Networking the seceder model: Group formation in social and economic systems

Intergroup networks as random threshold graphs

Similar-minded people tend to form social groups. Due to pluralistic homophily as well as a sort of heterophily, people also participate in a wide variety of groups. Thus, these groups generally overlap with each other; an overlap between two groups can be characterized by the number of common members. These common members can play a crucial role in the transmission of information between the groups. As a step towards understanding the information dissemination, we perceive the system as a pruned intergroup network and show that it maps to a very basic graph theoretic concept known as a threshold graph. We analyze several structural properties of this network such as degree distribution, largest component size, edge density, and local clustering coefficient. We compare the theoretical predictions with the results obtained from several online social networks (LiveJournal, Flickr, YouTube) and find a good match.

Local-global interaction and the emergence of scale-free networks with community structures

Understanding complex networks in the real world is a nontrivial task. In the study of community structures we normally encounter several examples of these networks, which makes any statistical inferencing a challenging endeavor. Researchers resort to computer-generated networks that resemble networks encountered in the real world as a means to generate many networks with different sizes, while maintaining the real-world characteristics of interest. The generation of networks that resemble the real world turns out in itself to be a complex search problem. We present a new rewiring algorithm for the generation of networks with unique characteristics that combine the scale-free effects and community structures encountered in the real world. The algorithm is inspired by social interactions in the real world, whereby people tend to connect locally while occasionally they connect globally. This local-global coupling turns out to be a powerful characteristics that is required for our proposed rewiring algorithm to generate networks with community structures, power law distributions both in degree and in community size, positive assortative mixing by degree, and the rich-club phenomenon.

Model validation of simple-graph representations of metabolism

The large-scale properties of chemical reaction systems, such as metabolism, can be studied with graph-based methods. To do this, one needs to reduce the information, lists of chemical reactions, available in databases. Even for the simplest type of graph representation, this reduction can be done in several ways. We investigate different simple network representations by testing how well they encode information about one biologically important network structure-network modularity (the propensity for edges to be clustered into dense groups that are sparsely connected between each other). To achieve this goal, we design a model of reaction systems where network modularity can be controlled and measure how well the reduction to simple graphs captures the modular structure of the model reaction system. We find that the network types that best capture the modular structure of the reaction system are substrate-product networks (where substrates are linked to products of a reaction) and substance networks (with edges between all substances participating in a reaction). Furthermore, we argue that the proposed model for reaction systems with tunable clustering is a general framework for studies of how reaction systems are affected by modularity. To this end, we investigate statistical properties of the model and find, among other things, that it recreates correlations between degree and mass of the molecules.

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.

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.

Identifying communities within energy landscapes

Potential energy landscapes can be represented as a network of minima linked by transition states. The community structure of such networks has been obtained for a series of small Lennard-Jones (LJ) clusters. This community structure is compared to the concept of funnels in the potential energy landscape. Two existing algorithms have been used to find community structure, one involving removing edges with high betweenness, the other involving optimization of the modularity. The definition of the modularity has been refined, making it more appropriate for networks such as these where multiple edges and self-connections are not included. The optimization algorithm has also been improved, using Monte Carlo methods with simulated annealing and basin hopping, both often used successfully in other optimization problems. In addition to the small clusters, two examples with known heterogeneous landscapes, the 13-atom cluster (LJ13) with one labeled atom and the 38-atom cluster (LJ38) , were studied with this approach. The network methods found communities that are comparable to those expected from landscape analyses. This is particularly interesting since the network model does not take any barrier heights or energies of minima into account. For comparison, the network associated with a two-dimensional hexagonal lattice is also studied and is found to have high modularity, thus raising some questions about the interpretation of the community structure associated with such partitions.

Growing network model for community with group structure

We propose a growing network model for a community with a group structure. The community consists of individual members and groups, gatherings of members. The community grows as a new member is introduced by an existing member at each time step. The new member then creates a new group or joins one of the groups of the introducer. We investigate the emerging community structure analytically and numerically. The group size distribution shows a power-law distribution for a variety of growth rules, while the activity distribution follows an exponential or a power law depending on the details of the growth rule. We also present an analysis of empirical data from online communities the "Groups" in http://www.yahoo.com and the "Cafe" in http://www.daum.net, which show a power-law distribution for a wide range of group sizes.

Networking genetic regulation and neural computation: directed network topology and its effect on the dynamics

Two different types of directed networks are investigated, transcriptional regulation networks and neural networks. The directed network structure is studied and is also shown to reflect the different processes taking place on the networks. The distribution of influence, identified as the the number of downstream vertices, are used as a tool for investigating random vertex removal. In the transcriptional regulation networks we observe that only a small number of vertices have a large influence. The small influences of most vertices limit the effect of a random removal to, in most cases, only a small fraction of vertices in the network. The neural network has a rather different topology with respect to the influence, which are large for most vertices. To further investigate the effect of vertex removal we simulate the biological processes taking place on the networks. Opposed to the presumed large effect of random vertex removal in the neural network, the high density of edges in conjunction with the dynamics used makes the change in the state of the system to be highly localized around the removed vertex.