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Non-dyadic higher-order interactions affect collective behavior in various networked dynamical systems. Here we discuss the properties of a novel Ising model with higher-order interactions and characterize its phase transitions between the ordered and the disordered phase. By a mean-field treatment, we show that the transition is continuous when only three-body interactions are considered but becomes abrupt when interactions of higher orders are introduced. Using a Georges-Yedidia expansion to go beyond a na\"ive mean-field approximation, we reveal a quantitative shift in the critical point of the phase transition, which does not affect the universality class of the model. Finally, we compare our results with traditional $p$-spin models with many-body interactions. Our work unveils new collective phenomena on complex interacting systems, revealing the importance of investigating higher-order systems beyond three-body interactions.

Higher-order properties of functional magnetic resonance imaging (fMRI) induced connectivity have been shown to unravel many exclusive topological and dynamical insights beyond pairwise interactions. Nonetheless, whether these fMRI-induced higher-order properties play a role in disentangling other neuroimaging modalities' insights remains largely unexplored and poorly understood. In this work, by analyzing fMRI data from the Human Connectome Project Young Adult dataset using persistent homology, we discovered that the volume-optimal persistence homological scaffolds of fMRI-based functional connectomes exhibited conservative topological reconfigurations from the resting state to attentional task-positive state. Specifically, while reflecting the extent to which each cortical region contributed to functional cycles following different cognitive demands, these reconfigurations were constrained such that the spatial distribution of cavities in the connectome is relatively conserved. Most importantly, such level of contributions covaried with powers of aperiodic activities mostly within the theta-alpha (4-12 Hz) band measured by magnetoencephalography (MEG). This comprehensive result suggests that fMRI-induced hemodynamics and MEG theta-alpha aperiodic activities are governed by the same functional constraints specific to each cortical morpho-structure. Methodologically, our work paves the way toward an innovative computing paradigm in multimodal neuroimaging topological learning., Comment: The code for our analyses is provided in https://github.com/ngcaonghi/scaffold_noise

Despite the widespread adoption of higher-order mathematical structures such as hypergraphs, methodological tools for their analysis lag behind those for traditional graphs. This work addresses a critical gap in this context by proposing two micro-canonical random null models for directed hypergraphs: the Directed Hypergraph Configuration Model (DHCM) and the Directed Hypergraph JOINT Model (DHJM). These models preserve essential structural properties of directed hypergraphs such as node in- and out-degree sequences and hyperedge head and tail size sequences, or their joint tensor. We also describe two efficient MCMC algorithms, NuDHy-Degs and NuDHy-JOINT, to sample random hypergraphs from these ensembles. To showcase the interdisciplinary applicability of the proposed null models, we present three distinct use cases in sociology, epidemiology, and economics. First, we reveal the oscillatory behavior of increased homophily in opposition parties in the US Congress over a 40-year span, emphasizing the role of higher-order structures in quantifying political group homophily. Second, we investigate non-linear contagion in contact hyper-networks, demonstrating that disparities between simulations and theoretical predictions can be explained by considering higher-order joint degree distributions. Last, we examine the economic complexity of countries in the global trade network, showing that local network properties preserved by NuDHy explain the main structural economic complexity indexes. This work advances the development of null models for directed hypergraphs, addressing the intricate challenges posed by their complex entity relations, and providing a versatile suite of tools for researchers across various domains., Comment: Accepted for publication in PRX

The richness of many complex systems stems from the interactions among their components. The higher-order nature of these interactions, involving many units at once, and their temporal dynamics constitute crucial properties that shape the behaviour of the system itself. An adequate description of these systems is offered by temporal hypergraphs, that integrate these features within the same framework. However, tools for their temporal and topological characterization are still scarce. Here we develop a series of methods specifically designed to analyse the structural properties of temporal hypergraphs at multiple scales. Leveraging the hyper-core decomposition of hypergraphs, we follow the evolution of the hyper-cores through time, characterizing the hypergraph structure and its temporal dynamics at different topological scales, and quantifying the multi-scale structural stability of the system. We also define two static hypercoreness centrality measures that provide an overall description of the nodes aggregated structural behaviour. We apply the characterization methods to several data sets, establishing connections between structural properties and specific activities within the systems. Finally, we show how the proposed method can be used as a model-validation tool for synthetic temporal hypergraphs, distinguishing the higher-order structures and dynamics generated by different models from the empirical ones, and thus identifying the essential model mechanisms to reproduce the empirical hypergraph structure and evolution. Our work opens several research directions, from the understanding of dynamic processes on temporal higher-order networks to the design of new models of time-varying hypergraphs., Comment: Main document: 21 pages, 10 figures; Supplementary Material: 52 pages, 47 figures

The interplay between causal mechanisms and emerging collective behaviors is a central aspect of understanding, controlling, and predicting complex networked systems. In our work, we investigate the relationship between higher-order mechanisms and higher-order behavioral observables in two representative models with group interactions: a simplicial Ising model and a social contagion model. In both systems, we find that group (higher-order) interactions show emergent synergistic (higher-order) behavior. The emergent synergy appears only at the group level and depends in a complex, non-linear way on the trade-off between the strengths of the low- and higher-order mechanisms and is invisible to low-order behavioral observables. Our work sets the basis for systematically investigating the relation between causal mechanisms and behavioral patterns in complex networked systems with group interactions, offering a robust methodological framework to tackle this challenging task.

We propose a cross-order Laplacian renormalization group (X-LRG) scheme for arbitrary higher-order networks. The renormalization group is a pillar of the theory of scaling, scale-invariance, and universality in physics. An RG scheme based on diffusion dynamics was recently introduced for complex networks with dyadic interactions. Despite mounting evidence of the importance of polyadic interactions, we still lack a general RG scheme for higher-order networks. Our approach uses a diffusion process to group nodes or simplices, where information can flow between nodes and between simplices (higher-order interactions). This approach allows us (i) to probe higher-order structures, defining scale-invariance at various orders, and (ii) to propose a coarse-graining scheme. We demonstrate our approach on controlled synthetic higher-order systems and then use it to detect the presence of order-specific scale-invariant profiles of real-world complex systems from multiple domains.

The bulk of the research effort on brain connectivity revolves around statistical associations among brain regions, which do not directly relate to the causal mechanisms governing brain dynamics. Here we propose the multiscale causal backbone (MCB) of brain dynamics, shared by a set of individuals across multiple temporal scales, and devise a principled methodology to extract it. Our approach leverages recent advances in multiscale causal structure learning and optimizes the trade-off between the model fit and its complexity. Empirical assessment on synthetic data shows the superiority of our methodology over a baseline based on canonical functional connectivity networks. When applied to resting-state fMRI data, we find sparse MCBs for both the left and right brain hemispheres. Thanks to its multiscale nature, our approach shows that at low-frequency bands, causal dynamics are driven by brain regions associated with high-level cognitive functions; at higher frequencies instead, nodes related to sensory processing play a crucial role. Finally, our analysis of individual multiscale causal structures confirms the existence of a causal fingerprint of brain connectivity, thus supporting the existing extensive research in brain connectivity fingerprinting from a causal perspective., Comment: Accepted at the 3rd conference on Causal Learning and Reasoning (CLeaR 2024)

A key challenge of nonlinear dynamics and network science is to understand how higher-order interactions influence collective dynamics. Although many studies have approached this question through linear stability analysis, less is known about how higher-order interactions shape the global organization of different states. Here, we shed light on this issue by analyzing the rich patterns supported by identical Kuramoto oscillators on hypergraphs. We show that higher-order interactions can have opposite effects on linear stability and basin stability: they stabilize twisted states (including full synchrony) by improving their linear stability, but also make them hard to find by dramatically reducing their basin size. Our results highlight the importance of understanding higher-order interactions from both local and global perspectives., Comment: Published version

We provide quantitative evidence suggesting social learning in sperm whales across socio-cultural boundaries, using acoustic data from the Pacific and Atlantic Oceans. Traditionally, sperm whale populations are categorized into clans based on their vocal repertoire: the rhythmically patterned click sequences (codas) that they use. Among these codas, identity codas function as symbolic markers for each clan, accounting for 35-60% of codas they produce. We introduce a computational method to model whale speech, which encodes rhythmic micro-variations within codas, capturing their vocal style. We find that vocal style-clans closely align with repertoire-clans. However, contrary to vocal repertoire, we show that sympatry increases vocal style similarity between clans for non-identity codas, i.e. most codas, suggesting social learning across cultural boundaries. More broadly, this subcoda structure model offers a framework for comparing communication systems in other species, with potential implications for deeper understanding of vocal and cultural transmission within animal societies.

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. Our framework is based on topology, discrete differential geometry as well as gradient flows and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models., Comment: 36 pages, 11 figures

Going beyond networks, to include higher-order interactions of arbitrary sizes, is a major step to better describe complex systems. In the resulting hypergraph representation, tools to identify structures and central nodes are scarce. We consider the decomposition of a hypergraph in hyper-cores, subsets of nodes connected by at least a certain number of hyperedges of at least a certain size. We show that this provides a fingerprint for data described by hypergraphs and suggests a novel notion of centrality, the hyper-coreness. We assess the role of hyper-cores and nodes with large hyper-coreness in higher-order dynamical processes: such nodes have large spreading power and spreading processes are localized in central hyper-cores. Additionally, in the emergence of social conventions very few committed individuals with high hyper-coreness can rapidly overturn a majority convention. Our work opens multiple research avenues, from comparing empirical data to model validation and study of temporally varying hypergraphs., Comment: Main document: 13 pages, 6 figures; Supplementary Material: 47 pages, 40 figures

Network theory has primarily focused on pairwise relationships, disregarding many-body interactions: neglecting them, however, can lead to misleading representations of complex systems. Hypergraphs represent an increasingly popular alternative for describing polyadic interactions: our innovation lies in leveraging the representation of hypergraphs based on the incidence matrix for extending the entropy-based framework to higher-order structures. In analogy with the Exponential Random Graphs, we name the members of this novel class of models Exponential Random Hypergraphs. Here, we focus on two explicit examples, i.e. the generalisations of the Erd\"os-R\'enyi Model and of the Configuration Model. After discussing their asymptotic properties, we employ them to analyse real-world configurations: more specifically, i) we extend the definition of several network quantities to hypergraphs, ii) compute their expected value under each null model and iii) compare it with the empirical one, in order to detect deviations from random behaviours. Differently from currently available techniques, ours is analytically tractable, scalable and effective in singling out the structural patterns of real-world hypergraphs differing significantly from those emerging as a consequence of simpler, structural constraints., Comment: 27 pages, 11 figures, 4 tables

Single contagion processes are known to display a continuous transition from an epidemic-free phase at low contagion rates to the epidemic state for rates above a critical threshold. This transition can become discontinuous when two simple contagion processes are coupled in a bi-directional symmetric way. However, in many cases, the coupling is not symmetric and the processes can be of a different nature. For example, risky social behaviors -- such as not wearing masks or engaging in large gatherings -- can affect the spread of a disease, and their adoption dynamics via social reinforcement mechanisms are better described by complex contagion models, rather than by the simple contagion paradigm, which is more appropriate for disease spreading phenomena. Motivated by this example, we consider a simplicial contagion (describing the adoption of a behavior) that uni-directionally drives a simple contagion (describing a disease propagation). We show that, above a critical driving strength, such driven simple contagion can exhibit both discontinuous transitions and bi-stability, which are instead absent in standard simple contagions. We provide a mean-field analytical description of the phase diagram of the system, and complement the results with Markov-chain simulations. Our results provide a novel route for a simple contagion process to display the phenomenology of a higher-order contagion, through a driving mechanism that may be hidden or unobservable in many practical instances., Comment: 10 pages, 7 figures. I.I. and M.L. contributed equally to this work. A.B and G.P. jointly supervised this work

Time series analysis has proven to be a powerful method to characterize several phenomena in biology, neuroscience and economics, and to understand some of their underlying dynamical features. Despite a plethora of methods have been proposed for the analysis of multivariate time series, most of them neglect the effect of non-pairwise interactions on the emerging dynamics. Here, we propose a novel framework to characterize the temporal evolution of higher-order dependencies within multivariate time series. Using network analysis and topology, we show that, unlike traditional tools based on pairwise statistics, our framework robustly differentiates various spatiotemporal regimes of coupled chaotic maps, including chaotic dynamical phases and various types of synchronization. Hence, using the higher-order co-fluctuation patterns in simulated dynamical processes as a guide, we highlight and quantify signatures of higher-order patterns in data from brain functional activity, financial markets, and epidemics. Overall, our approach sheds new light on the higher-order organization of multivariate time series, allowing a better characterization of dynamical group dependencies inherent to real-world data., Comment: 16 pages, 5 figures. Supplementary Information (16 figures, 2 tables)

Although ubiquitous, interactions of groups of individuals (e.g., modern messaging applications, group meetings, or even a parliament discussion) are not yet thoroughly studied. Frequently, single-groups are modeled as critical-mass dynamics, which is a widespread concept used not only by academics but also by politicians and the media. However, less explored questions are how a collection of groups will behave and how the intersection between these groups might change the global dynamics. Here, we formulate this process in terms of binary state dynamics on hypergraphs. We showed that our model has a very rich and unexpected behavior that goes beyond discontinuous transitions. In particular, we might have multistability and intermittency as a consequence of bimodal state distributions. By using artificial random models, we demonstrated that this phenomenology could be associated with community structures. Specifically, we might have multistability or intermittency by controlling the number of bridges between two communities with different densities. The introduction of bridges destroys multistability but creates an intermittent behavior. Furthermore, we provide an analytical formulation showing that the observed pattern for the order parameter and susceptibility are compatible with hybrid phase transitions. Our findings open new paths for research, ranging from physics, on the formal calculation of quantities of interest, to social sciences, where new experiments can be designed.

We formulate a general Kuramoto model on weighted simplicial complexes where phases oscillators are supported on simplices of any order $k$. Crucially, we introduce linear and non-linear frustration terms that are independent of the orientation of the $k+1$ simplices, providing a natural generalization of the Sakaguchi-Kuramoto model. In turn, this provides a generalized formulation of the Kuramoto higher-order parameter as a potential function to write the dynamics as a gradient flow. With a selection of simplicial complexes of increasingly complex structure, we study the properties of the dynamics of the simplicial Sakaguchi-Kuramoto model with oscillators on edges to highlight the complexity of dynamical behaviors emerging from even simple simplicial complexes. We place ourselves in the case where the vector of internal frequencies of the edge oscillators lies in the kernel of the Hodge Laplacian, or vanishing linear frustration, and, using the Hodge decomposition of the solution, we understand how the nonlinear frustration couples the dynamics in orthogonal subspaces. We discover various dynamical phenomena, such as the partial loss of synchronization in subspaces aligned with the Hodge subspaces and the emergence of simplicial phase re-locking in regimes of high frustration.

Complex networks have become the main paradigm for modelling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by higher-order interactions involving groups of three or more units. Higher-order structures, such as hypergraphs and simplicial complexes, are therefore a better tool to map the real organization of many social, biological and man-made systems. Here, we highlight recent evidence of collective behaviours induced by higher-order interactions, and we outline three key challenges for the physics of higher-order systems., Comment: pre-peer-reviewed version of the Nature Physics perspective, 7 pages, 4 figures

Several biological and social contagion phenomena, such as superspreading events or social reinforcement, are the results of multi-body interactions, for which hypergraphs offer a natural mathematical description. In this paper, we develop a novel mathematical framework based on approximate master equations to study contagions on random hypergraphs with a heterogeneous structure, both in terms of group size (hyperedge cardinality) and of membership of nodes to groups (hyperdegree). The characterization of the inner dynamics of groups provides an accurate description of the contagion process, without losing the analytical tractability. Using a contagion model where multi-body interactions are mapped onto a nonlinear infection rate, our two main results show how large groups are influential, in the sense that they drive both the early spread of a contagion and its endemic state (i.e., its stationary state). First, we provide a detailed characterization of the phase transition, which can be continuous or discontinuous with a bistable regime, and derive analytical expressions for the critical and tricritical points. We find that large values of the third moment of the membership distribution suppress the emergence of a discontinuous phase transition. Furthermore, the combination of heterogeneous group sizes and nonlinear contagion facilitates the onset of a mesoscopic localization phase, where contagion is sustained only by the largest groups, thereby inhibiting bistability as well. Second, we formulate the problem of optimal seeding for hypergraph contagion, and we compare two strategies: tuning the allocation of seeds according to either individual node or group properties. We find that, when the contagion is sufficiently nonlinear, groups are more effective seeds of contagion than individual nodes., Comment: 23 pages, 12 figures (main text); 6 pages, 3 figures (supplementary information)

The past decade has witnessed a groundbreaking rise of machine learning for human language analysis, with current methods capable of automatically accurately recovering various aspects of syntax and semantics - including sentence structure and grounded word meaning - from large data collections. Recent research showed the promise of such tools for analyzing acoustic communication in nonhuman species. We posit that machine learning will be the cornerstone of future collection, processing, and analysis of multimodal streams of data in animal communication studies, including bioacoustic, behavioral, biological, and environmental data. Cetaceans are unique non-human model species as they possess sophisticated acoustic communications, but utilize a very different encoding system that evolved in an aquatic rather than terrestrial medium. Sperm whales, in particular, with their highly-developed neuroanatomical features, cognitive abilities, social structures, and discrete click-based encoding make for an excellent starting point for advanced machine learning tools that can be applied to other animals in the future. This paper details a roadmap toward this goal based on currently existing technology and multidisciplinary scientific community effort. We outline the key elements required for the collection and processing of massive bioacoustic data of sperm whales, detecting their basic communication units and language-like higher-level structures, and validating these models through interactive playback experiments. The technological capabilities developed by such an undertaking are likely to yield cross-applications and advancements in broader communities investigating non-human communication and animal behavioral research.

How can minorities of individuals overturn social conventions? The theory of critical mass states that when a committed minority reaches a critical size, a cascade of behavioural changes can occur, overturning apparently stable social norms. Evidence comes from theoretical and empirical studies in which minorities of very different sizes, including extremely small ones, manage to bring a system to its tipping point. Here, we explore this diversity of scenarios by introducing group interactions as a crucial element of realism into a model for social convention. We find that the critical mass necessary to trigger behaviour change can be very small if individuals have a limited propensity to change their views. Moreover, the ability of the committed minority to overturn existing norms depends in a complex way on the group size. Our findings reconcile the different sizes of critical mass found in previous investigations and unveil the critical role of groups in such a process. This further highlights the importance of the emerging field of higher-order networks, beyond pairwise interactions, Comment: 10 pages, 5 figures, Supplementary Material (13 pages, 12 figures)

Networks can describe the structure of a wide variety of complex systems by specifying which pairs of entities in the system are connected. While such pairwise representations are flexible, they are not necessarily appropriate when the fundamental interactions involve more than two entities at the same time. Pairwise representations nonetheless remain ubiquitous, because higher-order interactions are often not recorded explicitly in network data. Here, we introduce a Bayesian approach to reconstruct latent higher-order interactions from ordinary pairwise network data. Our method is based on the principle of parsimony and only includes higher-order structures when there is sufficient statistical evidence for them. We demonstrate its applicability to a wide range of datasets, both synthetic and empirical., Comment: 13 pages, 7 figures. Code is available at https://graph-tool.skewed.de/

The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, in face-to-face human communication, chemical reactions and ecological systems, interactions can occur in groups of three or more nodes and cannot be simply described just in terms of simple dyads. Until recently, little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can greatly enhance our modeling capacities and help us to understand and predict their emerging dynamical behaviors. Here, we present a complete overview of the emerging field of networks beyond pairwise interactions. We first discuss the methods to represent higher-order interactions and give a unified presentation of the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review the measures designed to characterize the structure of these systems and the models proposed in the literature to generate synthetic structures, such as random and growing simplicial complexes, bipartite graphs and hypergraphs. We introduce and discuss the rapidly growing research on higher-order dynamical systems and on dynamical topology. We focus on novel emergent phenomena characterizing landmark dynamical processes, such as diffusion, spreading, synchronization and games, when extended beyond pairwise interactions. We elucidate the relations between higher-order topology and dynamical properties, and conclude with a summary of empirical applications, providing an outlook on current modeling and conceptual frontiers., Comment: Accepted for publication in Physics Reports. 109 pages, 47 figures

The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global features onto individual network components, unless one provides a principled way to make such a choice. In this paper, we apply recent advances in the computation of minimal homology bases to introduce a quasi-canonical version of the scaffold, called minimal, and employ it to analyze data both real and in silico. At the same time, we verify that, statistically, the standard scaffold is a good proxy of the minimal one for sufficiently complex networks., Comment: 25 pages, 9 figures

Our understanding of the dynamics of complex networked systems has increased significantly in the last two decades. However, most of our knowledge is built upon assuming pairwise relations among the system's components. This is often an oversimplification, for instance, in social interactions that occur frequently within groups. To overcome this limitation, here we study the dynamics of social contagion on hypergraphs. We develop an analytical framework and provide numerical results for arbitrary hypergraphs, which we also support with Monte Carlo simulations. Our analyses show that the model has a vast parameter space, with first and second-order transitions, bi-stability, and hysteresis. Phenomenologically, we also extend the concept of latent heat to social contexts, which might help understanding oscillatory social behaviors. Our work unfolds the research line of higher-order models and the analytical treatment of hypergraphs, posing new questions and paving the way for modeling dynamical processes on these networks., Comment: 17 pages, including 14 figures

Topological representations are rapidly becoming a popular way to capture and encode higher-order interactions in complex systems. They have found applications in disciplines as different as cancer genomics, brain function, and computational social science, in representing both descriptive features of data and inference models. While intense research has focused on the connectivity and homological features of topological representations, surprisingly scarce attention has been given to the investigation of the community structures of simplicial complexes. To this end, we adopt recent advances in symbolic embeddings to compute and visualize the community structures of simplicial complexes. We first investigate the stability properties of embedding obtained for synthetic simplicial complexes to the presence of higher order interactions. We then focus on complexes arising from social and brain functional data and show how higher order interactions can be leveraged to improve clustering detection and assess the effect of higher order interaction on individual nodes. We conclude delineating limitations and directions for extension of this work.

We study a general epidemic model with arbitrary recovery rate distributions. This simple deviation from the standard setup is sufficient to prove that heterogeneity in the dynamical parameters can be as important as the more studied structural heterogeneity. Our analytical solution is able to predict the shift in the critical properties induced by heterogeneous recovery rates. Additionally, we show that the critical value of infectivity tends to be smaller than the one predicted by quenched mean-field approaches in the homogeneous case and that it can be linked to the variance of the recovery rates. We then illustrate the role of dynamical--structural correlations, which allow for a complete change in the critical behavior. We show that it is possible for a power-law network topology to behave similarly to a homogeneous structure by an appropriate tuning of its recovery rates, and vice versa. Finally, we show how heterogeneity in recovery rates affects the network localization properties of the spreading process., Comment: Submitted for publication. 6 pages, including 4 figures

Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems., Comment: 32 pages, 4 figures, 1 table, Supplementary Information

Many complex systems find a convenient representation in terms of networks: structures made by pairwise interactions (links) of elements (nodes). For many biological and social systems, elementary interactions involve however more than two elements, and simplicial complexes are more adequate to describe such phenomena. Moreover, these interactions often change over time. Here, we propose a framework to model such an evolution: the Simplicial Activity Driven (SAD) model, in which the building block is a simplex of nodes representing a multi-agent interaction. We show analytically and numerically that the use of simplicial structures leads to crucial differences in the outcome of paradigmatic processes modelling disease propagation or social contagion, with respect to the activity-driven (AD) model, a paradigmatic temporal network model involving only binary interactions. In particular, fluctuations in the number of nodes involved in the interactions can affect the outcome of models of simple contagion processes, contrarily to what happens in the AD model. Moreover, social contagion models such as cascading processes present a much richer phenomenology and can become extremely slow when occurring on evolving simplicial complexes., Comment: 20 pages, 3 figures (11 figures in supplementary)

The ability to learn new tasks and generalize performance to others is one of the most remarkable characteristics of the human brain and of recent AI systems. The ability to perform multiple tasks simultaneously is also a signature characteristic of large-scale parallel architectures, that is evident in the human brain, and has been exploited effectively more traditional, massively parallel computational architectures. Here, we show that these two characteristics are in tension, reflecting a fundamental tradeoff between interactive parallelism that supports learning and generalization, and independent parallelism that supports processing efficiency through concurrent multitasking. We formally show that, while the maximum number of tasks that can be performed simultaneously grows linearly with network size, under realistic scenarios (e.g. in an unpredictable environment), the expected number that can be performed concurrently grows radically sub-linearly with network size. Hence, even modest reliance on shared representation strictly constrains the number of tasks that can be performed simultaneously, implying profound consequences for the development of artificial intelligence that optimally manages the tradeoff between learning and processing, and for understanding the human brains remarkably puzzling mix of sequential and parallel capabilities., Comment: version 4. Added SIs, 33 pages total, 4 figures + 14 figures in SI, major edits to text

This paper introduces a formal method to model the level of demand on control when executing cognitive processes. The cost of cognitive control is parsed into an intensity cost which encapsulates how much additional input information is required so as to get the specified response, and an interaction cost which encapsulates the level of interference between individual processes in a network. We develop a formal relationship between the probability of successful execution of desired processes and the control signals (additive control biases). This relationship is also used to specify optimal control policies to achieve a desired probability of activation for processes. We observe that there are boundary cases when finding such control policies which leads us to introduce the interaction cost. We show that the interaction cost is influenced by the relative strengths of individual processes, as well as the directionality of the underlying competition between processes., Comment: 6 pages, 3 figures, Conference paper

Simplicial complexes are now a popular alternative to networks when it comes to describing the structure of complex systems, primarily because they encode multi-node interactions explicitly. With this new description comes the need for principled null models that allow for easy comparison with empirical data. We propose a natural candidate, the simplicial configuration model. The core of our contribution is an efficient and uniform Markov chain Monte Carlo sampler for this model. We demonstrate its usefulness in a short case study by investigating the topology of three real systems and their randomized counterparts (using their Betti numbers). For two out of three systems, the model allows us to reject the hypothesis that there is no organization beyond the local scale., Comment: 6 pages, 4 figures

Infectious disease outbreaks recapitulate biology: they emerge from the multi-level interaction of hosts, pathogens, and their shared environment. As a result, predicting when, where, and how far diseases will spread requires a complex systems approach to modeling. Recent studies have demonstrated that predicting different components of outbreaks--e.g., the expected number of cases, pace and tempo of cases needing treatment, demand for prophylactic equipment, importation probability etc.--is feasible. Therefore, advancing both the science and practice of disease forecasting now requires testing for the presence of fundamental limits to outbreak prediction. To investigate the question of outbreak prediction, we study the information theoretic limits to forecasting across a broad set of infectious diseases using permutation entropy as a model independent measure of predictability. Studying the predictability of a diverse collection of historical outbreaks--including, chlamydia, dengue, gonorrhea, hepatitis A, influenza, measles, mumps, polio, and whooping cough--we identify a fundamental entropy barrier for infectious disease time series forecasting. However, we find that for most diseases this barrier to prediction is often well beyond the time scale of single outbreaks. We also find that the forecast horizon varies by disease and demonstrate that both shifting model structures and social network heterogeneity are the most likely mechanisms for the observed differences across contagions. Our results highlight the importance of moving beyond time series forecasting, by embracing dynamic modeling approaches, and suggest challenges for performing model selection across long time series. We further anticipate that our findings will contribute to the rapidly growing field of epidemiological forecasting and may relate more broadly to the predictability of complex adaptive systems.

Persistent homology analysis, a recently developed computational method in algebraic topology, is applied to the study of the phase transitions undergone by the so-called XY-mean field model and by the phi^4 lattice model, respectively. For both models the relationship between phase transitions and the topological properties of certain submanifolds of configuration space are exactly known. It turns out that these a-priori known facts are clearly retrieved by persistent homology analysis of dynamically sampled submanifolds of configuration space., Comment: 10 pages; 10 figures

Let $P$ be a finite poset. We will show that for any reasonable $P$-persistent object $X$ in the category of finite topological spaces, there is a $P-$ weighted graph, whose clique complex has the same $P$-persistent homology as $X$.

We present a method to find the best temporal partition at any time-scale and rank the relevance of partitions found at different time-scales. This method is based on random walkers coevolving with the network and as such constitutes a generalization of partition stability to the case of temporal networks. We show that, when applied to a toy model and real datasets, temporal stability uncovers structures that are persistent over meaningful time-scales as well as important isolated events, making it an effective tool to study both abrupt changes and gradual evolution of a network mesoscopic structures., Comment: 15 pages, 12 figures