Properties of asymmetrical evolving networks

Social networks generally display a positively skewed degree distribution and higher values for clustering coefficient and degree assortativity than would be expected from the degree sequence. For some types of simulation studies, these properties need to be varied in the artificial networks over which simulations are to be conducted. Various algorithms to generate networks have been described in the literature but their ability to control all three of these network properties is limited. We introduce a spatially constructed algorithm that generates networks with constrained but arbitrary degree distribution, clustering coefficient and assortativity. Both a general approach and specific implementation are presented. The specific implementation is validated and used to generate networks with a constrained but broad range of property values.

This paper studies a simple asymmetrically evolved community network with a combination of preferential attachment and random properties. An important issue about community networks is to discover the different utility increments of two nodes, where the utility is introduced to investigate the asymmetrical effect of connecting two nodes. On the other hand, the connection of two nodes in community networks can be classified as two nodes belonging to the same or to different communities. The simulation results show that the model can reproduce a power-law utility distribution , which can be obtained by using mean-field approximation methods. Furthermore, the model exhibits exponential behaviour with respect to small values of a parameter denoting the random effect in our model at the low-utility region and a power-law feature with respect to big values of this parameter at the high-utility region, which is in good agreement with theoretical analysis. This kind of community network can reproduce a unique utility distribution by theoretical and numerical analysis.

Employing the Kuramoto model as an illustrative example, we show how the use of the mean-field approximation can be applied to large networks of phase oscillators with assortativity. We then use the ansatz of Ott and Antonsen (Chaos, 19 (2008) 037113) to reduce the mean-field kinetic equations to a system of ordinary differential equations. The resulting formulation is illustrated by application to a network Kuramoto problem with degree assortativity and correlation between the node degrees and the natural oscillation frequencies. Good agreement is found between the solutions of the reduced set of ordinary differential equations obtained from our theory and full simulations of the system. These results highlight the ability of our method to capture all the phase transitions (bifurcations) and system attractors. One interesting result is that degree assortativity can induce transitions from a steady macroscopic state to a temporally oscillating macroscopic state through both (presumed) Hopf and SNIPER (saddle-node, infinite period) bifurcations. Possible use of these techniques to a broad class of phase oscillator network problems is discussed.

Wireless sensor network (WSN) is a classical self-organizing communication network, and its topology evolution currently becomes one of the attractive issues in this research field. Accordingly, the problem is divided into two subproblems: one is to design a new preferential attachment method and the other is to analyze the dynamics of the network topology evolution. To solve the first subproblem, a revised PageRank algorithm, called Con-rank, is proposed to evaluate the node importance upon the existing node contraction, and then a novel preferential attachment is designed based on the node importance calculated by the proposed Con-rank algorithm. To solve the second one, we firstly analyze the network topology evolution dynamics in a theoretical way and then simulate the evolution process. Theoretical analysis proves that the network topology evolution of our model agrees with power-law distribution, and simulation results are well consistent with our conclusions obtained from the theoretical analysis and simultaneously show that our topology evolution model is superior to the classic BA model in the average path length and the clustering coefficient, and the network topology is more robust and can tolerate the random attacks.

Importance:Early alcohol initiation (before age 15) is associated with adverse outcomes. Understanding mechanisms behind early alcohol initiation is essential for informing prevention efforts.Objective:To examine whether structural covariance network properties at ages 9–10 years predict early alcohol initiation.Design:Case-control, population-based study design.Setting:Data from the Adolescent Brain Cognitive Development study were used. Baseline structural brain imaging data (ages 9–10) were used for generation and comparison of structural covariance networks. Data from baseline to 4-year follow-up (≤age 15) assessments were used to determine alcohol initiation.Participants:Participants were excluded if they reported consuming a full drink of alcohol at baseline, or did not meet imaging inclusion criteria. Controls were excluded if they had not yet been assessed or were missing substance use data at 4-year follow-up. In total, 3,878 participants met study criteria, of which 182 participants initiated alcohol. Structural covariance network properties were compared between the full sample and a 1:1 propensity-matched sample based on age, sex, race, ethnicity, religion, parental education, prenatal alcohol exposure, and baseline alcohol sipping.Main Outcomes and Measures:Structural covariance networks were estimated using regional cortical thickness and volume measurements. Measures of network segregation (modularity, clustering coefficient), integration (characteristic path length, global efficiency), and resilience (degree assortativity) were compared between groups. Early alcohol initiation was defined as consuming a full drink between baseline and 4-year follow-upResults:Alcohol initiators (n=182, median[IQR] age, 10.3[9.9–10.8]; 101 female[55.5%]) demonstrated lower network segregation (modularity: area-under-the-curve[AUC] difference[95%CI]=-0.017[-0.017,-0.007], p=0.030; clustering coefficient: AUC[95%CI]=-0.026[-0.027,-0.012], p=0.0495) and higher network integration (characteristic path length: AUC[95%CI]=-0.106[-0.099,-0.046], p=0.020; global efficiency: AUC[95%CI]=0.011[0.005,0.011], p=0.010), compared to non-initiators (n=3,696, median[IQR] age, 9.9[9.4–10.4]; 1750 female[47.4%]) when controlling for age, sex, and mean cortical thickness. Within the matched sample, only differences in network integration were preserved (characteristic path length: AUC[95%CI]=-0.044[-0.032,0.035], p=0.010; global efficiency: AUC[95%CI]=0.003[-0.003,0.003], p=0.040). There were no differences between full or matched samples when comparing cortical volume structural covariance networks.Conclusions and Relevance:Differences in cortical thickness structural covariance network properties at ages 9–10 predicted alcohol initiation before age 15. These findings suggest cortical thickness network topology may reflect a neuroanatomical risk marker for early alcohol initiation.

Characterizing and predicting the robustness of power-law networks

In the Yule-Simon process, creation and selection of words follows the preferential attachment mechanism, resulting in a power-law growth in the cumulative number of individual word occurrences as well as the power-law population distribution of the vocabulary. This is derived using mean-field approximation, assuming a continuum limit of both the time and number of word occurrences. However, time and word occurrences are inherently discrete in the process, and it is natural to assume that the cumulative number of word occurrences has a certain fluctuation around the average behavior predicted by the mean-field approximation. We derive the exact and approximate forms of the probability distribution of such fluctuation analytically, and confirm that those probability distributions are well supported by the numerical experiments.

Modeling and understanding social network structure has interested researchers from many backgrounds including social science, computer science, theoretical physics and graph theory. Notable models include [1] and [2] achieving graphs with power-law degree distribution using preferential attachment and small-world characteristics using randomized rewiring of a regular ring lattice respectively. In contrast to a body of follow-up research which refine upon these seminal works to better capture the graph structure and characteristics (such as improving clustering coefficient by considering social triads along with preferential attachment [3]), this work aims additionally to model the geographic spread in social networks. With increased mobility in our society as well as enhanced communication opportunities social networks are increasingly spread all over the globe. Synthetic graphs imitating real-world social network characteristics are often used for driving simulations for planning and decision support. Incorporating geographic spread can facilitate better infrastructure provisioning in distributed systems supporting social and collaborative applications or model information of malware diffusion, word-of-mouth marketing, etc. The proposed model is tunable and modular. The model can be tuned to produce graphs with different geographic spread. The model is modular in the sense that existing geographic spread agnostic social network models can be plugged into our model to achieve desirable geographic spread in addition to other characteristics (such as degree distribution, clustering coefficient) that such a model would natively support.

Emerging evidence suggests regional neuroanatomical variability may be predictive of early alcohol use (before age 15). However, the relationship between whole-brain network organization and early alcohol initiation remains unknown. Using data from the Adolescent Brain Cognitive Development (ABCD) study, we conducted a structural covariance network (SCN) analysis to examine brain network features preceding early alcohol initiation. Structural MRI data collected at baseline (ages 9–10) were used to generate SCNs based on regional cortical thickness measurements. Early alcohol initiation was defined as consuming a full drink between baseline and 4-year follow-up ( ≤ age 15). Participants who reported a full drink of alcohol at baseline, did not participate in the 4-year follow-up, or did not meet imaging quality control criteria were excluded. The remaining participants were compared to a subsample matched at a 1:1 ratio (n = 160 per group). SCN properties, including network segregation (modularity, clustering coefficient), integration (characteristic path length, global efficiency), and resilience (degree assortativity), were compared between groups. While no differences in regional cortical thickness between groups were identified, early initiators demonstrated lower segregation and higher integration compared to non-initiators. These findings suggest that cortical thickness network topology at ages 9–10 may serve as a neuroanatomical risk marker for early adolescent alcohol initiation, independent of prior alcohol exposure, sociodemographic differences, and regional neuroanatomical variability.

Cryptocurrency is a well-developed blockchain technology application that is currently a heated topic throughout the world. The public availability of transaction histories offers an opportunity to analyze and compare different cryptocurrencies. In this paper, we present a dynamic network analysis of three representative blockchain-based cryptocurrencies: Bitcoin, Ethereum, and Namecoin. By analyzing the accumulated network growth, we find that, unlike most other networks, these cryptocurrency networks do not always densify over time, and they are changing all the time with relatively low node and edge repetition ratios. Therefore, we then construct separate networks on a monthly basis, trace the changes of typical network characteristics (including degree distribution, degree assortativity, clustering coefficient, and the largest connected component) over time, and compare the three. We find that the degree distribution of these monthly transaction networks cannot be well fitted by the famous power-law distribution, at the same time, different currency still has different network properties, e.g., both Bitcoin and Ethereum networks are heavy-tailed with disassortative mixing, however, only the former can be treated as a small world. These network properties reflect the evolutionary characteristics and competitive power of these three cryptocurrencies and provide a foundation for future research.

Weibo is the Twitter counterpart in China that has attracted hundreds of millions of users. We crawled an almost complete Weibo user network that contains 222 million users and 27 billion links in 2013. This paper analyzes the structural properties of this network, and compares it with a Twitter user network. The topological properties we studied include the degree distributions, reciprocity, clustering coefficient, PageRank centrality, and degree assortativity. We find that Weibo users have a higher diversity index, higher Gini index, but a lower reciprocity and clustering coefficient for most of the nodes. A surprising observation is that the reciprocity of Weibo is only about a quarter of the reciprocity of the Twitter user network. We also show that Weibo adoption rate correlates with economic development positively, and Weibo network can be used to quantify the connections between provinces and regions in China. In particular, point-wise mutual information is shown to be accurate in quantifying the strength of connections.

Negative correlation between power-law scaling and Hurst exponents in long-range connective sandpile models and real seismicity

There are conflicting reports in the literature regarding the finite-size scaling of the Liouvillian gap and dynamical fluctuations at discontinuous phase transitions, with various studies reporting either exponential or power-law behavior. We clarify this issue by employing large deviation theory. We distinguish two distinct classes of discontinuous phase transitions that have different dynamical properties. The first class is associated with phase coexistence, i.e., the presence of multiple stable attractors of the system dynamics (e.g., local minima of the free-energy functional) in a finite phase diagram region around the phase transition point. In that case, one observes asymptotic exponential scaling related to stochastic switching between attractors (though the onset of exponential scaling may sometimes occur for very large system sizes). In the second class, there is no phase coexistence away from the phase transition point, while at the phase transition point itself there are infinitely many attractors. In that case, one observes power-law scaling related to the diffusive nature of the system relaxation to the stationary state.

A random pseudofractal network (RPN) is generated by a recursive growing rule. The RPN is of the scale-free feature and small-world effect. We obtain the theoretical results of power-law exponent γ=3, clustering coefficient C=3π2-19≈ 0.74, and a proof that the mean distance increases no faster than ln N, where N is the network size. These results agree with the numerical simulation very well. In particular, we explain the property of growth and preferential attachment in RPNs. And the properties of a class of general RPNs are discussed in the end.

The study and analysis of real-world social, communication, information and citation networks for understanding their structure and identifying interesting patterns have cultivated the need for designing generative models for such networks. A generative model generates an artificial but a realistic-looking network with the same characteristics as that of a real network under study. In this paper, we propose a new generative model for generating realistic networks. Our proposed model is a blend of three key ideas namely preferential attachment, associativity of social links and randomness in real networks. We present a framework that first tests these ideas separately and then blends them into a mixed model based on the idea that a real-world graph could be formed by a mixture of these concepts. Our model can be used for generating static as well as time evolving graphs and this feature distinguishes it from previous approaches. We compare our model with previous methods for generating graphs and show that it outperforms in several aspects. We compare our graphs with real-world graphs across many metrics such as degree, clustering coefficient and path length distributions, assortativity, eigenvector centrality and modularity. In addition, we give both qualitative and quantitative results for clarity.