Growing network model for community with group structure

Kaleidoscopic reorganization of network communities across different scales

The notion of structural heterogeneity is pervasive in real networks, and their community organization is no exception. Still, a vast majority of community detection methods assume neatly hierarchically organized communities of a characteristic scale for a given hierarchical level. In this work, we demonstrate that the reality of scale-dependent community reorganization is convoluted with simultaneous processes of community splitting and merging, challenging the conventional understanding of community-scale adjustment. We provide a mathematical argument concerning the modularity function, the results from real-network analysis, and a simple network model for a comprehensive understanding of the nontrivial community reorganization process. The reorganization is characterized by a local drop in the number of communities as the resolution parameter varies. This study suggests a need for a paradigm shift in the study of network communities, which emphasizes the importance of considering scale-dependent reorganization to better understand the genuine structural organization of networks.

Fractal and multifractal analyses of bipartite networks

Bipartite networks have attracted considerable interest in various fields. Fractality and multifractality of unipartite (classical) networks have been studied in recent years, but there is no work to study these properties of bipartite networks. In this paper, we try to unfold the self-similarity structure of bipartite networks by performing the fractal and multifractal analyses for a variety of real-world bipartite network data sets and models. First, we find the fractality in some bipartite networks, including the CiteULike, Netflix, MovieLens (ml-20m), Delicious data sets and (u, v)-flower model. Meanwhile, we observe the shifted power-law or exponential behavior in other several networks. We then focus on the multifractal properties of bipartite networks. Our results indicate that the multifractality exists in those bipartite networks possessing fractality. To capture the inherent attribute of bipartite network with two types different nodes, we give the different weights for the nodes of different classes, and show the existence of multifractality in these node-weighted bipartite networks. In addition, for the data sets with ratings, we modify the two existing algorithms for fractal and multifractal analyses of edge-weighted unipartite networks to study the self-similarity of the corresponding edge-weighted bipartite networks. The results show that our modified algorithms are feasible and can effectively uncover the self-similarity structure of these edge-weighted bipartite networks and their corresponding node-weighted versions.

Country neighborhood network on territory and its geometrical model

The country neighborhood network, where nodes represent countries and two nodes are considered linked if the corresponding countries are neighbors on territory, is created and its giant component, the Asia, Europe, and Africa (AEA) cluster, is carefully studied in this paper. It is found that, as common, the degree distribution and the clustering function of the AEA cluster are both compatible with scale-free property, besides, the AEA cluster presents a little disassortativity, and its near power-law country area-degree relationship with the exponent close to 1.7 may imply a fractal dimension close to 1.2 of country borderlines in the AEA continent. It is also revealed that the average difference of population density between two countries obeys an approximately increasing function of the shortest path length between them, which may suggest a gradual consensus of population density in the AEA cluster. A simple unity rule is then adopted to model the AEA cluster and such model explains the AEA cluster very well in most aspects, e.g., power-law domain area distribution and fractal domain borderlines, etc., except that the network derived by the model has stronger disassortativity, which may be explained by the fact that, in the evolution history of countries, unbalanced neighbors are more likely to be united as one than balanced neighbors. Additionally, the network evolving process can be divided into three periods, defined as formation period, growth period, and combination period, and there are indications that the AEA cluster is in its third period.

Multicommunity weight-driven bipartite network model

Community structure and rewiring phenomena exist in many complex networks, particularly in bipartite networks. We construct a model for the degree distribution of the rewiring problem in a multicommunity weight-driven bipartite network (MCWBN). The network consists of many interconnected communities, each of which holds a bipartite graph. The bipartite graph consists of two sets of nodes. We name each node in one set a "producer" and each node in the other set a "consumer." A weight value matrix defining the trade barrier between any two communities is used to characterize the structure of the communities, which ensures the higher preferential attachment probability in intracommunity than in intercommunity. The size of one producer is defined as the number of consumers connected to it. We find that the nonlinear dynamics of the scale of production, or the total size of all producers in each community is dependent only on the initial scale of production in each community, and independent of the distribution of the producer size. Furthermore, if the nonlinear system of the scale of production in each community is at an equilibrium state, the distribution of the producer size in each community of the MCWBN model is equivalent to that in a one-community model.

Epidemic spreading in complex networks

The study of epidemic spreading in complex networks is currently a hot topic and a large body of results have been achieved. In this paper, we briefly review our contributions to this field, which includes the underlying mechanism of rumor propagation, the epidemic spreading in community networks, the influence of varying topology, and the influence of mobility of agents. Also, some future directions are pointed out.

Growth model for complex networks with hierarchical and modular structures

A hierarchical and modular network model is suggested by adding a growth rule along with the preferential attachment (PA) rule into Motter's modeling procedure. The proposed model has an increasing number of vertices but a fixed number of modules and hierarchical levels. The vertices form lowest-level modules which in turn constitute higher-level modules hierarchically. The creation of connections between two vertices in a single module or in two different modules of the same level obeys the PA rule. The structural characteristics of this model are investigated analytically and numerically. The results show that the degree distribution, the module size distribution, and the clustering function of the model possess a power-law property which is similar to that in many real-world networks. The model is then used to predict the growth trends of real-world networks with modular and hierarchical structures. By comparing this model with those real-world networks, an interesting conclusion is found: that many real-world networks are in their early stages of development, and when the growth time is large enough, the modules and levels of the networks will be ultimately merged.