Degree Assortativity Research Articles

The assortativity of scholars at a research-intensive university in Malaysia

Purpose – The purpose of this study was to provide quantitative indicators of intra-university assortative mixing patterns of scholars based on five parameters: degree of connections, faculty, professional position, gender and race. Design/methodology/approach – We conducted a case study of business and management scholars of University of Malaya, a research-intensive University in Malaysia, using co-authorship in papers indexed in the Digital Library Indexing System, Social Sciences Citation Index (SSCI) database of Web of Science, from 2006 to 2010. Background information of scholars was extracted from bibliometric records, online curriculum vitaes and other online sources. Assortativity coefficients were calculated for all parameters. Findings – The study found a degree assortativity coefficient of 0.195, which corresponds with other studies in the literature. Assortativity due to university faculty was strongly assortative at 0.649, whereas gender (0.28) and race (0.16) were weakly assortative. However, in contrast to the common belief that “similarity breeds connection”, the scholars were highly disassortative by professional position (−0.75). Practical implications – This study holds significance to policymakers in understanding the socio-academic factors that bring scholars together. It also adds to the body of knowledge in social network studies that are examining the various factors responsible in bringing the actors together. Originality/value – This is one of the first studies that empirically investigated mixing patterns of researchers at a prominent research-intensive university in Malaysia.

Incorporating Assortativity and Degree Dependence into Scalable Network Models

Due to the recent availability of large complex networks, considerable analysis has focused on understanding and characterizing the properties of these networks. Scalable generative graph models focus on modeling distributions of graphs that match real world network properties and scale to large datasets. Much work has focused on modeling networks with a power law degree distribution, clustering, and small diameter. In network analysis, the assortativity statistic is defined as the correlation between the degrees of linked nodes in the network. The assortativity measure can distinguish between types of networks---social networks commonly exhibit positive assortativity, in contrast to biological or technological networks that are typically disassortative. Despite this, little work has focused on scalable graph models that capture assortativity in networks. The contributions of our work are twofold. First, we prove that an unbounded number of pairs of networks exist with the same degree distribution and assortativity, yet different joint degree distributions. Thus, assortativity as a network measure cannot distinguish between graphs with complex (non-linear) dependence in their joint degree distributions. Motivated by this finding, we introduce a generative graph model that explicitly estimates and models the joint degree distribution. Our Binned Chung Lu method accurately captures both the joint degree distribution and assortativity, while still matching characteristics such as the degree distribution and clustering coefficients. Further, our method has subquadratic learning and sampling methods that enable scaling to large, real world networks. We evaluate performance compared to other scalable graph models on six real world networks, including a citation network with over 14 million edges.

Network of participants in European research: accepted versus rejected proposals

We investigate the network formed by the collaboration of researchers seeking funding by the European Commission by submitting research proposals. Institutions are network nodes and collaborations are links between the nodes. We constructed one network for the accepted proposals and one for the rejected ones, in order to look for any structural differences between them. To this end, first, we compare the size of the largest connected components and the resulting degree distributions. The latter show notable difference only in the region of relatively small degrees. We calculate the assortative mixing by participant type, i.e. a property which indicates whether the participant is a university/research institute, a company (non-profit included), or undefined. By aggregating the data of both networks into three geographical scales (city, region, country), we compare the degree assortativity and average node weight, in all scales. With respect to these two features the networks display similar behaviour. Finally, we compare a series of centrality measures and the Minimum Spanning Trees, at the country scale, to assess the relative performance of the countries. We find that five countries, France, Germany, the United Kingdom, Spain and Italy, play a central role in both networks, however, their relative significance is not the same.

Degree correlations in directed scale-free networks.

Scale-free networks, in which the distribution of the degrees obeys a power-law, are ubiquitous in the study of complex systems. One basic network property that relates to the structure of the links found is the degree assortativity, which is a measure of the correlation between the degrees of the nodes at the end of the links. Degree correlations are known to affect both the structure of a network and the dynamics of the processes supported thereon, including the resilience to damage, the spread of information and epidemics, and the efficiency of defence mechanisms. Nonetheless, while many studies focus on undirected scale-free networks, the interactions in real-world systems often have a directionality. Here, we investigate the dependence of the degree correlations on the power-law exponents in directed scale-free networks. To perform our study, we consider the problem of building directed networks with a prescribed degree distribution, providing a method for proper generation of power-law-distributed directed degree sequences. Applying this new method, we perform extensive numerical simulations, generating ensembles of directed scale-free networks with exponents between 2 and 3, and measuring ensemble averages of the Pearson correlation coefficients. Our results show that scale-free networks are on average uncorrelated across directed links for three of the four possible degree-degree correlations, namely in-degree to in-degree, in-degree to out-degree, and out-degree to out-degree. However, they exhibit anticorrelation between the number of outgoing connections and the number of incoming ones. The findings are consistent with an entropic origin for the observed disassortativity in biological and technological networks.

Mean-field theory of assortative networks of phase oscillators

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.

Degree-Degree Dependencies in Directed Networks with Heavy-Tailed Degrees

In network theory, Pearson’s correlation coefficients are most commonly used to measure the degree assortativity of a network. We investigate the behavior of these coefficients in the setting of directed networks with heavy-tailed degree sequences. We prove that for graphs where the in- and out-degree sequences satisfy a power law with realistic parameters, Pearson’s correlation coefficients converge to a nonnegative number in the infinite network size limit. We propose alternative measures for degree-degree dependencies in directed networks based on Spearman’s rho and Kendall’s tau. Using examples and calculations on the Wikipedia graphs for nine different languages, we show why these rank correlation measures are more suited for measuring degree assortativity in directed graphs with heavy-tailed degrees.

A k-population model to calculate the firing rate of neuronal networks with degree correlations

Revealing the interplay of structure and function of the brain is one of the most intriguing topics in neuroscience. The theory of complex networks is a promising approach to this aim, where one assumes that high cognitive processes arise as emergent properties of a network, in which many inane neurons are connected by a complex topology [1]. In this regard, we analyze analytically the emerging responses of networks with increasingly complex connectivity. We present a mathematical theory to calculate the firing rate of a network of leaky integrate-and-fire neurons, taking into account network features such as degree distributions and degree correlations (Figure ​(Figure1).1). Heterogeneous connectivity and degree correlations have been shown to heavily influence network function and dynamics [2,3]. Our method is to divide the neuronal network in k-populations according to the number k of afferent synaptic links that connect to the neuron. Then, the steady state firing rates for these coupled populations can be calculated self-consistently. One of our main findings is that the population heterogeneity yields substantial deviations from mean-field calculations, where one ignores the network properties [4]. Importantly, our analysis shows that networks with assortative degree correlations lead to firing patterns even for sub-threshold inputs, where an uncorrelated network would not fire and thus, to a much larger sensitivity to low stimuli (Figure ​(Figure2).2). Using information theory we further find an optimum in assortativity, with larger levels reducing again sensitivity for signal ensembles. Figure 1 Schematic of the complex neural network. In the uncorrelated network (A), highly connected neurons (red dots) and poorly connected neurons (blue dots) are joined randomly. In the network with assortative degree correlations (B), neurons with similar connectivity ... Figure 2 Firing rate of a heterogeneous network of integrate and fire neurons with in-degree correlations. Simulation results (dots) and theoretical predictions (lines). The assortative network shows sustained activity for very small stimuli.

Matching patterns in networks with multi-dimensional attributes: a machine learning approach

Assortative matching is a network phenomenon that arises when nodes exhibit a bias towards connections to others of similar characteristics. While mixing patterns in networks have been studied in the literature, and there are well-defined metrics that capture the degree of assortativity (e.g., assortativity coefficient), the latter deal only with single-dimensional enumerative or scalar features. Nevertheless, various complex behaviors of network entities—e.g., human behaviors in social networks—are captured through vector attributes. To date, no formal metric able to cope with similar situations has been defined. In this paper, we propose a novel, two-step process that extends the applicability of the assortativity coefficient to multi-dimensional attributes. In brief, we first apply clustering of the vertices on their vector characteristic. After clustering is completed, each network node is assigned a cluster label, which is an enumerative characteristic and we can compute the assortativity coefficient on the cluster labels. We further compare this method with an alternative baseline, which is an immediate extension of the assortativity coefficient, namely, the assortativity vector. The latter treats each element of the node’s attribute vector separately and then combines the independent results in a single value. Finally, we apply our method and the baseline on two different social network datasets. We also use synthetic network data to delve into the details of each metric/method. Our findings indicate that while the baseline of assortativity vector performs satisfactory when the variance of the elements of the vector attribute across the network population is kept low, it provides biased results as this variance increases. On the contrary, our approach appears to be robust in such scenarios.

Resolving Structural Variability in Network Models and the Brain

Large-scale white matter pathways crisscrossing the cortex create a complex pattern of connectivity that underlies human cognitive function. Generative mechanisms for this architecture have been difficult to identify in part because little is known in general about mechanistic drivers of structured networks. Here we contrast network properties derived from diffusion spectrum imaging data of the human brain with 13 synthetic network models chosen to probe the roles of physical network embedding and temporal network growth. We characterize both the empirical and synthetic networks using familiar graph metrics, but presented here in a more complete statistical form, as scatter plots and distributions, to reveal the full range of variability of each measure across scales in the network. We focus specifically on the degree distribution, degree assortativity, hierarchy, topological Rentian scaling, and topological fractal scaling—in addition to several summary statistics, including the mean clustering coefficient, the shortest path-length, and the network diameter. The models are investigated in a progressive, branching sequence, aimed at capturing different elements thought to be important in the brain, and range from simple random and regular networks, to models that incorporate specific growth rules and constraints. We find that synthetic models that constrain the network nodes to be physically embedded in anatomical brain regions tend to produce distributions that are most similar to the corresponding measurements for the brain. We also find that network models hardcoded to display one network property (e.g., assortativity) do not in general simultaneously display a second (e.g., hierarchy). This relative independence of network properties suggests that multiple neurobiological mechanisms might be at play in the development of human brain network architecture. Together, the network models that we develop and employ provide a potentially useful starting point for the statistical inference of brain network structure from neuroimaging data.

Differential Adhesion between Moving Particles as a Mechanism for the Evolution of Social Groups

The evolutionary stability of cooperative traits, that are beneficial to other individuals but costly to their carrier, is considered possible only through the establishment of a sufficient degree of assortment between cooperators. Chimeric microbial populations, characterized by simple interactions between unrelated individuals, restrain the applicability of standard mechanisms generating such assortment, in particular when cells disperse between successive reproductive events such as happens in Dicyostelids and Myxobacteria. In this paper, we address the evolutionary dynamics of a costly trait that enhances attachment to others as well as group cohesion. By modeling cells as self-propelled particles moving on a plane according to local interaction forces and undergoing cycles of aggregation, reproduction and dispersal, we show that blind differential adhesion provides a basis for assortment in the process of group formation. When reproductive performance depends on the social context of players, evolution by natural selection can lead to the success of the social trait, and to the concomitant emergence of sizeable groups. We point out the conditions on the microscopic properties of motion and interaction that make such evolutionary outcome possible, stressing that the advent of sociality by differential adhesion is restricted to specific ecological contexts. Moreover, we show that the aggregation process naturally implies the existence of non-aggregated particles, and highlight their crucial evolutionary role despite being largely neglected in theoretical models for the evolution of sociality.

Node Assortativity in Complex Networks: An Alternative Approach

Assortativity quantifies the tendency of nodes being connected to similar nodes in a complex network. Degree Assortativity can be quantified as a Pearson correlation. However, it is insufficient to explain assortative or disassortative tendencies of individual nodes or links, which may be contrary to the overall tendency of the network. A number of ‘local’ assortativity measures have been proposed to address this. In this paper we define and analyse an alternative formulation for node assortativity, primarily for undirected networks. The alternative approach is justified by some inherent shortcomings of existing local measures of assortativity. Using this approach, we show that most real world scale-free networks have disassortative hubs, though we can synthesise model networks which have assortative hubs. Highlighting the relationship between assortativity of the hubs and network robustness, we show that real world networks do display assortative hubs in some instances, particularly when high robustness to targeted attacks is a necessity.

A Model of Spatially Constrained Social Network Dynamics

Social networks characterize the set of relationships among a population of social agents. As such, their structure both constrains and is constrained by social processes such as partnership formation and the spread of information, opinions, and behavior. Models of these coevolutionary network dynamics exist, but they are generally limited to specific interaction types such as games on networks or opinion dynamics. Here we present a dynamic model of social network formation and maintenance that exhibits the characteristic features of real-world social networks such as community structure, high clustering, positive degree assortativity, and short characteristic path length. Although these macrostructural network properties are stable, the network microstructure undergoes continuous change at the level of relationships between individuals. Notably, the edges are weighted, allowing for gradual change in relationship strength in contrast to more abrupt mechanisms, such as rewiring, used in other models. We show how the structural features that characterize social networks can arise as the result of constraints placed on the interactions between individuals. Here we explore the relationship between structural properties and four idealized constraints placed on social interactions: space, affinity, time, and history. We show that spatial embedding and the subsequent constraints on possible interactions are crucial in this model for the emergence of the structures characterizing social networks.

Explosive synchronization transitions in complex neural networks

It has been recently reported that explosive synchronization transitions can take place in networks of phase oscillators [Gómez-Gardeñes et al. Phys. Rev. Lett. 106, 128701 (2011)] and chaotic oscillators [Leyva et al. Phys. Rev. Lett. 108, 168702 (2012)]. Here, we investigate the effect of a microscopic correlation between the dynamics and the interacting topology of coupled FitzHugh-Nagumo oscillators on phase synchronization transition in Barabási-Albert (BA) scale-free networks and Erdös-Rényi (ER) random networks. We show that, if natural frequencies of the oscillations are positively correlated with node degrees and the width of the frequency distribution is larger than a threshold value, a strong hysteresis loop arises in the synchronization diagram of BA networks, indicating the evidence of an explosive transition towards synchronization of relaxation oscillators system. In contrast to the results in BA networks, in more homogeneous ER networks, the synchronization transition is always of continuous type regardless of the width of the frequency distribution. Moreover, we consider the effect of degree-mixing patterns on the nature of the synchronization transition, and find that the degree assortativity is unfavorable for the occurrence of such an explosive transition.

Has Large-Scale Named-Entity Network Analysis Been Resting on a Flawed Assumption?

The assumption that a name uniquely identifies an entity introduces two types of errors: splitting treats one entity as two or more (because of name variants); lumping treats multiple entities as if they were one (because of shared names). Here we investigate the extent to which splitting and lumping affect commonly-used measures of large-scale named-entity networks within two disambiguated bibliographic datasets: one for co-author names in biomedicine (PubMed, 2003–2007); the other for co-inventor names in U.S. patents (USPTO, 2003–2007). In both cases, we find that splitting has relatively little effect, whereas lumping has a dramatic effect on network measures. For example, in the biomedical co-authorship network, lumping (based on last name and both initials) drives several measures down: the global clustering coefficient by a factor of 4 (from 0.265 to 0.066); degree assortativity by a factor of ∼13 (from 0.763 to 0.06); and average shortest path by a factor of 1.3 (from 5.9 to 4.5). These results can be explained in part by the fact that lumping artificially creates many intransitive relationships and high-degree vertices. This effect of lumping is much less dramatic but persists with measures that give less weight to high-degree vertices, such as the mean local clustering coefficient and log-based degree assortativity. Furthermore, the log-log distribution of collaborator counts follows a much straighter line (power law) with splitting and lumping errors than without, particularly at the low and the high counts. This suggests that part of the power law often observed for collaborator counts in science and technology reflects an artifact: name ambiguity.

Attack Robustness and Centrality of Complex Networks

Many complex systems can be described by networks, in which the constituent components are represented by vertices and the connections between the components are represented by edges between the corresponding vertices. A fundamental issue concerning complex networked systems is the robustness of the overall system to the failure of its constituent parts. Since the degree to which a networked system continues to function, as its component parts are degraded, typically depends on the integrity of the underlying network, the question of system robustness can be addressed by analyzing how the network structure changes as vertices are removed. Previous work has considered how the structure of complex networks change as vertices are removed uniformly at random, in decreasing order of their degree, or in decreasing order of their betweenness centrality. Here we extend these studies by investigating the effect on network structure of targeting vertices for removal based on a wider range of non-local measures of potential importance than simply degree or betweenness. We consider the effect of such targeted vertex removal on model networks with different degree distributions, clustering coefficients and assortativity coefficients, and for a variety of empirical networks.

Remodeling the network for microgroup detection on microblog

In this paper, we focus on the problem of community detection on Sina weibo, the most popular microblogging system in China. By characterizing the structure and content of microgroup (community) on Sina weibo in detail, we observe that different from ordinary social networks, the degree assortativity coefficients are negative on most microgroups. In addition, we find that users from the same microgroup tend to share some common attributes (e.g., followers, tags) and interests extracted from their published posts. Inspired by these new findings, we propose a united method to remodel the network for microgroup detection while maintaining the information of link structure and user content. Firstly, the link direction is concerned by assigning greater weight values to more surprising links, while the content similarity is measured by the Jaccard coefficient of common features and interest similarity based on Latent Dirichlet Allocation model. Then, both link direction and content similarity between two users are uniformly converted to the edge weight of a new remodeled network, which is undirected and weighted. Finally, multiple frequently used community detection algorithms that support weighted networks could be employed. Extensive experiments on real-world social networks show that both link structure and user content play almost equally important roles in microgroup detection on Sina weibo. Our method outperforms the traditional methods with average accuracy improvement up to 39 %, and the number of unrecognized users decreased by about 75 %.

Effects of frequency-degree correlation on synchronization transition in scale-free networks

Explosive synchronization in the scale-free network with a positive frequency-degree correlation has been reported recently (Gomez G. J. et al., Phys. Rev. Lett., 106 (2011) 128701). In this article, we generalize this study and find that the explosive synchronization is replaced by a kind of hierarchical synchronization if the microscopic correlation between the frequency and the interacting topology of the network becomes negative. A star network model is set to prove this novel behavior. We also find that the degree assortativity has significant influence on the explosive synchronization but slight impact on the hierarchical synchronization. These findings are meaningful for revealing unusual effects of correlations between dynamics and structure of complex networks.

Degree and correlates of sexual mixing in female sex workers in Karnataka, India

Background The degree of sexual mixing plays an important role in understanding disparities in sexually transmissible infections and HIV across social groups. This study examines the degree of sexual age mixing, and explores its individual and partnership level correlates among female sex workers (FSWs) in Karnataka, India. Data were drawn from special behavioural surveys conducted in 2006-07 among 577 FSWs in two districts of Karnataka: Belgaum and Bangalore. Sexual mixing in age was assessed as the difference in age between FSWs and their sexual partners, and the degree of assortativeness in sexual mixing was assessed using Newman's assortativity coefficient. A total of 577 FSWs were interviewed; 418 of whom reported two or more partnerships, resulting in 942 partnerships. In about half (52%) of these partnerships, the age difference between the FSW and her sexual partner was 5 years or more. The degree of assortativity in age mixing was 0.098, indicating minimally assortative mixing. The disassortativeness in age mixing was positively associated with young age and no formal education, and negatively with duration in sex work. Partnerships which were of a commercial nature were more likely to be disassortative than noncommercial partnerships. The minimally assortative age mixing indicates sexually transmissible infections can transfer from members of one age group to another. Efforts are required to limit the transmission of infection from one group to other by promoting safer sexual behaviour.

Homo Moralis--Preference Evolution Under Incomplete Information and Assortative Matching

What preferences will prevail in a society of rational individuals when preference evolution is driven by the resulting payoffs? We show that when individuals’ preferences are their private information, a convex combination of selfishness and morality stands out as evolutionarily stable. We call individuals with such preferences homo moralis .A t one end of the spectrum is homo oeconomicus, who acts so as to maximize his or her own payoff. At the opposite end is homo kantiensis, who does what would be “the right thing to do,” in terms of payoffs, if all others would do likewise. We show that the stable degree of morality—the weight placed on the moral goal—is determined by the degree of assortativity in the process whereby individuals are matched to interact.

Network function shapes network structure: the case of the Arabidopsis flower organ specification genetic network

The reconstruction of many biological networks has allowed detailed studies of their structural properties. Several features exhibited by these networks have been interpreted to be the result of evolutionary dynamics. For instance the degree distributions may follow from a preferential attachment of new genes to older ones during evolution. Here we argue that even in the absence of any evolutionary dynamics, the presence of atypical features may follow from the fact that the network implements certain functions. To examine this network function shapes network structure scenario, we focus on the Arabidopsis genetic network controlling early flower organogenesis in which gene expression dynamics has been modelled using a Boolean framework. Specifically, for a system with 15 master genes, the phenotype consists of 10 experimentally determined steady-state expression patterns, considered here as the functional constraints on the network. The space of genetic networks satisfying these constraints is sometimes referred to as the neutral or genotype network. We sample this space using Markov Chain Monte Carlo which allows us to demonstrate how the functional (phenotypic) constraints shape the gene network structure. We find that this shaping is strongest for the edge (interaction) usage, with effects that are functionally interpretable. In contrast, higher order features such as degree assortativity and network motifs are hardly shaped by the phenotypic constraints.