Hidden geometry of traffic jamming

Simplicial complex entropy for time series analysis

The complex behavior of many systems in nature requires the application of robust methodologies capable of identifying changes in their dynamics. In the case of time series (which are sensed values of a system during a time interval), several methods have been proposed to evaluate their irregularity. However, for some types of dynamics such as stochastic and chaotic, new approaches are required that can provide a better characterization of them. In this paper we present the simplicial complex approximate entropy, which is based on the conditional probability of the occurrence of elements of a simplicial complex. Our results show that this entropy measure provides a wide range of values with details not easily identifiable with standard methods. In particular, we show that our method is able to quantify the irregularity in simulated random sequences and those from low-dimensional chaotic dynamics. Furthermore, it is possible to consistently differentiate cardiac interbeat sequences from healthy subjects and from patients with heart failure, as well as to identify changes between dynamical states of coupled chaotic maps. Our results highlight the importance of the structures revealed by the simplicial complexes, which holds promise for applications of this approach in various contexts.

A simplicial analysis of the fMRI data from human brain dynamics under functional cognitive tasks

The topological analysis of fMRI time series data has recently been used to characterize the identification of patterns of brain activity seen during specific tasks carried out under experimentally controlled conditions. This study uses the methods of algebraic topology to characterize time series networks constructed from fMRI data measured for adult and children populations carrying out differentiated reading tasks. Our pilot study shows that our methods turn out to be capable of identifying distinct differences between the activity of adult and children populations carrying out identical reading tasks. We also see differences between activity patterns seen when subjects recognize word and nonword patterns. The results generalize across different populations, different languages and different active and inactive brain regions.

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.

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.

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 [Šuvakov 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.

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 (Tadić 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.

Characterizing the complexity of time series networks of dynamical systems: A simplicial approach

We analyze the time series obtained from different dynamical regimes of evolving maps and flows by constructing their equivalent time series networks, using the visibility algorithm. The regimes analyzed include periodic, chaotic, and hyperchaotic regimes, as well as intermittent regimes and regimes at the edge of chaos. We use the methods of algebraic topology, in particular, simplicial complexes, to define simplicial characterizers, which can analyze the simplicial structure of the networks at both the global and local levels. The simplicial characterizers bring out the hierarchical levels of complexity at various topological levels. These hierarchical levels of complexity find the skeleton of the local dynamics embedded in the network, which influence the global dynamical properties of the system and also permit the identification of dominant motifs. We also analyze the same networks using conventional network characterizers such as average path lengths and clustering coefficients. We see that the simplicial characterizers are capable of distinguishing between different dynamical regimes and can pick up subtle differences in dynamical behavior, whereas the usual characterizers provide a coarser characterization. However, the two taken in conjunction can provide information about the dynamical behavior of the time series, as well as the correlations in the evolving system. Our methods can, therefore, provide powerful tools for the analysis of dynamical systems.

Characterizing stochastic time series with ordinal networks

Approaches for mapping time series to networks have become essential tools for dealing with the increasing challenges of characterizing data from complex systems. Among the different algorithms, the recently proposed ordinal networks stand out due to their simplicity and computational efficiency. However, applications of ordinal networks have been mainly focused on time series arising from nonlinear dynamical systems, while basic properties of ordinal networks related to simple stochastic processes remain poorly understood. Here, we investigate several properties of ordinal networks emerging from random time series, noisy periodic signals, fractional Brownian motion, and earthquake magnitude series. For ordinal networks of random series, we present an approach for building the exact form of the adjacency matrix, which in turn is useful for detecting nonrandom behavior in time series and the existence of missing transitions among ordinal patterns. We find that the average value of a local entropy, estimated from transition probabilities among neighboring nodes of ordinal networks, is more robust against noise addition than the standard permutation entropy. We show that ordinal networks can be used for estimating the Hurst exponent of time series with accuracy comparable with state-of-the-art methods. Finally, we argue that ordinal networks can detect sudden changes in Earth's seismic activity caused by large earthquakes.

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.

Structural Entropy: Monitoring Correlation-Based Networks Over Time With Application To Financial Markets

The concept of "Structural Diversity" of a network refers to the level of dissimilarity between the various agents acting in the system, and it is typically interpreted as the number of connected components in the network. This key property of networks has been studied in multiple settings, including diffusion of ideas in social networks and functional diversity of regions in brain networks. Here, we propose a new measure, "Structural Entropy", as a revised interpretation to "Structural Diversity". The proposed measure relies on the finer-grained network communities (in contrast to the network's connected components), and takes into consideration both the number of communities and their sizes, generating a single representative value. We then propose an approach for monitoring the structure of correlation-based networks over time, which relies on the newly suggested measure. Finally, we illustrate the usefulness of the new approach, by applying it to the particular case of emergent organization of financial markets. This provides us a way to explore their underlying structural changes, revealing a remarkably high linear correlation between the new measure and the volatility of the assets' prices over time.

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 Šuvakov 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.

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.

Categorisation of polyphonic musical signals by using modularity community detection in audio-associated visibility network

This article proposes a method to numerically characterise the homogeneity of polyphonic musical signals through community detection in audio-associated visibility networks and to detect patterns that allow the categorisation of these signals into two types of grouping based on this numerical characterization. To implement this methodology, we first calculate the variance fluctuation series in fixed-size windows of an audio stretch. Next we map this series onto a visibility graph, where the nodes are the points of the series, and the edges are defined by the visibility between each pair of points. Then, we measure the quality of the partitions of the network using the modularity and Louvain optimisation. We observed that a greater or lesser homogeneity of the magnitudes of the signal transients is related to a higher or lower modularity of the audio-associated visibility network. We also note that these differences are related to musical choices that can establish important differences between musical styles. In this article, we show that the modularity is able to give relevant information to allow the categorisation of 120 musical signs labelled in percussive and symphonic music.

Traffic congestion and the lifetime of networks with moving nodes

For many power-limited networks, such as wireless sensor networks and mobile ad hoc networks, maximizing the network lifetime is the first concern in the related designing and maintaining activities. We study the network lifetime from the perspective of network science. In our model, nodes are initially assigned a fixed amount of energy moving in a square area and consume the energy when delivering packets. We obtain four different traffic regimes: no, slow, fast, and absolute congestion regimes, which are basically dependent on the packet generation rate. We derive the network lifetime by considering the specific regime of the traffic flow. We find that traffic congestion inversely affects network lifetime in the sense that high traffic congestion results in short network lifetime. We also discuss the impacts of factors such as communication radius, node moving speed, routing strategy, etc., on network lifetime and traffic congestion.

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.

Totally asymmetric simple exclusion process simulations of molecular motor transport on random networks with asymmetric exit rates

Using the totally asymmetric simple-exclusion-process and mean-field transport theory, we investigate the transport in closed random networks with simple crossing topology-two incoming, two outgoing segments, as a model for molecular motor motion along biopolymer networks. Inspired by in vitro observation of molecular motor motion, we model the motor behavior at the intersections by introducing different exit rates for the two outgoing segments. Our simulations of this simple network reveal surprisingly rich behavior of the transport current with respect to the global density and exit rate ratio. For asymmetric exit rates, we find a broad current plateau at intermediate motor densities resulting from the competition of two subnetwork populations. This current plateau leads to stabilization of transport properties within such networks.