Heavy-tailed Degree Distributions Research Articles

Complex systems of secrecy: the offshore networks of oligarchs.

Following the invasion of Ukraine, the USA, UK, and EU governments-among others-sanctioned oligarchs close to Putin. This approach has come under scrutiny, as evidence has emerged of the oligarchs' successful evasion of these punishments. To address this problem, we analyze the role of an overlooked but highly influential group: the secretive professional intermediaries who create and administer the oligarchs' offshore financial empires. Drawing on the Offshore Leaks Database provided by the International Consortium of Investigative Journalists (ICIJ), we examine the ties linking offshore expert advisors (lawyers, accountants, and other wealth management professionals) to ultra-high-net-worth individuals from four countries: Russia, China, the USA, and Hong Kong. We find that resulting nation-level "oligarch networks" share a scale-free structure characterized by a heterogeneity of heavy-tailed degree distributions of wealth managers; however, network topologies diverge across clients from democratic versus autocratic regimes. While generally robust, scale-free networks are fragile when targeted by attacks on highly connected nodes. Our "knock-out" experiments pinpoint this vulnerability to the small group of wealth managers themselves, suggesting that sanctioning these professional intermediaries may be more effective and efficient in disrupting dark finance flows than sanctions on their wealthy clients. This vulnerability is especially pronounced amongst Russian oligarchs, who concentrate their offshore business in a handful of boutique wealth management firms. The distinctive patterns we identify suggest a new approach to sanctions, focused on expert intermediaries to disrupt the finances and alliances of their wealthy clients. More generally, our research contributes to the larger body of work on complexity science and the structures of secrecy.

Influence maximization under limited network information: seeding high-degree neighbors

The diffusion of information, norms, and practices across a social network can be initiated by compelling a small number of seed individuals to adopt first. Strategies proposed in previous work either assume full network information or a large degree of control over what information is collected. However, privacy settings on the Internet and high non-response in surveys often severely limit available connectivity information. Here we propose a seeding strategy for scenarios with limited network information: Only the degrees and connections of some random nodes are known. This new strategy is a modification of ‘random neighbor sampling’ (or ‘one-hop’) and seeds the highest-degree neighbors of randomly selected nodes. Simulating a fractional threshold model, we find that this new strategy excels in networks with heavy tailed degree distributions such as scale-free networks and large online social networks. It outperforms the conventional one-hop strategy even though the latter can seed 50% more nodes, and other seeding possibilities including pure high-degree seeding and clustered seeding.

Random processes with high variance produce scale free networks

Real-world networks tend to be scale free, having heavy-tailed degree distributions with more hubs than predicted by classical random graph generation methods. Preferential attachment and growth are the most commonly accepted mechanisms leading to these networks and are incorporated in the Barabási–Albert (BA) model (Barabási, 2009 [1]). We provide an alternative model using a randomly stopped linking process inspired by a generalized Central Limit Theorem (CLT) for geometric distributions with widely varying parameters. The common characteristic of both the BA model and our randomly stopped linking model is the mixture of widely varying geometric distributions, suggesting the critical characteristic of scale free networks is high variance, not growth or preferential attachment. The limitation of classical random graph models is low variance in parameters, while scale free networks are the natural, expected result of real-world variance.

Modelling the impact of social distancing and targeted vaccination on the spread of COVID-19 through a real city-scale contact network.

We use mobile device data to construct empirical interpersonal physical contact networks in the city of Portland, Oregon, both before and after social distancing measures were enacted during the COVID-19 pandemic. These networks reveal how social distancing measures and the public’s reaction to the incipient pandemic affected the connectivity patterns within the city. We find that as the pandemic developed there was a substantial decrease in the number of individuals with many contacts. We further study the impact of these different network topologies on the spread of COVID-19 by simulating an SEIR epidemic model over these networks and find that the reduced connectivity greatly suppressed the epidemic. We then investigate how the epidemic responds when part of the population is vaccinated, and we compare two vaccination distribution strategies, both with and without social distancing. Our main result is that the heavy-tailed degree distribution of the contact networks causes a targeted vaccination strategy that prioritizes high-contact individuals to reduce the number of cases far more effectively than a strategy that vaccinates individuals at random. Combining both targeted vaccination and social distancing leads to the greatest reduction in cases, and we also find that the marginal benefit of a targeted strategy as compared to a random strategy exceeds the marginal benefit of social distancing for reducing the number of cases. These results have important implications for ongoing vaccine distribution efforts worldwide.

Non-parametric estimation of the preferential attachment function from one network snapshot

Abstract Preferential attachment is commonly invoked to explain the emergence of those heavy-tailed degree distributions characteristic of growing network representations of diverse real-world phenomena. Experimentally confirming this hypothesis in real-world growing networks is an important frontier in network science research. Conventional preferential attachment estimation methods require that a growing network be observed across at least two snapshots in time. Numerous publicly available growing network datasets are, however, only available as single snapshots, leaving the applied network scientist with no means of measuring preferential attachment in these cases. We propose a nonparametric method, called PAFit-oneshot, for estimating preferential attachment in a growing network from one snapshot. PAFit-oneshot corrects for a previously unnoticed bias that arises when estimating preferential attachment values only for degrees observed in the single snapshot. Our work provides a means of measuring preferential attachment in a large number of publicly available one-snapshot networks. As a demonstration, we estimated preferential attachment in three such networks, and found sublinear preferential attachment in all cases. PAFit-oneshot is implemented in the $\textsf{R}$ package $\texttt{PAFit}$.

Assortativity and bidegree distributions on Bernoulli random graph superpositions

AbstractA probabilistic generative network model with$n$nodes and$m$overlapping layers is obtained as a superposition of$m$mutually independent Bernoulli random graphs of varying size and strength. When$n$and$m$are large and of the same order of magnitude, the model admits a sparse limiting regime with a tunable power-law degree distribution and nonvanishing clustering coefficient. In this article, we prove an asymptotic formula for the joint degree distribution of adjacent nodes. This yields a simple analytical formula for the model assortativity and opens up ways to analyze rank correlation coefficients suitable for random graphs with heavy-tailed degree distributions. We also study the effects of power laws on the asymptotic joint degree distributions.

Learning function from structure in neuromorphic networks

The connection patterns of neural circuits in the brain form a complex network. Collective signalling within the network manifests as patterned neural activity and is thought to support human cognition and adaptive behaviour. Recent technological advances permit macroscale reconstructions of biological brain networks. These maps, termed connectomes, display multiple non-random architectural features, including heavy-tailed degree distributions, segregated communities and a densely interconnected core. Yet, how computation and functional specialization emerge from network architecture remains unknown. Here we reconstruct human brain connectomes using in vivo diffusion-weighted imaging and use reservoir computing to implement connectomes as artificial neural networks. We then train these neuromorphic networks to learn a memory-encoding task. We show that biologically realistic neural architectures perform best when they display critical dynamics. We find that performance is driven by network topology and that the modular organization of intrinsic networks is computationally relevant. We observe a prominent interaction between network structure and dynamics throughout, such that the same underlying architecture can support a wide range of memory capacity values as well as different functions (encoding or decoding), depending on the dynamical regime the network is in. This work opens new opportunities to discover how the network organization of the brain optimizes cognitive capacity. The relationship between brain organization, connectivity and computation is not well understood. The authors construct neuromorphic artificial neural networks endowed with biological connection patterns derived from diffusion-weighted imaging. The neuromorphic networks are trained to perform a memory task, revealing an interaction between network structure and dynamics.

Transitions between human functional brain networks reveal complex, cost-efficient and behaviorally-relevant temporal paths

Resting-state functional MRI activity is organized as a complex network. However, this coordinated brain activity changes with time, raising questions about its evolving temporal arrangement. Does the brain visit different configurations through time in a random or ordered way? Advances in this area depend on developing novel paradigms that would allow us to shed light on these issues. We here propose to study the temporal changes in the functional connectome by looking at transition graphs of network activity. Nodes of these graphs correspond to brief whole-brain connectivity patterns (or meta-states), and directed links to the temporal transition between consecutive meta-states. We applied this method to two datasets of healthy subjects (160 subjects and a replication sample of 54), and found that transition networks had several non-trivial properties, such as a heavy-tailed degree distribution, high clustering, and a modular organization. This organization was implemented at a low biological cost with a high cost-efficiency of the dynamics. Furthermore, characteristics of the subjects’ transition graphs, including global efficiency, local efficiency and their transition cost, were correlated with cognition and motor functioning. All these results were replicated in both datasets. We conclude that time-varying functional connectivity patterns of the brain in health progress in time in a highly organized and complex order, which is related to behavior.

Thresholding normally distributed data creates complex networks.

Network data sets are often constructed by some kind of thresholding procedure. The resulting networks frequently possess properties such as heavy-tailed degree distributions, clustering, large connected components, and short average shortest path lengths. These properties are considered typical of complex networks and appear in many contexts, prompting consideration of their universality. Here we introduce a simple model for correlated relational data and study the network ensemble obtained by thresholding it. We find that some, but not all, of the properties associated with complex networks can be seen after thresholding the correlated data, even though the underlying data are not "complex." In particular, we observe heavy-tailed degree distributions, a large numbers of triangles, and short path lengths, while we do not observe nonvanishing clustering or community structure.

Heterogeneous cooperative leadership structure emerging from random regular graphs.

This paper investigates the evolution of cooperation and the emergence of hierarchical leadership structure in random regular graphs. It is found that there exist different learning patterns between cooperators and defectors, and cooperators are able to attract more followers and hence more likely to become leaders. Hence, the heterogeneous distributions of reputation and leadership can emerge from homogeneous random graphs. The important directed game-learning skeleton is then studied, revealing some important structural properties, such as the heavy-tailed degree distribution and the positive in-in degree correlation.

Participation shifts explain degree distributions in a human communications network

Human interpersonal communications drive political, technological, and economic systems, placing importance on network link prediction as a fundamental problem of the sciences. These systems are often described at the network-level by degree counts —the number of communication links associated with individuals in the network—that often follow approximate Pareto distributions, a divergence from Poisson-distributed counts associated with random chance. A defining challenge is to understand the inter-personal dynamics that give rise to such heavy-tailed degree distributions at the network-level; primarily, these distributions are explained by preferential attachment, which, under certain conditions, can create power law distributions; preferential attachment’s prediction of these distributions breaks down, however, in conditions with no network growth. Analysis of an organization’s email network suggests that these degree distributions may be caused by the existence of individual participation-shift dynamics that are necessary for coherent communication between humans. We find that the email network’s degree distribution is best explained by turn-taking and turn-continuing norms present in most social network communication. We thus describe a mechanism to explain a long-tailed degree distribution in conditions with no network growth.

Towards attack tolerant networks: Concurrent multipath routing and the butterfly network.

It is crucial for large-scale communication networks such as the internet to be resilient against attacks such as censorship and surveillance, which pose a threat to free expression and free association. Self-organized networks such as the internet’s router network typically have heavy-tailed degree distributions, making them highly vulnerable to targeted attacks against central nodes. While cryptographic solutions exist, they fail to address the underlying topological problem, and remain vulnerable to man-in-the-middle attacks and coercion. Coercion-resistant, topological approaches to attack tolerance are needed to address the current vulnerability of communications infrastructure to censorship and surveillance. We present a novel concurrent multipath routing (CMR) algorithm for the wraparound butterfly network topology, as well as a highly attack-tolerant Structured Multipath Fault Tolerance (SMFT) architecture which incorporates the butterfly CMR algorithm. We also identify a previously unexplored relationship between network topology, trust transitivity, and attack-tolerance, and provide a framework for further exploration of this relationship. Our work is the first theoretical demonstration of a point-to-point communication network architecture that can resist coercion and other non-technical attacks, without requiring infinitely transitive trust. To address cases where the network structure cannot be fully controlled, we demonstrate how a snapshot of the internet’s router network can be partially rewired for greater attack-tolerance. More broadly, we hope that this work will serve as a starting point for the evelopment of additional topology-based attack-tolerant communication architectures to guard against the dangers of censorship and surveillance.

Evolution of cooperation on large networks with community structure.

Cooperation is a major factor in the evolution of human societies. The structure of social networks, which affects the dynamics of cooperation and other interpersonal phenomena, have common structural signatures. One of these signatures is the tendency to organize as groups. This tendency gives rise to networks with community structure, which are composed of distinct modules. In this paper, we study analytically the evolutionary game dynamics on large modular networks in the limit of weak selection. We obtain novel analytical conditions such that natural selection favours cooperation over defection. We calculate the transition point for each community to favour cooperation. We find that a critical inter-community link creation probability exists for given group density, such that the overall network supports cooperation even if individual communities inhibit it. As a byproduct, we present solutions for the critical benefit-to-cost ratio which perform with remarkable accuracy for diverse generative network models, including those with community structure and heavy-tailed degree distributions. We also demonstrate the generalizability of the results to arbitrary two-player games.

Assortative mixing in spatially-extended networks

We focus on spatially-extended networks during their transition from short-range connectivities to a scale-free structure expressed by heavy-tailed degree-distribution. In particular, a model is introduced for the generation of such graphs, which combines spatial growth and preferential attachment. In this model the transition to heterogeneous structures is always accompanied by a change in the graph’s degree-degree correlation properties: while high assortativity levels characterize the dominance of short distance couplings, long-range connectivity structures are associated with small amounts of disassortativity. Our results allow to infer that a disassortative mixing is essential for establishing long-range links. We discuss also how our findings are consistent with recent experimental studies of 2-dimensional neuronal cultures.

Characterization of the community structure in a large-scale production network in Japan

Inter-firm organizations, which play a driving role in the economy of a country, can be represented in the form of a customer–supplier network. Such a network exhibits a heavy-tailed degree distribution, disassortative mixing and a prominent community structure. We analyze a large-scale data set of customer–supplier relationships containing data from one million Japanese firms. Using a directed network framework, we show that the production network exhibits the characteristics listed above. We conduct detailed investigations to characterize the communities in the network. The topology within smaller communities is found to be very close to a tree-like structure but becomes denser as the community size increases. A large fraction (∼40%) of firms with relatively small in- or out-degrees have customers or suppliers solely from within their own communities, indicating interactions of a highly local nature. The interaction strengths between communities as measured by the inter-community link weights follow a highly heterogeneous distribution. We further present the statistically significant over-expressions of different prefectures and sectors within different communities.

Representation Learning for Scale-Free Networks

Network embedding aims to learn the low-dimensional representations of vertexes in a network, while structure and inherent properties of the network is preserved. Existing network embedding works primarily focus on preserving the microscopic structure, such as the first- and second-order proximity of vertexes, while the macroscopic scale-free property is largely ignored. Scale-free property depicts the fact that vertex degrees follow a heavy-tailed distribution (i.e., only a few vertexes have high degrees) and is a critical property of real-world networks, such as social networks. In this paper, we study the problem of learning representations for scale-free networks. We first theoretically analyze the difficulty of embedding and reconstructing a scale-free network in the Euclidean space, by converting our problem to the sphere packing problem. Then, we propose the "degree penalty" principle for designing scale-free property preserving network embedding algorithm: punishing the proximity between high-degree vertexes. We introduce two implementations of our principle by utilizing the spectral techniques and a skip-gram model respectively. Extensive experiments on six datasets show that our algorithms are able to not only reconstruct heavy-tailed distributed degree distribution, but also outperform state-of-the-art embedding models in various network mining tasks, such as vertex classification and link prediction.

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.

Modeling Spatial Social Complex Networks for Dynamical Processes

The study of social networks—where people are located, geographically, and how they might be connected to one another—is a current hot topic of interest, because of its immediate relevance to important applications, from devising efficient immunization techniques for the arrest of epidemics to the design of better transportation and city planning paradigms to the understanding of how rumors and opinions spread and take shape over time. We develop a Spatial Social Complex Network (SSCN) model that captures not only essential connectivity features of real-life social networks, including a heavy-tailed degree distribution and high clustering, but also the spatial location of individuals, reproducing Zipf’s law for the distribution of city populations as well as other observed hallmarks. We then simulate Milgram’s Small-World experiment on our SSCN model, obtaining good qualitative agreement with the known results and shedding light on the role played by various network attributes and the strategies used by the players in the game. This demonstrates the potential of the SSCN model for the simulation and study of the many social processes mentioned above, where both connectivity and geography play a role in the dynamics.

Properties of latent variable network models

AbstractWe derive properties of latent variable models for networks, a broad class of models that includes the widely used latent position models. We characterize several features of interest, with particular focus on the degree distribution, clustering coefficient, average path length, and degree correlations. We introduce the Gaussian latent position model, and derive analytic expressions and asymptotic approximations for its network properties. We pay particular attention to one special case, the Gaussian latent position model with random effects, and show that it can represent the heavy-tailed degree distributions, positive asymptotic clustering coefficients, and small-world behaviors that often occur in observed social networks. Finally, we illustrate the ability of the models to capture important features of real networks through several well-known datasets.

The structure and vulnerability of a drug trafficking collaboration network

Prior research has established that collaboration networks exhibit heavy-tailed degree distributions, assortative degree mixing, and large clustering coefficients. Using court record data, we assess these properties in a collaboration network among heroin traffickers. Consistent with prior research, we find an exponential degree distribution and strong local clustering. However, the traffickers mix dissortatively by degree rather than assortatively. Using a graph sampling method, we show that a consequence of dissortative mixing is that targeted vertex removals have a greater impact on the connectivity and cohesion of the trafficking network. We also note the importance of degree mixing for characterizing and identifying topological weaknesses.