Effect of hidden geometry and higher-order interactions on the synchronization and hysteresis behavior of phase oscillators on 5-clique simplicial assemblies.

The hidden geometry of simplicial complexes can influence the collective dynamics of nodes in different ways depending on the simplex-based interactions of various orders and competition between local and global structural features. We study a system of phase oscillators attached to nodes of four-dimensional simplicial complexes and interacting via positive/negative edges-based pairwise K_{1} and triangle-based triple K_{2}≥0 couplings. Three prototypal simplicial complexes are grown by aggregation of 5-cliques, controlled by the chemical affinity parameter ν, resulting in sparse, mixed, and compact architecture, all of which have 1-hyperbolic graphs but different spectral dimensions. By changing the interaction strength K_{1}∈[-4,2] along the forward and backward sweeps, we numerically determine individual phases of each oscillator and a global order parameter to measure the level of synchronization. Our results reveal how different architectures of simplicial complexes, in conjunction with the interactions and internal-frequency distributions, impact the shape of the hysteresis loop and lead to patterns of locally synchronized groups that hinder global network synchronization. Remarkably, these groups are differently affected by the size of the shared faces between neighboring 5-cliques and the presence of higher-order interactions. At K_{1}<0, partial synchronization is much higher in the compact community than in the assemblies of cliques sharing single nodes, at least occasionally. These structures also partially desynchronize at a lower triangle-based coupling K_{2} than the compact assembly. Broadening of the internal frequency distribution gradually reduces the synchronization level in the mixed and sparse communities, even at positive pairwise couplings. The order-parameter fluctuations in these partially synchronized states are quasicyclical with higher harmonics, described by multifractal analysis and broad singularity spectra.

Hysteresis and synchronization processes of Kuramoto oscillators on high-dimensional simplicial complexes with competing simplex-encoded couplings.

Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviors on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here we investigate the phase synchronization (Kuramoto) dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous four-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds d_{s}=4. By numerically solving the set of coupled equations for the phase oscillators associated with the network nodes, we determine the time-averaged system's order parameter to characterize the synchronization level. Our results reveal a variety of synchronization and desynchronization scenarios, including partially synchronized states and nonsymmetrical hysteresis loops, depending on the sign and strength of the pairwise interactions and the geometric frustrations promoted by couplings on triangle faces. For substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronization and the abrupt desynchronization transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronization processes.

Microscopic dynamics modeling unravels the role of asymptomatic virus carriers in SARS-CoV-2 epidemics at the interplay between biological and social factors.

The recent experience of SARS-CoV-2 epidemics spreading revealed the importance of passive forms of infection transmissions. Apart from the virus survival outside the host, the latent infection transmissions caused by asymptomatic and presymptomatic hosts represent major challenges for controlling the epidemics. In this regard, social mixing and various biological factors play their subtle, but often critical, role. For example, a life-threatening condition may result in the infection contracted from an asymptomatic virus carrier. Here, we use a new recently developed microscopic agent-based modelling framework to shed light on the role of asymptomatic hosts and unravel the interplay between the biological and social factors of these nonlinear stochastic processes at high temporal resolution. The model accounts for each human actor's susceptibility and the virus survival time, as well as traceability along the infection path. These properties enable an efficient dissection of the infection events caused by asymptomatic carriers from those which involve symptomatic hosts before they develop symptoms and become removed to a controlled environment. Consequently, we assess how their relative proportions in the overall infection curve vary with changing model parameters. Our results reveal that these proportions largely depend on biological factors in the process, specifically, the virus transmissibility and the critical threshold for developing symptoms, which can be affected by the virus pathogenicity. Meanwhile, social participation activity is crucial for the overall infection level, further modulated by the virus transmissibility.

Magnetisation Processes in Geometrically Frustrated Spin Networks with Self-Assembled Cliques.

Functional designs of nanostructured materials seek to exploit the potential of complex morphologies and disorder. In this context, the spin dynamics in disordered antiferromagnetic materials present a significant challenge due to induced geometric frustration. Here we analyse the processes of magnetisation reversal driven by an external field in generalised spin networks with higher-order connectivity and antiferromagnetic defects. Using the model in (Tadic et al. Arxiv:1912.02433), we grow nanonetworks with geometrically constrained self-assemblies of simplexes (cliques) of a given size n, and with probability p each simplex possesses a defect edge affecting its binding, leading to a tree-like pattern of defects. The Ising spins are attached to vertices and have ferromagnetic interactions, while antiferromagnetic couplings apply between pairs of spins along each defect edge. Thus, a defect edge induces n - 2 frustrated triangles per n-clique participating in a larger-scale complex. We determine several topological, entropic, and graph-theoretic measures to characterise the structures of these assemblies. Further, we show how the sizes of simplexes building the aggregates with a given pattern of defects affects the magnetisation curves, the length of the domain walls and the shape of the hysteresis loop. The hysteresis shows a sequence of plateaus of fractional magnetisation and multiscale fluctuations in the passage between them. For fully antiferromagnetic interactions, the loop splits into two parts only in mono-disperse assemblies of cliques consisting of an odd number of vertices n. At the same time, remnant magnetisation occurs when n is even, and in poly-disperse assemblies of cliques in the range n ∈ [ 2 , 10 ] . These results shed light on spin dynamics in complex nanomagnetic assemblies in which geometric frustration arises in the interplay of higher-order connectivity and antiferromagnetic interactions.

Large-scale influence of defect bonds in geometrically constrained self-assembly.

Recently, the importance of higher-order interactions in the physics of quantum systems and nanoparticle assemblies has prompted the exploration of new classes of networks that grow through geometrically constrained simplex aggregation. Based on the model of chemically tunable self-assembly of simplexes [Suvakov et al., Sci. Rep. 8, 1987 (2018)2045-232210.1038/s41598-018-20398-x], here we extend the model to allow the presence of a defect edge per simplex. Using a wide distribution of simplex sizes (from edges, triangles, tetrahedrons, etc., up to 10-cliques) and various chemical affinity parameters, we investigate the magnitude of the impact of defects on the self-assembly process and the emerging higher-order networks. Their essential characteristics are treelike patterns of defect bonds, hyperbolic geometry, and simplicial complexes, which are described using the algebraic topology method. Furthermore, we demonstrate how the presence of patterned defects can be used to alter the structure of the assembly after the growth process is complete. In the assemblies grown under different chemical affinities, we consider the removal of defect bonds and analyze the progressive changes in the hierarchical architecture of simplicial complexes and the hyperbolicity parameters of the underlying graphs. Within the framework of cooperative self-assembly of nanonetworks, these results shed light on the use of defects in the design of complex materials. They also provide a different perspective on the understanding of extended connectivity beyond pairwise interactions in many complex systems.

Modeling latent infection transmissions through biosocial stochastic dynamics.

The events of the recent SARS-CoV-2 epidemics have shown the importance of social factors, especially given the large number of asymptomatic cases that effectively spread the virus, which can cause a medical emergency to very susceptible individuals. Besides, the SARS-CoV-2 virus survives for several hours on different surfaces, where a new host can contract it with a delay. These passive modes of infection transmission remain an unexplored area for traditional mean-field epidemic models. Here, we design an agent-based model for simulations of infection transmission in an open system driven by the dynamics of social activity; the model takes into account the personal characteristics of individuals, as well as the survival time of the virus and its potential mutations. A growing bipartite graph embodies this biosocial process, consisting of active carriers (host) nodes that produce viral nodes during their infectious period. With its directed edges passing through viral nodes between two successive hosts, this graph contains complete information about the routes leading to each infected individual. We determine temporal fluctuations of the number of exposed and the number of infected individuals, the number of active carriers and active viruses at hourly resolution. The simulated processes underpin the latent infection transmissions, contributing significantly to the spread of the virus within a large time window. More precisely, being brought by social dynamics and exposed to the currently existing infection, an individual passes through the infectious state until eventually spontaneously recovers or otherwise is moves to a controlled hospital environment. Our results reveal complex feedback mechanisms that shape the dependence of the infection curve on the intensity of social dynamics and other sociobiological factors. In particular, the results show how the lockdown effectively reduces the spread of infection and how it increases again after the lockdown is removed. Furthermore, a reduced level of social activity but prolonged exposure of susceptible individuals have adverse effects. On the other hand, virus mutations that can gradually reduce the transmission rate by hopping to each new host along the infection path can significantly reduce the extent of the infection, but can not stop the spreading without additional social strategies. Our stochastic processes, based on graphs at the interface of biology and social dynamics, provide a new mathematical framework for simulations of various epidemic control strategies with high temporal resolution and virus traceability.

The topology of higher-order complexes associated with brain hubs in human connectomes.

Higher-order connectivity in complex systems described by simplexes of different orders provides a geometry for simplex-based dynamical variables and interactions. Simplicial complexes that constitute a functional geometry of the human connectome can be crucial for the brain complex dynamics. In this context, the best-connected brain areas, designated as hub nodes, play a central role in supporting integrated brain function. Here, we study the structure of simplicial complexes attached to eight global hubs in the female and male connectomes and identify the core networks among the affected brain regions. These eight hubs (Putamen, Caudate, Hippocampus and Thalamus-Proper in the left and right cerebral hemisphere) are the highest-ranking according to their topological dimension, defined as the number of simplexes of all orders in which the node participates. Furthermore, we analyse the weight-dependent heterogeneity of simplexes. We demonstrate changes in the structure of identified core networks and topological entropy when the threshold weight is gradually increased. These results highlight the role of higher-order interactions in human brain networks and provide additional evidence for (dis)similarity between the female and male connectomes.

Spectral properties of hyperbolic nanonetworks with tunable aggregation of simplexes.

Cooperative self-assembly is a ubiquitous phenomenon found in natural systems which is used for designing nanostructured materials with new functional features. Its origin and mechanisms, leading to improved functionality of the assembly, have attracted much attention from researchers in many branches of science and engineering. These complex structures often come with hyperbolic geometry; however, the relation between the hyperbolicity and their spectral and dynamical properties remains unclear. Using the model of aggregation of simplexes introduced by Suvakov et al. [Sci. Rep. 8, 1987 (2018)2045-232210.1038/s41598-018-20398-x], here we study topological and spectral properties of a large class of self-assembled structures or nanonetworks consisting of monodisperse building blocks (cliques of size n=3,4,5,6) which self-assemble via sharing the geometrical shapes of a lower order. The size of the shared substructure is tuned by varying the chemical affinity ν such that for significant positive ν sharing the largest face is the most probable, while for ν<0, attaching via a single node dominates. Our results reveal that, while the parameter of hyperbolicity remains δ_{max}=1 across the assemblies, their structure and spectral dimension d_{s} vary with the size of cliques n and the affinity when ν≥0. In this range, we find that d_{s}>4 can be reached for n≥5 and sufficiently large ν. For the aggregates of triangles and tetrahedra, the spectral dimension remains in the range d_{s}∈[2,4), as well as for the higher cliques at vanishing affinity. On the other end, for ν<0, we find d_{s}≂1.57 independently on n. Moreover, the spectral distribution of the normalized Laplacian eigenvalues has a characteristic shape with peaks and a pronounced minimum, representing the hierarchical architecture of the simplicial complexes. These findings show how the structures compatible with complex dynamical properties can be assembled by controlling the higher-order connectivity among the building blocks.

Functional Geometry of Human Connectomes.

Mapping the brain imaging data to networks, where nodes represent anatomical brain regions and edges indicate the occurrence of fiber tracts between them, has enabled an objective graph-theoretic analysis of human connectomes. However, the latent structure on higher-order interactions remains unexplored, where many brain regions act in synergy to perform complex functions. Here we use the simplicial complexes description of human connectome, where the shared simplexes encode higher-order relationships between groups of nodes. We study consensus connectome of 100 female (F-connectome) and of 100 male (M-connectome) subjects that we generated from the Budapest Reference Connectome Server v3.0 based on data from the Human Connectome Project. Our analysis reveals that the functional geometry of the common F&M-connectome coincides with the M-connectome and is characterized by a complex architecture of simplexes to the 14th order, which is built in six anatomical communities, and linked by short cycles. The F-connectome has additional edges that involve different brain regions, thereby increasing the size of simplexes and introducing new cycles. Both connectomes contain characteristic subjacent graphs that make them 3/2-hyperbolic. These results shed new light on the functional architecture of the brain, suggesting that insightful differences among connectomes are hidden in their higher-order connectivity.

The critical Barkhausen avalanches in thin random-field ferromagnets with an open boundary.

The interplay between the critical fluctuations and the sample geometry is investigated numerically using thin random-field ferromagnets exhibiting the field-driven magnetisation reversal on the hysteresis loop. The system is studied along the theoretical critical line in the plane of random-field disorder and thickness. The thickness is varied to consider samples of various geometry between a two-dimensional plane and a complete three-dimensional lattice with an open boundary in the direction of the growing thickness. We perform a multi-fractal analysis of the Barkhausen noise signals and scaling of the critical avalanches of the domain wall motion. Our results reveal that, for sufficiently small thickness, the sample geometry profoundly affects the dynamics by modifying the spectral segments that represent small fluctuations and promoting the time-scale dependent multi-fractality. Meanwhile, the avalanche distributions display two distinct power-law regions, in contrast to those in the two-dimensional limit, and the average avalanche shapes are asymmetric. With increasing thickness, the scaling characteristics and the multi-fractal spectrum in thicker samples gradually approach the hysteresis loop criticality in three-dimensional systems. Thin ferromagnetic films are growing in importance technologically, and our results illustrate some new features of the domain wall dynamics induced by magnetisation reversal in these systems.

Hidden geometries in networks arising from cooperative self-assembly.

Multilevel self-assembly involving small structured groups of nano-particles provides new routes to development of functional materials with a sophisticated architecture. Apart from the inter-particle forces, the geometrical shapes and compatibility of the building blocks are decisive factors. Therefore, a comprehensive understanding of these processes is essential for the design of assemblies of desired properties. Here, we introduce a computational model for cooperative self-assembly with the simultaneous attachment of structured groups of particles, which can be described by simplexes (connected pairs, triangles, tetrahedrons and higher order cliques) to a growing network. The model incorporates geometric rules that provide suitable nesting spaces for the new group and the chemical affinity of the system to accept excess particles. For varying chemical affinity, we grow different classes of assemblies by binding the cliques of distributed sizes. Furthermore, we characterize the emergent structures by metrics of graph theory and algebraic topology of graphs, and 4-point test for the intrinsic hyperbolicity of the networks. Our results show that higher Q-connectedness of the appearing simplicial complexes can arise due to only geometric factors and that it can be efficiently modulated by changing the chemical potential and the polydispersity of the binding simplexes.

Mechanisms of self-organized criticality in social processes of knowledge creation.

In online social dynamics, a robust scale invariance appears as a key feature of collaborative efforts that lead to new social value. The underlying empirical data thus offers a unique opportunity to study the origin of self-organized criticality (SOC) in social systems. In contrast to physical systems in the laboratory, various human attributes of the actors play an essential role in the process along with the contents (cognitive, emotional) of the communicated artifacts. As a prototypical example, we consider the social endeavor of knowledge creation via Questions and Answers (Q&A). Using a large empirical data set from one of such Q&A sites and theoretical modeling, we reveal fundamental characteristics of SOC by investigating the temporal correlations at all scales and the role of cognitive contents to the avalanches of the knowledge-creation process. Our analysis shows that the universal social dynamics with power-law inhomogeneities of the actions and delay times provides the primary mechanism for self-tuning towards the critical state; it leads to the long-range correlations and the event clustering in response to the external driving by the arrival of new users. In addition, the involved cognitive contents (systematically annotated in the data and observed in the model) exert important constraints that identify unique classes of the knowledge-creation avalanches. Specifically, besides determining a fine structure of the developing knowledge networks, they affect the values of scaling exponents and the geometry of large avalanches and shape the multifractal spectrum. Furthermore, we find that the level of the activity of the communities that share the knowledge correlates with the fluctuations of the innovation rate, implying that the increase of innovation may serve as the active principle of self-organization. To identify relevant parameters and unravel the role of the network evolution underlying the process in the social system under consideration, we compare the social avalanches to the avalanche sequences occurring in the field-driven physical model of disordered solids, where the factors contributing to the collective dynamics are better understood.

Algebraic Topology of Multi-Brain Connectivity Networks Reveals Dissimilarity in Functional Patterns during Spoken Communications.

Human behaviour in various circumstances mirrors the corresponding brain connectivity patterns, which are suitably represented by functional brain networks. While the objective analysis of these networks by graph theory tools deepened our understanding of brain functions, the multi-brain structures and connections underlying human social behaviour remain largely unexplored. In this study, we analyse the aggregate graph that maps coordination of EEG signals previously recorded during spoken communications in two groups of six listeners and two speakers. Applying an innovative approach based on the algebraic topology of graphs, we analyse higher-order topological complexes consisting of mutually interwoven cliques of a high order to which the identified functional connections organise. Our results reveal that the topological quantifiers provide new suitable measures for differences in the brain activity patterns and inter-brain synchronisation between speakers and listeners. Moreover, the higher topological complexity correlates with the listener's concentration to the story, confirmed by self-rating, and closeness to the speaker's brain activity pattern, which is measured by network-to-network distance. The connectivity structures of the frontal and parietal lobe consistently constitute distinct clusters, which extend across the listener's group. Formally, the topology quantifiers of the multi-brain communities exceed the sum of those of the participating individuals and also reflect the listener's rated attributes of the speaker and the narrated subject. In the broader context, the presented study exposes the relevance of higher topological structures (besides standard graph measures) for characterising functional brain networks under different stimuli.

Topology of Innovation Spaces in the Knowledge Networks Emerging through Questions-And-Answers.

The communication processes of knowledge creation represent a particular class of human dynamics where the expertise of individuals plays a substantial role, thus offering a unique possibility to study the structure of knowledge networks from online data. Here, we use the empirical evidence from questions-and-answers in mathematics to analyse the emergence of the network of knowledge contents (or tags) as the individual experts use them in the process. After removing extra edges from the network-associated graph, we apply the methods of algebraic topology of graphs to examine the structure of higher-order combinatorial spaces in networks for four consecutive time intervals. We find that the ranking distributions of the suitably scaled topological dimensions of nodes fall into a unique curve for all time intervals and filtering levels, suggesting a robust architecture of knowledge networks. Moreover, these networks preserve the logical structure of knowledge within emergent communities of nodes, labeled according to a standard mathematical classification scheme. Further, we investigate the appearance of new contents over time and their innovative combinations, which expand the knowledge network. In each network, we identify an innovation channel as a subgraph of triangles and larger simplices to which new tags attach. Our results show that the increasing topological complexity of the innovation channels contributes to network's architecture over different time periods, and is consistent with temporal correlations of the occurrence of new tags. The methodology applies to a wide class of data with the suitable temporal resolution and clearly identified knowledge-content units.

The dynamics of meaningful social interactions and the emergence of collective knowledge.

Collective knowledge as a social value may arise in cooperation among actors whose individual expertise is limited. The process of knowledge creation requires meaningful, logically coordinated interactions, which represents a challenging problem to physics and social dynamics modeling. By combining two-scale dynamics model with empirical data analysis from a well-known Questions &Answers system Mathematics, we show that this process occurs as a collective phenomenon in an enlarged network (of actors and their artifacts) where the cognitive recognition interactions are properly encoded. The emergent behavior is quantified by the information divergence and innovation advancing of knowledge over time and the signatures of self-organization and knowledge sharing communities. These measures elucidate the impact of each cognitive element and the individual actor's expertise in the collective dynamics. The results are relevant to stochastic processes involving smart components and to collaborative social endeavors, for instance, crowdsourcing scientific knowledge production with online games.

Hidden geometry of traffic jamming.

We introduce an approach based on algebraic topological methods that allow an accurate characterization of jamming in dynamical systems with queues. As a prototype system, we analyze the traffic of information packets with navigation and queuing at nodes on a network substrate in distinct dynamical regimes. A temporal sequence of traffic density fluctuations is mapped onto a mathematical graph in which each vertex denotes one dynamical state of the system. The coupling complexity between these states is revealed by classifying agglomerates of high-dimensional cliques that are intermingled at different topological levels and quantified by a set of geometrical and entropy measures. The free-flow, jamming, and congested traffic regimes result in graphs of different structure, while the largest geometrical complexity and minimum entropy mark the edge of the jamming region.

How the online social networks are used: dialogues-based structure of MySpace.

Quantitative study of collective dynamics in online social networks is a new challenge based on the abundance of empirical data. Conclusions, however, may depend on factors such as user's psychology profiles and their reasons to use the online contacts. In this study, we have compiled and analysed two datasets from MySpace. The data contain networked dialogues occurring within a specified time depth, high temporal resolution and texts of messages, in which the emotion valence is assessed by using the SentiStrength classifier. Performing a comprehensive analysis, we obtain three groups of results: dynamic topology of the dialogues-based networks have a characteristic structure with Zipf's distribution of communities, low link reciprocity and disassortative correlations. Overlaps supporting 'weak-ties' hypothesis are found to follow the laws recently conjectured for online games. Long-range temporal correlations and persistent fluctuations occur in the time series of messages carrying positive (negative) emotion; patterns of user communications have dominant positive emotion (attractiveness) and strong impact of circadian cycles and interactivity times longer than 1 day. Taken together, these results give a new insight into the functioning of online social networks and unveil the importance of the amount of information and emotion that is communicated along the social links. All data used in this study are fully anonymized.

Stability and chaos in coupled two-dimensional maps on gene regulatory network of bacterium E. coli.

The collective dynamics of coupled two-dimensional chaotic maps on complex networks is known to exhibit a rich variety of emergent properties which crucially depend on the underlying network topology. We investigate the collective motion of Chirikov standard maps interacting with time delay through directed links of gene regulatory network of bacterium Escherichia coli. Departures from strongly chaotic behavior of the isolated maps are studied in relation to different coupling forms and strengths. At smaller coupling intensities the network induces stable and coherent emergent dynamics. The unstable behavior appearing with increase of coupling strength remains confined within a connected subnetwork. For the appropriate coupling, network exhibits statistically robust self-organized dynamics in a weakly chaotic regime.

Modeling collective charge transport in nanoparticle assemblies.

Mapping the assembled patterns of nanoparticles onto networks (mathematical graphs) provides a way for quantitative analysis of the structure effects on the physical properties of the assembly. Here we review the network modeling of the conduction with single-electron tunneling mechanisms in the assembled nanoparticle films. Simulations of the conduction predict the nonlinear current-voltage curves in different classes of the nanoparticle networks. Furthermore, the numerical analysis reveals how the I(V) nonlinearity is related to the collective charge fluctuations along the conducting paths through the sample, and stresses the role of the topology and quenched charge disorder.

Spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities.

We study structure, eigenvalue spectra, and random-walk dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their internal structure as scale-free and correlated subgraphs, and the topology of connecting network. Within the exhaustive spectral analysis for both the adjacency matrix and the normalized Laplacian matrix we identify the spectral properties, which characterize the mesoscopic structure of sparse cyclic graphs and trees. The minimally connected nodes, the clustering, and the average connectivity affect the central part of the spectrum. The number of distinct modules leads to an extra peak at the lower part of the Laplacian spectrum in cyclic graphs. Such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs. We also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue lambda(L)=2 of the Laplacian. Further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs. The distribution of first-return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1/3 for trees and sigma approximately 2/3 for cyclic graphs, which is independent of their mesoscopic and global structure.