Network geometry with flavor: From complexity to quantum geometry

Effect of individual activity level heterogeneity on disease spreading in higher-order networks

The active state of individuals has a significant impact on disease spread dynamics. In addition, pairwise interactions and higher-order interactions coexist in complex systems, and the pairwise networks proved insufficient for capturing the essence of complex systems. Here, we propose a higher-order network model to study the effect of individual activity level heterogeneity on disease-spreading dynamics. Activity level heterogeneity radically alters the dynamics of disease spread in higher-order networks. First, the evolution equations for infected individuals are derived using the mean field method. Second, numerical simulations of artificial networks reveal that higher-order interactions give rise to a discontinuous phase transition zone where the coexistence of health and disease occurs. Furthermore, the system becomes more unstable as individual activity levels rise, leading to a higher likelihood of disease outbreaks. Finally, we simulate the proposed model on two real higher-order networks, and the results are consistent with the artificial networks and validate the inferences from theoretical analysis. Our results explain the underlying reasons why groups with higher activity levels are more likely to initiate social changes. Simultaneously, the reduction in group activity, characterized by measures such as "isolation," emerges as a potent strategy for disease control.

Probabilistic activity driven model of temporal simplicial networks and its application on higher-order dynamics

Network modeling characterizes the underlying principles of structural properties and is of vital significance for simulating dynamical processes in real world. However, bridging structure and dynamics is always challenging due to the multiple complexities in real systems. Here, through introducing the individual's activity rate and the possibility of group interaction, we propose a probabilistic activity-driven (PAD) model that could generate temporal higher-order networks with both power-law and high-clustering characteristics, which successfully links the two most critical structural features and a basic dynamical pattern in extensive complex systems. Surprisingly, the power-law exponents and the clustering coefficients of the aggregated PAD network could be tuned in a wide range by altering a set of model parameters. We further provide an approximation algorithm to select the proper parameters that can generate networks with given structural properties, the effectiveness of which is verified by fitting various real-world networks. Finally, we construct the co-evolution framework of the PAD model and higher-order contagion dynamics and derive the critical conditions for phase transition and bistable phenomenon using theoretical and numerical methods. Results show that tendency of participating in higher-order interactions can promote the emergence of bistability but delay the outbreak under heterogeneous activity rates. Our model provides a basic tool to reproduce complex structural properties and to study the widespread higher-order dynamics, which has great potential for applications across fields.

Persistent Dirac for molecular representation

Molecular representations are of fundamental importance for the modeling and analysing molecular systems. The successes in drug design and materials discovery have been greatly contributed by molecular representation models. In this paper, we present a computational framework for molecular representation that is mathematically rigorous and based on the persistent Dirac operator. The properties of the discrete weighted and unweighted Dirac matrix are systematically discussed, and the biological meanings of both homological and non-homological eigenvectors are studied. We also evaluate the impact of various weighting schemes on the weighted Dirac matrix. Additionally, a set of physical persistent attributes that characterize the persistence and variation of spectrum properties of Dirac matrices during a filtration process is proposed to be molecular fingerprints. Our persistent attributes are used to classify molecular configurations of nine different types of organic-inorganic halide perovskites. The combination of persistent attributes with gradient boosting tree model has achieved great success in molecular solvation free energy prediction. The results show that our model is effective in characterizing the molecular structures, demonstrating the power of our molecular representation and featurization approach.

Synchronization of phase oscillators on complex hypergraphs

We study the effect of structured higher-order interactions on the collective behavior of coupled phase oscillators. By combining a hypergraph generative model with dimensionality reduction techniques, we obtain a reduced system of differential equations for the system's order parameters. We illustrate our framework with the example of a hypergraph with hyperedges of sizes 2 (links) and 3 (triangles). For this case, we obtain a set of two coupled nonlinear algebraic equations for the order parameters. For strong values of coupling via triangles, the system exhibits bistability and explosive synchronization transitions. We find conditions that lead to bistability in terms of hypergraph properties and validate our predictions with numerical simulations. Our results provide a general framework to study the synchronization of phase oscillators in hypergraphs, and they can be extended to hypergraphs with hyperedges of arbitrary sizes, dynamic-structural correlations, and other features.

Anomalous finite-size scaling in higher-order processes with absorbing states

Here we study standard and higher-order birth-death processes on fully connected networks, within the perspective of large-deviation theory [also referred to as the Wentzel-Kramers-Brillouin (WKB) method in some contexts]. We obtain a general expression for the leading and next-to-leading terms of the stationary probability distribution of the fraction of "active" sites as a function of parameters and network size N. We reproduce several results from the literature and, in particular, we derive all the moments of the stationary distribution for the q-susceptible-infected-susceptible (q-SIS) model, i.e., a high-order epidemic model requiring q active ("infected") sites to activate an additional one. We uncover a very rich scenario for the fluctuations of the fraction of active sites, with nontrivial finite-size-scaling properties. In particular, we show that the variance-to-mean ratio diverges at criticality for [1≤q≤3], with a maximal variability at q=2, confirming that complex-contagion processes can exhibit peculiar scaling features including wild variability. Moreover, the leading order in a large-deviation approach does not suffice to describe them: next-to-leading terms are essential to capture the intrinsic singularity at the origin of systems with absorbing states. Some possible extensions of this work are also discussed.

Weighted simplicial complexes and their representation power of higher-order network data and topology

Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the adopted mathematical framework to describe higher-order networks starting from data of higher-order interactions. Unweighted simplicial complexes typically involve a loss of information of the data, though having the benefit to capture the higher-order topology of the data. In this work we show that weighted simplicial complexes allow one to circumvent all the limitations of unweighted simplicial complexes to represent higher-order interactions. In particular, weighted simplicial complexes can represent higher-order networks without loss of information, allowing one at the same time to capture the weighted topology of the data. The higher-order topology is probed by studying the spectral properties of suitably defined weighted Hodge Laplacians displaying a normalized spectrum. The higher-order spectrum of (weighted) normalized Hodge Laplacians is studied combining cohomology theory with information theory. In the proposed framework we quantify and compare the information content of higher-order spectra of different dimension using higher-order spectral entropies and spectral relative entropies. The proposed methodology is tested on real higher-order collaboration networks and on the weighted version of the simplicial complex model "Network Geometry with Flavor."

Dynamics on higher-order networks: a review

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.

Local topological moves determine global diffusion properties of hyperbolic higher-order networks

From social interactions to the human brain, higher-order networks are key to describe the underlying network geometry and topology of many complex systems. While it is well known that network structure strongly affects its function, the role that network topology and geometry has on the emerging dynamical properties of higher-order networks is yet to be clarified. In this perspective, the spectral dimension plays a key role since it determines the effective dimension for diffusion processes on a network. Despite its relevance, a theoretical understanding of which mechanisms lead to a finite spectral dimension, and how this can be controlled, still represents a challenge and is the object of intense research. Here, we introduce two nonequilibrium models of hyperbolic higher-order networks and we characterize their network topology and geometry by investigating the intertwined appearance of small-world behavior, δ-hyperbolicity, and community structure. We show that different topological moves, determining the nonequilibrium growth of the higher-order hyperbolic network models, induce tuneable values of the spectral dimension, showing a rich phenomenology which is not displayed in random graph ensembles. In particular, we observe that, if the topological moves used to construct the higher-order network increase the area/volume ratio, then the spectral dimension continuously decreases, while the opposite effect is observed if the topological moves decrease the area/volume ratio. Our work reveals a new link between the geometry of a network and its diffusion properties, contributing to a better understanding of the complex interplay between network structure and dynamics.

Spectral detection of simplicial communities via Hodge Laplacians

While the study of graphs has been very popular, simplicial complexes are relatively new in the network science community. Despite being a source of rich information, graphs are limited to pairwise interactions. However, several real-world networks such as social networks, neuronal networks, etc., involve interactions between more than two nodes. Simplicial complexes provide a powerful mathematical framework to model such higher-order interactions. It is well known that the spectrum of the graph Laplacian is indicative of community structure, and this relation is exploited by spectral clustering algorithms. Here we propose that the spectrum of the Hodge Laplacian, a higher-order Laplacian defined on simplicial complexes, encodes simplicial communities. We formulate an algorithm to extract simplicial communities (of arbitrary dimension). We apply this algorithm to simplicial complex benchmarks and to real higher-order network data including social networks and networks extracted using language or text processing tools. However, datasets of simplicial complexes are scarce, and for the vast majority of datasets that may involve higher-order interactions, only the set of pairwise interactions are available. Hence, we use known properties of the data to infer the most likely higher-order interactions. In other words, we introduce an inference method to predict the most likely simplicial complex given the community structure of its network skeleton. This method identifies as most likely the higher-order interactions inducing simplicial communities that maximize the adjusted mutual information measured with respect to ground-truth community structure. Finally, we consider higher-order networks constructed through thresholding the edge weights of collaboration networks (encoding only pairwise interactions) and provide an example of persistent simplicial communities that are sustained over a wide range of the threshold.

Persistent homology of fractional Gaussian noise

In this paper, we employ the persistent homology (PH) technique to examine the topological properties of fractional Gaussian noise (fGn). We develop the weighted natural visibility graph algorithm, and the associated simplicial complexes through the filtration process are quantified by PH. The evolution of the homology group dimension represented by Betti numbers demonstrates a strong dependency on the Hurst exponent (H). The coefficients of the birth and death curves of the k-dimensional topological holes (k-holes) at a given threshold depend on H which is almost not affected by finite sample size. We show that the distribution function of a lifetime for k-holes decays exponentially and the corresponding slope is an increasing function versus H and, more interestingly, the sample size effect completely disappears in this quantity. The persistence entropy logarithmically grows with the size of the visibility graph of a system with almost H-dependent prefactors. On the contrary, the local statistical features are not able to determine the corresponding Hurst exponent of fGn data, while the moments of eigenvalue distribution (M_{n}) for n≥1 reveal a dependency on H, containing the sample size effect. Finally, the PH shows the correlated behavior of electroencephalography for both healthy and schizophrenic samples.

Principles and open questions in functional brain network reconstruction

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network representation involves often covert theoretical assumptions and methodological choices which affect the way networks are reconstructed from experimental data, and ultimately the resulting network properties and their interpretation. Here, we review some fundamental conceptual underpinnings and technical issues associated with brain network reconstruction, and discuss how their mutual influence concurs in clarifying the organization of brain function.

MeSCoT: The tool for quantitative trait simulation through the mechanistic modelling of genes' regulatory interactions

This work represents a novel mechanistic approach to simulate and study genomic networks with accompanying regulatory interactions and complex mechanisms of quantitative trait formation. The approach implemented in MeSCoT software is conceptually based on the omnigenic genetic model of quantitative (complex) trait, and closely imitates the basic in vivo mechanisms of quantitative trait realization. The software provides a framework to study molecular mechanisms of gene-by-gene and gene-by-environment interactions underlying quantitative trait’s realization and allows detailed mechanistic studies of impact of genetic and phenotypic variance on gene regulation. MeSCoT performs a detailed simulation of genes’ regulatory interactions for variable genomic architectures and generates complete set of transcriptional and translational data together with simulated quantitative trait values. Such data provide opportunities to study, for example, verification of novel statistical methods aiming to integrate intermediate phenotypes together with final phenotype in quantitative genetic analyses or to investigate novel approaches for exploiting gene-by-gene and gene-by-environment interactions.

Growth strategy determines the memory and structural properties of brain networks

The interplay between structure and function affects the emerging properties of many natural systems. Here we use an adaptive neural network model that couples activity and topological dynamics and reproduces the experimental temporal profiles of synaptic density observed in the brain. We prove that the existence of a transient period of relatively high synaptic connectivity is critical for the development of the system under noise circumstances, such that the resulting network can recover stored memories. Moreover, we show that intermediate synaptic densities provide optimal developmental paths with minimum energy consumption, and that ultimately it is the transient heterogeneity in the network that determines its evolution. These results could explain why the pruning curves observed in actual brain areas present their characteristic temporal profiles and they also suggest new design strategies to build biologically inspired neural networks with particular information processing capabilities.

Covering Problems and Core Percolations on Hypergraphs

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

Explosive Higher-Order Kuramoto Dynamics on Simplicial Complexes

The higher-order interactions of complex systems, such as the brain, are captured by their simplicial complex structure and have a significant effect on dynamics. However, the existing dynamical models defined on simplicial complexes make the strong assumption that the dynamics resides exclusively on the nodes. Here we formulate the higher-order Kuramoto model which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on. We show that higher-order Kuramoto dynamics can lead to an explosive synchronization transition by using an adaptive coupling dependent on the solenoidal and the irrotational component of the dynamics.

Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in nonamenable two-dimensional branching simplicial and cell complexes, i.e., simplicial and cell complexes in which the boundary scales like the volume. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with interlayer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition.

Renormalization group for link percolation on planar hyperbolic manifolds

Network geometry is currently a topic of growing scientific interest, as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However, the field is still in its infancy. In this work we investigate the role of network geometry in determining the nature of the percolation transition in planar hyperbolic manifolds. Boettcher et al. [Nat. Comm. 3, 787 (2012)2041-172310.1038/ncomms1774] have shown that a special type of two-dimensional hyperbolic manifolds, the Farey graphs, display a discontinuous transition for ordinary link percolation. Here we investigate using the renormalization group the critical properties of link percolation on a wider class of two-dimensional hyperbolic deterministic and random manifolds constituting the skeletons of two-dimensional cell complexes. These hyperbolic manifolds are built iteratively by subsequently gluing m-polygons to single edges. We show that when the size m of the polygons is drawn from a distribution q_{m} with asymptotic power-law scaling q_{m}≃Cm^{-γ} for m≫1, different universality classes can be observed depending on the value of the power-law exponent γ. Interestingly, the percolation transition is hybrid for γ∈(3,4) and becomes continuous for γ∈(2,3].

Shape of shortest paths in random spatial networks

In the classic model of first-passage percolation, for pairs of vertices separated by a Euclidean distance L, geodesics exhibit deviations from their mean length L that are of order L^{χ}, while the transversal fluctuations, known as wandering, grow as L^{ξ}. We find that when weighting edges directly with their Euclidean span in various spatial network models, we have two distinct classes defined by different exponents ξ=3/5 and χ=1/5, or ξ=7/10 and χ=2/5, depending only on coarse details of the specific connectivity laws used. Also, the travel-time fluctuations are Gaussian, rather than Tracy-Widom, which is rarely seen in first-passage models. The first class contains proximity graphs such as the hard and soft random geometric graph, and the k-nearest neighbor random geometric graphs, where via Monte Carlo simulations we find ξ=0.60±0.01 and χ=0.20±0.01, showing a theoretical minimal wandering. The second class contains graphs based on excluded regions such as β skeletons and the Delaunay triangulation and are characterized by the values ξ=0.70±0.01 and χ=0.40±0.01, with a nearly theoretically maximal wandering exponent. We also show numerically that the so-called Kardar-Parisi-Zhang (KPZ) relation χ=2ξ-1 is satisfied for all these models. These results shed some light on the Euclidean first-passage process but also raise some theoretical questions about the scaling laws and the derivation of the exponent values and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space.

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.

Fermi-Bose Mixtures and BCS-BEC Crossover in High-Tc Superconductors

In this review article we consider theoretically and give experimental support to the models of the Fermi-Bose mixtures and the BCS-BEC (Bardeen Cooper Schrieffer–Bose Einstein) crossover compared with the strong-coupling approach, which can serve as the cornerstones on the way from high-temperature to room-temperature superconductivity in pressurized metallic hydrides. We discuss some key theoretical ideas and mechanisms proposed for unconventional superconductors (cuprates, pnictides, chalcogenides, bismuthates, diborides, heavy-fermions, organics, bilayer graphene, twisted graphene, oxide hetero-structures), superfluids and balanced or imbalanced ultracold Fermi gases in magnetic traps. We build a bridge between unconventional superconductors and recently discovered pressurized hydrides superconductors H3S and LaH10 with the critical temperature close to room temperature. We discuss systems with a line of nodal Dirac points close to the Fermi surface and superconducting shape resonances, and hyperbolic superconducting networks which are very important for the development of novel topological superconductors, for the energetics, for the applications in nano-electronics and quantum computations.

Synchronization in network geometries with finite spectral dimension

Recently there is a surge of interest in network geometry and topology. Here we show that the spectral dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the spectral dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for spectral dimensions above four and that phase entrainment of the oscillators can only be found for spectral dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called complex network manifolds, which displays a tunable spectral dimension.

The multidimensional brain

Brain activity takes place in three spatial-plus time dimensions. This rather obvious claim has been recently questioned by papers that, taking into account the big data outburst and novel available computational tools, are starting to unveil a more intricate state of affairs. Indeed, various brain activities and their correlated mental functions can be assessed in terms of trajectories embedded in phase spaces of dimensions higher than the canonical ones. In this review, I show how further dimensions may not just represent a convenient methodological tool that allows a better mathematical treatment of otherwise elusive cortical activities, but may also reflect genuine functional or anatomical relationships among real nervous functions. I then describe how to extract hidden multidimensional information from real or artificial neurodata series, and make clear how our mind dilutes, rather than concentrates as currently believed, inputs coming from the environment. Finally, I argue that the principle "the higher the dimension, the greater the information" may explain the occurrence of mental activities and elucidate the mechanisms of human diseases associated with dimensionality reduction.

Simplicial Activity Driven Model

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 model, in which the building block is a simplex of nodes representing a multiagent interaction. We show analytically and numerically that the use of simplicial structures leads to crucial structural differences with respect to the activity driven model, a paradigmatic temporal network model involving only binary interactions. It also impacts the outcome of paradigmatic processes modeling disease propagation or social contagion. 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 activity driven model.

Complex Network Geometry and Frustrated Synchronization

The dynamics of networks of neuronal cultures has been recently shown to be strongly dependent on the network geometry and in particular on their dimensionality. However, this phenomenon has been so far mostly unexplored from the theoretical point of view. Here we reveal the rich interplay between network geometry and synchronization of coupled oscillators in the context of a simplicial complex model of manifolds called Complex Network Manifold. The networks generated by this model combine small world properties (infinite Hausdorff dimension) and a high modular structure with finite and tunable spectral dimension. We show that the networks display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network.

Dense power-law networks and simplicial complexes

There is increasing evidence that dense networks occur in on-line social networks, recommendation networks and in the brain. In addition to being dense, these networks are often also scale-free, i.e., their degree distributions follow P(k)∝k^{-γ} with γ∈(1,2]. Models of growing networks have been successfully employed to produce scale-free networks using preferential attachment, however these models can only produce sparse networks as the numbers of links and nodes being added at each time step is constant. Here we present a modeling framework which produces networks that are both dense and scale-free. The mechanism by which the networks grow in this model is based on the Pitman-Yor process. Variations on the model are able to produce undirected scale-free networks with exponent γ=2 or directed networks with power-law out-degree distribution with tunable exponent γ∈(1,2). We also extend the model to that of directed two-dimensional simplicial complexes. Simplicial complexes are generalization of networks that can encode the many body interactions between the parts of a complex system and as such are becoming increasingly popular to characterize different data sets ranging from social interacting systems to the brain. Our model produces dense directed simplicial complexes with power-law distribution of the generalized out-degrees of the nodes.

Sparse Power-Law Network Model for Reliable Statistical Predictions Based on Sampled Data

A projective network model is a model that enables predictions to be made based on a subsample of the network data, with the predictions remaining unchanged if a larger sample is taken into consideration. An exchangeable model is a model that does not depend on the order in which nodes are sampled. Despite a large variety of non-equilibrium (growing) and equilibrium (static) sparse complex network models that are widely used in network science, how to reconcile sparseness (constant average degree) with the desired statistical properties of projectivity and exchangeability is currently an outstanding scientific problem. Here we propose a network process with hidden variables which is projective and can generate sparse power-law networks. Despite the model not being exchangeable, it can be closely related to exchangeable uncorrelated networks as indicated by its information theory characterization and its network entropy. The use of the proposed network process as a null model is here tested on real data, indicating that the model offers a promising avenue for statistical network modelling.

Complex network view of evolving manifolds

We study complex networks formed by triangulations and higher-dimensional simplicial complexes representing closed evolving manifolds. In particular, for triangulations, the set of possible transformations of these networks is restricted by the condition that at each step, all the faces must be triangles. Stochastic application of these operations leads to random networks with different architectures. We perform extensive numerical simulations and explore the geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties. This characterization includes the Hausdorff and spectral dimensions of the resulting networks, their degree distributions, and various structural correlations. Our results reveal a rich zoo of architectures and geometries of these networks, some of which appear to be small worlds while others are finite dimensional with Hausdorff dimension equal or higher than the original dimensionality of their simplices. The range of spectral dimensions of the evolving triangulations turns out to be from about 1.4 to infinity. Our models include simplicial complexes representing manifolds with evolving topologies, for example, an h-holed torus with a progressively growing number of holes. This evolving graph demonstrates features of a small-world network and has a particularly heavy-tailed degree distribution.

Relaxation dynamics of maximally clustered networks

We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics-the double-edge swap, corresponding to degree-preserving randomization of the configuration model, and single edge replacement, corresponding to full randomization of the Erdős-Rényi random graph. We derive expressions for the time evolution of the degree distribution, edge multiplicity distribution and clustering coefficient. We show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed. We calculate the position of the phase transition analytically using the Erdős-Rényi phenomenology.

Centralities in simplicial complexes. Applications to protein interaction networks

Complex networks can be used to represent complex systems which originate in the real world. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplices of the complex. We extend the concept of node centrality to that of simplicial centrality and study several mathematical properties of degree, closeness, betweenness, eigenvector, Katz, and subgraph centrality for simplicial complexes. We study the degree distributions of these centralities at the different levels. We also compare and describe the differences between the centralities at the different levels. Using these centralities we study a method for detecting essential proteins in PPI networks of cells and explain the varying abilities of the centrality measures at the different levels in identifying these essential proteins.

Construction of and efficient sampling from the simplicial configuration model

Simplicial complexes are now a popular alternative to networks when it comes to describing the structure of complex systems, primarily because they encode multinode 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.

Generative Models for Global Collaboration Relationships

When individuals interact with each other and meaningfully contribute toward a common goal, it results in a collaboration. The artifacts resulting from collaborations are best captured using a hypergraph model, whereas the relation of who has collaborated with whom is best captured via an abstract simplicial complex (SC). We propose a generative algorithm GENESCs for SCs modeling fundamental collaboration relations. The proposed network growth process favors attachment that is preferential not to an individual's degree, i.e., how many people has he/she collaborated with, but to his/her facet degree, i.e., how many maximal groups or facets has he/she collaborated within. Based on our observation that several real-world facet size distributions have significant deviation from power law-mainly since larger facets tend to subsume smaller ones-we adopt a data-driven approach. We prove that the facet degree distribution yielded by GENESCs is power law distributed for large SCs and show that it is in agreement with real world co-authorship data. Finally, based on our intuition of collaboration formation in domains such as collaborative scientific experiments and movie production, we propose two variants of GENESCs based on clamped and hybrid preferential attachment schemes, and show that they perform well in these domains.

Navigability of Random Geometric Graphs in the Universe and Other Spacetimes

Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter. We find these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology.

Weighted growing simplicial complexes

Simplicial complexes describe collaboration networks, protein interaction networks, and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial complexes are weighted. Here we propose a nonequilibrium model for weighted growing simplicial complexes. The proposed dynamics is able to generate weighted simplicial complexes with a rich interplay between weights and topology emerging not just at the level of nodes and links, but also at the level of faces of higher dimension.

Emergent Hyperbolic Network Geometry

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a discretization of spacetime. Here, by extending our knowledge of growing complex networks to growing simplicial complexes we investigate the nature of the emergent geometry of complex networks and explore whether this geometry is hyperbolic. Specifically we show that an hyperbolic network geometry emerges spontaneously from models of growing simplicial complexes that are purely combinatorial. The statistical and geometrical properties of the growing simplicial complexes strongly depend on their dimensionality and display the major universal properties of real complex networks (scale-free degree distribution, small-world and communities) at the same time. Interestingly, when the network dynamics includes an heterogeneous fitness of the faces, the growing simplicial complex can undergo phase transitions that are reflected by relevant changes in the network geometry.

Random holographic "large worlds" with emergent dimensions

I propose a random network model governed by a Gaussian weight corresponding to Ising link antiferromagnetism as a model for emergent quantum space-time. In this model, discrete space is fundamental, not a regularization; its spectral dimension d_{s} is not a model input but is, rather, completely determined by the antiferromagnetic coupling constant. Perturbative terms suppressing triangles and favoring squares lead to locally Euclidean ground states that are Ricci flat "large worlds" with power-law extension. I then consider the quenched graphs of lowest energy for d_{s}=2 and d_{s}=3, and I show how quenching leads to the spontaneous emergence of embedding spaces of Hausdorff dimension d_{H}=4 and d_{H}=5, respectively. One of the additional, spontaneous dimensions can be interpreted as time, causality being an emergent property that arises in the large N limit (with N the number of vertices). For d_{s}=2, the quenched graphs constitute a discrete version of a 5D-space-filling surface with a number of fundamental degrees of freedom scaling like N^{2/5}, a graph version of the holographic principle. These holographic degrees of freedom can be identified with the squares of the quenched graphs, which, being triangle-free, are the fundamental area (or loop) quanta.

Eigenvalue tunneling and decay of quenched random network

We consider the canonical ensemble of N-vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity μ for each closed triple of bonds. We claim complete defragmentation of large-N graphs into the collection of [p^{-1}] almost full subgraphs (cliques) above critical fugacity, μ_{c}, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing μ leads to the formation of a multizonal support for μ>μ_{c}. Eigenvalue tunneling from the central zone to the side one means formation of a new clique in the defragmentation process. The adjacency matrix of the network ground state has a block-diagonal form, where the number of vertices in blocks fluctuates around the mean value Np. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.

Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes

Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complexes describe a large variety of complex interacting systems ranging from brain networks to social and collaboration networks. Here we characterize the structure of simplicial complexes using their generalized degrees that capture fundamental properties of one, two, three, or more linked nodes. Moreover, we introduce the configuration model and the canonical ensemble of simplicial complexes, enforcing, respectively, the sequence of generalized degrees of the nodes and the sequence of the expected generalized degrees of the nodes. We evaluate the entropy of these ensembles, finding the asymptotic expression for the number of simplicial complexes in the configuration model. We provide the algorithms for the construction of simplicial complexes belonging to the configuration model and the canonical ensemble of simplicial complexes. We give an expression for the structural cutoff of simplicial complexes that for simplicial complexes of dimension d=1 reduces to the structural cutoff of simple networks. Finally, we provide a numerical analysis of the natural correlations emerging in the configuration model of simplicial complexes without structural cutoff.