Degree Assortativity Research Articles

Special issue on Big Data Security and Intelligent Data in Clouds (BDS‐IDC)

Special issue on Big Data Security and Intelligent Data in Clouds (<scp>BDS‐IDC</scp>)

Inferring and analysis of social networks using RFID check-in data in China.

Social networks play an important role in our daily lives. However, social ties are rather elusive to quantify, especially for large groups of subjects over prolonged periods of time. In this work, we first propose a methodology for extracting social ties from long spatio-temporal data streams, where the subjects are 17,795 undergraduates from a university of China and the data streams are the 9,147,106 time-stamped RFID check-in records left behind by them during one academic year. By several metrics mentioned below, we then analyze the structure of the social network. Our results center around three main observations. First, we characterize the global structure of the network, and we confirm the small-world phenomenon on a global scale. Second, we find that the network shows clear community structure. And we observe that younger students at lower levels tend to form large communities, while students at higher levels mostly form smaller communities. Third, we characterize the assortativity patterns by studying the basic demographic and network properties of users. We observe clear degree assortativity on a global scale. Furthermore, we find a strong effect of grade and school on tie formation preference, but we do not find any strong region homophily. Our research may help us to elucidate the structural characteristics and the preference of the formation of social ties in college students’ social network.

Two-walks degree assortativity in graphs and networks

Degree assortativity is the tendency for nodes of high degree (resp. low degree) in a graph to be connected to high degree nodes (resp. to low degree ones). It is usually quantified by the Pearson correlation coefficient of the degree–degree correlation. Here we extend this concept to account for the effect of second neighbours to a given node in a graph. That is, we consider the two-walks degree of a node as the sum of all the degrees of its adjacent nodes. The two-walks degree assortativity of a graph is then the Pearson correlation coefficient of the two-walks degree–degree correlation. We found here analytical expression for this two-walks degree assortativity index as a function of contributing subgraphs. We then study all the 261,000 connected graphs with 9 nodes and observe the existence of assortative–assortative and disassortative–disassortative graphs according to degree and two-walks degree, respectively. More surprisingly, we observe a class of graphs which are degree disassortative and two-walks degree assortative. We explain the existence of some of these graphs due to the presence of certain topological features, such as a node of low-degree connected to high-degree ones. More importantly, we study a series of 49 real-world networks, where we observe the existence of the disassortative–assortative class in several of them. In particular, all biological networks studied here were in this class. We also conclude that no graphs/networks are possible with assortative–disassortative structure.

Robustness and efficiency in interconnected networks with changes in network assortativity

In this study, the effect of assortativity on the robustness and efficiency of interconnected networks was investigated. This involved constructing a network that possessed the desired degree of assortativity. Additionally, an interconnected network was constructed wherein the assortativity between component networks possessed the desired value. With respect to single networks, the results indicated that a decrease in assortativity provided low hop length, high information diffusion efficiency, and distribution of communication load on edges. The study also revealed that excessive assortativity led to poor network performance. In the study, the assortativity between networks was defined and the following results were demonstrated: assortative connections between networks lowered the average hop length and enhanced information diffusion efficiency, whereas disassortative connections between networks distributed the communication loads of internetwork links and enhanced robustness. Furthermore, it is necessary to carefully adjust assortativity based on the node degree distribution of networks. Finally, the application of the results to the design of robust and efficient information networks was discussed.

Effects of contact network structure on epidemic transmission trees: implications for data required to estimate network structure.

Understanding the dynamics of disease spread is key to developing effective interventions to control or prevent an epidemic. The structure of the network of contacts over which the disease spreads has been shown to have a strong influence on the outcome of the epidemic, but an open question remains as to whether it is possible to estimate contact network features from data collected in an epidemic. The approach taken in this paper is to examine the distributions of epidemic outcomes arising from epidemics on networks with particular structural features to assess whether that structure could be measured from epidemic data and what other constraints might be needed to make the problem identifiable. To this end, we vary the network size, mean degree, and transmissibility of the pathogen, as well as the network feature of interest: clustering, degree assortativity, or attribute-based preferential mixing. We record several standard measures of the size and spread of the epidemic, as well as measures that describe the shape of the transmission tree in order to ascertain whether there are detectable signals in the final data from the outbreak. The results suggest that there is potential to estimate contact network features from transmission trees or pure epidemic data, particularly for diseases with high transmissibility or for which the relevant contact network is of low mean degree. Copyright © 2017 John Wiley & Sons, Ltd.

Unraveling the contact patterns and network structure of pig shipments in the United States and its association with porcine reproductive and respiratory syndrome virus (PRRSV) outbreaks

The analysis of the pork value chain is becoming key to understanding the risk of infectious disease dissemination in the swine industry. In this study, we used social network analysis to characterize the swine shipment network structure and properties in a typical multisite swine production system in the US. We also aimed to evaluate the association between network properties and porcine respiratory and reproductive syndrome virus (PRRSV) transmission between production sites. We analyzed the 109,868 swine shipments transporting over 93 million swine between more than 500 production sites from 2012 to 2014. A total of 248 PRRSV positive occurrences were reported from 79 production sites during those 3 years. The temporal dynamics of swine shipments was evaluated by computing network properties in one-month and three-month networks. The association of PRRS occurrence in sow farms with centrality properties from one-month and three-month networks was assessed by using the multilevel logistic regression. All monthly networks showed a scale-free network topology with positive degree assortativity. The regression model revealed that out-degree centrality had a negative association with PRRS occurrence in sow farms in both one-month and three-month networks [OR=0.79 (95% CI, 0.63–0.99) in one-month network and 0.56 (95% CI, 0.36, 0.88) in three-month network] and in-closeness centrality model was positively associated with PRRS occurrence in sow farms in the three-month network [OR=2.45 (95% CI, 1.14–5.26)]. We also describe how the occurrence of porcine epidemic diarrheac (PED) outbreaks severely affected the network structure as well as the PRRS occurrence reports and its association with centrality measures in sow farms. The structure of the swine shipment network and the connectivity between production sites influenced on the PRRSV transmission. The use of network topology and characteristics combining with spatial analysis based on fine scale geographical location of production sites will be useful to inform the design of more cost-efficient, risk-based surveillance and control measures for PRRSV as well as other diseases in the US swine industry.

Spreading dynamics in heterogeneous graphs: Beyond the assortativity coefficient

Abstract We study spreading dynamics of a reaction–diffusion process in a special class of heterogeneous graphs with Poissonian degree distribution and composed of both local and long range links. The behavior of the spreading dynamics on such networks are investigated by relating them to the topological features of graphs. We find that the degree of assortativity can give just some indication about the large scale behavior of the spreading dynamics while a detailed description of the process can be addressed by introducing new, more appropriate, topological quantities linked to the distance between nodes.

Spatial associations in global household bicycle ownership

The interest in bicycling and its determining factors is growing within the public health, transportation and geography communities. Ownership is one factor affecting bicycle usage, but work is still ongoing to not only quantify its effects but also to understand patterns in its growth and influence. In recent work, we mined and discovered patterns in global bicycle ownership that showed the existence of four characteristic country groups and their trends. Building on these results, we show in this paper that the ownership dataset can be modeled as a network. First, we observe mixing tendencies that indicate neighboring countries are more likely to be in the same ownership group and we map the likelihoods for cross-group mixings. Further, we define the strength of connections between countries by their proximity in ownership levels. We then determine the weighted degree assortative coefficient for the network and for each group relative to the network. We find that while the weighted degree assortativity of the ownership network is statistically insignificant, the highest and lowest ownership groups exhibit disassortative behavior with respect to the entire network. The second and third ranked groups, however, are strongly assortative. Our model serves as a step toward further work in studying the relationship between proximity and bicycle ownership among nations and unearthing possible patterns of influence. Considering further developments, this work can inform policy-relevant recommendations toward regional planning. This effort also contributes to expanding research in assortativity analyses, especially in weighted networks.

The influence of social status and network structure on consensus building in collaboration networks.

In this paper, we analyze the influence of social status on opinion dynamics and consensus building in collaboration networks. To that end, we simulate the diffusion of opinions in empirical networks and take into account both the network structure and the individual differences of people reflected through their social status. For our simulations, we adapt a well-known Naming Game model and extend it with the Probabilistic Meeting Rule to account for the social status of individuals participating in a meeting. This mechanism is sufficiently flexible and allows us to model various society forms in collaboration networks, as well as the emergence or disappearance of social classes. In particular, we are interested in the way how these society forms facilitate opinion diffusion. Our experimental findings reveal that (i) opinion dynamics in collaboration networks is indeed affected by the individuals’ social status and (ii) this effect is intricate and non-obvious. Our results suggest that in most of the networks the social status favors consensus building. However, relying on it too strongly can also slow down the opinion diffusion, indicating that there is a specific setting for an optimal benefit of social status on the consensus building. On the other hand, in networks where status does not correlate with degree or in networks with a positive degree assortativity consensus is always reached quickly regardless of the status.

Assortativity Anomalies in a Large Test System

This letter compares the degree assortativity of the various test power systems that compose the NESTA archive. One outlier is detected, whose network structure is implausible. A new visualization of this network supports this conclusion.

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.

Assortment and the evolution of cooperation in a Moran process with exponential fitness

We study the evolution of cooperation in a finite population interacting according to a simple model of like-with-like assortment. Evolution proceeds as a Moran process, and payoffs from the underlying cooperator–defector game are translated to positive fitnesses by an exponential transformation. These evolutionary dynamics can arise, for example, in a nest-structured population with rare migration. The use of the exponential transformation, rather than the usual linear one, is appropriate when interactions have multiplicative fitness effects, and allows for a tractable characterisation of the effect of assortment on the evolution of cooperation. We define two senses in which a greater degree of assortment can favour the evolution of cooperation, the first stronger than the second: (i) greater assortment increases, at all population states, the probability that the number of cooperators increases, relative to the probability that the number of defectors increases; and (ii) greater assortment increases the fixation probability of cooperation, relative to that of defection. We show that, by the stronger definition, greater assortment favours the evolution of cooperation for a subset of cooperative dilemmas: prisoners' dilemmas, snowdrift games, stag-hunt games, and some prisoners' delight games. For other cooperative dilemmas, greater assortment favours cooperation by the weak definition, but not by the strong definition. We also show that increasing assortment expands the set of games in which cooperation dominates the evolutionary dynamics. Our results hold for any strength of selection.

Evolution of Cooperation in Social Dilemmas on Complex Networks.

Cooperation in social dilemmas is essential for the functioning of systems at multiple levels of complexity, from the simplest biological organisms to the most sophisticated human societies. Cooperation, although widespread, is fundamentally challenging to explain evolutionarily, since natural selection typically favors selfish behavior which is not socially optimal. Here we study the evolution of cooperation in three exemplars of key social dilemmas, representing the prisoner’s dilemma, hawk-dove and coordination classes of games, in structured populations defined by complex networks. Using individual-based simulations of the games on model and empirical networks, we give a detailed comparative study of the effects of the structural properties of a network, such as its average degree, variance in degree distribution, clustering coefficient, and assortativity coefficient, on the promotion of cooperative behavior in all three classes of games.

A geometric graph model for coauthorship networks

Modeling coauthorship networks helps to understand the emergence and propagation of thoughts in academic society. A random geometric graph is proposed to model coauthorship networks, the connection mechanism of which expresses the effects of the academic influences and homophily of authors, and the collaborations between research teams. Our analysis reveals that the modeled networks have a range of features of empirical coauthorship networks, namely, the degree distribution made up of a mixture Poisson distribution with a power-law tail, clear community structure, small-world, high clustering, and degree assortativity. Moreover, the underlying formulae of the tail and forepart of the degree distribution, and the tail of the scaling relation between local clustering coefficient and degree are derived for the modeled networks, and are also applicable to the empirical networks.

A comparative analysis on Weibo and Twitter

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, connected components, distance 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. We developed an interactive analyzing software framework for this study, and released the data and code online.

Degree distribution and assortativity in line graphs of complex networks

Topological characteristics of links of complex networks influence the dynamical processes executed on networks triggered by links, such as cascading failures triggered by links in power grids and epidemic spread due to link infection. The line graph transforms links in the original graph into nodes. In this paper, we investigate how graph metrics in the original graph are mapped into those for its line graph. In particular, we study the degree distribution and the assortativity of a graph and its line graph. Specifically, we show, both analytically and numerically, the degree distribution of the line graph of an Erdős–Rényi graph follows the same distribution as its original graph. We derive a formula for the assortativity of line graphs and indicate that the assortativity of a line graph is not linearly related to its original graph. Additionally, line graphs of various graphs, e.g. Erdős–Rényi graphs, scale-free graphs, show positive assortativity. In contrast, we find certain types of trees and non-trees whose line graphs have negative assortativity.

Complex network approach to fractional time series.

In order to extract correlation information inherited in stochastic time series, the visibility graph algorithm has been recently proposed, by which a time series can be mapped onto a complex network. We demonstrate that the visibility algorithm is not an appropriate one to study the correlation aspects of a time series. We then employ the horizontal visibility algorithm, as a much simpler one, to map fractional processes onto complex networks. The degree distributions are shown to have parabolic exponential forms with Hurst dependent fitting parameter. Further, we take into account other topological properties such as maximum eigenvalue of the adjacency matrix and the degree assortativity, and show that such topological quantities can also be used to predict the Hurst exponent, with an exception for anti-persistent fractional Gaussian noises. To solve this problem, we take into account the Spearman correlation coefficient between nodes' degrees and their corresponding data values in the original time series.

Degree Correlations Optimize Neuronal Network Sensitivity to Sub-Threshold Stimuli.

Information processing in the brain crucially depends on the topology of the neuronal connections. We investigate how the topology influences the response of a population of leaky integrate-and-fire neurons to a stimulus. We devise a method to calculate firing rates from a self-consistent system of equations taking into account the degree distribution and degree correlations in the network. We show that assortative degree correlations strongly improve the sensitivity for weak stimuli and propose that such networks possess an advantage in signal processing. We moreover find that there exists an optimum in assortativity at an intermediate level leading to a maximum in input/output mutual information.

Characterization of Complex Networks for Epidemics Modeling

Complex networks have certain properties that characterize them. These inherent properties can be measured and applied in other fields of research. To this end, we characterize several complex networks from different domains using concepts from graph theory. In particular, the node degrees, graph spectral radius, degree assortativity, and the entire topological structure of selected complex networks are studied on the Susceptible, Infectious, Recovered (SIR) epidemic model. The results show that various complex networks properties affect the epidemic model differently. For instance nodes with high average degrees with corresponding high clustering coefficients are seen to be effective in spreading epidemics. Also the degree distribution patterns have an effect on the spreading rate of epidemic, that is dis-assortative networks are good conduits for the epidemic spreading if low degree nodes are in turn connected to high degree nodes. These results have given us a new dimension on how epidemic spreadings can be studied using complex networks as these networks possessed in them certain properties that resemble that of human society. To the best of our knowledge, this is the first work that simulated the SIR model using heterogenous complex network properties in order to study their cumulative effects on epidemic spreading.

A new form of assortativity in online social networks

The term assortativity indicates the tendency, for a network node, to be directly connected to other nodes that are someway similar. In more technical terms, a given feature is assortative in a network if the probability that an arc exists between two nodes having this feature is greater than the probability that an arc exists between two generic nodes. The role of assortativity in real-world and online social networks has been largely investigated in the literature, in which, starting from degree assortativity, several forms of assortativity have been analyzed. When moving from a single-social-network to a multiple-social-network perspective, new specific traits can be studied, also under the assortativity magnifying glass. This is the case of membership overlap among networks (i.e., the fact that people belong to more online social networks) as expression of different traits of users׳ personality. In this paper, we deal with the above issue, by defining two different measures of membership overlap assortativity, called Loose and Constrained Inter-social-network Assortativity, respectively and by observing that in two of the most representative online social networks, namely Facebook and Twitter, membership overlap is assortative.