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Simplicial Activity Driven Model
Giovanni Petri, Alain Barrat
TL;DR
The paper addresses how to model evolving group interactions that are not adequately represented by pairwise temporal networks. It introduces the Simplicial Activity Driven model and finds that simplex-size fluctuations affect simple contagion, while social cascades can display richer and extremely slow dynamics.
Problem
Pairwise temporal-network models do not fully represent biological and social interactions involving groups, whose changing sizes may affect dynamical processes.
Method
The paper introduces the Simplicial Activity Driven model, using time-evolving simplices as building blocks for multi-agent interactions and studying dynamics analytically and numerically.
Results
Simplicial-size variability can drive the epidemic threshold toward 0 as 1/⟨s2⟩, while cascade dynamics can change sharply with s and become much faster under strong size variability.
Takeaways & Limitations
Simplicial structure and evolving group size materially change contagion outcomes compared with binary-interaction activity-driven models.
Abstract
from arXiv · showhide
Many complex systems find a convenient representation in terms of networks: structures made by pairwise interactions (links) of elements (nodes). For many biological and social systems, elementary interactions involve however more than two elements, and simplicial complexes are more adequate to describe such phenomena. Moreover, these interactions often change over time. Here, we propose a framework to model such an evolution: the Simplicial Activity Driven (SAD) model, in which the building block is a simplex of nodes representing a multi-agent interaction. We show analytically and numerically that the use of simplicial structures leads to crucial differences in the outcome of paradigmatic processes modelling disease propagation or social contagion, with respect to the activity-driven (AD) model, a paradigmatic temporal network model involving only binary interactions. In particular, fluctuations in the number of nodes involved in the interactions can affect the outcome of models of simple contagion processes, contrarily to what happens in the AD model. Moreover, social contagion models such as cascading processes present a much richer phenomenology and can become extremely slow when occurring on evolving simplicial complexes.
I. CO-AUTHORSHIP DATA
APS co-authorship data show that collaboration-group sizes became progressively more heterogeneous over time, motivating simplicial representations and related structural definitions.
- I. CO-AUTHORSHIP DATA: The APS dataset contains 587675 papers by 413513 authors, with co-author distributions broadening from the 1940s through recent decades.The broadening indicates increasing variation in collaboration-group size.
- I. CO-AUTHORSHIP DATA: The ratio βs = ⟨s2⟩/⟨s⟩ increases over time, highlighting greater heterogeneity in co-author counts.
- II. SIMPLICES AND SIMPLICIAL COMPLEXES: A simplicial complex contains every subface of each simplex, so group interactions include their lower-order sub-interactions.
- II. SIMPLICES AND SIMPLICIAL COMPLEXES: The 1-skeleton retains only edges, whereas the clique complex fills every clique with a corresponding simplex.
- II. SIMPLICES AND SIMPLICIAL COMPLEXES: Simplicial connected components join simplices through shared faces, extending connectivity concepts beyond the underlying graph.
III. DETAILED COMPUTATION OF THE AGGREGATED DEGREE
The aggregated SAD degree is derived from cumulative interactions and compared with node-matched and edge-matched AD models, revealing dependence on simplex-size heterogeneity.
- III. DETAILED COMPUTATION OF THE AGGREGATED DEGREE: Aggregated degree calculations use cumulative node interactions over T steps, with approximations valid for large N and small T/N.
- III. DETAILED COMPUTATION OF THE AGGREGATED DEGREE: The average aggregated degree grows quadratically with simplex size s in SAD and eAD, but linearly in nAD.
- III. DETAILED COMPUTATION OF THE AGGREGATED DEGREE: The SAD and node-matched AD models have the same temporal growth of the aggregated giant connected component, whereas eAD produces systematically larger components.
- III. DETAILED COMPUTATION OF THE AGGREGATED DEGREE: With fixed average simplex size, fluctuations in s affect the SAD aggregated degree through the second moment of the simplex-size distribution.
IV. NUMBER OF 2-SIMPLICES
The paper derives the number of genuine 2-simplices involving each node and validates the prediction, while showing that network projections overcount simplicial structure and connectivity.
- IV. NUMBER OF 2-SIMPLICES: k2(i, T) counts distinct genuine 2-simplices involving node i, computed from repeated group participation and shared node pairs.
- IV. NUMBER OF 2-SIMPLICES: The predicted k2(i, T) agrees with empirical values across nodes and aggregation times, with correlations increasing for longer T.The comparison uses simulations averaged over 50 realizations.
- IV. NUMBER OF 2-SIMPLICES: The average number of genuine 2-simplices grows more slowly than all triangles in the aggregated 1-skeleton because triangles can combine edges formed in different simplices.
- IV. NUMBER OF 2-SIMPLICES: The largest 3-clique component in the 1-skeleton grows much faster than the largest connected 2-simplex component.The 1-skeleton cannot distinguish genuine simplices from triangles assembled across different interactions, creating more possible paths.
V. SIMPLICIAL LAPLACIAN
The simplicial Laplacian extends graph-based dynamical analysis to every simplex dimension, while preserving the graph Laplacian at dimension zero. Higher-dimensional spectra distinguish a simplicial complex from its 1-skeleton and clique complex through their different holes.
- V. SIMPLICIAL LAPLACIAN: L_0 operates on nodes as the conventional N × N graph Laplacian, so its eigenspectrum is identical for a simplicial complex and its 1-skeleton.This equivalence does not generally extend to higher-dimensional Laplacians.
- V. SIMPLICIAL LAPLACIAN: The simplicial Laplacian L_k is defined for each simplex dimension k using boundary maps and maps k-chains C_k to themselves.Its up and down components correspond to moving between adjacent simplex dimensions.
- V. SIMPLICIAL LAPLACIAN: The nullity of L_k equals the dimension of the corresponding homology group H_k, linking zero eigenvalues to holes of dimension k.For L_0, this reduces to the number of connected components.
- V. SIMPLICIAL LAPLACIAN: Higher-dimensional spectra distinguish the original complex from its 1-skeleton and clique complex because their numbers of holes differ.The 1-skeleton lacks higher simplices and therefore counts graph loops as holes, whereas the clique complex contains at least as many 2-simplices as the original complex.
B. Case of a distribution of clique sizes P(s)
Allowing clique sizes to fluctuate changes the SAD epidemic threshold through the second moment of the size distribution. At fixed mean size, diverging size fluctuations drive the threshold toward zero.
- B. Case of a distribution of clique sizes P(s): Introducing a clique-size distribution p(s) adds an integration over p(s) to the infectious-node evolution equation when size and activity are uncorrelated.The resulting equations are analyzed through a matrix whose largest eigenvalue determines the threshold condition.
- B. Case of a distribution of clique sizes P(s): The epidemic threshold λSAD_c decreases as 1/⟨s^2⟩ when clique-size fluctuations diverge at fixed average clique size ⟨s⟩.Figure S6 illustrates this inverse dependence for several average simplex sizes.
- B. Case of a distribution of clique sizes P(s): The predicted SAD-to-eAD epidemic-threshold ratio in APS co-authorship data increases from 1900 to 2015.Activity fluctuations drive the increase until the 1950s, followed by clique-size fluctuations.
C. Ratio between SAD and eAD epidemic thresholds
For meaningful activity and clique-size moments, the SAD epidemic threshold is at least as high as the corresponding eAD threshold. The ratio varies across empirical temporal interaction data as activity and group-size fluctuations change.
- C. Ratio between SAD and eAD epidemic thresholds: The SAD critical threshold is always at least as high as the corresponding eAD threshold for s ≥ 2 and ⟨s^2⟩ ≥ ⟨s⟩^2.The eAD comparison uses m = s(s−1)/2 edges per activation.
- C. Ratio between SAD and eAD epidemic thresholds: The predicted threshold ratio changes significantly across APS years, driven first by activity fluctuations and later by co-authorship-size fluctuations.The activity contribution dominates through the 1950s, while the size contribution dominates afterward.
- C. Ratio between SAD and eAD epidemic thresholds: Table S2 computes SAD-to-eAD threshold ratios from empirical activity and simplex-size distributions across SocioPatterns datasets and temporal resolutions.The datasets cover face-to-face contacts in contexts including offices, conferences, hospitals, and schools.
D. Case of a uniform activity
With uniform activity, SAD and eAD produce the same SIS epidemic threshold when each activation creates the same number of edges. Their mean-field evolution equations then coincide.
- D. Case of a uniform activity: The SIS epidemic thresholds are identical in SAD and eAD when all nodes have the same activity.Uniform activity leaves one node class, so the dynamics reduce to a mean-field equation.
- D. Case of a uniform activity: For uniform activity, SAD and eAD have the same mean-field equations because each creates the same number of edges when m = s(s−1)/2.The distinct SAD contagion terms become equivalent after averaging over nodes.
- D. Case of a uniform activity: Each activation creates m links in AD and s(s−1)/2 links in SAD, yielding 2m contagion opportunities when links connect susceptible and infectious nodes.Under uniform activity, all such links contribute equivalently to the SIS dynamics.
- D. Case of a uniform activity: In the SAD evolution equation, the additional term represents links between nonactive nodes brought into contact by the node activating a simplex.These links add contagion opportunities absent from the eAD equation before the uniform-activity simplification.
A. Definition
The temporal cascade model extends threshold-based adoption dynamics to evolving contacts by using a memory window θ. Because contacts change, cascades need not become blocked and may continue until all agents adopt.
- The model represents adopters and non-adopters, with non-adopters switching when the fraction of adopting contacts exceeds threshold φ.
- Temporal contacts introduce a memory parameter θ, defining the interval over which each non-adopter evaluates adopting contacts.
- Because new adopter contacts can continually arise, partially adopted configurations are not necessarily blocked and cascades may continue until all agents adopt.
- The study only illustrates selected cascade behaviors across φ, θ, and SAD network parameters rather than fully investigating their entire parameter space.
B. Cascades on AD and SAD temporal networks
Cascades slow only modestly with increasing φ on AD networks, whereas SAD networks can exhibit extremely slow dynamics because adoption requires multiple adopters within the same clique.
- As φ increases, SAD cascades become extremely slow, unlike the limited slowing observed in AD cascades.
- In AD networks, an activated adopter can directly convert a contacted non-adopter regardless of φ, preserving cascade continuation.
- In SAD networks, a non-adopter adopts only when its clique contains more than (s −1)φ adopters, producing threshold-dependent slowdowns.
C. Effect of the memory span θ
Increasing the temporal memory span θ slows cascades on both AD and SAD networks because a longer contact window can dilute adopters’ fraction among observed interactions.
- As θ increases, cascade dynamics become slower on both AD and SAD temporal networks.
- Figure S8 reports average adopter fractions over time for AD and SAD networks at θ = 1 across fixed clique sizes s.
- A longer window increases counted contacts while typically reducing the early-time adopter fraction, making the threshold φ harder to exceed.
D. Effect of the parameter s
The effect of clique size s differs sharply between AD and SAD networks: AD cascades accelerate with s, while SAD cascades show threshold-dependent, non-monotonic behavior and can respond strongly to size variability.
- In AD networks, increasing s makes cascades faster because an activated adopter can spread to a number of contacts growing quadratically with s.
- In SAD networks, increasing s speeds spreading within threshold intervals but causes a sharp slowdown when the required adopter count increases by one.
- For φ = 0.3, s = 4 is faster than s = 3, s = 5 slows at short times, s = 6 is faster than s = 5, and s = 8 is much slower.
- For φ = 0.5, s = 4 is faster than s = 3, whereas s = 5 is much slower.
- Overall, cascade velocity varies non-monotonically with s on SAD temporal networks.
- Variability in clique sizes slightly speeds AD cascades but can produce a much stronger speed-up in SAD cascades.