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

The seceder model illustrates how the desire to be different from the average can lead to formation of groups in a population. We turn the original, agent based, seceder model into a model of network evolution. We find that the structural characteristics of our model closely match empirical social networks. Statistics for the dynamics of group formation are also given. Extensions of the model to networks of companies are also discussed.

Figure 1

Illustration of the networked seceder model. (a) In step 1 three vertices, $i_1$, $i_2$, and $i_3$, are chosen at random. (b) In step 2 the least central of the three vertices is relabeled to $\widehat{i}$. In step 3 a vertex $j$ is selected at random and (c) the edges of $j$ are rewired to $\widehat{i}$ and $\widehat{i}$s neighborhood (and to a set of random other vertices if necessary). Note that, in (c), $j$ is moved to the cluster it is rewired to. In step 4 $j$’s edges are rewired with a probability $p$. The shaded areas represent tightly connected subgraphs.

Figure 2

One realization of the networked seceder model. The model parameters are $N=50$, $M=150$, and $p=0$. The indicated groups are indentified with Newman’s clustering algorithm (see Sec. 2C). This realization has modularity $Q=0.575$, clustering coefficient $C=0.530$, and assortative mixing coefficient $r=0.0456$.

Figure 3

The modularity $Q$ as a function of the model parameters. (a) shows $Q$ as a function of $N$ with $M=3N$ and $p=0.1$. (b) displays $Q$ for different $M$ for $N=600$ and $p=0.1$. In (c) we plot the $p$ dependence of $Q$ for $N=200$ and $M=600$. The gray line in (a) is a fit to an exponential. All error bars are smaller than the symbol size.

Figure 4

The number of groups $b$ as a function of the system size $N$. The other parameter values are $M=3N$ and $p=0.1$. The line is a fit to a power-law $a{b}^{\beta }$. For this set of parameters $\beta =0.400\left(6\right)$ for the seceder model and $0.193\left(6\right)$ for the reference graphs of the conditional uniform graph test. All error bars are smaller than the symbol size. Note the double-logarithmic scale.

Figure 5

Average distance between, and within, clusters (as identified by the algorithm described in Sec. 2C). The gray lines are fits to an exponential form. The slope of the original is the same in (a) and (b) [also the rewired line has the same slope in (a) and (b)]. All error bars are smaller than the symbol size.

Figure 6

Degree distribution of the networked seceder model. The model parameters are $N=1800$, $M=5400$, and $p=0.1$. The squares indicate the degree distribution of a random graph with the sizes ($N$ and $M$), i.e., the initial network before the iterations of the seceder model commence.

Figure 7

Common structural measures. (a) shows the clustering coefficient $C$ as a function of the number of vertices for the seceder model and rewired networks. (b) shows the local clustering coefficient $C_v$ as a function of the degree $k$ of $v$. The gray line is inversely proportional to $k$. (Note the double logarithmic scale.) (c) displays the corresponding plot of the assortative mixing coefficient. (d) shows the average degree of the neighbors of a vertex as a function of the vertex’s degree. The network parameters are $M=3N$ and $p=0.1$, in (b) and (d) $N=1800$. Error bars are shown if they are larger than the symbol size.

Figure 8

Illustration of the identification of clusters at consecutive time steps. The vertex set is represented by the horizontal line. The vertical tics demarcate cluster boundaries. The communities at consecutive time step are matched so that the overlap (the horizontal sum of shaded segments) is maximized.

Figure 9

Community dynamics for a typical run with the parameter values $N=100$, $M=300$, and $p=0$ and 100 iterations of the networked seceder model. The different panels show different statistics for one single run of the algorithm. (a) shows the time evolution of the assortative mixing coefficient. (b) shows the clustering coefficient $C$. (c) shows the modularity $Q$. (d) shows the maximal overlap $y$ between consecutive time steps. (e) illustrates the time evolution of the communities. A vertical cross section of (e) gives the respective relative sizes of the different clusters. The clusters are sorted horizontally according to age—the oldest clusters are in the top of the panel.