Signed Simplicial Contagion Model

• SSCM is a mathematical framework that models contagion using signed simplicial complexes, capturing both pairwise and group interactions in networks.
• The model employs Heider’s structural balance theory to distinguish balanced and unbalanced simplices, enabling analysis of continuous and discontinuous phase transitions.
• Empirical simulations show that increasing distrust shifts dynamics from bistable, group-driven outbreaks to gradual, pairwise contagion, highlighting the impact of network trust.

The signed simplicial contagion model (SSCM) is a mathematical framework for describing contagion processes that operate not only via pairwise interactions but also through higher-order group structures within networks populated by both trust and distrust relationships. SSCM captures the combined and divergent effects of dyadic (edge-based) and group (simplicial) mechanisms in signed networks, providing a rigorous basis for analyzing both gradual and abrupt shifts in collective behaviors in social systems (Ma et al., 2024, Kemmeter et al., 2023).

1. Signed Simplicial Complex Structure

The underlying network is encoded as a random signed simplicial complex. The vertex set $V = \{v_1, \ldots, v_N\}$ represents $N$ individuals. Each unordered pair $(i, j)$ may form an edge $s_{ij} \in \{+1, -1\}$, indicating trust or distrust, partitioning the edge set into positive (trust) edges $E_+ = \{ij: s_{ij} = +1\}$ and negative (distrust) edges $E_- = \{ij: s_{ij} = -1\}$. A $k$-simplex is any $(k+1)$-tuple of nodes $\{i_0, ..., i_k\}$ fully connected by edges—each edge signed as above. In particular, 2-simplices (triangles) are central to modeling group effects.

Heider's structural balance theory is encoded by classifying each simplex $\Delta$ by the product of its edge signs:

$N$0

with $N$1 deemed "balanced" if $N$2 and "unbalanced" if $N$3. The probability that a randomly sampled triangle is balanced is

$N$4

where $N$5 is the fraction of negative (distrust) edges.

2. Contagion Dynamics: States, Transitions, and Rates

Each node holds a binary state $N$6, with $N$7 (susceptible) and $N$8 (infectious). Dynamics proceed via three processes:

• Pairwise (Edge-Based) Infection: An $N$9-$(i, j)$0 pair linked by $(i, j)$1 (trust) transmits infection at rate $(i, j)$2, i.e., $(i, j)$3. Negative edges do not transmit contagion.
• Group (Triangle-Based) Infection: In any balanced triangle $(i, j)$4 with $(i, j)$5, if exactly two are $(i, j)$6 and one is $(i, j)$7, the susceptible node becomes infectious at rate $(i, j)$8: $(i, j)$9.
• Recovery: Each $s_{ij} \in \{+1, -1\}$0 node recovers to $s_{ij} \in \{+1, -1\}$1 at rate $s_{ij} \in \{+1, -1\}$2: $s_{ij} \in \{+1, -1\}$3.

The macroscopic observable is the infection density:

$s_{ij} \in \{+1, -1\}$4

3. Mean-Field Theory and Phase Transitions

Assuming homogeneous mixing, one defines the average degree $s_{ij} \in \{+1, -1\}$5, the trusted degree $s_{ij} \in \{+1, -1\}$6, and the average number of 2-simplices per node $s_{ij} \in \{+1, -1\}$7. The number of balanced triangles per node is $s_{ij} \in \{+1, -1\}$8.

The mean-field evolution for $s_{ij} \in \{+1, -1\}$9 is:

$E_+ = \{ij: s_{ij} = +1\}$0

Defining $E_+ = \{ij: s_{ij} = +1\}$1 and $E_+ = \{ij: s_{ij} = +1\}$2 yields

$E_+ = \{ij: s_{ij} = +1\}$3

Steady-state roots are

$E_+ = \{ij: s_{ij} = +1\}$4

Phase transition regimes:

• For $E_+ = \{ij: s_{ij} = +1\}$5, transitions are continuous (SIS-like); the threshold occurs at $E_+ = \{ij: s_{ij} = +1\}$6.
• For $E_+ = \{ij: s_{ij} = +1\}$7, a saddle-node bifurcation occurs at $E_+ = \{ij: s_{ij} = +1\}$8: there is bistability and a discontinuous jump in $E_+ = \{ij: s_{ij} = +1\}$9 at $E_- = \{ij: s_{ij} = -1\}$0, with a hysteresis loop as $E_- = \{ij: s_{ij} = -1\}$1 is swept.

As $E_- = \{ij: s_{ij} = -1\}$2 increases, $E_- = \{ij: s_{ij} = -1\}$3 drops nonlinearly, while $E_- = \{ij: s_{ij} = -1\}$4 decreases linearly, shifting the system from a group-dominated, discontinuous regime to a pairwise-dominated, continuous regime at a critical distrust level $E_- = \{ij: s_{ij} = -1\}$5.

4. Numerical Simulation Results

Simulations are performed on random signed simplicial complexes generated from Erdős–Rényi base graphs ($E_- = \{ij: s_{ij} = -1\}$6, $E_- = \{ij: s_{ij} = -1\}$7) with random assignment of negative edges (varied $E_- = \{ij: s_{ij} = -1\}$8). Every triple receives a 2-simplex with probability chosen to yield prescribed $E_- = \{ij: s_{ij} = -1\}$9.

Key empirical findings:

• Discontinuous transitions: For low $k$0 (e.g., $k$1), high $k$2 yields abrupt outbreaks and bistability, with a prominent hysteresis loop in $k$3.
• Effect of distrust: As $k$4 increases, discontinuity and bistability widths decrease; above $k$5, all transitions are continuous, even at high $k$6.
• Group vs. pairwise sensitivity: The group contribution $k$7 decreases nonlinearly with $k$8 (steep drop at small $k$9 followed by a plateau and slow decay), while pairwise transmission $(k+1)$0 decreases linearly.

5. Differential Sensitivity and Mechanistic Switch

Pairwise and group channels respond asymmetrically to distrust:

Channel Type	Dependence on $(k+1)$1	Functional Form
Pairwise (edges)	Linear decrease	$(k+1)$2
Group (triangles)	Nonlinear, steeper	$(k+1)$3

For small $(k+1)$4 (high trust), both infection channels can operate, leading to the possibility of abrupt, group-driven epidemics. As $(k+1)$5 increases (growing distrust), group effects collapse more rapidly, and only pairwise (SIS) transmission persists. The critical point $(k+1)$6 marks the switch from group-dominated to pairwise-dominated contagion regimes.

A plausible implication is that the trust topology of a network controls not just epidemic thresholds but also the qualitative nature of collective shifts: trust-rich networks are susceptible to sudden, group-driven phase transitions, while distrust-rich structures allow only gradual, pairwise-driven spreading.

6. Relationship to Broader Context and Other Models

The SSCM extends conventional network contagion models by integrating signed edge information and higher-order simplicial (group) interactions into a single analytical framework. It operationalizes structural balance via Heider theory, connecting contagion outcomes to the global arrangement of trust and distrust. Comparative analyses with more general signed simplicial contagion frameworks (Kemmeter et al., 2023) demonstrate that the specific balance and sign distribution of group structures further modulate not only epidemic thresholds and phase transition types but also the potentially non-monotonic dependence of prevalence on mean connectivity and sign-bias parameters.

The conceptual lineage includes higher-order contagion models on unsigned simplicial complexes, classical SIS/SIR dynamics, and the theory of signed social networks. SSCM provides a robust foundation for exploring novel empirical phenomena such as collective emotional outbreaks, the collapse of coordination in the presence of antagony, and the resilience or fragility of multiplexed social systems under structural imbalance.