Activity-Driven Networks Overview

• Activity-driven networks are temporal models where nodes create edges based on intrinsic activity, effectively capturing the fluctuating dynamics of interactions.
• They employ analytical methods like hidden variable mapping and percolation analysis to relate individual activity rates to emergent properties such as degree distribution and clustering.
• Extensions incorporating memory effects and higher-order interactions enhance simulations of contagion, diffusion, and consensus in complex, real-world systems.

Activity-driven networks are a class of temporal network models in which the microscopic rule governing the creation of edges is based on the intrinsic “activity” of individual nodes. These frameworks were developed to address limitations of static or connectivity-driven models, particularly the inability of the latter to capture the highly dynamic, fluctuating, and temporally heterogeneous nature of interactions observed in real-world systems such as social, biological, and technological networks. Activity-driven models have since been analytically extended to reproduce a broad spectrum of empirically observed structural properties, and serve as the foundation for rigorous paper of dynamical processes—including diffusion, contagion, consensus, and percolation—on time-varying network topologies.

1. Fundamental Principles of Activity-Driven Network Models

The activity-driven network (ADN) paradigm assigns to each node $i$ an activity potential $x_i$, representing the normalized propensity of that agent to initiate interactions per unit time. The standard definition is

$$x_i = \frac{\text{number of interactions initiated by } i \text{ in } \Delta t}{\text{total number of interactions by all nodes in } \Delta t}$$

From this, the activity rate is defined as $a_i = \eta x_i$, where $\eta$ is a rescaling factor ensuring that the expected number of active nodes per unit time is $\eta \langle x \rangle N$.

The model evolves in discrete time steps:

• Each node is initially disconnected.
• At each time $t$, node $i$ becomes active with probability $a_i \Delta t$. When active, the node creates $m$ links to $m$ randomly chosen nodes (including possibly itself). These links exist only for duration $\Delta t$ and are removed before the next time step.
• The process is repeated at every time increment, resulting in a dynamic sequence of star-like subgraphs over the population.

This approach fundamentally differs from static or preferential-attachment-based models by divorcing the appearance of highly connected hubs from cumulative degree advantage, instead attributing them to inherent activity heterogeneities, as directly observed in real data (Perra et al., 2012).

2. Structural Features: Degree Distribution, Correlations, and Clustering

A key analytical finding is that the integrated network over $T$ time steps presents a degree distribution mirroring the activity potential distribution $F(a)$, with high-activity nodes acquiring high integrated degree:

$$k_i(T) = N \left[1 - \exp\left(-\frac{T m \eta x_i}{N}\right)\right]$$

For large $N$ and $k/T \ll 1$, the leading-order integrated degree distribution is $P_T(k) \sim F(k/(T m \eta))$ (Perra et al., 2012, Starnini et al., 2013). This result establishes a direct and quantitative link between agent-level activity heterogeneity and emergent hub formation.

Degree correlations and clustering are also analytically tractable within a mapping to the hidden variable formalism. For example, for large degrees,

$$\overline{k}^{nn}_T(k) \simeq 2 \langle a \rangle + \sigma_a^2 \frac{T}{k}$$

showing that the integrated network generated by the baseline model is disassortative. Clustering, as measured by the average local clustering coefficient, is found to be low (of order $T/N$), indicating weak triangle closure and little community structure—again in contrast to many empirical networks (Starnini et al., 2013).

These findings have motivated subsequent papers to introduce memory effects (such as triadic closure), leading to models that recover high clustering and positive (assortative) degree correlations characteristic of real social systems (Medus et al., 2013).

3. Modeling Extensions: Memory Effects and Higher-Order Interactions

Limitations of the “memoryless” ADN—particularly its inability to generate both large clustering and degree assortativity—have led to a series of significant extensions:

• Memory and Triadic Closure: By including a probability $q$ that an activated node will link to a second-neighbor (closing a triangle) rather than to a random node, the model naturally produces both high clustering and, in population-growth regimes, scale-free degree distributions of the form $n_{k,a} \sim k^{-1-2(\gamma+1)/q}$. Analytical and simulation results reveal that the degree distribution’s tail becomes independent of the activity distribution for large $k$, depending solely on $q$ and the population growth rate $\gamma$ (Medus et al., 2013).
• Attractiveness and Asymmetric Edge Formation: The attractiveness-driven (ADA) model extends the framework by endowing each node $i$ not only with an activity $a_i$ but also with an attractiveness $b_i$, which governs the likelihood the node is chosen as a target for incoming links. For a master equation describing SIS epidemics, the epidemic threshold is

$$\frac{\beta}{\mu} > \frac{2\langle a\rangle \langle b\rangle}{\langle ab\rangle + \sqrt{\langle a^2\rangle\langle b^2\rangle}}$$

with $\langle \cdot \rangle$ denoting averages over the joint $(a, b)$ distribution. Positive correlations between activity and attractiveness lower the epidemic threshold and increase the contagion potential, while negative correlations suppress spreading (Pozzana et al., 2017).

• Simplicial Interactions and Higher-Order Dynamics: Moving beyond pairwise links, the Simplicial Activity-Driven (SAD) and Probabilistic Activity-Driven (PAD) models employ group (simplex) interactions. Upon activation, nodes may form $k$-simplexes representing group events, with the statistical properties of these group sizes and frequencies controlling both the degree distribution and the aggregate network clustering. These models reveal that outbreaks in higher-order contagion models can be bistable and that the critical threshold depends on both the activity and the higher-order interaction probability (Petri et al., 2018, Han et al., 2022).

4. Analytical Approaches: Mapping, Percolation, and Scaling

Activity-driven models permit a range of analytical methods:

• Hidden Variable Mapping: The instantaneous and time-integrated networks are analyzed as hidden variable (activity) models, enabling the derivation of conditional degree distributions, correlations, and clustering measures via generating functions (Starnini et al., 2013).
• Temporal Percolation: The formation of a giant connected component as the network is integrated over time is characterized by a percolation threshold,

$$T_p = \frac{2\langle a \rangle}{\langle a^2 \rangle + 2 \langle a \rangle^2}$$

or, accounting for degree correlations, $$T_p = 1/(\sqrt{\langle a^2 \rangle} + \langle a \rangle)$$ (Starnini et al., 2013). This approach precisely links the network's evolving macroscopic connectivity to microscopic node activity statistics.

• Analytical Dynamics: Mean-field, branching process, and rate-equation techniques yield explicit results for contagion thresholds, spreading speeds, and the time to consensus. Crucially, these analyses demonstrate that temporal and activity heterogeneities fundamentally alter both the critical points and the scaling properties of dynamical processes (Perra et al., 2012, Pozzana et al., 2017).
• Glassy Dynamics and Aging: For random walks, the broad activity distribution induces extremely slow relaxation and aging effects, which can be mapped to Bouchaud’s trap model. The scaling of observables such as occupation probability and two-time correlation exhibits “simple aging” for $\gamma < 1$ and multi-scale relaxation for $\gamma > 1$ (Mata et al., 2014).

5. Dynamical Implications and Real-World Application

Dynamical processes on activity-driven networks display distinctive features not evident in time-aggregated or static models:

• Epidemic Thresholds and Spreading: The temporal organization and activity heterogeneity significantly raise the epidemic threshold compared to the time-aggregated network. Burstiness or heavy-tailed inter-event time distributions lower the threshold, facilitating outbreaks even for diseases with low transmission rates. However, burstiness can have a twofold impact, boosting prevalence near the threshold but reducing it above it, due to longer inter-activation times (Mancastroppa et al., 2019).
• Random Walks and Diffusion: Random walks in activity-driven frameworks yield steady-state occupation probabilities proportional to $a^{-1}$ for activity $a$, unlike the degree-proportional results of random walks on static networks. With non-Poissonian activations, walkers can experience effective homogenization of the network, due to the divergence of average waiting times (Moinet et al., 2019).
• Consensus and Agreement: Distributed consensus protocols converge at rates determined by the temporal network’s switching statistics and the spectrum of appropriately averaged Laplacian matrices, with tractable eigenvalue bounds in both sparse and fast-switching regimes (Ogura et al., 2018).
• Epidemic Control and Adaptation: Extensions incorporating adaptive behavior (e.g., quarantine, reduced activity, or selective contact avoidance) reveal that epidemic thresholds and outbreak sizes are highly sensitive to both the form of adaptation and its correlation structure. Notably, reducing either the activity of infected or susceptible individuals can equivalently elevate the epidemic threshold (Mancastroppa et al., 2020, Gozzi et al., 2020). In contexts where only a minority of high-activity individuals fail to adapt, disease resurgence can persist despite widespread adoption of precautionary behavior.

6. Structural Biases and Limitations of Time-Aggregated Representations

A core insight is the quantitative misrepresentation arising from the use of time-aggregated networks:

• Time-aggregated topologies artificially conflate temporally disjoint interactions, leading to overestimated connectivity, underestimated critical thresholds, and incorrect predictions for dynamical outcomes.
• Analytical treatments based on the activity-driven instantaneous network yield more accurate characterizations for processes whose timescales are comparable to network evolution or shorter (Perra et al., 2012, Starnini et al., 2013).
• The standard activity-driven model’s inability to capture empirically observed assortativity and high clustering remains a structural limitation, motivating ongoing extensions via memory mechanisms, higher-order interactions, and adaptations of the node activation or link selection rules.

7. Broader Impact and Directions for Research

Activity-driven models provide a versatile analytic and generative framework for time-varying networks, with direct application to:

• Modeling empirical dynamical networks, including social interactions, communication, and information spreading,
• Assessing and optimizing intervention strategies for epidemic containment,
• Simulation and analysis of higher-order processes in systems with group interactions, and
• Studying the interplay between memory, temporality, and emergent network structure.

Ongoing research addresses the enrichment of these models with adaptive behavior, inter-layer coupling, more realistic memory and higher-order effects, and rigorous validation against high-resolution empirical temporal data.

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References:

• "Activity driven modeling of time varying networks" (Perra et al., 2012)
• "Topological properties of a time-integrated activity driven network" (Starnini et al., 2013)
• "Memory effects induce structure in social networks with activity-driven agents" (Medus et al., 2013)
• "Temporal percolation in activity driven networks" (Starnini et al., 2013)
• "Slow relaxation dynamics and aging in random walks on activity driven temporal networks" (Mata et al., 2014)
• "Contagion processes on the static and activity driven coupling networks" (Lei et al., 2015)
• "Random walks on activity-driven networks with attractiveness" (Alessandretti et al., 2017)
• "Epidemic Spreading on Activity-Driven Networks with Attractiveness" (Pozzana et al., 2017)
• "Dynamic topologies of activity-driven temporal networks with memory" (Kim et al., 2017)
• "Distributed Agreement on Activity Driven Networks" (Ogura et al., 2018)
• "Optimal Containment of Epidemics over Temporal Activity-Driven Networks" (Ogura et al., 2018)
• "Generalized Voter-like model on activity driven networks with attractiveness" (Moinet et al., 2018)
• "Simplicial Activity Driven Model" (Petri et al., 2018)
• "Burstiness in activity-driven networks and the epidemic threshold" (Mancastroppa et al., 2019)
• "Random walks in non-Poissoinan activity driven temporal networks" (Moinet et al., 2019)
• "Active and inactive quarantine in epidemic spreading on adaptive activity-driven networks" (Mancastroppa et al., 2020)
• "Self-initiated behavioural change and disease resurgence on activity-driven networks" (Gozzi et al., 2020)
• "Activity networks determine project performance" (Vazquez et al., 2022)
• "Probabilistic activity driven model of temporal simplicial networks and its application on higher-order dynamics" (Han et al., 2022)