Describing a complex system is in many ways a problem akin to identifying an object, in that it involves defining boundaries, constituent parts and their relationships by the use of grouping laws. Here we propose a novel method which extends the use of complex networks theory to a generalized class of non-Gestaltic systems, taking the form of collections of isolated, possibly heterogeneous, scalars, e.g. sets of biomedical tests. The ability of the method to unveil relevant information is illustrated for the case of gene expression in the response to osmotic stress of {\it Arabidopsis thaliana}. The most important genes turn out to be the nodes with highest centrality in appropriately reconstructed networks. The method allows predicting a set of 15 genes whose relationship with such stress was previously unknown in the literature. The validity of such predictions is demonstrated by means of a target experiment, in which the predicted genes are one by one artificially induced, and the growth of the corresponding phenotypes turns out to feature statistically significant differences when compared to that of the wild-type.

Cluster synchronisation is a key phenomenon observed in networks of coupled dynamical units. Its presence has been linked to symmetry and, more generally, to equability of the underlying network pattern of interactions between identical dynamical units. In this article, we clarify once and for all the relation between equitability and cluster synchronisation on a very general dynamical system which allows multi-layer and higher-order interactions. Namely, we show that equitability is a necessary, and sufficient, condition for the existence of independent cluster synchronised solutions. As an important consequence, our results explain the ubiquity of explosive synchronisation, as opposed to cluster synchronisation, in multi-layer and higher-order networks: equitability imposes additional constraints that must be simultaneously satisfied on the same set of nodes. Our results have important implications for the design of complex dynamical systems of coupled dynamical units with arbitrary cluster synchronisation patterns and coupling functions.

The increasing power of computer technology does not dispense with the need to extract meaningful in- formation out of data sets of ever growing size, and indeed typically exacerbates the complexity of this task. To tackle this general problem, two methods have emerged, at chronologically different times, that are now commonly used in the scientific community: data mining and complex network theory. Not only do complex network analysis and data mining share the same general goal, that of extracting information from complex systems to ultimately create a new compact quantifiable representation, but they also often address similar problems too. In the face of that, a surprisingly low number of researchers turn out to resort to both methodologies. One may then be tempted to conclude that these two fields are either largely redundant or totally antithetic. The starting point of this review is that this state of affairs should be put down to contingent rather than conceptual differences, and that these two fields can in fact advantageously be used in a synergistic manner. An overview of both fields is first provided, some fundamental concepts of which are illustrated. A variety of contexts in which complex network theory and data mining have been used in a synergistic manner are then presented. Contexts in which the appropriate integration of complex network metrics can lead to improved classification rates with respect to classical data mining algorithms and, conversely, contexts in which data mining can be used to tackle important issues in complex network theory applications are illustrated. Finally, ways to achieve a tighter integration between complex networks and data mining, and open lines of research are discussed.

Monoclonal antibodies are among the most promising therapeutic agents in modern medicine, yet their formulation into high-concentration solutions for subcutaneous self-administration poses a major challenge. A key obstacle is the marked increase in viscosity often observed under these conditions. To gain deeper insights into this phenomenon, coarse-grained models derived from soft matter physics have been widely employed. However, these models have yet to be fully leveraged for analyzing the rheological collective properties of such systems. In this study, using molecular dynamics simulations, we directly compute the antibody solution viscosity by starting from commonly used models in which electrostatic interactions are treated through effective screened Coulomb potentials. We demonstrate that this approach fails to reproduce experimental evidence and we show, by analyzing stress correlations in the system, that it is necessary to treat the heterogeneously charged domains explicitly, also including counterions and salt ions, and to properly account for the long-ranged nature of Coulomb interactions. By thoroughly analyzing the microscopic structure of the system, we further reveal the presence of transient strongly correlated antibodies which would not be present if charges were treated implicitly, thus pointing to a prominent role of electrostatics in determining the increase in viscosity at high concentrations. By taking advantage of our realistic treatment, new approaches can be devised to ensure that antibody solutions exhibit the desired characteristics for their intended broad use and effective deployment.

Thermal fluctuations constantly and evenly excite all vibrational modes in an equilibrium crystal. As the temperature rises, these fluctuations promote the formation of defects and eventually melting. In active solids, the self-propulsion of "atomic" units provides another source of strong non-equilibrium fluctuations whose effect on the melting scenario is still largely unexplored. Here we show that when a colloidal crystal is activated by a bath of swimming bacteria, solvent temperature and active temperature cooperate to define dynamic and thermodynamic properties. Our system consists of repulsive paramagnetic particles confined in two dimensions and immersed in a bath of light-driven E. coli. The relative balance between fluctuations and interactions can be adjusted in two ways: by changing the strength of the magnetic field and by tuning activity with light. When the persistence time of active fluctuations is short, a single effective temperature controls both the amplitudes of vibrational modes and the melting transition. For more persistent active noise, energy equipartition is broken and multiple temperatures emerge, whereas melting occurs before the Lindemann parameter reaches its equilibrium critical value. We show that this phenomenology is fully confirmed by numerical simulations and can be framed within a minimal model of a single active particle in a periodic potential.

The Covid-19 pandemic caused disruptive effects for individuals, firms, and societies. In this paper, we offer insights on the major issues and challenges firms are facing in the Covid-19 pandemic, as well as their concerns for Corporate Social Responsibility (CSR) themes. To do so, we investigate large Italian firms' discussion on Twitter in the first nine months of the pandemic. We downloaded all Twitter posts from 1st of March, 2020, to 17th of November, 2020 by the accounts of the largest Italian firms, i.e. those with 250 or more employees. We then built the bipartite network of accounts and hashtags and, using an entropy-based null model as a benchmark, we projected the information contained in the network into the accounts layers, identifying a network of accounts in which a link indicates a non trivial similarity in terms of their usage of hashtags. We find that the conversation is focused around 13 communities, 10 of which include Covid-19 themes. The core of the network is formed of 5 communities, which deal with environmental sustainability, digital innovation and safety. Firms' ownership type does not seem to influence the conversation. 10 communities out of 13 mention hashtags related to CSR, with the environmental and social dimensions as the prevalent ones. Interestingly enough, the social dimension seems more relevant in the communities dealing with digital innovation and safety. However, the relevance of CSR hashtags is very small at the single message level, but with some peculiarities arising in specific communities. Overall, our paper highlights the role of network methods on Twitter data as a tool which can support managers and policy makers to design their strategies and decision making, capturing firms' emerging issues and relevant themes.

Synchronization of networked oscillators is known to depend fundamentally on the interplay between the dynamics of the graph's units and the microscopic arrangement of the network's structure. For non identical elements, the lack of quantitative tools has hampered so far a systematic study of the mechanisms behind such a collective behavior. We here propose an effective network whose topological properties reflect the interplay between the topology and dynamics of the original network. On that basis, we are able to introduce the "synchronization centrality", a measure which quantifies the role and importance of each network's node in the synchronization process. In particular, we use such a measure to assess the propensity of a graph to synchronize explosively, thus indicating a unified framework for most of the different models proposed so far for such an irreversible transition. Taking advantage of the predicting power of this measure, we furthermore discuss a strategy to induce the explosive behavior in a generic network, by acting only upon a small fraction of its nodes.

Percolation and synchronization are two phase transitions that have been extensively studied since already long ago. A classic result is that, in the vast majority of cases, these transitions are of the second-order type, i.e. continuous and reversible. Recently, however, explosive phenomena have been reported in com- plex networks' structure and dynamics, which rather remind first-order (discontinuous and irreversible) transitions. Explosive percolation, which was discovered in 2009, corresponds to an abrupt change in the network's structure, and explosive synchronization (which is concerned, instead, with the abrupt emergence of a collective state in the networks' dynamics) was studied as early as the first models of globally coupled phase oscillators were taken into consideration. The two phenomena have stimulated investigations and de- bates, attracting attention in many relevant fields. So far, various substantial contributions and progresses (including experimental verifications) have been made, which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations, as well as have greatly enhanced our understanding of phase transitions in networked systems. Our intention is to offer here a monographic review on the main-stream literature, with the twofold aim of summarizing the existing results and pointing out possible directions for future research.

Adaptation plays a fundamental role in shaping the structure of a complex network and improving its functional fitting. Even when increasing the level of synchronization in a biological system is considered as the main driving force for adaptation, there is evidence of negative effects induced by excessive synchronization. This indicates that coherence alone can not be enough to explain all the structural features observed in many real-world networks. In this work, we propose an adaptive network model where the dynamical evolution of the node states towards synchronization is coupled with an evolution of the link weights based on an anti-Hebbian adaptive rule, which accounts for the presence of inhibitory effects in the system. We found that the emergent networks spontaneously develop the structural conditions to sustain explosive synchronization. Our results can enlighten the shaping mechanisms at the heart of the structural and dynamical organization of some relevant biological systems, namely brain networks, for which the emergence of explosive synchronization has been observed.

All interesting and fascinating collective properties of a complex system arise from the intricate way in which its components interact. Various systems in physics, biology, social sciences and engineering have been successfully modelled as networks of coupled dynamical systems, where the graph links describe pairwise interactions. This is, however, too strong a limitation, as recent studies have revealed that higher-order many-body interactions are present in social groups, ecosystems and in the human brain, and they actually affect the emergent dynamics of all these systems. Here, we introduce a general framework that allows to study coupled dynamical systems accounting for the precise microscopic structure of their interactions at any possible order. We consider the most general ensemble of identical dynamical systems, organized on the nodes of a simplicial complex, and interacting through synchronization-non-invasive coupling function. The simplicial complex can be of any dimension, meaning that it can account, at the same time, for pairwise interactions, three-body interactions and so on. In such a broad context, we show that complete synchronization exists as an invariant solution, and we give the necessary condition for it to be observed as a stable state in terms of a Master Stability Function. This generalizes the existing results valid for pairwise interactions (i.e. graphs) to the case of complex systems with the most general possible architecture. Moreover, we show how the approach can be simplified for specific, yet frequently occurring, instances, and we verify all our theoretical predictions in synthetic and real-world systems. Given the completely general character of the method proposed, our results contribute to the theory of dynamical systems with many-body interactions and can find applications in an extremely wide range of practical cases.

A wealth of evidence shows that real world networks are endowed with the small-world property i.e., that the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. In addition, most social networks are organized so that no individual is more than six connections apart from any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultra-small world organization, whereby the graph's diameter is independent of the network size over several orders of magnitude, is still unknown. We show that the 'six degrees of separation' are the property featured by the equilibrium state of any network where individuals weigh between their aspiration to improve their centrality and the costs incurred in forming and maintaining connections. We show, moreover, that the emergence of such a regularity is compatible with all other features, such as clustering and scale-freeness, that normally characterize the structure of social networks. Thus, our results show how simple evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for the emergence of one of the most intriguing attributes of social networks.

Synchronization is a widespread phenomenon observed across natural and artificial networked systems. It often manifests itself by clusters of units exhibiting coincident dynamics. These clusters are a direct consequence of the organization of the Laplacian matrix eigenvalues into spectral localized blocks. We show how the concept of spectral blocks can be leveraged to design straightforward yet powerful controllers able to fully manipulate cluster synchronization of a generic network, thus shaping at will its parallel functioning. Specifically, we demonstrate how to induce the formation of spectral blocks in networks where such structures would not exist, and how to achieve precise mastering over the synchronizability of individual clusters by dictating the sequence in which each of them enters or exits the synchronization stability region as the coupling strength varies. Our results underscore the pivotal role of cluster synchronization control in shaping the parallel operation of networked systems, thereby enhancing their efficiency and adaptability across diverse applications.

We consider a dense assembly of repulsive particles whose fluctuating sizes are subject to an energetic landscape that defines three species: two distinct states of particles with a finite size, and point particles as an intermediate state between the two previous species. We show that the non-equilibrium synchronization of sizes systematically leads to a homogeneous configuration associated with the survival of a single species. Remarkably, the relaxation towards such a configuration features a transient phase separation. By delineating and analyzing the dominant kinetic factors at play during relaxation, we recapitulate the phase diagram of species survival in terms of the parameters of the size landscape. Finally, we obtain a hydrodynamic mapping to equilibrium by coarse-graining the microscopic dynamics, which leads to predicting the nature of the transitions between various regimes where distinct species survive.

Hollow microgels, consisting of a pNIPAM polymer network with a central cavity, have significant potential due to their tunable softness and encapsulation capabilities. Using molecular dynamics simulations, we thoroughly characterise the swelling behaviour of neutral hollow microgels across the Volume Phase Transition (VPT) upon varying crosslinker concentration, shell thickness, and size. In particular, we examine in detail the onset of cavity filling and its relation to the VPT, detecting the presence of a discontinuity in the radius of gyration of the microgels, if an appropriate balance between shell stiffness and thermoresposiveness is reached. The discontinuity is, however, absent in the behaviour of the hydrodynamic radius, in agreement with experimental observations. We then test our numerical model by direct comparison of form factors with available measurements in the literature and also establish a minimal-size, stable hollow microgel for future computationally feasible bulk investigations. Overall, our findings provide valuable insights into the fundamental swelling properties of hollow microgels that can be useful to control the opening and closing of the cavity for application purposes.

Microgels made of poly(N-isopropylacrylamide) are the prototype of soft, thermoresponsive particles widely used to study fundamental problems in condensed matter physics. However, their internal structure is far from homogeneous, and existing mean-field approaches, such as Flory-Rehner theory, provide only qualitative descriptions of their thermoresponsive behavior. Here, we combine machine learning and numerical simulations to accurately predict the concentration and spatial distribution of crosslinkers, the latter hitherto unknown experimentally, as well as the full swelling behavior of microgels, using only polymer density profiles. Our approach provides unprecedented insight into structural and thermodynamic properties of any standard microgel, including experimental ones.

In distributed systems, knowledge of the network structure of the connections among the unitary components is often a requirement for an accurate prediction of the emerging collective dynamics. However, in many real-world situations, one has, at best, access to partial connectivity data, and therefore the entire graph structure needs to be reconstructed from a limited number of observations of the dynamical processes that take place on it. While existing studies predominantly focused on reconstructing traditional pairwise networks, higher-order interactions remain largely unexplored. Here, we introduce three methods to reconstruct a simplicial complex structure of connection from observations of evolutionary games that take place on it, and demonstrate their high accuracy and excellent overall performance in synthetic and empirical complexes. The methods have different requirements and different complexity, thereby constituting a series of approaches from which one can pick the most appropriate one given the specific circumstances of the application under study.

We study a system of globally coupled FitzHugh-Nagumo oscillators, showing that the presence of higher-order interactions affects the character of the transition between synchronous and asynchronous states. In particular, we demonstrate that, around the synchronization transition, solitary states emerge due to the presence of second-order interactions. In difference to the phenomenology observed in systems of phase oscillators, we show that, at low coupling strengths, solitary states appear for both transition directions, whereas for higher couplings they only occur in the forward direction, with the backwards one characterized by explosive desynchronization.

We investigate the search of a target with a given spatial distribution in a finite one-dimensional domain. The searcher follows Brownian dynamics and is always reset to its initial position when reaching the boundaries of the domain (boundary resetting). In addition, the searcher may be reset to its initial position from any internal point of the domain (bulk resetting). Specifically, we look for the optimal strategy for bulk resetting, i.e., the spatially dependent bulk resetting rate that minimizes the average search time. The best search strategy exhibits a second-order transition from vanishing to non-vanishing bulk resetting when varying the target distribution. The obtained mathematical criteria are further analyzed for a monoparametric family of distributions, to shed light on the properties that control the optimal strategy for bulk resetting. Our work paves new research lines in the study of search processes, emphasizing the relevance of the target distribution for the optimal search strategy, and identifies a successful framework to address these questions.

Recent research has focused on understanding how cooperation is fostered through various mechanisms in cognitive settings, particularly through pairwise interactions. However, real-world interactions often extend beyond simple dyads, including multiple cliques with both pairwise and higher-order interactions. These complex interactions influence how individuals perceive and adapt their strategies based on social norms. We here introduce a model that explores the evolution of collective strategies and social norms within a heterogeneous environment, encompassing both dyadic and three-body interactions. We find that social norms play a crucial role in promoting cooperation in comparison to simply imitating the most successful neighbor. We also show that the rise of prosocial norms leads to increased cooperation across various social dilemmas, often resulting in shifts from defective to cooperative behavior. Additionally, we observe that a moderate level of information privacy helps sustaining prosocial norms and curtails antisocial tendencies, even in situations where mutual defection might seem advantageous. Our research thus offers insights into the evolution of cooperation through the lens of social norm diffusion in higher-order networks.