Phase Transitions of Diversity in Stochastic Block Model Dynamics

TL;DR

The paper develops a two-community weighted stochastic block model with dynamics in which recruitment follows preferential attachment based on weighted degree and nodes recruit from their own community. A deterministic approximation is derived, yielding a one-dimensional update with fixed points at $0$, $\tfrac{1}{2}$, and $1$, and a key threshold $\rho = \frac{a\alpha}{b\beta}$ that governs diversity outcomes. The main finding is a phase transition: when $\rho>1$ the minority vanishes, when $\rho<1$ parity is approached, and when $\rho=1$ the minority remains constant; this depends on the balance of cross- and same-community edge probabilities and their weights. The work provides theoretical insight into how collaboration value and cross-community integration influence long-term diversity, with potential implications for organizational design and policy to sustain minority representation.

Abstract

This paper proposes a stochastic block model with dynamics where the population grows using preferential attachment. Nodes with higher weighted degree are more likely to recruit new nodes, and nodes always recruit nodes from their own community. This model can capture how communities grow or shrink based on their collaborations with other nodes in the network, where an edge represents collaboration on a project. Focusing on the case of two communities, we derive a deterministic approximation to the dynamics and characterize the phase transitions for diversity, i.e. the parameter regimes in which either one of the communities dies out or the two communities reach parity over time. In particular, we find that the minority may vanish when the probability of cross-community edges is low, even when cross-community projects are more valuable than projects with collaborators from the same community.

Figures (5)

• Figure 1: Initial steps of the stochastic block model dynamics from Definition \ref{['def:stochastic']} illustrated. Suppose $\mathbf{p} = [[0.75, 0.25], [0.25, 0.75]]$, the weights are $\boldsymbol{\zeta} = [[2, 3], [3, 2]]$, and $\lambda = 3/4$. Figure (a) shows the initial set of nodes $V_0 = \{1, 2, 3, 4\}$. Figure (b) shows a realization of the random graph $G_0^+$ generated using the stochastic block model on $V_0$ with the parameters. Figure (c) shows the samples $(1, 3, 3)$ drawn from the vertices of $G_0^+$ according to the weighted degree distribution of $G_0^+$. Figure (d) shows how each sample $v_i$ brings in a new node $u_i$, which copies $v_i$'s color and they form an edge. The graph obtained is $G_1$. Figure (e) shows graph $G_1^+$, generated using the same parameters on the vertices of $G_1$.
• Figure 2: Trial runs of the stochastic block dynamics from Definition \ref{['def:stochastic']} with $11$ initial nodes. Red is the initial minority, starting with $5$ nodes in both cases. The $X$ axis shows time and the $Y$ axis shows the value of the fraction of red nodes. In (a) the fraction of red nodes reaches 50% in the limit, while in (b) it goes to zero in the limit. The probability matrix is $\mathbf{p} = [[0.75, 0.25], [0.25, 0.75]]$ in both cases (a) and (b). The fraction of nodes arriving in each round is $\lambda = 0.1$. The weight matrix is $\boldsymbol{\zeta} = [[1, 100], [100, 1]]$ in (a) and $\boldsymbol{\zeta} = [[100, 1], [1, 100]]$ in (b).
• Figure 3: Figure (a) shows the random system with $n=70$ initial nodes such that $n_R = 5$ are red. The probability matrix is $\mathbf{p}= {0.75}{0.25}{0.25}{0.75}$ and the weight matrix is $\mathbf{\omega}= 11001001$. We have $\rho \approx 0.03$. The fraction of nodes arriving in each round is $\lambda = 0.1$. Figure (b) shows the corresponding deterministic system, for the same initial parameters, as $n \to \infty$.
• Figure 4: Figure (a) shows the random system with $n=70$ initial nodes such that $n_R = 32$ are red. The probability matrix is $\mathbf{p}= {0.95}{0.05}{0.05}{0.95}$ and the weight matrix is $\mathbf{\omega}= 11.21.21$. We have $\rho \approx 15.8$. The fraction of nodes arriving in each round is $\lambda = 0.1$. Figure (b) shows the corresponding deterministic system, for the same initial parameters, as $n \to \infty$.
• Figure 5: The function $f$ and its derivative $f'$ for two values of $\rho$. In both cases $\lambda = 2$. Note here we are choosing a $\lambda > 1$ to better illustrate the curvature of the function, but the shape remains similar for all $\lambda > 0$.

Theorems & Definitions (20)

• Definition 1: Stochastic Block Model
• Definition 2: Stochastic Block Model Dynamics
• Lemma 1
• proof
• Lemma 2
• Definition 3: Deterministic System
• Lemma 3
• proof
• Lemma 4
• proof