A typical complex system should be described by a supernetwork or a network of networks, in which the networks are coupled to some other networks. As the first step to understanding the complex systems on such more systematic level, scientists studie
d interdependent multilayer networks. In this letter, we introduce a new kind of interdependent multilayer networks, i.e., interconnecting networks, for which the component networks are coupled each other by sharing some common nodes. Based on the empirical investigations, we revealed a common feature of such interconnecting networks, namely, the networks with smaller averaged topological differences of the interconnecting nodes tend to share more nodes. A very simple node sharing mechanism is proposed to analytically explain the observed feature of the interconnecting networks.
Systems with simultaneous cooperation and competition among the elements are ubiquitous. In spite of their practical importance, knowledge on the evolution mechanism of this class of complex system is still very limit. In this work, by conducting ext
ensive empirical survey to a large number of cooperation-competition systems which cover wide categories and contain the information of network topology, cooperation-competition gain, and the evolution time, we try to get some insights to the universal mechanism of their evolutions. Empirical investigations show that the distributions of the cooperation-competition gain interpolates between power law function and exponential function. Particularly, we found that the cooperation-competition systems with longer evolution durations tend to have more heterogeneous distributions of the cooperation-competition gain. Such an empirical observation can be well explained by an analytic model in which the evolution of the systems are mainly controlled by the Matthew effect, and the marginal heterogeneity of the initial distribution is amplified by the Matthew effect with similar speed in spite of the diversity of the investigated systems.
This manuscript serves as an online supplement of a preprint, which presents a study on a kind of bilayer networks where some nodes (called interconnecting nodes) in two layers merge. A model showing an important general property of the bilayer netwo
rks is proposed. Then the analytic discussion of the model is compared with empirical conclusions. We present all the empirical observations in this online supplement.
We present a model, in which some nodes (called interconnecting nodes) in two networks merge and play the roles in both the networks. The model analytic and simulation discussions show a monotonically increasing dependence of interconnecting node top
ological position difference and a monotonically decreasing dependence of the interconnecting node number on function difference of both networks. The dependence function details do not influence the qualitative relationship. This online manuscript presents the details of the model simulation and analytic discussion, as well as the empirical investigations performed in eight real world bilayer networks. The analytic and simulation results with different dependence function forms show rather good agreement with the empirical conclusions.
Recently, our group quantitatively defined two quantities, competition ability and uniqueness (Chin. Phys. Lett. 26 (2009) 058901) for a kind of cooperation-competition bipartite networks, where producers produce some products and output them to a ma
rket to make competition. Factories, universities or restaurants can serve as the examples. In the letter we presented an analytical conclusion that the competition ability was linearly dependent on the uniqueness in the trivial cases, where both the input quality and competition gain obey normal distributions. The competition between Chinese regional universities was taken as examples. In this article we discuss the abnormal cases where competition gains show the distributions near to power laws. In addition, we extend the study onto all the cooperation-competition bipartite networks and therefore redefine the competition ability. The empirical investigation of the competition ability dependence on the uniqueness in 15 real world collaboration-competition systems is presented, 14 of which belong to the general nontrivial cases. We find that the dependence generally follows the so-called shifted power law (SPL), but very near to power laws. The empirically obtained heterogeneity indexes of the distributions of competition ability and uniqueness are also presented. These empirical investigations will be used as a supplementary of a future paper, which will present the comparison and further discussions about the competition ability dependence on the uniqueness in the abnormal collaboration-competition systems and the relationship between the dependence and the competition ability and uniqueness heterogeneity.
Recently, we introduced a quantity, node weight, to describe the collaboration sharing or competition gain of the elements in the collaboration-competition networks, which can be well described by bipartite graphs. We find that the node weight distri
butions of all the networks follow the so-called shifted power law (SPL). The common distribution function may indicate that the evolution of the collaboration and competition in very different systems obeys a general rule. In order to set up a base of the further investigations on the universal system evolution dynamics, we now present the definition of the networks and their node weights, the node weight distributions, as well as the evolution durations of 15 real world collaboration-competition systems which are belonging to diverse fields.
