Assortative mixing in spatially-extended networks
В. В. МакаровDaniil KirsanovNikita FrolovVladimir MaksimenkoXuelong LiZhen WangAlexander E. HramovStefano Boccaletti
9
Citation
41
Reference
10
Related Paper
Citation Trend
Abstract:
We focus on spatially-extended networks during their transition from short-range connectivities to a scale-free structure expressed by heavy-tailed degree-distribution. In particular, a model is introduced for the generation of such graphs, which combines spatial growth and preferential attachment. In this model the transition to heterogeneous structures is always accompanied by a change in the graph's degree-degree correlation properties: while high assortativity levels characterize the dominance of short distance couplings, long-range connectivity structures are associated with small amounts of disassortativity. Our results allow to infer that a disassortative mixing is essential for establishing long-range links. We discuss also how our findings are consistent with recent experimental studies of 2-dimensional neuronal cultures.Keywords:
Assortativity
Degree distribution
Mixing patterns
Degree (music)
Preferential attachment
Scale-free network
Uncorrelated scale-free networks are necessarily small world (and, in fact, smaller than small world). Nonetheless, for scale-free networks with correlated degree distribution this may not be the case. We describe a mechanism to generate highly assortative scale-free networks which are not small world. We show that it is possible to generate scale-free networks, with arbitrary degree exponent gamma>1 , such that the average distance between nodes in the network is large. To achieve this, nodes are not added to the network with preferential attachment. Instead, we greedily optimize the assortativity of the network. The network generation scheme is physically motivated, and we show that the recently observed global network of Avian Influenza outbreaks arises through a mechanism similar to what we present here. Simulations show that this network exhibits very similar physical characteristics (very high assortativity, clustering, and path length).
Assortativity
Scale-free network
Degree distribution
Small-world network
Average path length
Exponent
Degree (music)
Clustering coefficient
Evolving networks
Cite
Citations (43)
Many networks generated by nature have two generic properties: they are formed in the process of preferential attachment and they are scale-free. Considering these features, by interfering with mechanism of the preferential attachment, we propose a generalisation of the Barabási–Albert model—the ’Fractional Preferential Attachment’ (FPA) scale-free network model—that generates networks with time-independent degree distributions p ( k ) ∼ k − γ with degree exponent 2 < γ ≤ 3 (where γ = 3 corresponds to the typical value of the BA model). In the FPA model, the element controlling the network properties is the f parameter, where f ∈ ( 0 , 1 ⟩ . Depending on the different values of f parameter, we study the statistical properties of the numerically generated networks. We investigate the topological properties of FPA networks such as degree distribution, degree correlation (network assortativity), clustering coefficient, average node degree, network diameter, average shortest path length and features of fractality. We compare the obtained values with the results for various synthetic and real-world networks. It is found that, depending on f, the FPA model generates networks with parameters similar to the real-world networks. Furthermore, it is shown that f parameter has a significant impact on, among others, degree distribution and degree correlation of generated networks. Therefore, the FPA scale-free network model can be an interesting alternative to existing network models. In addition, it turns out that, regardless of the value of f, FPA networks are not fractal.
Preferential attachment
Degree distribution
Degree (music)
Scale-free network
Assortativity
Exponent
Average path length
Clustering coefficient
Network model
Cite
Citations (9)
The combination of growth and preferential attachment is responsible for the power-law distribution of vertices' degree in random networks, however there are many kinds of growing ways of real networks. Based on preferential attachment and exponential growth, the paper presents the analytical results which the vertices' degree of scale-free network follows power-law distribution ( ) and parameter satisfies. At same time we find that the preferential attachment is taken place in a dynamic local world and the size of the dynamic local world is in direct proportion to the size of whole networks. The paper also gives the analytical results of no-preferential attachment and exponential growth on random networks. At last, computer simulated results of the model illustrate these analytical results.
Preferential attachment
Scale-free network
Degree distribution
Degree (music)
Cite
Citations (0)
Preferential attachment
Assortativity
Degree distribution
Degree (music)
Scale-free network
Cite
Citations (7)
One of the most important properties of self- organized networks is their scale-free property. Prior research proved empirically and theoretically that scale-free networks emerge under the preferential attachment rule. However, a few empirical studies also show that empirical networks diverge from the structure of scale-free networks. Empirical networks exhibit a lower exponent of the power law distribution than constructed scale-free networks. Our research aims at establishing a simple evolutionary network model that explains this difference. The results of our model suggest that there are two reasons for this discrepancy. First, as already known, additional links between existing nodes distort the scale-free feature. Second, boundaries between subgroups (groups of network nodes) distort the degree distribution. In general, we believe that our evolutionary model may be applicable not only to describe the structural evolution of networks but also to make network design recommendations in a variety of areas such as WWW-hyperlink networks, business collaboration networks, Peer-To-Peer Networks, and Web2.0 service networks.
Scale-free network
Assortativity
Preferential attachment
Hierarchical network model
Degree distribution
Evolving networks
Network Formation
Hyperlink
Exponent
Empirical Research
Interdependent networks
Cite
Citations (6)
Assortativity
Preferential attachment
Degree distribution
Degree (music)
Clustering coefficient
Hierarchical network model
Scale-free network
Hierarchical clustering
Cite
Citations (42)
The combination of growth and preferential attachment is responsible for the power-law distribution of vertices" degree in random networks; however there are many kinds of growing ways of real networks.Based on preferential attachment and exponential growth, the paper presents the analytical results which the vertices" degree of scale-free network follows power-law distribution) and parameter satisfies 0.5 1.At same time we find that the preferential attachment is taken place in a dynamic local world and the size of the dynamic local world is in direct proportion to the size of whole networks.The paper also gives the analytical results of no-preferential attachment and exponential growth on random networks.At last, computer simulated results of the model illustrate these analytical results.
Preferential attachment
Scale-free network
Degree distribution
Degree (music)
Cite
Citations (0)
Abstract Networks are useful representations for analyzing and modeling real-world complex systems. They are often both scale-free and dense: their degree distribution follows a power-law and their average degree grows over time. So far, it has been argued that producing such networks is difficult without externally imposing a suitable cutoff for the scale-free regime. Here, we propose a new growing network model that produces dense scale-free networks with dynamically generated cutoffs. The link formation rule is based on a weak form of preferential attachment depending only on order relations between the degrees of nodes. By this mechanism, our model yields scale-free networks whose scaling exponents can take arbitrary values greater than 1. In particular, the resulting networks are dense when scaling exponents are 2 or less. We analytically study network properties such as the degree distribution, the degree correlation function, and the local clustering coefficient. All analytical calculations are in good agreement with numerical simulations. These results show that both sparse and dense scale-free networks can emerge through the same self-organizing process.
Preferential attachment
Scale-free network
Degree distribution
Degree (music)
Clustering coefficient
Cut-off
Ordinal Scale
Cite
Citations (5)
One of the most important properties of self-organized networks is their scale-free property. Prior research proved empirically and theoretically that scale-free networks emerge under the preferential attachment rule. However, a few empirical studies also show that empirical networks diverge from the structure of scale-free networks. Empirical networks exhibit a lower exponent of the power law distribution than constructed scale-free networks. Our research aims at establishing a simple evolutionary network model that explains this difference. The results of our model suggest that there are two reasons for this discrepancy. First, as already known, additional links between existing nodes distort the scale-free feature. Second, boundaries between subgroups (groups of network nodes) distort the degree distribution. In general, we believe that our evolutionary model may be applicable not only to describe the structural evolution of networks but also to make network design recommendations in a variety of areas such as WWW-hyperlink networks, business collaboration networks, Peer-To-Peer Networks, and Web2.0 service networks.
Scale-free network
Assortativity
Degree distribution
Preferential attachment
Hierarchical network model
Network Formation
Evolving networks
Hyperlink
Exponent
Network model
Empirical Research
Cite
Citations (0)
Many networks exhibit scale free behavior where their degree distribution obeys a power law for large vertex degrees. Models constructed to explain this phenomena have relied on preferential attachment where the networks grow by the addition of both vertices and edges, and the edges attach themselves to a vertex with a probability proportional to its degree. Simulations hint, though not conclusively, that both growth and preferential attachment are necessary for scale free behavior. We derive analytic expressions for degree distributions for networks that grow by the addition of edges to a fixed number of vertices, based on both linear and non-linear preferential attachment, and show that they fall off exponentially as would be expected for purely random networks. From this we conclude that preferential attachment alone might be necessary but is certainly not a sufficient condition for generating scale free networks.
Preferential attachment
Scale-free network
Degree distribution
Degree (music)
Cite
Citations (4)