover, many empirical networks are also strongly non-normal. Inspired by this latter fact in this work, we address the role of the non-normality in the synchronization dynamics by pointing out that standard techniques, such as the MSF, may fail to predict the stability of synchronized states. We demonstrate that, due to a transient growth that is induced by the structure’s non-normality, the system might lose synchronization, contrary to the spectral prediction. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.

@article{Asllani_Modularity_2020,
  title = {Role of modularity in self-organisation dynamics in biological networks},
  author = {Siebert, B. and Hall, C. L. and Gleeson, J. P. and Asllani M.},
  journal = {Physical Review E},
  volume = {102},
  pages = {052306},
  year = {2020},
  doi = {10.1103/PhysRevE.102.052306}
}

Interconnected ensembles of biological entities are perhaps some of the most complex systems that modern science has encountered so far. In particular, scientists have concentrated on understanding how the complexity of the interacting structure between different neurons, proteins, or species influences the functioning of their respective systems. It is well established that many biological networks are constructed in a highly hierarchical way with two main properties: short average paths that join two apparently distant nodes (neuronal, species, or protein patches) and a high proportion of nodes in modular aggregations. Although several hypotheses have been proposed so far, still little is known about the relation of the modules with the dynamical activity in such biological systems. Here we show that network modularity is a key ingredient for the formation of self-organizing patterns of functional activity, independently of the topological peculiarities of the structure of the modules. In particular, we propose a self-organizing mechanism which explains the formation of macroscopic spatial patterns, which are homogeneous within modules. This may explain how spontaneous order in biological networks follows their modular structural organization. We test our results on real-world networks to confirm the important role of modularity in creating macroscale patterns.

@article{Asllani_Pattern_2020,
  title = {A universal route to pattern formation in multicellular systems},
  author = {Asllani M. and Carletti, T. and Fanelli, D. and Maini, P. K.},
  journal = {The European Physical Journal B},
  volume = {93},
  pages = {1--11},
  year = {2020},
  doi = {10.1140/epjb/e2020-10206-3}
}

A general framework for the generation of long wavelength patterns in multi-cellular (discrete) systems is proposed, which extends beyond conventional reaction-diffusion (continuum) paradigms. The standard partial differential equations of reaction-diffusion framework can be considered as a mean-field like ansatz which corresponds, in the biological setting, to sending to zero the size (or volume) of each individual cell. By relaxing this approximation and, provided a directionality in the flux is allowed for, we demonstrate here that instability leading to spatial pattern formation can always develop if the (discrete) system is large enough, namely, composed of sufficiently many cells, the units of spatial patchiness. The macroscopic patterns that follow the onset of the instability are robust and show oscillatory or steady state behavior.

@article{Asllani_Nonlinear_Walkers_2020,
  title = {Nonlinear walkers and efficient exploration of a crowded network},
  author = {Carletti, T. and Asllani M. and Fanelli, D. and Latora, V.},
  journal = {Physical Review Research},
  volume = {2},
  pages = {033012},
  year = {2020},
  doi = {10.1103/PhysRevResearch.2.033012}
}

Random walks are the simplest way to explore or search a graph and have revealed a very useful tool to investigate and characterize the structural properties of complex networks from the real world. For instance, they have been used to identify the modules of a given network, its most central nodes and paths, or to determine the typical times to reach a target. Although various types of random walks whose motion is biased on node properties, such as the degree, have been proposed, which are still amenable to analytical solution, most if not all of them rely on the assumption of linearity and independence of the walkers. In this work we introduce a class of nonlinear stochastic processes describing a system of interacting random walkers moving over networks with finite node capacities. The transition probabilities that rule the motion of the walkers in our model are modulated by nonlinear functions of the available space at the destination node, with a bias parameter that allows to tune the tendency of the walkers to avoid nodes occupied by other walkers. First, we derive the master equation governing the dynamics of the system, and we determine an analytical expression for the occupation probability of the walkers at equilibrium in the most general case and under different level of network congestions. Then we study different types of synthetic and real-world networks, presenting numerical and analytical results for the entropy rate, a proxy for the network exploration capacities of the walkers. We find that, for each level of the nonlinear bias, there is an optimal crowding that maximizes the entropy rate in a given network topology. The analysis suggests that a large fraction of real-world networks are organized in such a way as to favor exploration under congested conditions. Our work provides a general and versatile framework to model nonlinear stochastic processes whose transition probabilities vary in time depending on the current state of the system.

@article{Asllani_Detectability_2020,
  title = {Dynamics imposes limits on detectability of network structures},
  author = {Asllani M. and da Cunha, B. R. and Estrada, E. and Gleeson, J. P.},
  journal = {New Journal of Physics},
  volume = {22},
  pages = {063037},
  year = {2020},
  doi = {10.1088/1367-2630/ab8ef9}
}

Networks are universally considered as complex structures of interactions of large multi-component systems. To determine the role that each node has inside a complex network, several centrality measures have been developed. Such topological features are also crucial for their role in the dynamical processes occurring in networked systems. In this paper, we argue that the dynamical activity of the nodes may strongly reshape their relevance inside the network, making centrality measures in many cases, misleading. By proposing a generalisation of the communicability function, we show that when the dynamics taking place at the local level of the node is slower than the global one between the nodes, then the system may lose track of the structural features. On the contrary, hidden global properties such as the shortest path distances can be recovered only in the limit where network-level dynamics are negligible compared to node-level dynamics. From the perspective of network inference, this constitutes an uncertainty condition, in the sense that it limits the extraction of multi-resolution information about the structure, particularly in the presence of noise. For illustration purposes, we show that for networks with different time-scale structures such as strong modularity, the existence of fast global dynamics can imply that precise inference of the community structure is impossible.

@article{Asllani_Resilience_2019,
  title = {Resilience for stochastic systems interacting via a quasi-degenerate network},
  author = {Nicoletti, S. and Fanelli, D. and Zagli, N. and Asllani M. and Battistelli, G. and Carletti, T. and Chisci, L. and Inocenti, G. and Livi, R.},
  journal = {Chaos},
  volume = {29},
  pages = {083123},
  year = {2019},
  doi = {10.1063/1.5099538}
}

A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network, two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process, which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasidegenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, can develop seemingly regular patterns in the concentration amount. Non-normality and quasidegenerate networks may, therefore, amplify the inherent stochasticity and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.

@article{Asllani_Non_Normality_2019,
  title = {Patterns of non-normality in networked systems},
  author = {Muolo, R. and Asllani M. and Fanelli, D. and Maini, P. K. and Carletti, T.},
  journal = {Journal of Theoretical Biology},
  volume = {480},
  pages = {81--91},
  year = {2019},
  doi = {10.1016/j.jtbi.2019.07.004}
}

Several mechanisms have been proposed to explain the spontaneous generation of self-organised patterns, hypothesised to play a role in the formation of many of the magnificent patterns observed in Nature. In several cases of interest, the system under scrutiny displays a homogeneous equilibrium, which is destabilised via a symmetry breaking instability which reflects the specificity of the problem being inspected. The Turing instability is among the most celebrated paradigms for pattern formation. In its original form, the diffusion constants of the two mobile species need to be quite different from each other for the instability to develop. Unfortunately, this condition limits the applicability of the theory. To overcome this impediment, and with the ambitious long term goal to eventually reconcile theory and experiments, we here propose an alternative mechanism for promoting the onset of pattern. To this end a multi-species reactive model is studied, assuming a generalized transport on a discrete and directed network-like support: the instability is triggered by the non-normality of the embedding network. The non-normal character of the dynamics instigates a short time amplification of the imposed perturbation, thus making the system unstable for a choice of parameters that would yield stability under the conventional scenario. In other words, non-normality promotes the emergence of patterns in cases where a classical linear analysis would not predict them. The importance of our result relies also on the fact that non-normal networks are pervasively found, motivating the general interest of the mechanism discussed here.

@article{Asllani_Non_Normal_2018,
  title = {Structure and dynamical behavior of non-normal networks},
  author = {Asllani M. and Lambiotte, R. and Carletti, T.},
  journal = {Science Advances},
  volume = {4},
  pages = {eaau9403},
  year = {2018},
  doi = {10.1126/sciadv.aau9403}
}

We analyze a collection of empirical networks in a wide spectrum of disciplines and show that strong non-normality is ubiquitous in network science. Dynamical processes evolving on non-normal networks exhibit a peculiar behavior, as initial small disturbances may undergo a transient phase and be strongly amplified in linearly stable systems. In addition, eigenvalues may become extremely sensible to noise and have a diminished physical meaning. We identify structural properties of networks that are associated with non-normality and propose simple models to generate networks with a tunable level of non-normality. We also show the potential use of a variety of metrics capturing different aspects of non-normality and propose their potential use in the context of the stability of complex ecosystems.

@article{Asllani_Neurostimulation_2018,
  title = {A minimally invasive neurostimulation method for controlling epilepsy seizures},
  author = {Asllani M. and Expert, P. and Carletti, T.},
  journal = {PLoS Computational Biology},
  volume = {14},
  pages = {e1006296},
  year = {2018},
  doi = {10.1371/journal.pcbi.1006296}
}

Many collective phenomena in Nature emerge from the partial synchronization of the units comprising a system. In the case of the brain, this self-organised process allows groups of neurons to fire in highly intricate partially synchronised patterns and eventually lead to high-level cognitive outputs and control over the human body. However, when the synchronisation patterns are altered and hypersynchronisation occurs, undesirable effects can arise. This is particularly striking and well documented in the case of epileptic seizures and tremors in neurodegenerative diseases such as Parkinson’s disease. In this paper, we propose an innovative, minimally invasive, control method that can effectively desynchronise misfiring brain regions and thus mitigate and even eliminate the symptoms of these diseases. The control strategy, grounded in the Hamiltonian control theory, is applied to ensembles of neurons modelled via the Kuramoto or the Stuart-Landau models and allows for heterogeneous coupling among the interacting units. The theory has been complemented with dedicated numerical simulations performed using the small-world Newman-Watts network and the random Erdős-Rényi network. Finally, the method has been compared with the gold-standard Proportional-Differential Feedback control technique. Our method is shown to achieve equivalent levels of desynchronisation using lesser control strength and/or fewer controllers, making it minimally invasive.

@article{Asllani_Hopping_2018,
  title = {Hopping in the crowd to unveil network topology},
  author = {Asllani M. and Carletti, T. and Patti, F. Di and Fanelli, D. and Piazza, F.},
  journal = {Physical Review Letters},
  volume = {120},
  pages = {158301},
  year = {2018},
  doi = {10.1103/PhysRevLett.120.158301}
}

We introduce a nonlinear operator to model diffusion on a complex undirected network under crowded conditions. We show that the asymptotic distribution of diffusing agents is a nonlinear function of the nodes’ degree and saturates to a constant value for sufficiently large connectivities, at variance with standard diffusion in the absence of excluded-volume effects. Building on this observation, we define and solve an inverse problem, aimed at reconstructing the a priori unknown connectivity distribution. The method gathers all the necessary information by repeating a limited number of independent measurements of the asymptotic density at a single node, which can be chosen randomly. The technique is successfully tested against both synthetic and real data and is also shown to estimate with great accuracy the total number of nodes.

@article{Asllani_Resilience_2018,
  title = {Topological resilience in non-normal networked systems},
  author = {Asllani M., T. Carletti},
  journal = {Physical Review E},
  volume = {97},
  number = {},
  pages = {042302 },
  year = {2018},
  doi = {10.1103/PhysRevE.97.042302}
}

The network of interactions in complex systems strongly influences their resilience and the system capability to resist external perturbations or structural damages and to promptly recover thereafter. The phenomenon manifests itself in different domains, e.g., parasitic species invasion in ecosystems or cascade failures in human-made networks. Understanding the topological features of the networks that affect the resilience phenomenon remains a challenging goal for the design of robust complex systems. We hereby introduce the concept of non-normal networks, namely networks whose adjacency matrices are non-normal, propose a generating model, and show that such a feature can drastically change the global dynamics through an amplification of the system response to exogenous disturbances and eventually impact the system resilience. This early stage transient period can induce the formation of inhomogeneous patterns, even in systems involving a single diffusing agent, providing thus a new kind of dynamical instability complementary to the Turing one. We provide, first, an illustrative application of this result to ecology by proposing a mechanism to mute the Allee effect and, second, we propose a model of virus spreading in a population of commuters moving using a non-normal transport network, the London Tube.

@article{Asllani_Kuramoto_2017,
  title = {Using Hamiltonian control to desynchronize Kuramoto oscillators},
  author = {Gjata, O. and Asllani M. and Barletti, L. and Carletti, T.},
  journal = {Physical Review E},
  volume = {95},
  pages = {022209},
  year = {2017},
  doi = {10.1103/PhysRevE.95.022209}
}

Many coordination phenomena are based on a synchronization process, whose global behavior emerges from the interactions among the individual parts. Often in nature, such self-organized mechanism allows the system to behave as a whole and thus grounding its very first existence, or expected functioning, on such process. There are, however, cases where synchronization acts against the stability of the system; for instance in some neurodegenerative diseases or epilepsy or the famous case of Millennium Bridge where the crowd synchronization of the pedestrians seriously endangered the stability of the structure. In this paper we propose an innovative control method to tackle the synchronization process based on the application of the Hamiltonian control theory, by adding a small control term to the system we are able to impede the onset of the synchronization. We present our results on a generalized class of the paradigmatic Kuramoto model.

@article{Asllani_Tune_Topology_2016,
  title = {Tune the topology to create or destroy patterns},
  author = {Asllani M., T. Carletti, D. Fanelli},
  journal = {The European Physical Journal B},
  volume = {89},
  number = {},
  pages = {260},
  year = {2016},
  doi = {10.1140/epjb/e2016-70248-6}
}

We consider the dynamics of a reaction-diffusion system on a multigraph. The species share the same set of nodes but can access different links to explore the embedding spatial support. By acting on the topology of the networks we can control the ability of the system to self-organise in macroscopic patterns, emerging as a symmetry breaking instability of an homogeneous fixed point. Two different cases study are considered: on the one side, we produce a global modification of the networks, starting from the limiting setting where species are hosted on the same graph. On the other, we consider the effect of inserting just one additional single link to differentiate the two graphs. In both cases, patterns can be generated or destroyed, as follows the imposed, small, topological perturbation. Approximate analytical formulae allow to grasp the essence of the phenomenon and can potentially inspire innovative control strategies to shape the macroscopic dynamics on multigraph networks.

@article{Asllani_Delayed_Processes_2016,
  title = {Pattern formation in a two-component reaction-diffusion system with delayed processes on a network},
  author = {Petit, J. and Asllani M. and Fanelli, D. and Lauwens, B. and Carletti, T.},
  journal = {Physica A: Statistical Mechanics and its Applications},
  volume = {462},
  pages = {230},
  year = {2016},
  doi = {10.1016/j.physa.2016.06.003}
}

Reaction–diffusion systems with time-delay defined on complex networks have been studied in the framework of the emergence of Turing instabilities. The use of the Lambert W-function allowed us to get explicit analytic conditions for the onset of patterns as a function of the main involved parameters, the time-delay, the network topology and the diffusion coefficients. Depending on these parameters, the analysis predicts whether the system will evolve towards a stationary Turing pattern or rather to a wave pattern associated to a Hopf bifurcation. The possible outcomes of the linear analysis overcome the respective limitations of the single-species case with delay, and that of the classical activator–inhibitor variant without delay. Numerical results gained from the Mimura–Murray model support the theoretical approach.

@article{Asllani_Turing_2015,
  title = {Delay-induced Turing-like waves for one-species reaction-diffusion model on a network},
  author = {Petit, J. and Carletti, T. and Asllani M. and Fanelli, D.},
  journal = {EPL (Europhysics Letters)},
  volume = {111},
  pages = {58002},
  year = {2015},
  doi = {10.1209/0295-5075/111/58002}
}

A one-species time-delay reaction-diffusion system defined on a complex network is studied. Traveling waves are predicted to occur following a symmetry-breaking instability of a homogeneous stationary stable solution, subject to an external nonhomogeneous perturbation. These are generalized Turing-like waves that materialize in a single-species populations dynamics model, as the unexpected byproduct of the imposed delay in the diffusion part. Sufficient conditions for the onset of the instability are mathematically provided by performing a linear stability analysis adapted to time-delayed differential equations. The method here developed exploits the properties of the Lambert W-function. The prediction of the theory are confirmed by direct numerical simulation carried out for a modified version of the classical Fisher model, defined on a Watts-Strogatz network and with the inclusion of the delay.

@article{Asllani_Anisotropic_Diffusion_2015,
  title = {Pattern formation for reactive species undergoing anisotropic diffusion},
  author = {Busiello, D. M. and Planchon, G. and Asllani M. and Carletti, T. and Fanelli, D.},
  journal = {The European Physical Journal B},
  volume = {88},
  pages = {222},
  year = {2015},
  doi = {10.1140/epjb/e2015-60269-0}
}

Turing instabilities for a two species reaction-diffusion system is studied under anisotropic diffusion. More specifically, the diffusion constants which characterize the ability of the species to relocate in space are direction sensitive. Under this working hypothesis, the conditions for the onset of the instability are mathematically derived and numerically validated. Patterns which closely resemble those obtained in the classical context of isotropic diffusion, develop when the usual Turing condition is violated, along one of the two accessible directions of migration. Remarkably, the instability can also set in when the activator diffuses faster than the inhibitor, along the direction for which the usual Turing conditions are not matched.

@article{Asllani_Turing_Cartesian_2015,
  title = {Turing instabilities on Cartesian product networks},
  author = {Asllani M. and Busiello, D. M. and Carletti, T. and Fanelli, D. and Planchon, G.},
  journal = {Scientific Reports},
  volume = {5},
  pages = {12927},
  year = {2015},
  doi = {10.1038/srep12927}
}

The problem of Turing instabilities for a reaction-diffusion system defined on a complex Cartesian product network is considered. To this end we operate in the linear regime and expand the time dependent perturbation on a basis formed by the tensor product of the eigenvectors of the discrete Laplacian operators, associated to each of the individual networks that build the Cartesian product. The dispersion relation which controls the onset of the instability depends on a set of discrete wavelengths, the eigenvalues of the aforementioned Laplacians. Patterns can develop on the Cartesian network, if they are supported on at least one of its constitutive sub-graphs. Multiplex networks are also obtained under specific prescriptions. In this case, the criteria for the instability reduce to compact explicit formulae. Numerical simulations carried out for the Mimura-Murray reaction kinetics confirm the adequacy of the proposed theory.

@article{Asllani_Turing_Multiplex_2014,
  title = {Turing Patterns in Multiplex Networks},
  author = {Asllani M. and Busiello, D. M. and Carletti, T. and Fanelli, D. and Planchon, G.},
  journal = {Physical Review E},
  volume = {90},
  number = {4},
  pages = {042814},
  year = {2014},
  doi = {10.1103/PhysRevE.90.042814}
}

The theory of pattern formation for a reaction-diffusion system defined on a multiplex is developed via a perturbative approach. The interlayer diffusion constants serve as a small parameter in the expansion, with the unperturbed state corresponding to decoupled multiplex layers. The interaction between adjacent layers can induce the instability of a homogeneous fixed point, leading to self-organized patterns that are otherwise suppressed in decoupled layers. Patterns on individual layers may also dissipate due to cross-layer interactions. Analytical results are compared with direct simulations.

@article{Asllani_Directed_Networks_2014,
  title = {The Theory of Pattern Formation on Directed Networks},
  author = {Asllani M. and Challenger, J. D. and Pavone, F. S. and Sacconi, L. and Fanelli, D.},
  journal = {Nature Communications},
  volume = {5},
  pages = {4517},
  year = {2014},
  doi = {10.1038/ncomms5517}
}

Dynamical processes on networks have garnered significant interest recently. The theory of pattern formation in reaction–diffusion systems on symmetric networks has been well-explored due to its wide applications. This study extends the theory to directed networks, commonly found in fields such as neuroscience, computer networks, and traffic systems. Due to the network Laplacian’s structure, the dispersion relation exhibits both real and imaginary parts, unlike symmetric, undirected networks. Network topology can destabilize the homogeneous fixed point, leading to a new class of instabilities not possible on undirected graphs. Numerical simulations show traveling waves or quasi-stationary patterns depending on the underlying graph characteristics.

@article{Asllani_Stochastic_2013,
  title = {Stochastic Amplification of Spatial Modes in a System with One Diffusing Species},
  author = {Cantini, L. and Cianci, C. and Fanelli, D. and Masi, E. and Barletti, L. and Asllani M.},
  journal = {Journal of Mathematical Biology},
  volume = {69},
  pages = {1585--1608},
  year = {2013},
  doi = {10.1007/s00285-013-0743-x}
}

Pattern formation in a two-species reaction-diffusion model is studied under the condition that only one species can diffuse. In such systems, classical Turing instability does not occur. However, in a stochastic framework, spatially organized patterns can emerge, driven by finite-size effects. General conditions for stochastic patterns are provided, with theoretical predictions tested in a specific case study.

@article{Asllani_Linear_Noise_2013,
  title = {The Linear Noise Approximation for Reaction-Diffusion Systems on Networks},
  author = {Asllani M. and Biancalani, T. and Fanelli, D. and McKane, A. J.},
  journal = {The European Physical Journal B},
  volume = {86},
  pages = {476},
  year = {2013},
  doi = {10.1140/epjb/e2013-40570-8}
}

Stochastic reaction-diffusion models on complex networks can be analytically studied using the linear noise approximation. This is demonstrated with a specific stochastic model that exhibits traveling waves in its deterministic limit. Stochastic fluctuations are shown to induce stochastic waves beyond the instability region predicted by deterministic theory. Simulations validate the theoretical results, analyzed through a generalized Fourier transform algorithm based on the eigenvectors of the network’s discrete Laplacian. A peak in the power spectrum of fluctuations matches the theoretical predictions.

@article{Asllani_Stochastic_Network_2012,
  title = {Stochastic Patterns on a Network},
  author = {Asllani M. and Di Patti, F. and Fanelli, D.},
  journal = {Physical Review E},
  volume = {86},
  number = {4},
  pages = {046105},
  year = {2012},
  doi = {10.1103/PhysRevE.86.046105}
}

The stochastic Turing instability on a scale-free network is analyzed using the stochastic Brusselator model. The system spontaneously differentiates into activator-rich and activator-poor nodes outside the classical deterministic Turing instability parameters. This phenomenon is explained analytically, traced back to finite-size corrections due to the inherent granularity of the medium, and supported by direct stochastic simulations.

@article{Asllani_Statistical_2012,
  title = {Statistical Theory of Quasi-Stationary States Beyond the Single Water-Bag Case Study},
  author = {Asllani M. and Fanelli, D. and Turchi, A. and Carletti, T. and Leoncini, X.},
  journal = {Physical Review E},
  volume = {85},
  number = {2},
  pages = {021148},
  year = {2012},
  doi = {10.1103/PhysRevE.85.021148}
}

An analytical solution for out-of-equilibrium quasi-stationary states of the Hamiltonian mean field (HMF) model is derived using a maximum entropy principle. The theory, previously applied to the single-level water-bag case, is extended to multiple overlapping water bags. The theory’s accuracy is validated against direct microcanonical simulations for a two-level water-bag case, showing excellent agreement.