Power-law Degree Distribution Research Articles

Generating complex networks through a vertex merging mechanism: Empirical and analytical analysis

One of the most well-known mechanisms contributing to the emergence of networks with a power-law degree distribution is preferential attachment. In this study, we examined a family of network evolution models based on the merging of two arbitrary vertices, which is shown to also lead to the creation of power-law distributed networks. These models simultaneously apply rules for both node addition and merging, which reflects that many real systems exhibit the processes of growth and shrink. At each iteration, when two vertices merge, the neighbors of one of the vertices become neighbors of the other, and the vertex itself is removed from the network. In addition, at each iteration, a new vertex appears that is attached to randomly selected nodes. As an enhancement, we incorporate a triadic closure mechanism into the evolution to increase the clustering coefficient, a key characteristic of real social networks. We show that in the process of evolution any initial network converges to a stationary state with a power law degree distribution, while the number of edges, the average degree, and the average clustering coefficient saturate to a certain limit values depending on the model parameters.

Gaussian level-set percolation on complex networks.

We present a solution of the problem of level-set percolation for multivariate Gaussians defined in terms of weighted graph Laplacians on complex networks. It is achieved using a cavity or message passing approach, which allows one to obtain the essential ingredient required for the solution, viz. a self-consistent determination of locally varying percolation probabilities. The cavity solution can be evaluated both for single large instances of locally treelike graphs, and in the thermodynamic limit of random graphs of finite mean degree in the configuration model class. The critical level h_{c} of the percolation transition is obtained through the condition that the largest eigenvalue of a weighted version B of a nonbacktracking matrix satisfies λ_{max}(B)|_{h_{c}}=1. We present level-dependent distributions of local percolation probabilities for Erdős-Rényi networks and and for networks with degree distributions described by power laws. We find that there is a strong correlation between marginal single-node variances of a massless multivariate Gaussian and local percolation probabilities at a given level h, which is nearly perfect at negative values h, but weakens as h↗0 for the system with power-law degree distribution, and generally also for negative values of h, if the multivariate Gaussian acquires a nonzero mass. The theoretical analysis simplifies in the case of random regular graphs with uniform edge weights of the weighted graph Laplacian of the system and uniform mass parameter of the Gaussian field. An asymptotic analysis reveals that for edge weights K=K(c)≡1 the critical percolation threshold h_{c} decreases to 0, as the degree c of the random regular graph diverges. For K=(c)=1/c, however, the critical percolation threshold h_{c} is shown to diverge as c→∞.

Maximal cliques in scale-free random graphs

AbstractWe investigate the number of maximal cliques, that is, cliques that are not contained in any larger clique, in three network models: Erdős–Rényi random graphs, inhomogeneous random graphs (IRGs) (also called Chung–Lu graphs), and geometric inhomogeneous random graphs (GIRGs). For sparse and not-too-dense Erdős–Rényi graphs, we give linear and polynomial upper bounds on the number of maximal cliques. For the dense regime, we give super-polynomial and even exponential lower bounds. Although (G)IRGs are sparse, we give super-polynomial lower bounds for these models. This comes from the fact that these graphs have a power-law degree distribution, which leads to a dense subgraph in which we find many maximal cliques. These lower bounds seem to contradict previous empirical evidence that (G)IRGs have only few maximal cliques. We resolve this contradiction by providing experiments indicating that, even for large networks, the linear lower-order terms dominate, before the super-polynomial asymptotic behavior kicks in only for networks of extreme size.

Nonuniversal critical dynamics on planar random lattices with heterogeneous degree distributions

The weighted planar stochastic (WPS) lattice introduces a topological disorder that emerges from a multifractal structure. Its dual network has a power-law degree distribution and is embedded in a two-dimensional space, forming a planar network. We modify the original recipe to construct WPS networks with degree distributions interpolating smoothly between the original power-law tail, P(q)∼q−α with exponent α≈5.6, and a square lattice. We analyze the role of the disorder in the modified WPS model, considering the critical behavior of the contact process (CP). We report a critical scaling depending on the network degree distribution. The scaling exponents differ from the standard mean-field behavior reported for CP on infinite-dimensional (random) graphs with power-law degree distribution. Furthermore, the disorder present in the WPS lattice model is in agreement with the Luck-Harris criterion for the relevance of disorder in critical dynamics. However, despite the same wandering exponent ω=1/2, the disorder effects observed for the WPS lattice are weaker than those found for uncorrelated disorder.

Integrate-and-fire model of disease transmission.

We create an epidemiological susceptible-infected-susceptible model of disease transmission using integrate-and-fire nodes on a network, allowing memory of previous interactions and infections. Agents in the network sum infectious matter from their nearest neighbors at every time step, until they exceed their infection threshold, at which point they "fire" and become infected for as long as the recovery time. The model has memory of previous interactions by tracking the amount of infectious matter carried by agents as well as just binary infected or susceptible states, and the model has memory of previous infections by modeling immunity as increasing the infection threshold after recovery. Creating a simulation of the model on networks with a power-law degree distribution and homogeneous agent parameters, we find a single strain version of the model matches well with the England COVID-19 case data, with a root-mean-squared error of 0.014%. A simulation of a multistrain version of the model (where there is cross-strain immunity) matches well with the influenza strain A and strain B case numbers in Canada, with a root-mean-squared error of 0.002% and 0.0012%, respectively, though due to the coupling in the model, both strains peak in phase. Since the dynamics of the model successfully capture real-life transmission dynamics, we test interventions to study their effect on case numbers, with both quarantining and social gathering restrictions lowering the peak. Since the model has memory, the stricter the intervention, the higher the secondary peak when the restriction is removed, showing that interventions change only the shape of the curves and not the overall number infected in the population.

Accuracy of discrete- and continuous-time mean-field theories for epidemic processes on complex networks.

Discrete- and continuous-time approaches are frequently used to model the role of heterogeneity on dynamical interacting agents on the top of complex networks. While, on the one hand, one does not expect drastic differences between these approaches, and the choice is usually based on one's expertise or methodological convenience, on the other hand, a detailed analysis of the differences is necessary to guide the proper choice of one or another approach. We tackle this problem by investigating both discrete- and continuous-time mean-field theories for the susceptible-infected-susceptible (SIS) epidemic model on random networks with power-law degree distributions. We compare the discrete epidemic link equations(ELE) and continuous pair quenched mean-field (PQMF) theories with the corresponding stochastic simulations, both theories that reckon pairwise interactions explicitly. We show that ELE converges to the PQMF theory when the time step goes to zero. We performed an epidemic localization analysis considering the inverse participation ratio (IPR). Both theories present the same localization dependence on network degree exponent γ: for γ<5/2 the epidemics are localized on the maximum k-core of networks with a vanishing IPR in the infinite-size limit while, for γ>5/2, the localization happens on hubs that do not form a densely connected set and leads to a finite value of the IPR. However, the IPR and epidemic threshold of ELE depend on the time-step discretization such that a larger time step leads to more localized epidemics. A remarkable difference between discrete- and continuous-time approaches is revealed in the epidemic prevalence near the epidemic threshold, in which the discrete-time stochastic simulations indicate a mean-field critical exponent θ=1 instead of the value θ=1/(3-γ) obtained rigorously and verified numerically for the continuous-time SIS on the same networks.

Node-Level Community Detection within Edge Exchangeable Models for Interaction Processes

Scientists are increasingly interested in discovering community structure from modern relational data arising on large-scale social networks. While many methods have been proposed for learning community structure, few account for the fact that these modern networks arise from processes of interactions in the population. We introduce block edge exchangeable models (BEEM) for the study of interaction networks with latent node-level community structure. The block vertex components model (B-VCM) is derived as a canonical example. Several theoretical and practical advantages over traditional vertex-centric approaches are highlighted. In particular, BEEMs allow for sparse degree structure and power-law degree distributions within communities. Our theoretical analysis bounds the misspecification rate of block assignments while supporting simulations show the properties of the network can be recovered. A computationally tractable Gibbs algorithm is derived. We demonstrate the proposed model using post-comment interaction data from Talklife, a large-scale online peer-to-peer support network, and contrast the learned communities from those using standard algorithms including degree-corrected stochastic block models, popularity-adjusted block models, and weighted stochastic block models. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Simulation system design for critical-node detection and robustness analysis of supply chains

ABSTRACT A well-developed manufacturer requires a stable supply chain network that is robust to crises. In this study, a simulation system was developed to build a specialised supply chain model and analyse the critical nodes and risks. It makes three major theoretical contributions to literature. First, a novel supply chain model and cascading failure model are developed with adjustable power-law degree distributions for companies to identify suitable models. Second, a novel approach to the critical-node detection problem, considering cascading failure, is proposed. Finally, based on the proposed algorithm, supply chain vulnerability is analysed under two types of attacks.

Network Evolution Model with Preferential Attachment at Triadic Formation Step

It is recognized that most real systems and networks exhibit a much higher clustering with comparison to a random null model, which can be explained by a higher probability of the triad formation—a pair of nodes with a mutual neighbor have a greater possibility of having a link between them. To catch the more substantial clustering of real-world networks, the model based on the triadic closure mechanism was introduced by P. Holme and B. J. Kim in 2002. It includes a “triad formation step” in which a newly added node links both to a preferentially chosen node and to its randomly chosen neighbor, therefore forming a triad. In this study, we propose a new model of network evolution in which the triad formation mechanism is essentially changed in comparison to the model of P. Holme and B. J. Kim. In our proposed model, the second node is also chosen preferentially, i.e., the probability of its selection is proportional to its degree with respect to the sum of the degrees of the neighbors of the first selected node. The main goal of this paper is to study the properties of networks generated by this model. Using both analytical and empirical methods, we show that the networks are scale-free with power-law degree distributions, but their exponent γ is tunable which is distinguishable from the networks generated by the model of P. Holme and B. J. Kim. Moreover, we show that the degree dynamics of individual nodes are described by a power law.

Assessing mutualistic metacommunity capacity by integrating spatial and interaction networks

We develop a spatially realistic model of mutualistic metacommunities that exploits the joint structure of spatial and interaction networks. Assuming that all species have the same colonisation and extinction parameters, this model exhibits a sharp transition between stable non-null equilibrium states and a global extinction state. This behaviour allows defining a threshold on colonisation/extinction parameters for the long-term metacommunity persistence. This threshold, the ‘metacommunity capacity’, extends the metapopulation capacity concept and can be calculated from the spatial and interaction networks without needing to simulate the whole dynamics. In several applications we illustrate how the joint structure of the spatial and the interaction networks affects metacommunity capacity. It results that a weakly modular spatial network and a power-law degree distribution of the interaction network provide the most favourable configuration for the long-term persistence of a mutualistic metacommunity. Our model that encodes several explicit ecological assumptions should pave the way for a larger exploration of spatially realistic metacommunity models involving multiple interaction types.

Higher-order random network models

Most existing random network models that describe complex systems in nature and society are developed through connections that indicate a binary relationship between two nodes. However, real-world networks are so complicated that we can only identify many critical hidden structural properties through higher-order structures such as network motifs. Here we propose a framework in which we define higher-order stubs, higher-order degrees, and generating functions for developing higher-order complex network models. Then we develop higher-order random networks with arbitrary higher-order degree distributions. The developed higher-order random networks share critical structural properties with real-world networks, but traditional connection-based random networks fail to exhibit these structural properties. For example, as opposed to connection-based random network models, the proposed higher-order random network models can generate networks with power-law higher-order degree distributions, right-skewed degree distributions, and high average clustering coefficients simultaneously. These properties are also observed on the Internet, the Amazon product co-purchasing network, and collaboration networks. Thus, the proposed higher-order random networks are necessary supplements to traditional connection-based random networks.

A network model of social contacts with small-world and scale-free features, tunable connectivity, and geographic restrictions.

Small-world networks and scale-free networks are well-known theoretical models within the realm of complex graphs. These models exhibit "low" average shortest-path length; however, key distinctions are observed in their degree distributions and average clustering coefficients: in small-world networks, the degree distribution is bell-shaped and the clustering is "high"; in scale-free networks, the degree distribution follows a power law and the clustering is "low". Here, a model for generating scale-free graphs with "high" clustering is numerically explored, since these features are concurrently identified in networks representing social interactions. In this model, the values of average degree and exponent of the power-law degree distribution are both adjustable, and spatial limitations in the creation of links are taken into account. Several topological metrics are calculated and compared for computer-generated graphs. Unexpectedly, the numerical experiments show that, by varying the model parameters, a transition from a power-law to a bell-shaped degree distribution can occur. Also, in these graphs, the degree distribution is most accurately characterized by a pure power-law for values of the exponent typically found in real-world networks.

Network Analysis of Ship Domestic Sewage Discharge—The Yangtze River Case

Water transportation has always occupied a large proportion of China’s transportation and has become the key to China’s economic development. Water transportation is called “green transportation” in the industry due to the advantages of low transportation cost, high safety factor, and large capacity. However, water transportation has caused a great impact on the ecological environment of the waters for a long time, and solving the problem of sewage pollution from ships has also become an unavoidable problem. To control pollution from ships, it is essential to analyze the characteristics of ship domestic sewage discharge. In this study, a ship domestic sewage discharge complex network (SDCN) is established based on the ship voyage data to analyze the discharge characteristics in the Yangtze River. According to the topological analysis, the SDCN is a small-world network with the power-law degree distribution and superliner betweenness-degree correlation as distinguishing characteristics. The top five MDs with the highest in-vertex strength are Chaotianmen MD (S = 165,129,561), Tongling MD (S = 66,426,616), Maanshan MD (S = 62,087,158), Baqu MD (S = 59,964,550), and Shashi MD (S = 55,569,399), which indicates that these MDs receive a large amount of sewage. And the volume of domestic sewage between Chaotianmen MD and Baqu MD is the largest. The results can help us understand the discharge characteristics of ship domestic sewage and how they can be targeted to develop control measures for the countermeasures of “zero discharge” for inland ships.

Modular tipping points: How local network structure impacts critical transitions in networked spin systems

Critical transitions describe a phenomenon where a system abruptly shifts from one stable state to an alternative, often detrimental, stable state. Understanding and possibly preventing the occurrence of a critical transition is thus highly relevant to many ecological, sociological, and physical systems. In this context, it has been shown that the underlying network structure of a system heavily impacts the transition behavior of that system. In this paper, we study a crucial but often overlooked aspect in critical transitions: the modularity of the system’s underlying network topology. In particular, we investigate how the transition behavior of a networked system changes as we alter the local network structure of the system through controlled changes of the degree assortativity. We observe that systems with high modularity undergo cascading transitions, while systems with low modularity undergo more unified transitions. We also observe that networked systems that consist of nodes with varying degrees of connectivity tend to transition earlier in response to changes in a control parameter than one would anticipate based solely on the average degree of that network. However, in rare cases, such as when there is both low modularity and high degree disassortativity, the transition behavior aligns with what we would expected given the network’s average degree. Results are confirmed for a diverse set of degree distributions including stylized two-degree networks, uniform, Poisson, and power-law degree distributions. On the basis of these results, we argue that to understand critical transitions in networked systems, they must be understood in terms of individual system components and their roles within the network structure.

Degree-preserving graph dynamics: a versatile process to construct random networks

Abstract Real-world networks evolve over time via the addition or removal of vertices and edges. In current network evolution models, vertex degree varies or grows arbitrarily. A recently introduced degree-preserving network growth (DPG) family of models preserves vertex degree, resulting in structures significantly different from and more diverse than previous models ([Nature Physics 2021, DOI:10.1038/s41567-021-01417-7]). Despite its degree preserving property, the DPG model is able to replicate the output of several well-known real-world network growth models. Simulations showed that many real-world networks can also be constructed from small seed graphs via the DPG process. Here, we start the development of a rigorous mathematical theory underlying the DPG family of network growth models. We prove that the degree sequence of the output of some of the well-known, real-world network growth models can be reconstructed via the DPG process, using proper parametrization. We also show that the general problem of deciding whether a simple graph can be obtained via the DPG process from a small seed (DPG feasibility) is, however, NP-complete. It is an intriguing open problem to uncover whether there is a structural reason behind the DPG-constructability of real-world networks. Keywords: network growth models; degree-preserving growth (DPG); matching theory; synthetic networks; power-law degree distribution.

Scale-free networks beyond power-law degree distribution

Complex networks across various fields are often considered to be scale free—a statistical property usually solely characterized by a power-law distribution of the nodes’ degree k. However, this characterization is incomplete. In real-world networks, the distribution of the degree–degree distance η, a simple link-based metric of network connectivity similar to k, appears to exhibit a stronger power-law distribution than k. While offering an alternative characterization of scale-freeness, the discovery of η raises a fundamental question: do the power laws of k and η represent the same scale-freeness? To address this question, here we investigate the exact asymptotic relationship between the distributions of k and η, proving that every network with a power-law distribution of k also has a power-law distribution of η, but not vice versa. This prompts us to introduce two network models as counterexamples that have a power-law distribution of η but not k, constructed using the preferential attachment and fitness mechanisms, respectively. Both models show promising accuracy by fitting only one model parameter each when modeling real-world networks. Our findings suggest that η is a more suitable indicator of scale-freeness and can provide a deeper understanding of the universality and underlying mechanisms of scale-free networks.

Structural Identity Representation Learning for Blockchain-Enabled Metaverse Based on Complex Network Analysis

The metaverse and its underlying blockchain technology have attracted extensive attention in the past few years. How to mine, process, and analyze the tremendous data generated by the metaverse systems has posed a number of challenges. Aiming to address them, we mainly focus on modeling and understanding the blockchain transaction network from a structural identity perspective, which represents the entire network structure and reveals the relations among multiple entities. In this article, we analyze three metaverse-related systems: non-fungible token (NFT), Ethereum (ETH), and Bitcoin (BTC) from the structural-identity perspective. First, we conduct the complex network analysis of the metaverse network and obtain several new insights (i.e., power-law degree distribution, disconnection, disassortativity, preferential attachment, and non-rich-club effect). Secondly, based on such findings, we propose a novel representation learning method named structure-to-vector with random pace (SVRP) for learning both the latent representation and structural identity of the network. Thirdly, we conduct node classification and link prediction tasks with the integration of graph neural networks (GNNs). Empirical results on three real-world datasets demonstrate that our proposed SVRP outperforms other existing methods in multiple tasks. In particular, our SVRP achieves the highest node classification accuracy (Acc) (99.3 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\%$</tex-math> </inline-formula> ) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$</tex-math> </inline-formula> 1-score (96.7 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\%$</tex-math> </inline-formula> ) while only requiring original non-attributed graphs.

Combination of bioaffinity ultrafiltration-UFLC-ESI-Q/TOF-MS/MS, in silico docking and multiple complex networks to explore antitumor mechanism of topoisomerase I inhibitors from Artemisiae Scopariae Herba

BackgroundArtemisiae Scopariae Herba (ASH) has been widely used as plant medicine in East Asia with remarkable antitumor activity. However, the underlying mechanisms have not been fully elucidated.MethodsThis study aimed to construct a multi-disciplinary approach to screen topoisomerase I (topo I) inhibitors from ASH extract, and explore the antitumor mechanisms. Bioaffinity ultrafiltration-UFLC-ESI-Q/TOF-MS/MS was used to identify chemical constitution of ASH extract as well as the topo I inhibitors, and in silico docking coupled with multiple complex networks was applied to interpret the molecular mechanisms.ResultsCrude ASH extract exhibited toxicogenetic and antiproliferative activities on A549 cells. A series of 34 ingredients were identified from the extract, and 6 compounds were screened as potential topo I inhibitors. Docking results showed that the formation of hydrogen bond and π-π stacking contributed most to their binding with topo I. Interrelationships among the 6 compounds, related targets and pathways were analyzed by multiple complex networks model. These networks displayed power-law degree distribution and small-world property. Statistical analysis indicated that isorhamnetin and quercetin were main active ingredients, and that chemical carcinogenesis-reactive oxygen species was the critical pathway. Electrophoretic results showed a therapeutic effect of ASH extract on the conversion of supercoiled DNA to relaxed forms, as well as potential synergistic effect of isorhamnetin and quercetin.ConclusionsThe results improved current understanding of Artemisiae Scopariae Herba on the treatment of tumor. Moreover, the combination of multi-disciplinary methods provided a new strategy for the study of bioactive constituents in medicinal plants.

Trust based attachment.

In social systems subject to indirect reciprocity, a positive reputation is key for increasing one's likelihood of future positive interactions [1-13]. The flow of gossip can amplify the impact of a person's actions on their reputation depending on how widely it spreads across the social network, which leads to a percolation problem [14]. To quantify this notion, we calculate the expected number of individuals, the "audience", who find out about a particular interaction. For a potential donor, a larger audience constitutes higher reputational stakes, and thus a higher incentive, to perform "good" actions in line with current social norms [7, 15]. For a receiver, a larger audience therefore increases the trust that the partner will be cooperative. This idea can be used for an algorithm that generates social networks, which we call trust based attachment (TBA). TBA produces graphs that share crucial quantitative properties with real-world networks, such as high clustering, small-world behavior, and powerlaw degree distributions [16-21]. We also show that TBA can be approximated by simple friend-of-friend routines based on triadic closure, which are known to be highly effective at generating realistic social network structures [19, 22-25]. Therefore, our work provides a new justification for triadic closure in social contexts based on notions of trust, gossip, and social information spread. These factors are thus identified as potential significant influences on how humans form social ties.

Bootstrap percolation in inhomogeneous random graphs

AbstractA bootstrap percolation process on a graph with n vertices is an ‘infection’ process evolving in rounds. Let $r \ge 2$ be fixed. Initially, there is a subset of infected vertices. In each subsequent round, every uninfected vertex that has at least r infected neighbors becomes infected as well and remains so forever.We consider this process in the case where the underlying graph is an inhomogeneous random graph whose kernel is of rank one. Assuming that initially every vertex is infected independently with probability $p \in (0,1]$ , we provide a law of large numbers for the size of the set of vertices that are infected by the end of the process. Moreover, we investigate the case $p = p(n) = o(1)$ , and we focus on the important case of inhomogeneous random graphs exhibiting a power-law degree distribution with exponent $\beta \in (2,3)$ . The first two authors have shown in this setting the existence of a critical $p_c =o(1)$ such that, with high probability, if $p =o(p_c)$ , then the process does not evolve at all, whereas if $p = \omega(p_c)$ , then the final set of infected vertices has size $\Omega(n)$ . In this work we determine the asymptotic fraction of vertices that will eventually be infected and show that it also satisfies a law of large numbers.