Time-Varying Hypergraph Diffusion

• Time-varying hypergraph diffusion is a framework that models information flow over dynamically changing hyperedges and vertex availabilities with temporal causality.
• It employs a rigorous mathematical formulation using presence, latency, and vertex functions to capture realistic, time-respecting diffusion processes.
• Empirical studies, such as those on code review communications, reveal that static models overestimate diffusion compared to algorithms that respect time constraints.

Time-varying hypergraph diffusion concerns the propagation of information, heat, or analogous entities over structures where group interactions (hyperedges) and their availabilities evolve with time. Unlike static or time-aggregated networks, this approach provides a mathematically rigorous framework that incorporates both the combinatorial complexity of hypergraphs and temporal causality, yielding quantitatively distinct, more realistic diffusion outcomes.

1. Mathematical Formulation of Time-Varying Hypergraphs

A time-varying hypergraph is defined as a quintuple

$$\mathcal H=(V,\;\mathcal E,\;\rho,\;\xi,\;\psi),\ \mathcal T\subseteq\mathbb R_+\ \text{or}\ \mathbb N,$$

where:

• $V$ is the set of vertices (e.g., participants, nodes).
• $\mathcal E$ is a multiset of hyperedges, each representing a group interaction (e.g., a code review discussion).
• $$\rho\colon\mathcal{E}\times\mathcal{T}\rightarrow\{0,1\}$$ is the hyperedge‐presence function: $\rho(e,t)=1$ if $e$ is active at time $t$.
• $$\xi\colon\mathcal{E}\times\mathcal{T}\rightarrow\mathbb{R}_+$$ is a latency function, encoding the "in flight" time for information crossing hyperedges.
• $\psi\colon V\times\mathcal{T}\rightarrow\{0,1\}$ is the vertex‐presence function, indicating node availability.

In many scenarios of interest, a simplified variant is used: $\psi\equiv 1$ and all edges correspond to completed group events at specific timestamps, giving

$$\mathcal{H} = (V,E,T),\quad E\subseteq 2^V \times T,$$

where each $(H,t)\in E$ represents a hyperedge $H$ active at time $t$ (Dorner et al., 2021).

2. Diffusion Dynamics and Time-Respecting Journeys

A fundamental difference between static and time-varying hypergraph diffusion lies in causal constraints. An "information unit" originates with a vertex $u$ at time $t_0$. It can traverse a hyperedge $e$ at time $t$ \emph{only if} $u\in e$ and $e$ is active at that instant. After a latency $\xi(e, t)$, all other vertices in $e$ gain access to the information.

A journey from $u$ to $v$ is defined as a finite sequence $\bigl(e_1, t_1\bigr),\ldots, \bigl(e_k, t_k\bigr)$ such that

• $u\in e_1$, $v\in e_k$
• $\rho(e_i, t_i)=1$,
• $t_{i+1}>t_i+\xi(e_i,t_i)$,
• $e_i\cap e_{i+1}\neq\emptyset$ (to ensure the information can "hop" between events by overlapping participant).

The horizon of $u$ is then the set of vertices $v$ such that there exists a journey $u\leadsto v$ (Dorner et al., 2021).

3. Algorithmic Approaches

Let $\mathcal{H}=(V,\mathcal{E},\rho,\xi,\psi)$. The computation of reachable sets—static versus time-respecting—uses a breadth-first search-like algorithm, parameterized by a mode flag:

function REACHABLE(ℋ, s, mode): let Q ← empty queue mark s as reachable; enqueue s into Q while Q not empty: v ← dequeue Q if mode == "ignore time": N ← all vertices w that share a hyperedge e with v (∃e∈𝓔: v∈e, w∈e) else # mode == "respect time" N ← all w for which ∃(e,t) with ρ(e,t)=1, v∈e, w∈e, and t > time_when_v_became_reachable + ξ(e, time_when_v_became_reachable) for each n in N: if n not yet marked reachable: mark n reachable; enqueue n return {all marked vertices}

The per-source worst-case complexity is $O(|V|+\sum_{e\in\mathcal{E}}|e|)$, dominated in practice by repeated scans of the edge set for large hypergraphs (Dorner et al., 2021).

4. Structural and Empirical Consequences

Ignoring temporal order systematically overestimates the number of reachable nodes. In a study of code-review communications at Microsoft, with $|V|=37,103$ and $|\mathcal{E}|=309,740$ hyperedges, static models produced a mean reachable $R_\text{static}(u)\approx 29,660$ and median $\approx 33,172$ ($\sim$89% coverage). The time-respecting model yielded mean $R_\text{temporal}(u)\approx 10,907$, median $\approx 11,652$, and max $26,216$ ($\sim$71%). The average per-node overestimation was $\Delta(u)\approx18,752$ (Dorner et al., 2021).

Metric	Static Model (time-ignoring)	Temporal Model (time-respecting)
Mean reachable participants	$\approx$29,660	$\approx$10,907
Median reachable participants	$\approx$33,172	$\approx$11,652
Max reachable participants	33,173	26,216
% of network covered (max)	89%	71%

This demonstrates that time-aggregation artificially introduces diffusion paths unavailable in practice.

5. Hypergraph Diffusion Equations and Control

The continuous-time diffusion of a scalar (e.g., "heat") on a weighted hypergraph is described by a nonlinear evolution equation involving the hypergraph $p$-Laplacian: $x'(t) + L_{G,p}(x(t)) \ni h(t), \quad x(0) = x_0,$ where $L_{G,p}$ is defined as the subdifferential of the convex energy

$$\varphi_{G,p}(x) = \frac{1}{p}\sum_{e\in E} w(e)[f_e(x)]^p, \quad f_e(x)=\max_{u,v\in e}(x(u)-x(v)).$$

Internal control is implemented by constraining a subset of the vertices to follow given functions, leading to a nonautonomous subdifferential inclusion: $$x'(t) + \partial\varphi_{G,p}(x(t)) + \partial I_{K_a(t)}(x(t)) \ni h(t), \quad x(0)\in K_a(0).$$ Existence and uniqueness are obtained for $a\in W^{1,2}$, $h\in L^2$, by reducing to a regularized problem via Yosida approximation, extracting limits with compactness arguments, and relying on the monotonicity of the operators involved (Fukao et al., 2022).

6. Long-Time Behavior and Continuous Dependence

For the diffusion problem with moving constraints, solutions depend continuously (in a $1/2$–Hölder sense in $a$) on the data, as established by Gronwall-type arguments on energy estimates over the "free" vertices. Global-in-time solutions exist under local regularity for $a$ and $h$.

• With constant (in time) control and source, solutions converge to a stationary state determined by the (possibly constrained) equilibrium.
• When both forcing and source decay to zero (with their derivatives integrable), solutions decay, with explicit rates depending on $p$.
• For convergent (but not necessarily constant) forcing and source and $1\le p<3$, solutions converge to a corresponding stationary state in the set $K_{a_\infty}$ (Fukao et al., 2022).

7. Extensions, Limitations, and Future Directions

Time-varying hypergraph models accommodate natural extensions:

• Probabilistic edge and node presence functions, introducing stochasticity into the diffusion process.
• Multi-layer hypergraphs for modeling layered information flows (e.g., code review, email, meetings).
• Ageing and decay mechanisms via time-dependent utility scores, reflecting information obsolescence (Dorner et al., 2021).

For the general hypergraph diffusion equation, the lack of closed-form solutions (except in trivial cases) necessitates implicit-time discretization or operator-splitting numerical approaches. The framework generalizes to any compact network or metric-measure space where the Laplacian is a subdifferential of a suitable convex energy plus a time-variant convex indicator. The non-autonomous nature of the operator prevents the existence of a contraction semigroup, though resolvent schemes provide convergent approximations (Fukao et al., 2022).

A plausible implication is that future research will increasingly employ these frameworks to model group-based, temporally constrained processes in empirical domains beyond code review, such as collaboration, epidemiology, and distributed systems, leveraging the structural clarity and computational tractability of time-respecting hypergraph models.