Robustness of coupled networks with multiple support from functional components at different scales.

Robustness is an essential component of modern network science. Here, we investigate the robustness of coupled networks where the functionality of a node depends not only on its connectivity, here measured by the size of its connected component in its own network, but also the support provided by at least M links from another network. We here develop a theoretical framework and investigate analytically and numerically the cascading failure process when the system is under attack, deriving expressions for the proportion of functional nodes in the stable state, and the critical threshold when the system collapses. Significantly, our results show an abrupt phase transition and we derive the minimum inner and inter-connectivity density necessary for the system to remain active. We also observe that the system necessitates an increased density of links inside and across networks to prevent collapse, especially when conditions on the coupling between the networks are more stringent. Finally, we discuss the importance of our results in real-world settings and their potential use to aid decision-makers design more resilient infrastructure systems.

Percolation Theories for Quantum Networks.

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network's indirect connectivity. This realization leads to the emergence of an alternative theory called "concurrence percolation", which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design.

Multiscale mixing patterns in networks.

Assortative mixing in networks is the tendency for nodes with the same attributes, or metadata, to link to each other. It is a property often found in social networks, manifesting as a higher tendency of links occurring between people of the same age, race, or political belief. Quantifying the level of assortativity or disassortativity (the preference of linking to nodes with different attributes) can shed light on the organization of complex networks. It is common practice to measure the level of assortativity according to the assortativity coefficient, or modularity in the case of categorical metadata. This global value is the average level of assortativity across the network and may not be a representative statistic when mixing patterns are heterogeneous. For example, a social network spanning the globe may exhibit local differences in mixing patterns as a consequence of differences in cultural norms. Here, we introduce an approach to localize this global measure so that we can describe the assortativity, across multiple scales, at the node level. Consequently, we are able to capture and qualitatively evaluate the distribution of mixing patterns in the network. We find that, for many real-world networks, the distribution of assortativity is skewed, overdispersed, and multimodal. Our method provides a clearer lens through which we can more closely examine mixing patterns in networks.

Dynamical exploration of the repertoire of brain networks at rest is modulated by psilocybin.

Growing evidence from the dynamical analysis of functional neuroimaging data suggests that brain function can be understood as the exploration of a repertoire of metastable connectivity patterns ('functional brain networks'), which potentially underlie different mental processes. The present study characterizes how the brain's dynamical exploration of resting-state networks is rapidly modulated by intravenous infusion of psilocybin, a tryptamine psychedelic found in "magic mushrooms". We employed a data-driven approach to characterize recurrent functional connectivity patterns by focusing on the leading eigenvector of BOLD phase coherence at single-TR resolution. Recurrent BOLD phase-locking patterns (PL states) were assessed and statistically compared pre- and post-infusion of psilocybin in terms of their probability of occurrence and transition profiles. Results were validated using a placebo session. Recurrent BOLD PL states revealed high spatial overlap with canonical resting-state networks. Notably, a PL state forming a frontoparietal subsystem was strongly destabilized after psilocybin injection, with a concomitant increase in the probability of occurrence of another PL state characterized by global BOLD phase coherence. These findings provide evidence of network-specific neuromodulation by psilocybin and represent one of the first attempts at bridging molecular pharmacodynamics and whole-brain network dynamics.

Graph partitions and cluster synchronization in networks of oscillators.

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.

Sufficient conditions of endemic threshold on metapopulation networks.

In this paper, we focus on susceptible-infected-susceptible dynamics on metapopulation networks, where nodes represent subpopulations, and where agents diffuse and interact. Recent studies suggest that heterogeneous network structure between elements plays an important role in determining the threshold of infection rate at the onset of epidemics, a fundamental quantity governing the epidemic dynamics. We consider the general case in which the infection rate at each node depends on its population size, as shown in recent empirical observations. We first prove that a sufficient condition for the endemic threshold (i.e., its upper bound), previously derived based on a mean-field approximation of network structure, also holds true for arbitrary networks. We also derive an improved condition showing that networks with the rich-club property (i.e., high connectivity between nodes with a large number of links) are more prone to disease spreading. The dependency of infection rate on population size introduces a considerable difference between this upper bound and estimates based on mean-field approximations, even when degree-degree correlations are considered. We verify the theoretical results with numerical simulations.

The Non-linear Health Consequences of Living in Larger Cities.

Urbanization promotes economy, mobility, access, and availability of resources, but on the other hand, generates higher levels of pollution, violence, crime, and mental distress. The health consequences of the agglomeration of people living close together are not fully understood. Particularly, it remains unclear how variations in the population size across cities impact the health of the population. We analyze the deviations from linearity of the scaling of several health-related quantities, such as the incidence and mortality of diseases, external causes of death, wellbeing, and health care availability, in respect to the population size of cities in Brazil, Sweden, and the USA. We find that deaths by non-communicable diseases tend to be relatively less common in larger cities, whereas the per capita incidence of infectious diseases is relatively larger for increasing population size. Healthier lifestyle and availability of medical support are disproportionally higher in larger cities. The results are connected with the optimization of human and physical resources and with the non-linear effects of social networks in larger populations. An urban advantage in terms of health is not evident, and using rates as indicators to compare cities with different population sizes may be insufficient.

Uncovering space-independent communities in spatial networks.

Many complex systems are organized in the form of a network embedded in space. Important examples include the physical Internet infrastructure, road networks, flight connections, brain functional networks, and social networks. The effect of space on network topology has recently come under the spotlight because of the emergence of pervasive technologies based on geolocalization, which constantly fill databases with people's movements and thus reveal their trajectories and spatial behavior. Extracting patterns and regularities from the resulting massive amount of human mobility data requires the development of appropriate tools for uncovering information in spatially embedded networks. In contrast with most works that tend to apply standard network metrics to any type of network, we argue in this paper for a careful treatment of the constraints imposed by space on network topology. In particular, we focus on the problem of community detection and propose a modularity function adapted to spatial networks. We show that it is possible to factor out the effect of space in order to reveal more clearly hidden structural similarities between the nodes. Methods are tested on a large mobile phone network and computer-generated benchmarks where the effect of space has been incorporated.

Multirelational organization of large-scale social networks in an online world.

The capacity to collect fingerprints of individuals in online media has revolutionized the way researchers explore human society. Social systems can be seen as a nonlinear superposition of a multitude of complex social networks, where nodes represent individuals and links capture a variety of different social relations. Much emphasis has been put on the network topology of social interactions, however, the multidimensional nature of these interactions has largely been ignored, mostly because of lack of data. Here, for the first time, we analyze a complete, multirelational, large social network of a society consisting of the 300,000 odd players of a massive multiplayer online game. We extract networks of six different types of one-to-one interactions between the players. Three of them carry a positive connotation (friendship, communication, trade), three a negative (enmity, armed aggression, punishment). We first analyze these types of networks as separate entities and find that negative interactions differ from positive interactions by their lower reciprocity, weaker clustering, and fatter-tail degree distribution. We then explore how the interdependence of different network types determines the organization of the social system. In particular, we study correlations and overlap between different types of links and demonstrate the tendency of individuals to play different roles in different networks. As a demonstration of the power of the approach, we present the first empirical large-scale verification of the long-standing structural balance theory, by focusing on the specific multiplex network of friendship and enmity relations.

Exploring QSAR models for activity-cliff prediction.

INTRODUCTION AND METHODOLOGY: Pairs of similar compounds that only differ by a small structural modification but exhibit a large difference in their binding affinity for a given target are known as activity cliffs (ACs). It has been hypothesised that QSAR models struggle to predict ACs and that ACs thus form a major source of prediction error. However, the AC-prediction power of modern QSAR methods and its quantitative relationship to general QSAR-prediction performance is still underexplored. We systematically construct nine distinct QSAR models by combining three molecular representation methods (extended-connectivity fingerprints, physicochemical-descriptor vectors and graph isomorphism networks) with three regression techniques (random forests, k-nearest neighbours and multilayer perceptrons); we then use each resulting model to classify pairs of similar compounds as ACs or non-ACs and to predict the activities of individual molecules in three case studies: dopamine receptor D2, factor Xa, and SARS-CoV-2 main protease. RESULTS AND CONCLUSIONS: Our results provide strong support for the hypothesis that indeed QSAR models frequently fail to predict ACs. We observe low AC-sensitivity amongst the evaluated models when the activities of both compounds are unknown, but a substantial increase in AC-sensitivity when the actual activity of one of the compounds is given. Graph isomorphism features are found to be competitive with or superior to classical molecular representations for AC-classification and can thus be employed as baseline AC-prediction models or simple compound-optimisation tools. For general QSAR-prediction, however, extended-connectivity fingerprints still consistently deliver the best performance amongs the tested input representations. A potential future pathway to improve QSAR-modelling performance might be the development of techniques to increase AC-sensitivity.

The discovery of population differences in network community structure: new methods and applications to brain functional networks in schizophrenia.

The modular organization of the brain network can vary in two fundamental ways. The amount of inter- versus intra-modular connections between network nodes can be altered, or the community structure itself can be perturbed, in terms of which nodes belong to which modules (or communities). Alterations have previously been reported in modularity, which is a function of the proportion of intra-modular edges over all modules in the network. For example, we have reported that modularity is decreased in functional brain networks in schizophrenia: There are proportionally more inter-modular edges and fewer intra-modular edges. However, despite numerous and increasing studies of brain modular organization, it is not known how to test for differences in the community structure, i.e., the assignment of regional nodes to specific modules. Here, we introduce a method based on the normalized mutual information between pairs of modular networks to show that the community structure of the brain network is significantly altered in schizophrenia, using resting-state fMRI in 19 participants with childhood-onset schizophrenia and 20 healthy participants. We also develop tools to show which specific nodes (or brain regions) have significantly different modular communities between groups, a subset that includes right insular and perisylvian cortical regions. The methods that we propose are broadly applicable to other experimental contexts, both in neuroimaging and other areas of network science.

Unifying information propagation models on networks and influence maximization.

Information propagation on networks is a central theme in social, behavioral, and economic sciences, with important theoretical and practical implications, such as the influence maximization problem for viral marketing. Here we consider a model that unifies the classical independent cascade models and the linear threshold models, and generalize them by considering continuous variables and allowing feedback in the dynamics. We then formulate its influence maximization as a mixed integer nonlinear programming problem and adopt derivative-free methods. Furthermore, we show that the problem can be exactly solved in the special case of linear dynamics, where the selection criterion is closely related to the Katz centrality, and propose a customized direct search method with local convergence. We then demonstrate the close to optimal performance of the customized direct search numerically on both synthetic and real networks.

Flow stability for dynamic community detection.

Many systems exhibit complex temporal dynamics due to the presence of different processes taking place simultaneously. An important task in these systems is to extract a simplified view of their time-dependent network of interactions. Community detection in temporal networks usually relies on aggregation over time windows or consider sequences of different stationary epochs. For dynamics-based methods, attempts to generalize static-network methodologies also face the fundamental difficulty that a stationary state of the dynamics does not always exist. Here, we derive a method based on a dynamical process evolving on the temporal network. Our method allows dynamics that do not reach a steady state and uncovers two sets of communities for a given time interval that accounts for the ordering of edges in forward and backward time. We show that our method provides a natural way to disentangle the different dynamical scales present in a system with synthetic and real-world examples.

Brexit and bots: characterizing the behaviour of automated accounts on Twitter during the UK election.

Online Social Networks (OSNs) offer new means for political communications that have quickly begun to play crucial roles in political campaigns, due to their pervasiveness and communication speed. However, the OSN environment is quite slippery and hides potential risks: many studies presented evidence about the presence of d/misinformation campaigns and malicious activities by genuine or automated users, putting at severe risk the efficiency of online and offline political campaigns. This phenomenon is particularly evident during crucial political events, as political elections. In the present paper, we provide a comprehensive description of the networks of interactions among users and bots during the UK elections of 2019. In particular, we focus on the polarised discussion about Brexit on Twitter, analysing a data set made of more than 10 millions tweets posted for over a month. We found that the presence of automated accounts infected the debate particularly in the days before the UK national elections, in which we find a steep increase of bots in the discussion; in the days after the election day, their incidence returned to values similar to the ones observed few weeks before the elections. On the other hand, we found that the number of suspended users (i.e. accounts that were removed by the platform for some violation of the Twitter policy) remained constant until the election day, after which it reached significantly higher values. Remarkably, after the TV debate between Boris Johnson and Jeremy Corbyn, we observed the injection of a large number of novel bots whose behaviour is markedly different from that of pre-existing ones. Finally, we explored the bots' political orientation, finding that their activity is spread across the whole political spectrum, although in different proportions, and we studied the different usage of hashtags and URLs by automated accounts and suspended users, targeting the formation of common narratives in different sides of the debate.

Gromov centrality: A multiscale measure of network centrality using triangle inequality excess.

Centrality measures quantify the importance of a node in a network based on different geometric or diffusive properties, and focus on different scales. Here, we adopt a geometrical viewpoint to define a multiscale centrality in networks. Given a metric distance between the nodes, we measure the centrality of a node by its tendency to be close to geodesics between nodes in its neighborhood, via the concept of triangle inequality excess. Depending on the size of the neighborhood, the resulting Gromov centrality defines the importance of a node at different scales in the graph, and it recovers as limits well-known concepts such as the clustering coefficient and closeness centrality. We argue that Gromov centrality is affected by the geometric and boundary constraints of the network, and illustrate how it can help distinguish different types of nodes in random geometric graphs and empirical transportation networks.

Consensus from group interactions: An adaptive voter model on hypergraphs.

We study the effect of group interactions on the emergence of consensus in a spin system. Agents with discrete opinions {0,1} form groups. They can change their opinion based on their group's influence (voter dynamics), but groups can also split and merge (adaptation). In a hypergraph, these groups are represented by hyperedges of different sizes. The heterogeneity of group sizes is controlled by a parameter ß. To study the impact of ß on reaching consensus, we provide extensive computer simulations and compare them with an analytic approach for the dynamics of the average magnetization. We find that group interactions amplify small initial opinion biases, accelerate the formation of consensus, and lead to a drift of the average magnetization. The conservation of the initial magnetization, known for basic voter models, is no longer obtained.

Metastable oscillatory modes emerge from synchronization in the brain spacetime connectome.

A rich repertoire of oscillatory signals is detected from human brains with electro- and magnetoencephalography (EEG/MEG). However, the principles underwriting coherent oscillations and their link with neural activity remain under debate. Here, we revisit the mechanistic hypothesis that transient brain rhythms are a signature of metastable synchronization, occurring at reduced collective frequencies due to delays between brain areas. We consider a system of damped oscillators in the presence of background noise - approximating the short-lived gamma-frequency oscillations generated within neuronal circuits - coupled according to the diffusion weighted tractography between brain areas. Varying the global coupling strength and conduction speed, we identify a critical regime where spatially and spectrally resolved metastable oscillatory modes (MOMs) emerge at sub-gamma frequencies, approximating the MEG power spectra from 89 healthy individuals at rest. Further, we demonstrate that the frequency, duration, and scale of MOMs - as well as the frequency-specific envelope functional connectivity - can be controlled by global parameters, while the connectome structure remains unchanged. Grounded in the physics of delay-coupled oscillators, these numerical analyses demonstrate how interactions between locally generated fast oscillations in the connectome spacetime structure can lead to the emergence of collective brain rhythms organized in space and time.

Characterization of the anterior cingulate's role in the at-risk mental state using graph theory.

The onset of positive symptoms in schizophrenia is often preceded by a prodromal phase characterized by neurocognitive abnormalities as well as changes in brain structure and function. Increasing efforts have been made to identify individuals at elevated risk of developing schizophrenia, as early intervention may help prevent progression towards psychosis. The present study uses functional MRI and graph theoretical analysis to characterize the organization of a functional brain network in at-risk mental state patients with varying symptoms assessed with the PANSS and healthy volunteers during performance of a verbal fluency task known to recruit frontal lobe networks and to be impaired in psychosis. We first examined between-groups differences in total network connectivity and global network compactness/efficiency. We then addressed the role of specific brain regions in the network organization by calculating the node-specific "betweeness centrality", "degree centrality" and "local average path length" metrics; different ways of assessing a region's importance in a network. We focused our analysis on the anterior cingulate cortex (ACC); a region known to support executive function that is structurally and functionally impaired in at-risk mental state patients. Although global network connectivity and efficiency were maintained in at-risk patients relative to the controls, we report a significant decrease in the contribution of the ACC to task-relevant network organization in at risk subjects with elevated symptoms (PANSS ≥ 45) relative to both the controls and the less symptomatic at-risk subjects, as reflected by a reduction in the topological centrality of the ACC. These findings provide evidence of network abnormalities and anterior cingulate cortex dysfunction in people with prodromal signs of schizophrenia.

Dynamics of majority rule on hypergraphs.

A broad range of dynamical systems involve multibody interactions, or group interactions, which may not be encoded in traditional graphical structures. In this work, we focus on a canonical example from opinion dynamics, namely the majority rule, and we investigate the possibility to represent and analyze the system by means of hypergraphs. We explore the formation of consensus, and we restrict our attention to interaction groups of size 3 in order to simplify our analysis from a combinatorial perspective. We propose different types of hypergraph models, incorporating modular structure or mean-field heterogeneity, and we recast the dynamics in terms of Fokker-Planck equations, which allows us to predict the transient dynamics toward consensus. Numerical simulations show a very good agreement between the stochastic dynamics and theoretical predictions for large population sizes.