Scale-free Degree Distribution Research Articles

The contact process on scale-free geometric random graphs

We study the contact process on a class of geometric random graphs with scale-free degree distribution, defined on a Poisson point process on Rd. This class includes the age-dependent random connection model and the soft Boolean model. In the ultrasmall regime of these random graphs we provide exact asymptotics for the non-extinction probability when the rate of infection spread is small and show for a finite version of these graphs that the extinction time is of exponential order in the size of the graph.

Metastatic cells exploit their stoichiometric niche in the network of cancer ecosystems

Metastasis is a nonrandom process with varying degrees of organotropism—specific source-acceptor seeding. Understanding how patterns between source and acceptor tumors emerge remains a challenge in oncology. We hypothesize that organotropism results from the macronutrient niche of cells in source and acceptor organs. To test this, we constructed and analyzed a metastatic network based on 9303 records across 28 tissue types. We found that the topology of the network is nested and modular with scale-free degree distributions, reflecting organotropism along a specificity/generality continuum. The variation in topology is significantly explained by the matching of metastatic cells to their stoichiometric niche. Specifically, successful metastases are associated with higher phosphorus content in the acceptor compared to the source organ, due to metabolic constraints in proliferation crucial to the invasion of new tissues. We conclude that metastases are codetermined by processes at source and acceptor organs, where phosphorus content is a limiting factor orchestrating tumor ecology.

4Ward: A relayering strategy for efficient training of arbitrarily complex directed acyclic graphs

Thanks to their ease of implementation, multilayer perceptrons (MLPs) have become ubiquitous in deep learning applications. The graph underlying an MLP is indeed multipartite, i.e. each layer of neurons only connects to neurons belonging to the adjacent layer. In contrast, in vivo brain connectomes at the level of individual synapses suggest that biological neuronal networks are characterized by scale-free degree distributions or exponentially truncated power law strength distributions, hinting at potentially novel avenues for the exploitation of evolution-derived neuronal networks. In this paper, we present “4Ward”, a method and Python library capable of generating flexible and efficient neural networks (NNs) from arbitrarily complex directed acyclic graphs. 4Ward is inspired by layering algorithms drawn from the graph drawing discipline to implement efficient forward passes, and provides significant time gains in computational experiments with various Erdős-Rényi graphs. 4Ward not only overcomes the sequential nature of the learning matrix method, by parallelizing the computation of activations, but also addresses the scalability issues encountered in the current state-of-the-art and provides the designer with freedom to customize weight initialization and activation functions. Our algorithm can be of aid for any investigator seeking to exploit complex topologies in a NN design framework at the microscale.

Identifying bias in network clustering quality metrics.

We study potential biases of popular network clustering quality metrics, such as those based on the dichotomy between internal and external connectivity. We propose a method that uses both stochastic and preferential attachment block models construction to generate networks with preset community structures, and Poisson or scale-free degree distribution, to which quality metrics will be applied. These models also allow us to generate multi-level structures of varying strength, which will show if metrics favour partitions into a larger or smaller number of clusters. Additionally, we propose another quality metric, the density ratio. We observed that most of the studied metrics tend to favour partitions into a smaller number of big clusters, even when their relative internal and external connectivity are the same. The metrics found to be less biased are modularity and density ratio.

A matrix completion bootstrap method for estimating scale-free network degree distribution

Scale-free networks are very common in real practice. The key to analyze scale-free networks is the statistical inference of degree distribution. However, one observed network only allows us to calculate network statistics such as nodal degree, but does not provide enough information for further inference such as constructing confidence intervals. Borrowing the spirit of bootstrap, by generating network samples as bootstrap samples, we are then able to quantify statistical accuracy of various network statistics. In this paper, we propose a novel network bootstrap method named 1-BNB where bootstrap samples are generated via 1-bit matrix completion. We focus on constructing confidence interval for network degree distribution. Extensive simulation studies are conducted to demonstrate the finite sample performance of our newly propose method. A collaboration network among statisticians, a social network and an electrical grid network are studied for illustration purpose.

Effects of quadrilateral clustering on complex contagion

Clustering is one of the most important properties that determine the function of complex networks. But the conventional clustering coefficient considers only triangles without a clear basis. To examine the role of higher-order clustering beyond the conventional triangular clustering, we propose the quadrilateral clustering coefficient that counts the number of cycles of length 4. We also present algorithms to generate quadrilateral clustered networks with regular and scale-free degree distributions. We study the complex contagion model, where clustering promotes spreading. We show that quadrilateral clustered networks have a significant clustering effect, despite negligible conventional clustering coefficient. Moreover, we demonstrate that the clustering effect is stronger in the square lattice with zero conventional clustering coefficient than in the kagome lattice with a sizable conventional clustering coefficient, counterintuitively. Therefore, we conclude that the clustering by quadrilaterals is critical as well as the classical triangular clustering at least in complex contagion.

Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution

We study geometric random graphs defined on the points of a Poisson process in d-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centered in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.

Hyperbolic trees for efficient routing computation

Complex system theory is increasingly applied to develop control protocols for distributed computational and networking resources. The paper deals with the important subproblem of finding complex connected structures having excellent navigability properties using limited computational resources. Recently, the two-dimensional hyperbolic space turned out to be an efficient geometry for generative models of complex networks. The networks generated using the hyperbolic metric space share their basic structural properties (like small diameter or scale-free degree distribution) with several real networks. In the paper, a new model is proposed for generating navigation trees for complex networks embedded in the two-dimensional hyperbolic plane. The generative model is not based on known hyperbolic network models: the trees are not inferred from the existing links of any network; they are generated from scratch instead and based purely on the hyperbolic coordinates of nodes. We show that these hyperbolic trees have scale-free degree distributions and are present to a large extent both in synthetic hyperbolic complex networks and real ones (Internet autonomous system topology, US flight network) embedded in the hyperbolic plane. As the main result, we show that routing on the generated hyperbolic trees is optimal in terms of total memory usage of forwarding tables.

Spatial Networks and Percolation

The classical percolation problem is to find whether there is an infinite connected component in a random set created by removing edges from a $d$-dimensional lattice, independently at random. Since its introduction into the mathematical literature by Broadbent and Hammersley (1957) the subject of percolation has developed in many ways and is now one of the most exciting and active research areas in probability and statistical mechanics. In this workshop, we focused on current trends, including percolation on point sets with correlations, on spatial random graphs and networks with scale-free degree distribution or long-range edge distribution, percolation of random sets like level set of Gaussian fields or the vacant set of interlacements, conformally invariant percolation structures in the plane, and random walks or information diffusion on percolation clusters. The workshop brought together more than sixty experts and promising young researchers from probability, statistical mechanics and computer science working on all aspects of percolation and spatial networks for an unprecedented exchange of new ideas and methods, with outstanding high quality online talks.

Generalised popularity-similarity optimisation model for growing hyperbolic networks beyond two dimensions

Hyperbolic network models have gained considerable attention in recent years, mainly due to their capability of explaining many peculiar features of real-world networks. One of the most widely known models of this type is the popularity-similarity optimisation (PSO) model, working in the native disk representation of the two-dimensional hyperbolic space and generating networks with small-world property, scale-free degree distribution, high clustering and strong community structure at the same time. With the motivation of better understanding hyperbolic random graphs, we hereby introduce the dPSO model, a generalisation of the PSO model to any arbitrary integer dimension d>2. The analysis of the obtained networks shows that their major structural properties can be affected by the dimension of the underlying hyperbolic space in a non-trivial way. Our extended framework is not only interesting from a theoretical point of view but can also serve as a starting point for the generalisation of already existing two-dimensional hyperbolic embedding techniques.

Recurrence versus transience for weight-dependent random connection models

We investigate random graphs on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random mark and given marks and positions of the points we form an edge between two points independently with a probability depending via a kernel on the two marks and the distance of the points. Different kernels allow the mark to play different roles, like weight, radius or birth time of a vertex. The kernels depend on a parameter γ, which determines the power-law exponent of the degree distributions. A further independent parameter δ characterises the decay of the connection probabilities of vertices as their distance increases. We prove transience of the infinite cluster in the entire supercritical phase in regimes given by the parameters γ and δ, and complement these results by recurrence results if d=2. Our results are particularly interesting for the soft Boolean graph model discussed in the preprint [arXiv:2108:11252] and the age-dependent random connection model recently introduced by Gracar et al. [Queueing Syst. 93.3-4 (2019)]

Extremal linkage networks

We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with max-stable distribution. The fitness determines the node attractiveness w.r.t. incoming edges as well as the spatial range for outgoing edges. For max-stable fitness distributions, we thus obtain a complex spatial network, which we coin extremal linkage network.

Percolation phase transition in weight-dependent random connection models

AbstractWe investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.

Trust-based Friend Selection Algorithm for navigability in social Internet of Things

The Internet of Things (IoT) is a growing area of millions of heterogeneous objects denoted by intensive interactions. The social Internet of Things (SIoT) is a promising approach for expediting object interaction issues by integrating the IoT’s social networking notion. The object discovery and service search are promoted with the right friend selection and trust management on that ground. The number of friends and complexity of devoting relation types affect the navigability in a network. On the other hand, constraints in the storage area and battery life of SIoT devices are two crucial challenges that highlighted the importance of mechanisms to increase the lifetime and robustness of devices. Therefore, this scenario addressed an intelligent friend selection strategy by considering objects’ characteristics, typology, and associated functionalities. This issue is leveraged by trust value to eliminate interception of untrustworthy aspects. We designed a generic reference model and utilized optimization decision theory for optimal friend choice to reduce resource consumption to this effect. The simulation results illuminated that a rational selection of friends per service exploration fortifies global navigability in measuring average path length, degree distribution, and the number of links. The efficiency of our solution was revealed by lessening the average distance and promoting the number of links to achieve a well-tied network and benefiting scale-free degree distribution, which hinted existence of hubs or high degree nodes.

Scale-free networks may not necessarily witness cooperation

Networks with a scale-free degree distribution are widely thought to promote cooperation in various games. Herein, by studying the well-known prisoner's dilemma game, we demonstrate that this need not necessarily be true. For the very same degree sequence and degree distribution, we present a variety of possible behaviour. We reassess the perceived importance of hubs in a network towards the maintenance of cooperation. We also reevaluate the dependence of cooperation on network clustering and assortativity.

Optimisation of the coalescent hyperbolic embedding of complex networks

Several observations indicate the existence of a latent hyperbolic space behind real networks that makes their structure very intuitive in the sense that the probability for a connection is decreasing with the hyperbolic distance between the nodes. A remarkable network model generating random graphs along this line is the popularity-similarity optimisation (PSO) model, offering a scale-free degree distribution, high clustering and the small-world property at the same time. These results provide a strong motivation for the development of hyperbolic embedding algorithms, that tackle the problem of finding the optimal hyperbolic coordinates of the nodes based on the network structure. A very promising recent approach for hyperbolic embedding is provided by the noncentered minimum curvilinear embedding (ncMCE) method, belonging to the family of coalescent embedding algorithms. This approach offers a high-quality embedding at a low running time. In the present work we propose a further optimisation of the angular coordinates in this framework that seems to reduce the logarithmic loss and increase the greedy routing score of the embedding compared to the original version, thereby adding an extra improvement to the quality of the inferred hyperbolic coordinates.

Growing scale-free simplices

The past two decades have seen significant successes in our understanding of networked systems, from the mapping of real-world networks to the establishment of generative models recovering their observed macroscopic patterns. These advances, however, are restricted to pairwise interactions and provide limited insight into higher-order structures. Such multi-component interactions can only be grasped through simplicial complexes, which have recently found applications in social, technological, and biological contexts. Here we introduce a model to grow simplicial complexes of order two, i.e., nodes, links, and triangles, that can be straightforwardly extended to structures containing hyperedges of larger order. Specifically, through a combination of preferential and/or nonpreferential attachment mechanisms, the model constructs networks with a scale-free degree distribution and an either bounded or scale-free generalized degree distribution. We arrive at a highly general scheme with analytical control of the scaling exponents to construct ensembles of synthetic complexes displaying desired statistical properties.

Scale-free degree distributions, homophily and the glass ceiling effect in directed networks

Abstract Preferential attachment, homophily, and their consequences such as scale-free (i.e. power-law) degree distributions, the glass ceiling effect (the unseen, yet unbreakable barrier that keeps minorities and women from rising to the upper rungs of the corporate ladder, regardless of their qualifications or achievements) and perception bias are well-studied in undirected networks. However, such consequences and the factors that lead to their emergence in directed networks (e.g. author–citation graphs, Twitter) are yet to be coherently explained in an intuitive, theoretically tractable manner using a single dynamical model. To this end, we present a theoretical and numerical analysis of the novel Directed Mixed Preferential Attachment model in order to explain the emergence of scale-free degree distributions and the glass ceiling effect in directed networks with two groups (minority and majority). Specifically, we first derive closed-form expressions for the power-law exponents of the in-degree and out-degree distributions of each of the two groups and then compare the derived exponents with each other to obtain useful insights. These insights include answers to questions such as: when does the minority group have an out-degree (or in-degree) distribution with a heavier tail compared to the majority group? what factors cause the tail of the out-degree distribution of a group to be heavier than the tail of its own in-degree distribution? what effect does frequent addition of edges between existing nodes have on the in-degree and out-degree distributions of the majority and minority groups? Answers to these questions shed light on the interplay between structure (i.e. the in-degree and out-degree distributions of the two groups) and dynamics (characterized collectively by the homophily, preferential attachment, group sizes and growth dynamics) of various real-world directed networks. We also provide a novel definition of the glass ceiling faced by a group via the number of individuals with large out-degree (i.e. those with many followers) normalized by the number of individuals with large in-degree (i.e. those who follow many others) and then use it to characterize the conditions that cause the glass ceiling effect to emerge in a directed network. Our analytical results are supported by detailed numerical experiments. The DMPA model and its theoretical and numerical analysis provided in this article are useful for analysing various phenomena on directed networks in fields such as network science and computational social science.

Emergence of nonlinear crossover under epidemic dynamics in heterogeneous networks.

Potential diffusion processes of real-world systems are relevant to the underlying network structure and dynamical mechanisms. The vast majority of the existing work on spreading dynamics, in response to a large-scale network, is built on the condition of the infinite initial state, i.e., the extremely small seed size. The impact of an increasing seed size on the persistent diffusion has been less investigated. Based on classical epidemic models, this paper offers a framework for studying such impact through observing a crossover phenomenon in a two-diffusion-process dynamical system. The two diffusion processes are triggered by nodes with a high and low centrality, respectively. Specifically, given a finite initial state in networks with scale-free degree distributions, we demonstrate analytically and numerically that the diffusion process triggered by low centrality nodes pervades faster than that triggered by high centrality nodes from a certain point. The presence of the crossover phenomenon reveals that the dynamical process under the finite initial state is far more than the vertical scaling of the spreading curve under an infinite initial state. Further discussion emphasizes the persistent infection of individuals in epidemic dynamics as the essential reason rooted in the crossover, while the finite initial state is the catalyst directly leading to the emergence of this phenomenon. Our results provide valuable implications for studying the diversity of hidden dynamics on heterogeneous networks.

Dynamical Network Analysis of the South Korean Dialects Compared to Traditional Dialect Classification

In this paper, we investigate the network properties of the South Korean dialects. Among 153 words in “The Linguistic Atlas of Korea”, 33 words satisfying our criteria were surveyed. We set 138 basic administrative districts that make up South Korea as nodes and construct a language network by using a defined procedure. The network shows neither a small-world (small-world coefficient < 5) nor a scale-free (degree distribution forms like a unimodal Poisson) property in the general complex network. Also, the assortativity is seen to show transition from negative to positive at about the threshold value of 15. The number of Korean linguistic groups, which is the most import result in this study, is constantly four with s-curved Newman modularity in the threshold range 13 ∼ 22. The Korean dialect division by using a network modularity method is almost identical to the classification of traditional linguists (Central, Southwest, Southeast, and Jeju). Furthermore, subgroups can be obtained by performing a hierarchical dialect division. As a result, South Korean dialects are classified into seven minor groups, and some regions (e.g., Southern Gang-won) belong to groups different from those of the traditional classification method.