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In this article, we construct a triangle-growing network with tunable clusters and study the social balance dynamics in this network. The built network, which could reflect more features of real communities, has more triangle relations than the ordinary random-growing network. Then we apply the local triad social dynamics to the built network. The effects of the different cluster coefficients and the initial states to the final stationary states are discussed. Some new features of the sparse networks are found as well.

Since balance theory was first proposed by Heider [_{ij} is +1, it means the persons i and j are friendly with each other. If the edge s_{ij} is −1, it means the two persons are hostile towards each other. At every step, they choose a triangle randomly from the network. If the product of the three edges is +1, the triangle is stable. Otherwise, if the product is −1, the triangle is unstable. For the stable triangle, it satisfies 1) the friend of my friend being my friend; 2) the enemy of my friend being my enemy; 3) the friend of my enemy being my enemy; and 4) the enemy of my enemy being my friend. The unstable triangles always try to become stable, but the final state of the network depends on the edge flipping probability p, which is set manually. If p > 1/2, the network will reach the state of “paradise”, with all relations being friendly. Several studies around the balance dynamics have been carried out, including the studies of the university class of triad dynamics [

The researches on the dynamics of balance [6-8], which were based on the complete graph or the regular lattice, have revealed some interesting and important phenomena in the certain networks. Antal et al. [

Although the significant contributions have been made in the work mentioned above, the social balance dynamics in some real communities has seldom been considered. Real communities have the features as follows, firstly, not all the people know each other, secondly, the person will know more and more people as he/she is growing, finally, most of the real networks are not completely connected. The aim of us is to study the social balance behavior closed to the reality. In article [

In this article, we construct a random network with tunable clusters to reflect the features of the real communities mentioned above. Then we study the social balance dynamics in the network. In the first part of the article, the statistical properties of the network are studied. We focus on the difference between the network and E-R random network. The degree distribution and cluster coefficient are discussed, and we find the network constructed in the paper is suitable for the issue of the dynamics of the social balance. Based on the network, the dynamics is studied, and some interesting results are found.

An important aspect which is always present in social dynamics is the topology of the network. When applying social balance dynamics on specific topologies several nontrivial effects may arise [

Our inspiration of solving this problem comes from the random graph proposed by Eröds and Rényi [

where p is connection probability and is the average degree. Bollobás [

The random-growing network is the limiting case of the scale-free network [

1) To start with, the network consists of m_{0} vertices which are linked completely (This rule represents that all people know each other in a small community or a company).

2) Every step, one vertex v with 2 edges is added. One of the edges is attached to an existing vertex w randomly, and the other edge is attached to a neighbor of the vertex w: After N-m_{0} steps, there are N vertices in the network (This rule shows that a new individual will know a person randomly and know another person by the acquaintance of the random selected person).

3) Then we let the existing vertices grow two links with p_{A}, the links will find the targets. The finding rule is the same as step 2. By repeating this step, we may get a network which approximates to the completely graph with very large cluster coefficient (This rule means that the old members in the company will know others randomly by the acquaintance).

The 2nd step is the key of the rules. Every added point will engender more than one triangle in the graph. In sociology the clustering coefficient can be defined as the fraction of transitive triples [

the triangle-growing network is added triangles in every step, while random one is only added links in each step.

If the edges are added according to the step 3, the cluster coefficient and the triangles will vary with the increasing average degree.

Another important statistical feature of networks is degree distribution. The random-growing network and the triangle-growing one are very different. Especially, the triangle-growing network undergoes vary of the to pology. If there are only 2 edges of an added node, an arbitrary vertex w increases its degree with rate

where II_{R}(k_{w}) denotes the probability of the random link, and II_{N}(k_{w}) is proportional to the probability that a vertex in the neighborhood w is linked in the random link step before. So we can get

Where Γ_{w} is the neighborhood of w: We can get the conclusion that the degree distribution which is similar with the BA network. In

In this section, we will study the local triad dynamics (LTD), which has been introduced in article [_{ij}s_{jk}s_{ki} > 0), nothing happens. If the selected triad is imbalanced (s_{ij}s_{jk}s_{ki} < 0), one of the links of the triad will be changed form s to −s. The rules of the evolution are as follows:

1) If all the links are −1, we choose a link randomly and change it.

2) If there is only one −1 link, we change the link of −1 to 1 with the probability p, or change one of the other two links to −1 with the probability 1 − p.

Antal et al. [

The rules of the evolution are the same as in article [_{+}, which is defined as the ratio of friendly links to all the links. First, we let the network have the same quantity of the friendly and enemy links. That is the initial state ρ_{+0} = 0.5. Then we let the network evolve as the dynamics above. After a sufficiently long time, the system will get to the stationary state. Our interest is to study the influence of the topology to the evolution of the dynamics. Via increasing the adding probability p_{A}, the networks can be changed from sparse heterogeneous network with small degree to the homogeneous nearly fully connected ones.

_{+} as a function of p. From the figure, we can see that the system never reach the Utopia state without enemy relations in any p when the adding probability p_{A} is small. However, with the increase of p_{A}, the terminal stationary state will have more and more friendly links. Especially, when p_{A} is greater than a certain value, the system shows the same feature that the network undergoes a dynamical phase transition to an absorbing, paradise state for p ≥ 1/2. This value of p_{A} may be 0.5 from our calculations. The results indicate that for a certain scale network, the system is easier to get friendly state when the average degree is larger. This is because larger degree implies a link may belong to quite a number of triads. An enemy relation may not lead to imbalance in a triad, but other triads, which share this enemy relation, may be imbalanced because of the relation. So this results in a evolution in the whole network and the system is stable at the end. The best stationary state likes more friendly links.

Another feature we concern is whether the initial condition affects the result of the evolution. We simulate the dynamics in different networks with adding probability p_{A}=0.08 and p_{A}=0.5. The different initial conditions are considered and the results are drawn in Figures 5 and 6. It is found that when p_{A} is small, the final states de pend on the initial states. However, when p_{A} is large, the final states are the same in different initial states. For p_{A}=0.08, the relations between the individuals of the network are sparse. The network has lots of independent clusters.

The evolution of a cluster affects the others difficultly. So the different initial conditions lead to the different final states. For p_{A} = 0.5, most of the clusters are adjacent to each other. The change of one link will affect lots of cluster of the network. So no matter the initial state is, the final state will be the same for the certain p_{A}.

The dynamics of the social balance is discussed in this article. The dynamics depends on the relations of three persons, so we construct a triangle-growing network, which has large cluster coefficient and lots of triangles. Another interesting feature of the network is the network has the power-law degree distribution although the growth is random. With the increase of the links, the network can get to the completely linked graph in which the dynamics has been researched in article [

The work was supported by Scientific Research Program Funded by Shaanxi Provincial Education Commission (NO.2010JK675) and the Peiyu Foundation of Xi’an University of Science and Technology (No. 200844).