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Higher-order network adaptivity: co-evolution of higher-order structure and spreading dynamics

This paper introduces the concept of higher-order network adaptivity to model the co-evolution of higher-order structure and spreading dynamics, revealing that while both pairwise and higher-order adaptivity increase spreading thresholds, they exert fundamentally opposing effects on phase transition behaviors, with higher-order adaptivity uniquely eliminating bistability and shifting transitions from discontinuous to continuous.

Original authors: Longzhao Liu, Hongwei Zheng, Zhihao Han, Xin Wang, Shaoting Tang

Published 2026-07-08

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CC BY 4.0

Original authors: Longzhao Liu, Hongwei Zheng, Zhihao Han, Xin Wang, Shaoting Tang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a crowded room where a rumor (or a virus) is spreading. In the old way of thinking about this, we only looked at how two people talk to each other. If Person A hears the rumor, they might tell Person B. If Person B gets scared, they might stop talking to Person A. This is called "pairwise" interaction—it's a simple one-on-one relationship.

But real life is messier. We don't just talk in pairs; we hang out in groups, teams, and families. This paper introduces a new way of looking at these situations called "Higher-Order Adaptivity."

Here is the story of what they found, explained simply:

1. The Two Ways Groups React

The researchers looked at how groups (like a family dinner or a work team) change when some members get "infected" (with a virus or a bad idea). They compared two scenarios:

Scenario A: The "Simple" Reaction (Pairwise-like Adaptivity)
Imagine a group of friends. If one person gets sick, the whole group gets scared and breaks up immediately. Everyone runs away from the group, regardless of how many are sick.
- The Result: This makes it harder for the disease to spread at first, but it creates a weird "tipping point." Once the disease gets a foothold, it explodes. It also creates a "bistable" situation: the disease either dies out completely or takes over the whole room, with no middle ground. It's like a light switch that is either fully OFF or fully ON.
Scenario B: The "Smart" Reaction (Higher-Order Adaptivity)
Now, imagine the same group, but they are smarter. If one person is sick, the group stays together because the risk is low. But if three or four people get sick, the group realizes, "This is dangerous!" and breaks up.
- The Result: This is the paper's big surprise. Even though the group is still breaking up to avoid the disease, the way they break up changes everything. Instead of making the explosion worse, this "smart" reaction calms things down. It prevents the "light switch" effect. The disease doesn't just die or explode; it can settle into a steady, manageable level. It turns the "explosive" switch into a smooth "dimmer" switch.

2. The "Breaking" Rule

The paper uses a mathematical rule to describe this.

In the Simple version, the group breaks apart at a constant rate once any infection is found.
In the Higher-Order version, the rate at which the group breaks apart speeds up as more people get sick. It's like a smoke alarm: a little smoke might not trigger it, but thick smoke sets it off immediately.

3. The Counter-Intuitive Discovery

Usually, we think that if people try harder to avoid a disease (by breaking up groups), the disease should just get harder to spread, but the type of behavior should stay the same.

The researchers found the opposite.

Old Thinking: Avoiding contact makes the "explosive" outbreaks even more likely to happen suddenly.
New Finding: When groups react based on how many people are sick (Higher-Order Adaptivity), it actually stops those sudden explosions. It eliminates the chaotic "all-or-nothing" behavior and makes the system more stable.

4. The "Fitness" Factor

The paper also looked at how well people can find "safe" people to join.

If people are blind to who is sick (random selection), the groups form randomly.
If people are smart and can perfectly find healthy people to join new groups, it helps reduce the chaotic "explosions" even more, though it doesn't change the point at which the disease starts spreading.

The Big Picture

Think of the old model as a room full of people where everyone panics and leaves the moment one person sneezes. This causes chaos: the room empties, then suddenly fills up again with a new wave of sneezing.

The new model is a room where people only panic and leave when half the room is sneezing. This allows the group to stay together longer when the risk is low, but break up effectively when the risk is high. The result? The room doesn't swing wildly between empty and full. It finds a balance.

In short: The paper shows that when groups react to the size of a problem (not just the existence of it), it fundamentally changes the rules of how things spread, turning chaotic, explosive events into smoother, more predictable ones.

Technical Summary: Higher-Order Network Adaptivity

Problem Statement

Complex systems, ranging from social to natural domains, are often modeled as networks where structure and dynamics co-evolve, a phenomenon known as adaptivity. Previous research has predominantly focused on pairwise adaptivity, assuming all interactions are dyadic (between two nodes). In these models, structural changes (e.g., avoiding infected contacts) are driven by the state of individual pairs.

However, real-world systems frequently exhibit higher-order interactions involving multiple units simultaneously (e.g., group interactions in social settings or transmission within a household). These interactions cannot be decomposed into pairwise links and require characterization via higher-order structures like hypergraphs. While higher-order spreading mechanisms (nonlinear infection rates) and temporal higher-order networks have been studied separately, the co-evolution of higher-order structure and spreading dynamics (higher-order adaptivity) remains largely unexplored. Specifically, the fundamental rules governing how groups break down or reform based on the number of infected members, and how this feedback loop affects epidemic thresholds and phase transitions, are unclear.

Methodology

1. The Higher-Order Adaptive Spreading Model

The authors propose a model describing the co-evolution of higher-order networks and spreading dynamics on hypergraphs. The model consists of two mutually interacting processes:

Higher-Order Spreading Dynamics:
- Nodes exist in states: Susceptible ( $S$ ) or Infected ( $I$ ).
- Infection occurs via hyperedges of size $m$ .
- The infection rate follows a power-law mechanism: $\theta(j; \beta, v) = \beta j^v$ , where $j$ is the number of infected nodes in a hyperedge, $\beta$ is the basic infection rate, and $v$ controls nonlinearity.
- Infected nodes recover at rate $\mu$ .
Higher-Order Adaptivity (Structural Evolution):
- Hyperedge Breaking: Groups (hyperedges) containing infected members may break due to risk-avoidance behaviors. The breaking rate is defined as $\pi(j; r, h) = r j^h$ $π (j; r, h) = r j^{h}$ , where $j$ $j$ is the number of infected nodes in the hyperedge, $r$ $r$ is the basic breaking rate, and $h$ $h$ represents the dependence on the infection level.
  - If $h=0$ , the breaking rate is independent of $j$ , reducing to pairwise-like adaptivity.
  - If $h \neq 0$ , it captures higher-order adaptivity, where the likelihood of a group breaking increases nonlinearly with the number of infected members.
- Hyperedge Formation: To prevent network fragmentation and ensure analytical tractability, when a hyperedge breaks, a new $m$ -hyperedge is formed. A susceptible node from the broken group selects $m-1$ other nodes to form a new group.
- Selection Fitness: The probability of a susceptible node being chosen is governed by a fitness parameter $\lambda$ $λ$ : $\Gamma(N_s; \lambda) = \frac{\lambda N_s}{\lambda N_s + (N - N_s)}$ $Γ (N_{s}; λ) = \frac{λ N _{s}}{λ N _{s} + ( N - N _{s} )}$ .
  - $\lambda = 1$ : Random selection (no information).
  - $\lambda \to \infty$ : Perfect selection of susceptible nodes.

2. Simulation and Validation

Real Data: Simulations were performed on three human co-location datasets (Thiers13, etc.) converted into hypergraphs using the Gillespie algorithm.
Synthetic Data: Theoretical analysis was conducted on synthetic uniform hypergraphs ( $m$ -uniform) with Poisson-distributed hyperdegrees.

3. Theoretical Framework

The authors developed a hyperedge-based approximation method (inspired by pair approximation) to derive a self-contained system of differential equations.

Variables track the number of infected nodes ( $N_i$ ) and the number of hyperedges containing $k$ infected nodes ( $l_{m,k}$ ).
The system is reduced to a 4-dimensional ODE system for $m=3$ .
Threshold Analysis: The outbreak threshold ( $\beta_c$ ) is derived by linearizing the system around the trivial fixed point and analyzing the Jacobian matrix eigenvalues.

Key Contributions and Results

1. Quantitative Impact on Thresholds

Both pairwise-like adaptivity ( $h=0$ ) and higher-order adaptivity ( $h \neq 0$ ) consistently increase the epidemic spreading thresholds. This indicates that adaptive behaviors generally make it harder for an epidemic to take off.

2. Qualitative Divergence: Opposing Effects on Bistability

The most significant finding is the completely opposing qualitative effects of pairwise-like versus higher-order adaptivity:

Pairwise-like Adaptivity ( $h=0$ ): Enhances or induces bistability (co-existence of epidemic extinction and outbreak depending on initial density) and discontinuous phase transitions (explosive growth). This aligns with previous findings on pairwise networks.
Higher-Order Adaptivity ( $h \neq 0$ ): Counter-intuitively reduces or eliminates the bistable region and shifts phase transitions from discontinuous to continuous.
- As the parameter $h$ (dependence on infection level) increases, the bistable region shrinks and eventually disappears.
- The system transitions from discontinuous transitions to continuous ones, often passing through a cusp point where transition types coexist.

3. Microscopic Mechanism

The authors provide a microscopic explanation for this divergence:

Pairwise-like adaptivity breaks hyperedges containing any infected node with equal probability. This creates a scenario where, at low infection densities, susceptible and infected nodes become completely separated (leading to extinction), while at high densities, separation is impossible (leading to persistence), thus creating bistability.
Higher-order adaptivity breaks hyperedges with a rate that scales with the number of infected nodes ( $j^h$ $j^{h}$ ). Hyperedges with only one infected node are less likely to break than those with multiple infected nodes. This mechanism:
1. Disrupts the structural foundation required for higher-order spreading (which relies on multiple infected nodes).
2. Preserves connections between susceptible and infected nodes (since single-infected groups remain intact).
- Both factors hinder the emergence of bistability.

4. Role of Fitness ( $\lambda$ )

The fitness parameter $\lambda$ (representing the accuracy of information about infection status) does not influence the outbreak threshold $\beta_c$ . However, it plays a pivotal role in reducing the size of the bistable region and influencing the type of phase transitions.

Significance and Claims

The paper claims to introduce the concept of higher-order network adaptivity and demonstrates that it possesses fundamental differences from the well-studied pairwise adaptivity.

Theoretical Advancement: The work provides a theoretical framework that predicts outbreak sizes and thresholds for adaptive higher-order systems, validated by both synthetic and real-world data.
Paradigm Shift: It challenges the assumption that adaptivity always enhances bistability or leads to explosive transitions. Instead, it shows that the nonlinearity inherent in higher-order structural evolution (dependence on group infection levels) can suppress these phenomena.
Generalizability: While focused on epidemic spreading, the authors state that the idea and framework of higher-order adaptivity can be extended to other adaptive higher-order systems, such as opinion formation and evolutionary dynamics.
Future Directions: The paper notes that while the proposed mechanism is representative, other higher-order adaptive mechanisms (e.g., threshold-like relationships or flow between hyperedges of different scales) warrant further exploration.

The findings highlight that ignoring higher-order characteristics in adaptive systems may lead to incorrect predictions about critical thresholds and the nature of phase transitions in complex systems.