Dense power-law networks and simplicial complexes

A scaling law to model the effectiveness of identification techniques

AI techniques are increasingly being used to identify individuals both offline and online. However, quantifying their effectiveness at scale and, by extension, the risks they pose remains a significant challenge. Here, we propose a two-parameter Bayesian model for exact matching techniques and derive an analytical expression for correctness (κ), the fraction of people accurately identified in a population. We then generalize the model to forecast how κ scales from small-scale experiments to the real world, for exact, sparse, and machine learning-based robust identification techniques. Despite having only two degrees of freedom, our method closely fits 476 correctness curves and strongly outperforms curve-fitting methods and entropy-based rules of thumb. Our work provides a principled framework for forecasting the privacy risks posed by identification techniques, while also supporting independent accountability efforts for AI-based biometric systems.

Network science: Ising states of matter

Network science provides very powerful tools for extracting information from interacting data. Although recently the unsupervised detection of phases of matter using machine learning has raised significant interest, the full prediction power of network science has not yet been systematically explored in this context. Here we fill this gap by providing an in-depth statistical, combinatorial, geometrical, and topological characterization of 2D Ising snapshot networks (IsingNets) extracted from Monte Carlo simulations of the 2D Ising model at different temperatures, going across the phase transition. Our analysis reveals the complex organization properties of IsingNets in both the ferromagnetic and paramagnetic phases and demonstrates the significant deviations of the IsingNets with respect to randomized null models. In particular percolation properties of the IsingNets reflect the existence of the symmetry between configurations with opposite magnetization below the critical temperature and the very compact nature of the two emerging giant clusters revealed by our persistent homology analysis of the IsingNets. Moreover, the IsingNets display a very broad degree distribution and significant degree-degree correlations and weight-degree correlations demonstrating that they encode relevant information present in the configuration space of the 2D Ising model. The geometrical organization of the critical IsingNets is reflected in their spectral properties deviating from the one of the null model. This work reveals the important insights that network science can bring to the characterization of phases of matter. The set of tools described hereby can be applied as well to numerical and experimental data.

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.

Study on the regional risk classification method for the prevention and control of emerging infectious diseases based on directed graph theory

Background

Emerging infectious diseases are a class of diseases that are spreading rapidly and are highly contagious. It seriously affects social stability and poses a significant threat to human health, requiring urgent measures to deal with them. Its outbreak will very easily lead to the large-scale spread of the virus, causing social problems such as work stoppages and traffic control, thereby causing social panic and psychological unrest, affecting human activities and social stability, and even endangering lives. It is essential to prevent and control the spread of infectious diseases effectively.

Purpose

We aim to propose an effective method to classify the risk level of a new epidemic region by using graph theory and risk classification methods to provide a theoretical reference for the comprehensive evaluation and determination of epidemic prevention and control, as well as risk level classification.

Methods

Using the graph theory method, we first define the network structure of social groups and construct the risk transmission network of the new epidemic region. Then, combined with the risk classification method, the classification of high, medium, and low risk levels of the new epidemic region is discussed from two cases with common and looped graph nodes, respectively. Finally, the reasonableness of the classification method is verified by simulation data.

Results

The directed weighted scale-free network can better describe the transmission law of an epidemic. Moreover, the proposed method of classifying the risk level of a region by using the correlation function between two regions and the risk value of the regional nodes can effectively evaluate the risk level of different regions in the new epidemic region. The experiments show that the number of medium and high risk nodes shows no increasing trend. The number of high-risk regions is relatively small compared to medium-risk regions, and the number of low-risk regions is the largest.

Conclusions

It is necessary to distinguish scientifically between the risk level of the epidemic area and the neighboring regions so that the constructed social network model of the epidemic region's spread risk can better describe the spread of the epidemic risk in the social network relations.

Modeling and analysis of affiliation networks with preferential attachment and subsumption

Preferential attachment describes a variety of graph-based models in which a network grows incrementally via the sequential addition of new nodes and edges, and where existing nodes acquire new neighbors at a rate proportional to their degree. Some networks, however, are better described as groups of nodes rather than a set of pairwise connections. These groups are called affiliations, and the corresponding networks affiliation networks. When viewed as graphs, affiliation networks do not necessarily exhibit the power law distribution of node degrees that is typically associated with preferential attachment. We propose a preferential attachment mechanism for affiliation networks that highlights the power law characteristic of these networks when presented as hypergraphs and simplicial complexes. The two representations capture affiliations in similar ways, but the latter offers an intrinsic feature of the model called subsumption, where an affiliation cannot be a subset of another. Our model of preferential attachment has interesting features, both algorithmic and analytic, including implicit preferential attachment (node sampling does not require knowledge of node degrees), a locality property where the neighbors of a newly added node are also neighbors, the emergence of a power law distribution of degrees (defined in hypergraphs and simplicial complexes rather than at a graph level), implicit deletion of affiliations (through subsumption in the case of simplicial complexes), and to some extent a control over the affiliation size distribution. By varying the parameters of the model, the generated affiliation networks can resemble different types of real-world examples, so the framework also serves as a synthetic generation algorithm for simulation and experimental studies.

Complex hypergraphs

Providing an abstract representation of natural and human complex structures is a challenging problem. Accounting for the system heterogenous components while allowing for analytical tractability is a difficult balance. Here I introduce complex hypergraphs (chygraphs), bringing together concepts from hypergraphs, multilayer networks, simplicial complexes, and hyperstructures. To illustrate the applicability of this combinatorial structure I calculate the component sizes statistics and identify the transition to a giant component. To this end I introduce a vectorization technique that tackles the multilevel nature of chygraphs. I conclude that chygraphs are a unifying representation of complex systems allowing for analytical insight.

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.

An Experimental Study on the Scalability of Recent Node Centrality Metrics in Sparse Complex Networks

Node centrality measures are among the most commonly used analytical techniques for networks. They have long helped analysts to identify “important” nodes that hold power in a social context, where damages could have dire consequences for transportation applications, or who should be a focus for prevention in epidemiology. Given the ubiquity of network data, new measures have been proposed, occasionally motivated by emerging applications or by the ability to interpolate existing measures. Before analysts use these measures and interpret results, the fundamental question is: are these measures likely to complete within the time window allotted to the analysis? In this paper, we comprehensively examine how the time necessary to run 18 new measures (introduced from 2005 to 2020) scales as a function of the number of nodes in the network. Our focus is on giving analysts a simple and practical estimate for sparse networks. As the time consumption depends on the properties in the network, we nuance our analysis by considering whether the network is scale-free, small-world, or random. Our results identify that several metrics run in the order of O(nlogn) and could scale to large networks, whereas others can require O(n²) or O(n³) and may become prime targets in future works for approximation algorithms or distributed implementations.

Analysis and synthesis of a growing network model generating dense scale-free networks via category theory

We propose a growing network model that can generate dense scale-free networks with an almost neutral degree−degree correlation and a negative scaling of local clustering coefficient. The model is obtained by modifying an existing model in the literature that can also generate dense scale-free networks but with a different higher-order network structure. The modification is mediated by category theory. Category theory can identify a duality structure hidden in the previous model. The proposed model is built so that the identified duality is preserved. This work is a novel application of category theory for designing a network model focusing on a universal algebraic structure.

Scale-free networks with invariable diameter and density feature: Counterexamples

Here, we propose a class of scale-free networks G(t;m) with intriguing properties, which cannot be simultaneously held by all the theoretical models with power-law degree distribution in the existing literature, including the following: (i) average degrees 〈k〉 of all the generated networks are no longer constant in the limit of large graph size, implying that they are not sparse but dense; (ii) power-law parameters γ of these networks are precisely calculated equal to 2; and (iii) their diameters D are all invariant in the growth process of models. While our models have deterministic structure with clustering coefficients equivalent to zero, we might be able to obtain various candidates with nonzero clustering coefficients based on original networks using reasonable approaches, for instance, randomly adding new edges under the premise of keeping the three important properties above unchanged. In addition, we study the trapping problem on networks G(t;m) and then obtain a closed-form solution to mean hitting time 〈H〉_{t}. As opposed to other previous models, our results show an unexpected phenomenon that the analytic value for 〈H〉_{t} is approximately close to the logarithm of the vertex number of networks G(t;m). From the theoretical point of view, these networked models considered here can be thought of as counterexamples for most of the published models obeying power-law distribution in current study.

A Maximum Entropy Method for the Prediction of Size Distributions

We propose a method to derive the stationary size distributions of a system, and the degree distributions of networks, using maximisation of the Gibbs-Shannon entropy. We apply this to a preferential attachment-type algorithm for systems of constant size, which contains exit of balls and urns (or nodes and edges for the network case). Knowing mean size (degree) and turnover rate, the power law exponent and exponential cutoff can be derived. Our results are confirmed by simulations and by computation of exact probabilities. We also apply this entropy method to reproduce existing results like the Maxwell-Boltzmann distribution for the velocity of gas particles, the Barabasi-Albert model and multiplicative noise systems.

Ordinal Preferential Attachment: A Self-Organizing Principle Generating Dense Scale-Free Networks

Networks are useful representations for analyzing and modeling real-world complex systems. They are often both scale-free and dense: their degree distribution follows a power-law and their average degree grows over time. So far, it has been argued that producing such networks is difficult without externally imposing a suitable cutoff for the scale-free regime. Here, we propose a new growing network model that produces dense scale-free networks with dynamically generated cutoffs. The link formation rule is based on a weak form of preferential attachment depending only on order relations between the degrees of nodes. By this mechanism, our model yields scale-free networks whose scaling exponents can take arbitrary values greater than 1. In particular, the resulting networks are dense when scaling exponents are 2 or less. We analytically study network properties such as the degree distribution, the degree correlation function, and the local clustering coefficient. All analytical calculations are in good agreement with numerical simulations. These results show that both sparse and dense scale-free networks can emerge through the same self-organizing process.

Persistent homology of unweighted complex networks via discrete Morse theory

Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced filtration scheme based on the subsequence of the corresponding critical weights, thereby leading to a significant increase in computational efficiency. We have employed our filtration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect differences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between different model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks.

A network approach to topic models

One of the main computational and scientific challenges in the modern age is to extract useful information from unstructured texts. Topic models are one popular machine-learning approach that infers the latent topical structure of a collection of documents. Despite their success-particularly of the most widely used variant called latent Dirichlet allocation (LDA)-and numerous applications in sociology, history, and linguistics, topic models are known to suffer from severe conceptual and practical problems, for example, a lack of justification for the Bayesian priors, discrepancies with statistical properties of real texts, and the inability to properly choose the number of topics. We obtain a fresh view of the problem of identifying topical structures by relating it to the problem of finding communities in complex networks. We achieve this by representing text corpora as bipartite networks of documents and words. By adapting existing community-detection methods (using a stochastic block model (SBM) with nonparametric priors), we obtain a more versatile and principled framework for topic modeling (for example, it automatically detects the number of topics and hierarchically clusters both the words and documents). The analysis of artificial and real corpora demonstrates that our SBM approach leads to better topic models than LDA in terms of statistical model selection. Our work shows how to formally relate methods from community detection and topic modeling, opening the possibility of cross-fertilization between these two fields.