For most complex networks, the connection between a pair of nodes is the result of their mutual affinity and attachment. In this letter, we will propose a mutual attraction model to characterize weighted evolving networks. By introducing the initial attractiveness A and the general mechanism of mutual attraction ͑controlled by parameter m͒, our model can naturally reproduce scale-free distributions of degree, weight, and strength, as found in many real systems. Also, simulation results are consistent with theoretical predictions. Interestingly, we obtain nontrivial clustering coefficient C and tunable degree assortativity r, depending on the values of m and A. Our model appears as a more general one that unifies the characterization of both assortative and disassortative weighted networks. DOI: 10.1103/PhysRevE.73.016133 PACS number͑s͒: 89.75.Hc, 05.65.ϩb, 87.23.Ge, 87.23.Kg The past few years have witnessed a great deal of interest from the physics community to understand and characterize the underlying mechanisms that govern the evolution of complex networks ͓1-3͔. Prototypical examples cover as diverse as the Internet ͓4͔, the World Wide Web ͓5͔, the scientific collaboration networks ͑SCN͒ ͓6,7͔, and worldwide airport networks ͑WAN͒ ͓8,9͔. As a landmark, Barabási and Albert ͑BA͒ proposed their seminal model that introduced the linear preferential linking to mimic the topological evolution of complex networks ͓10͔. However, networks are far from a Boolean structure. The purely topological characterization will miss important attributes often encountered in real systems. For instance, the amount of traffic characterizing the connections of communication systems or large transport infrastructure is fundamental for a full description of these networks ͓11͔. This thus calls for the use of weighted network representation, which is often denoted by a weighted adjacency matrix with element w ij representing the weight on the edge connecting vertices i and j. In the case of undirected graphs, weights are symmetric w ij = w ji , as we will focus on. A natural generalization of connectivity in the case of weighted networks is the vertex strength defined as s i = ͚ j⌫͑i͒ w ij , where the sum runs over the set ⌫͑i͒ ͑neigh-bors of node i͒. This quantity is a natural measure of the importance or centrality of a vertex in the network. Most recently, the access to more complete empirical data and higher computation capability allowed scientists to study the variation of the connection weights of many real graphs. As confirmed by measurements, complex networks not only exhibit a scale-free degree distribution P͑k͒ϳk −␥ with 2 ഛ ␥ ഛ 3 ͓8,9͔, but also the power-law weight distribution P͑w͒ϳw − ͓12͔ and the strength distribution P͑s͒ϳs −␣ ͓9͔. Highly correlated with the degree, the strength usually displays scale-free property s ϳ k ␤ with ␤ ജ 1 ͓9,13,14͔. Motivated by all those findings, Barrat et al. presented a model ͑BBV for short͒ to study the growth of weighted networks ͓15͔. Controlled by a single parameter ␦, the BBV model can produce scale-free properties of degree, weight, and strength. But its disassortative property ͓15,16͔ ͑i.e., the hubs are primarily connected to less connected nodes͒, as observed in real technological and biological networks, can hardly give satisfying interpretations to social networks like the SCN where the hubs are very likely to be linked together ͑i.e., assortative mixing͒. Previous models, to the best of our knowledge, can generate either assortative networks ͓17-19͔ or disassortative ones ͓15-17,20͔, but rarely both. Thus, some questions arise here: why are social networks all assortative, while all biological and technological networks opposite? Is there a generic explanation for the observed incompatible patterns, or does it represent a feature that needs to be addressed in each network individually? Our work may shed some light to these questions. Former network models often impress on people such an evolution picture: preexisting nodes are passively attached by newly added ones according to the preferential linking mechanism. This scenario, however, lacks the other side of the fact that old nodes will choose the young at the same time. In addition, this evolution picture also ignores the universal mutual attraction between existing components, which leads to the creation and reinforcement of connections. This idea has been partly reflected in the studies of Dorogovtsev and Mendes ͑DM͒ ͓22͔ who proposed a class of undirected and unweighted models where new edges are added between old sites and existing edges can be removed. In this paper, we will present a model to study the weighted network evolution under the general mechanism of mutual attraction between nodes. In contrast with previous models where weights are assigned statically ͓23,24͔ or rearranged locally ͓15,16͔, our model allows weights to be widely updated. It can mimic the reinforcement and creation of internal links as well as the evolution of many infrastructure networks. Specifically, the model can generate a diversity of scale-free quantities, nontrivial clustering property, and tunable assortativity coefficient, in good accord with the features of various real networks. The model starts from N 0 = m isolated nodes, each with initial attractiveness A. At each time step, a new isolated node n is introduced into the system. Then, every existing *Electronic
WX Wang et al. REF proposed a model based on the mutual attraction.
Mutual attraction model for both assortative and disassortative weighted networks
Physical review. E, Statistical, nonlinear, and soft matter physics
Physical review. E, Statistical, nonlinear, and soft matter physics
Medicine Mathematics Physics