Assigning Entities to Teams as a Hypergraph Discovery Problem

TL;DR

This work reframes the Team Formation Problem as an edge-dependent vertex-weighted (EDVW) hypergraph optimization, with the objective of maximizing the algebraic connectivity $\mu_2(L^H)$ to promote resilience and fast diffusion of information among agents. A constrained simulated annealing (CSA) algorithm is developed to maximize $\mu_2(L^H)$ under energy- and budget-based feasibility constraints, using guided perturbations and penalty terms to steer the search toward feasible, high-connectivity solutions. The approach is validated on real-world collaboration datasets (MAG and APS), showing substantial improvements in algebraic connectivity and patching resilience compared to greedy baselines, and comparable performance to bipartite representations but at lower computational cost. The findings suggest that higher-order hypergraph modeling captures resilience and diffusion dynamics more effectively than traditional pairwise representations, with practical implications for robust task deployment in scientific collaborations and potentially finance-like networks.

Abstract

We propose a team assignment algorithm based on a hypergraph approach focusing on resilience and diffusion optimization. Specifically, our method is based on optimizing the algebraic connectivity of the Laplacian matrix of an edge-dependent vertex-weighted hypergraph. We used constrained simulated annealing, where we constrained the effort agents can exert to perform a task and the minimum effort a task requires to be completed. We evaluated our methods in terms of the number of unsuccessful patches to drive our solution into the feasible region and the cost of patching. We showed that our formulation provides more robust solutions than the original data and the greedy approach. We hope that our methods motivate further research in applying hypergraphs to similar problems in different research areas and in exploring variations of our methods.

Figures (17)

• Figure 1: Graphical representation of the task-assignment problem. In (a), the task assignment is represented by $\mathcal{B}$ as well as an exemplary case of budgets, $B_i$'s, and energies, $E_k$'s, in (b) and (c), the hypergraph and bipartite representations of the same task assignment.
• Figure 2: Examples of small hypergraphs. From left to right, in decreasing order of algebraic connectivity. In the top row, the graphical representation of the transposed incidence matrix is used for visualization, where the rows represent the hyperedges (tasks) and the columns represent the nodes (agents). The bottom row shows an example of the diffusion process defined by $L^H$. All processes start with the same initial condition, $\mathcal{B}_{ik} = 1$ for all assignments (see incidence matrices), $B_i =\sum_k \mathcal{B}_{ik}$, and $E_k =\sum_i \mathcal{B}_{ik}$.
• Figure 3: Hypergraph assignment swapping with different structures, focusing on preserving constraints on agent budgets and task requirements. In this example, we start with a set of isolated communities, each formed by $6$ nodes sharing $6$ hyperedges, i.e., $6$ agents sharing $6$ tasks. We introduce different connections to these isolated communities with different structures to explore properties favorable to the algebraic connectivity function. For simplicity, we let $N_c$ be the number of communities in $\mathcal{H}$, where $n_i$ is the number of nodes in community $i$, and $m_i$ is the number of hyperedges in the community $i$.
• Figure 4: Hypergraph assignment swapping on isolated communities. Here, we tested four different systems: (i) random, (ii) connected by one node, (iii) connected by one hyperedge, and (iv) head to tail (see Fig. \ref{['fig:Schematic']}). We notice that the algebraic connectivity scales with the system size as $\mu_2\sim N_c^{-a}$, where, in this example, $N=n_c \times N_c, K= m_c \times N_c, N_c=n_{c_i}=m_{c_i}, \forall c_i \in \text{the set of communities}$. This figure is the result of $30$ independent repetitions.
• Figure 5: The effect of the constraints on the algebraic connectivity for different budget multipliers, $\beta$. We show the average and standard deviation of the algebraic connectivity for the optimized sampled graphs under different budget relaxations after $10$ independent runs. We assigned the number of agents to each sampled hypergraph to be four times the number of tasks in the hypergraph. The behavior of the algebraic connectivity follows $\mu_2 \sim N^{-a}$, where $N$ is the hypergraph size and $a$ is the scaling parameter.