Negative Degree Correlation Research Articles

Properties of the connected components in projections of random bipartite networks: effects of clique size fluctuations

Projection is a helpful description for treating bipartite networks as (monopartite) networks with pairwise interactions. Projections induce correlation spontaneously, avoiding negative degree correlation, even if bipartite networks are entirely random. In this study, we examined the structure of projections of random bipartite networks characterized by the degree distribution of individual and group nodes through the generating function method. We decomposed a projection into two subgraphs, the giant component, and finite components and analyzed their degree correlation. We demonstrate that positive degree correlations in projections originating from the clique size fluctuation remain after the decomposition at the set of finite components, although the values of their clustering coefficient are still finite. The giant component can exhibit either positive or negative degree correlations based on the structure of the projection. However, they are positively correlated in most cases. In addition, we found that a projection removed the giant component coincides with one in the subcritical phase, i.e., the discrete duality relation, when the degree distributions for group and individual are of Poisson.

Coupled Information-Epidemic Spreading Dynamics with Selective Mass Media.

As a pandemic emerges, information on epidemic prevention disseminates among the populace, and the propagation of that information interacts with the proliferation of the disease. Mass media serve a pivotal function in facilitating the dissemination of epidemic-related information. Investigating coupled information-epidemic dynamics, while accounting for the promotional effect of mass media in information dissemination, is of significant practical relevance. Nonetheless, in the extant research, scholars predominantly employ an assumption that mass media broadcast to all individuals equally within the network: this assumption overlooks the practical constraint imposed by the substantial social resources required to accomplish such comprehensive promotion. In response, this study introduces a coupled information-epidemic spreading model with mass media that can selectively target and disseminate information to a specific proportion of high-degree nodes. We employed a microscopic Markov chain methodology to scrutinize our model, and we examined the influence of the various model parameters on the dynamic process. The findings of this study reveal that mass media broadcasts directed towards high-degree nodes within the information spreading layer can substantially reduce the infection density of the epidemic, and raise the spreading threshold of the epidemic. Additionally, as the mass media broadcast proportion increases, the suppression effect on the disease becomes stronger. Moreover, with a constant broadcast proportion, the suppression effect of mass media promotion on epidemic spreading within the model is more pronounced in a multiplex network with a negative interlayer degree correlation, compared to scenarios with positive or absent interlayer degree correlation.

Synergistic epidemic spreading in correlated networks.

We investigate the effect of degree correlation on a susceptible-infected-susceptible (SIS) model with a nonlinear cooperative effect (synergy) in infectious transmissions. In a mean-field treatment of the synergistic SIS model on a bimodal network with tunable degree correlation, we identify a discontinuous transition that is independent of the degree correlation strength unless the synergy is absent or extremely weak. Regardless of synergy (absent or present), a positive and negative degree correlation in the model reduces and raises the epidemic threshold, respectively. For networks with a strongly positive degree correlation, the mean-field treatment predicts the emergence of two discontinuous jumps in the steady-state infected density. To test the mean-field treatment, we provide approximate master equationsof the present model. We quantitatively confirm that the approximate master equationsagree with not only all qualitative predictions of the mean-field treatment but also corresponding Monte Carlo simulations.

The Effects Of Gas Changes In The Shelter In The Summer Period On The Milk yield And Dry Material Consumption Of Anatolian Water Buffalo (Bubalus bubalis)

In this study, the effects of in-shelter gas concentration on milk yield and total dry matter intake (TDMI) of buffaloes during the spring and summer periods were investigated. The research was carried out in a shelter with 20 main Anatolian buffaloes between March and July. Values for temperature, relative humidity, air velocity, and CH4, NH3, and CO2 gases were recorded in the shelter. Data records were collected continuously for 24 hours for 4 days. Milk yield and DMI of buffaloes were also determined. As it was observed, higher air velocity in the house reduces the methane gas levels (P<0.01). The regression equation between milk yield and relative humidity was Y= 6.011-0.03RH and showed a negative and low degree correlation. It was illustrated that TDMI varied between 11.00 ± 0.12-13.20 ± 0.06 kg during the summer. The difference observed between months in terms of feed intake was found to be statistically significant (P<0.05). Although the milk yield of water buffalo was low in March, it increased in April and May. However, there was a decrease in DMI (0.50 kg/day) and milk yield (264 mL/day) for an increase of + 1 °C in air temperature. The recorded values for CO2 concentration in the buffalo shelter during the summer period varied between 620-1120 ppm. Considering the obtained results, NH3 and CO2 gas levels in the shelter were below the higher limits and can be considered as not dangerous for animal and human health. It was determined that regression equation between feed intake and temperature was Y= 9.901 + 0.089T**, regression coefficient was R2= 0.19 and correlation coefficient was r= 0.44 (P<0.001).

The role of degree correlation in shaping filter bubbles in social networks

Filter bubbles shelter people from unconcerned but important information, which is a critical problem in modern online social networks. Although a quantitative model of filter bubbles is still missing, the identification and impact of filter bubbles are widely debated both at a scientific and political level. To shed light on this gap, we introduce a theoretical directed network model of filter bubbles with degree correlations and mathematically analyze information diffusion dynamics on the model. We find that the internal structure of filter bubbles can be modeled by the directed scale-free network with both negative (a node tend to possess high in-degree and low out-degree, or vice versa) and assortative (two nodes with similar degrees tend to be connected) degree correlation. Traditionally, filter bubbles are usually associated with the community structure and emphasize the sparseness of external connections to isolate the spreading of diverse information. However, the negative-assortative degree correlation shows that the filter bubble can spontaneously resist the spreading of non-preferred information (i.e., information with relatively lower transmissibility). Moreover, we study the competition epidemic of two information on the negative-assortative networks, and find that both of the information can coexist only if all nodes prefer the same information.

Analytic solution of the two-star model with correlated degrees.

Exponential random graphs are important to model the structure of real-world complex networks. Here we solve the two-star model with degree-degree correlations in the sparse regime. The model constraints the average correlation between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end-points of two-stars (next nearest neighbors). We compute exactly the network free energy and show that this model undergoes a first-order transition to a condensed phase. For non-negative degree correlations between next nearest neighbors, the degree distribution inside the condensed phase has a single peak at the largest degree, while for negative degree correlations between next nearest neighbors the condensed phase is characterized by a bimodal degree distribution. We calculate the degree assortativities and show they are nonmonotonic functions of the model parameters, with a discontinuous behavior at the first-order transition. The first-order critical line terminates at a second-order critical point, whose location in the phase diagram can be accurately determined. Our results can help to develop more detailed models of complex networks with correlated degrees.

Emergence of long-range correlations in random networks

We perform an analysis of the long-range degree correlation of the giant component (GC) in an uncorrelated random network by employing generating functions. By introducing a characteristic length, we find that a pair of nodes in the GC is negatively degree-correlated within the characteristic length and uncorrelated otherwise. At the critical point, where the GC becomes fractal, the characteristic length diverges and the negative long-range degree correlation emerges. We further propose a correlation function for degrees of two nodes separated by the shortest path length l, which behaves as an exponentially decreasing function of distance in the off-critical region. The correlation function obeys a power-law with an exponential cutoff near the critical point. The Erdős-Rényi random graph is employed to confirm this critical behavior.

Effect of degree correlation on the thermal transport in complex networks

In this work, we present a complex network model to study thermal transport in the nanotube and nanowire networks, with nodes forming a random network. It is shown that the temperature distribution in complex networks can be manipulated by the degrees of two heat source nodes, and the change of network topology can induce a transition between negative (disassortative) and positive (assortative) degree correlations. A positive degree correlation (assortativity) will enhance thermal transport, in contrast to the uncorrelated networks (zero degree correlation), while a negative degree correlation (disassortativity) will weaken it. The underlying mechanisms have been analyzed, and a possible realization of the model in a nanoscale system is briefly discussed.

Maximizing synchronizability of duplex networks.

We study the synchronizability of duplex networks formed by two randomly generated network layers with different patterns of interlayer node connections. According to the master stability function, we use the smallest nonzero eigenvalue and the eigenratio between the largest and the second smallest eigenvalues of supra-Laplacian matrices to characterize synchronizability on various duplexes. We find that the interlayer linking weight and linking fraction have a profound impact on synchronizability of duplex networks. The increasingly large inter-layer coupling weight is found to cause either decreasing or constant synchronizability for different classes of network dynamics. In addition, negative node degree correlation across interlayer links outperforms positive degree correlation when most interlayer links are present. The reverse is true when a few interlayer links are present. The numerical results and understanding based on these representative duplex networks are illustrative and instructive for building insights into maximizing synchronizability of more realistic multiplex networks.

Fractality and degree correlations in scale-free networks

Fractal scale-free networks are empirically known to exhibit disassortative degree mixing. It is, however, not obvious whether a negative degree correlation between nearest neighbor nodes makes a scale-free network fractal. Here we examine the possibility that disassortativity in complex networks is the origin of fractality. To this end, maximally disassortative (MD) networks are prepared by rewiring edges while keeping the degree sequence of an initial uncorrelated scale-free network that is guaranteed to become fractal by rewiring edges. Our results show that most of MD networks with different topologies are not fractal, which demonstrates that disassortativity does not cause the fractal property of networks. In addition, we suggest that fractality of scale-free networks requires a long-range repulsive correlation in similar degrees.

Local clustering in scale-free networks with hidden variables.

We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges. We focus on the regime where the hidden variables follow a power law with exponent τ∈(2,3), so that the degrees have infinite variance. The natural cutoff h_{c} characterizes the largest degrees in the hidden variable models, and a structural cutoff h_{s} introduces negative degree correlations (disassortative mixing) due to the infinite-variance degrees. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient C scales with the network size N, as a function of h_{s} and h_{c}. For scale-free networks with exponent 2<τ<3 and the default choices h_{s}∼N^{1/2} and h_{c}∼N^{1/(τ-1)} this gives C∼N^{2-τ}lnN for the universality class at hand. We characterize the extremely slow decay of C when τ≈2 and show that for τ=2.1, say, clustering starts to vanish only for networks as large as N=10^{9}.

Robustness analysis of bimodal networks in the whole range of degree correlation.

We present an exact analysis of the physical properties of bimodal networks specified by the two peak degree distribution fully incorporating the degree-degree correlation between node connections. The structure of the correlated bimodal network is uniquely determined by the Pearson coefficient of the degree correlation, keeping its degree distribution fixed. The percolation threshold and the giant component fraction of the correlated bimodal network are analytically calculated in the whole range of the Pearson coefficient from -1 to 1 against two major types of node removal, which are the random failure and the degree-based targeted attack. The Pearson coefficient for next-nearest-neighbor pairs is also calculated, which always takes a positive value even when the correlation between nearest-neighbor pairs is negative. From the results, it is confirmed that the percolation threshold is a monotonically decreasing function of the Pearson coefficient for the degrees of nearest-neighbor pairs increasing from -1 and 1 regardless of the types of node removal. In contrast, the node fraction of the giant component for bimodal networks with positive degree correlation rapidly decreases in the early stage of random failure, while that for bimodal networks with negative degree correlation remains relatively large until the removed node fraction reaches the threshold. In this sense, bimodal networks with negative degree correlation are more robust against random failure than those with positive degree correlation.

Nonlinear growth and condensation in multiplex networks.

Different types of interactions coexist and coevolve to shape the structure and function of a multiplex network. We propose here a general class of growth models in which the various layers of a multiplex network coevolve through a set of nonlinear preferential attachment rules. We show, both numerically and analytically, that by tuning the level of nonlinearity these models allow us to reproduce either homogeneous or heterogeneous degree distributions, together with positive or negative degree correlations across layers. In particular, we derive the condition for the appearance of a condensed state in which one node in each layer attracts an extensive fraction of all the edges.

Observability transitions in correlated networks

Yang, Wang, and Motter [Phys. Rev. Lett. 109, 258701 (2012)] analyzed a model for network observability transitions in which a sensor placed on a node makes the node and the adjacent nodes observable. The size of the connected components comprising the observable nodes is a major concern of the model. We analyze this model in random heterogeneous networks with degree correlation. With numerical simulations and analytical arguments based on generating functions, we find that negative degree correlation makes networks more observable. This result holds true both when the sensors are placed on nodes one by one in a random order and when hubs preferentially receive the sensors. Finally, we numerically optimize networks with a fixed degree sequence with respect to the size of the largest observable component. Optimized networks have negative degree correlation induced by the resulting hub-repulsive structure; the largest hubs are rarely connected to each other, in contrast to the rich-club phenomenon of networks.

Spontaneous scale-free structure in adaptive networks with synchronously dynamical linking

Inspired by the anti-Hebbian learning rule in neural systems, we study how the feedback from dynamical synchronization shapes network structure by adding new links. Through extensive numerical simulations, we find that an adaptive network spontaneously forms scale-free structure, as confirmed in many real systems. Moreover, the adaptive process produces two nontrivial power-law behaviors of deviation strength from mean activity of the network and negative degree correlation, which exists widely in technological and biological networks. Importantly, these scalings are robust to variation of the adaptive network parameters, which may have meaningful implications in the scale-free formation and manipulation of dynamical networks. Our study thus suggests an alternative adaptive mechanism for the formation of scale-free structure with negative degree correlation, which means that nodes of high degree tend to connect, on average, with others of low degree and vice versa. The relevance of the results to structure formation and dynamical property in neural networks is briefly discussed as well.

A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon

A random network model which allows for tunable, quite general forms of clustering, degree correlation and degree distribution is defined. The model is an extension of the configuration model, in which stubs (half-edges) are paired to form a network. Clustering is obtained by forming small completely connected subgroups, and positive (negative) degree correlation is obtained by connecting a fraction of the stubs with stubs of similar (dissimilar) degree. An SIR (Susceptible --> Infective --> Recovered) epidemic model is defined on this network. Asymptotic properties of both the network and the epidemic, as the population size tends to infinity, are derived: the degree distribution, degree correlation and clustering coefficient, as well as a reproduction number R(*), the probability of a major outbreak and the relative size of such an outbreak. The theory is illustrated by Monte Carlo simulations and numerical examples. The main findings are that (1) clustering tends to decrease the spread of disease, (2) the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and (3) disease spread broadly increases with degree correlation ρ when R(*) is just above its threshold value of one and decreases with ρ when R(*) is well above one.

Features and heterogeneities in growing network models

Many complex networks from the World Wide Web to biological networks grow taking into account the heterogeneous features of the nodes. The feature of a node might be a discrete quantity such as a classification of a URL document such as personal page, thematic website, news, blog, search engine, social network, etc., or the classification of a gene in a functional module. Moreover the feature of a node can be a continuous variable such as the position of a node in the embedding space. In order to account for these properties, in this paper we provide a generalization of growing network models with preferential attachment that includes the effect of heterogeneous features of the nodes. The main effect of heterogeneity is the emergence of an "effective fitness" for each class of nodes, determining the rate at which nodes acquire new links. The degree distribution exhibits a multiscaling behavior analogous to the the fitness model. This property is robust with respect to variations in the model, as long as links are assigned through effective preferential attachment. Beyond the degree distribution, in this paper we give a full characterization of the other relevant properties of the model. We evaluate the clustering coefficient and show that it disappears for large network size, a property shared with the Barabási-Albert model. Negative degree correlations are also present in this class of models, along with nontrivial mixing patterns among features. We therefore conclude that both small clustering coefficients and disassortative mixing are outcomes of the preferential attachment mechanism in general growing networks.

Network Structure of Japanese Firms - Scale-Free, Hierarchy, and Degree Correlation: Analysis from 800,000 Firms

We analyze fundamental characteristics of the inter-firm transaction network through the data of 800,000 Japanese firms. We find that there exists a hierarchical structure and a negative degree correlation in this transaction network. We also find that this undirected network is a scale-free network. We bring to light these characteristics of the network and discuss why there is an important need to conduct research work on the actual network structure. --

Network Structure of Japanese Firms. Scale-Free, Hierarchy, and Degree Correlation: Analysis from 800,000 Firms

Abstract We analyze fundamental characteristics of the inter-firm transaction network through the data of 800,000 Japanese firms. We find that there exists a hierarchical structure and a negative degree correlation in this transaction network. We also find that this undirected network is a scale-free network. We bring to light these characteristics of the network and discuss why there is an important need to conduct research work on the actual network structure.

Influences of degree inhomogeneity on average path length and random walks in disassortative scale-free networks.

Various real-life networks exhibit degree correlations and heterogeneous structure, with the latter being characterized by power-law degree distribution , where the degree exponent describes the extent of heterogeneity. In this paper, we study analytically the average path length (APL) of and random walks (RWs) on a family of deterministic networks, recursive scale-free trees (RSFTs), with negative degree correlations and various , with an aim to explore the impacts of structure heterogeneity on the APL and RWs. We show that the degree exponent has no effect on the APL of RSFTs: In the full range of , behaves as a logarithmic scaling with the number of network nodes (i.e., ), which is in sharp contrast to the well-known double logarithmic scaling previously obtained for uncorrelated scale-free networks with . In addition, we present that some scaling efficiency exponents of random walks are reliant on the degree exponent .