A generative model for community types in directed networks

TL;DR

This work addresses how directed networks can exhibit four pairwise community structures among two groups: assortative, core–periphery, disassortative, and source–basin. It proposes a minimal two-group rewiring framework where connectivity evolves via swap moves that control assortativity and change moves that regulate in-degree, analyzed with a mean-field density matrix $\boldsymbol{\omega}$ to identify long-run structure types. The analysis reveals that CP arises from asymmetry in swap preferences while SB requires a degree difference between groups, and shows how the SB regime expands when swap influence weakens (lower $P^S$) or when group sizes/degrees differ. These findings illuminate the mechanisms behind observed directed-community motifs and offer a foundation for extending the framework to more groups and to empirical data, including alternative density normalizations that can suppress certain types.

Abstract

Large complex networks are often organized into groups or communities. In this paper, we introduce and investigate a generative model of network evolution that reproduces all four pairwise community types that exist in directed networks: assortative, core-periphery, disassortative, and the newly introduced source-basin type. We fix the number of nodes and the community membership of each node, allowing node connectivity to change through rewiring mechanisms that depend on the community membership of the involved nodes. We determine the dependence of the community relationship on the model parameters using a mean-field solution. It reveals that a difference in the swap probabilities of the two communities is a necessary condition to obtain a core-periphery relationship and that a difference in the average in-degree of the communities is a necessary condition for a source-basin relationship. More generally, our analysis reveals multiple possible scenarios for the transition between the different structure types, and sheds light on the mechanisms underlying the observation of the different types of communities in network data.

Figures (6)

• Figure 1: Four types of communities in directed graphs with two groups $g$, red (0) and blue (1). (Top) Representative network showing the community-structure type. (Middle) Block Network in which each node represents one group. (Bottom) Density matrix $\omega_{rs}$ defined in Eq. (\ref{['eq.omegaU']}), with the two largest matrix entries coloured in gray (see legend).. The classification of the four types is given in Eq. (\ref{['eq.inequalities']}), as proposed in Ref. liu2023nonassortative.
• Figure 3: The network evolution model represented as a decision tree. The picture depicts the possible one-step alterations in the connectivity of the network around the randomly chosen focal node (highlighted in the centre of the top panel). The groups of nodes are represented in the picture by colours $g(i)=0$ in red and $g(i)=1$ in blue. The swap move allows users to give preference to edges (influence) from a user of the same group (assortative choice with probability $P^A$) or from the different group (disassortative choice with probability $1-P^A$), while the change move controls the number of incoming edges users in a group will have on average (controlled by parameter $\alpha$). The algorithmic implementation of this model is described in Appendix \ref{['algo:general']}.
• Figure 4: Density of edges $\omega_{rs}$ between nodes in groups $0$ and $1$ as a function of time $t$. The curves were obtained performing a direct simulation of our model with parameters $P_0^S = 0.7,P_1^S = 0.5, P^A_0 = 0.9, P^A_1 = 0.8,\alpha_0 = 0.2,\alpha_1 = 0.1,N_0 = 67, N_1 =33$. The horizontal straight line correspond to the expected value for $t\rightarrow \infty$ and $N\rightarrow\infty$ obtained from Eq. (\ref{['eq:wswap']}).
• Figure 5: Parameter space for Eq. \ref{['eq:wswap']}, which is valid in the limit $P^S\rightarrow 1_-$ with swap and change moves. The group size ratio $c =\frac{N_0}{N_1} =2$ and the average in-degree ratio $b= \frac{\langle z\rangle_0}{\langle z\rangle_1} = 0.5$. Insets: examples of networks obtained at the indicated parameters through direct numerical simulations of our model, plotted using the NetworkX package. The node color indicates the group membership of the node and the edge color indicates information source (same as source node color).
• Figure 6: Effect of group-asymmetry in group size $N$ (y axis) and in-degree $\langle z \rangle$ (x axis) in the parameter space of assortative linking shown in Fig. (\ref{['fig:four']}). The boundary of all structure types is computed from Eqs. (\ref{['eq:wswap']}), (\ref{['eq:Pr_sovled']}), and (\ref{['eq.inequalities']}) in the limit of $t\rightarrow \infty$, $N \rightarrow \infty$, and $P^S\rightarrow 1_-$.