Source-linked AI summary
Time varying networks and the weakness of strong ties
Márton Karsai, Nicola Perra, Alessandro Vespignani
TL;DR
Time-varying social networks couple changing connectivity with ongoing information processes, challenging models based on aggregated interactions. The paper analyzes mobile-call data, characterizes egocentric-network evolution, and encodes memory in a reinforcing network model. Across synthetic and real-world rumor-spreading processes, strong ties constrain diffusion within localized groups and weaken information spreading.
Problem
Aggregated network representations neglect time-varying connectivity, limiting understanding of how dynamic processes unfold on evolving social networks.
Method
The paper analyzes a large-scale mobile-call dataset, characterizes egocentric-network evolution, and introduces a non-Markovian reinforcement model with memory for time-varying networks.
Results
Strong ties constrain rumor diffusion within localized groups of individuals and weaken the spreading of information across networks.
Takeaways & Limitations
Recurrent communication through strong ties can act as a bottleneck controlling the global extent of rumor spreading.
Abstract
from arXiv · showhide
In most social and information systems the activity of agents generates rapidly evolving time-varying networks. The temporal variation in networks' connectivity patterns and the ongoing dynamic processes are usually coupled in ways that still challenge our mathematical or computational modelling. Here we analyse a mobile call dataset and find a simple statistical law that characterize the temporal evolution of users' egocentric networks. We encode this observation in a reinforcement process defining a time-varying network model that exhibits the emergence of strong and weak ties. We study the effect of time-varying and heterogeneous interactions on the classic rumour spreading model in both synthetic, and real-world networks. We observe that strong ties severely inhibit information diffusion by confining the spreading process among agents with recurrent communication patterns. This provides the counterintuitive evidence that strong ties may have a negative role in the spreading of information across networks.
Results
The mobile-call data reveal temporal and egocentric-network patterns that aggregated representations miss. A memory-reinforced time-varying model reproduces key empirical heterogeneities, while recurrent strong ties slow and localize rumor spreading compared with memoryless dynamics.
- Empirical network structure: Aggregated network measures show heavy-tailed degree and link-weight distributions, while activity is also heavy-tailed and time invariant across aggregation windows.Degree counts contacted individuals; link weight counts calls between connected pairs.
- Empirical network structure: Snapshots from different hours reveal dynamical interaction patterns that are invisible in the time-aggregated network representation.The aggregated network cannot identify the processes driving network dynamics.
- Memory-reinforced model: A non-Markovian reinforcement process reproduces empirical data and generates more skewed degree distributions and heterogeneous weights than the memoryless model.The memoryless model produces exponential weight distributions far from the empirical data, whereas the reinforced model matches real heterogeneity.
- Egocentric dynamics: The probability of forming a new tie decreases with egocentric network size, and rescaling collapses degree-group data onto one curve.This suggests a common mechanism drives egonet evolution independently of final connection count.
- Rumor spreading: The reinforced process makes the largest connected component grow more slowly and causes spreading processes to evolve more slowly than in memoryless networks.For epidemic spreading, memory shifts the threshold upward and reduces the final number of infected nodes.
- Rumor spreading: Strong ties constrain rumor diffusion within localized groups: one simulation reached 6 nodes beyond the seed in the reinforced network versus 92 in the memoryless network.The effect is linked to pair annihilation on strong ties and can produce up to ∼45% relative difference in the population reached.
- Rumor spreading: Time-varying networks spread rumors much more slowly than time-aggregated networks, with the time to reach a consistent fraction differing by four orders of magnitude.Static representations can introduce strong biases when process and network-evolution timescales are comparable.
- Real time-varying network: In the real mobile-call sequence, fewer than 40% of the network became aware, whereas the null model ended with everyone as a stifler.Repeated interactions produce local spreading, while removing memory and repetition eliminates the initial survival effect.
Discussion
The study characterizes memory in mobile-call interactions, builds a time-varying network model, and finds that strong ties constrain rumor diffusion within localized groups. It also emphasizes that dynamical-process behavior depends on the particular processes and networks considered.
- The authors empirically characterize memory effects in the microscopic evolution of social interactions.
- They define a generative time-varying network model with memory that reproduces degree and weight heterogeneities and produces strong and weak ties.
- Strong ties constrain rumor diffusion within localized groups of individuals, weakening information spreading across the network.
- No one-size-fits-all picture exists: classifying dynamical-process behavior requires analyzing each process and network separately.
- The framework can be extended to node-node correlations, heterogeneous dynamics, and bursty node behavior.
- Memory plays a determinant role in how connectivity patterns evolve and slows dynamical processes on time-varying networks.
Methods
The methods use a large timestamped mobile-call dataset filtered for mutual interactions, then run rumor-spreading simulations over the recorded temporal sequence. Simulations reuse the sequence cyclically after reaching its endpoint and do not reuse events within a run.
- The dataset contains 633,986,311 timestamped mobile-phone call events.
- The data come from a single operator with 20% market share in an undisclosed European country.
- The analysis retains interactions between users with at least one pair of mutual interactions to focus on social communication.
- Data-driven rumor spreading starts from a randomly selected call event of a randomly selected user and runs for the recorded period.
- After reaching the last event, simulations continue from the first event using a periodic temporal boundary condition, without reusing an event within a run.
Supplementary Materials
The supplementary material is titled “Time varying networks and the weakness of strong ties” and lists M. Karsai, N. Perra, and A. Vespignani.
- The supplementary material is titled “Time varying networks and the weakness of strong ties.”
- The listed authors are M. Karsai, N. Perra, and A. Vespignani.
1 Measures of egocentric network evolutions by directed communications
Directed and undirected communications show similar egocentric-network patterns. Larger personal networks are associated with greater recurrence toward existing neighbors, and the measured probabilities fit the paper’s proposed functional form.
- The supplementary analysis repeats measurements separately for directed outgoing communication sequences.
- Directed characteristic distributions of degree P(k) and activity P(a) are very similar to the undirected case.
- Events on existing links increase edge weight and activity but do not increase the ego’s degree.
- The conditional probability p(n) measures whether an individual’s next event targets one of n existing neighbors or a new person.
- The analysis computes p(n) across degree groups while restricting n to a common range so probabilities use the same number of users.
- Directed and undirected communication sequences exhibit similar behavior and can be fitted with functions of the form given in Eq.1.
- As the observed personal social network becomes larger, the probability of calling or receiving a call from an existing neighbor increases.
2 Degree evolution of reinforced activity driven networks
The reinforced activity-driven model captures heterogeneous network evolution through repeated interactions, producing distinct degree–strength relations and tunable structural heterogeneity. Reinforcement changes how degree and edge weights emerge compared with memoryless dynamics.
- Degree evolution: Memoryless networks reproduce the activity distribution in their integrated degree distribution, with γ = ν = 2.8 in simulations.Increasing integration time produces finite-size effects as networks approach full connectivity and small-degree nodes disappear.
- Model formulation: The reinforced activity-driven process models egocentric network evolution through reinforced interactions and a probability p(n) of creating new or repeating an existing connection.The parameter c controls this choice, while the model fixes c = 1 for every agent in the presented calculations.
- Degree–strength correlations: In memoryless networks, node strength and degree exhibit a linear relation across activity-exponent values.The measured correlation in Fig. 11 confirms the expected correspondence between activity, strength, and degree.
- Degree–strength correlations: In reinforced networks, the strength–degree relation is characterized by s ∼ k^2 independently of the γ exponent.The correlation is more dispersed than in memoryless networks, but the quadratic dependence remains apparent across exponent values.
3 Spreading rate dependencies
The spreading simulations examine how relative infection and recovery rates affect the equilibrium contagious level. Curves are generally similar across infection rates, with larger discrepancies when spreading is slow and the observation window is finite.
- Rate dependence: Only the relative values of λ and α matter for the equilibrium contagious level in the tested spreading processes.Curves for λ = 1.0, 0.8, 0.6, and 0.4 are very similar after averaging 1000 realizations.
- Rate dependence: The largest discrepancy occurs for λ = 0.2 at small α values because the rumour spreads slowly within the finite simulation window.At equilibrium, the corresponding processes can reach the same contagious level despite this transient difference.
- Simulation design: The simulations vary λ across 1.0, 0.8, 0.6, 0.4, and 0.2 while sweeping α/λ from 0 to 1.Results were averaged over 1000 surviving simulations with N = 10,000, m = 1, ϵ = 0.001, and T = 50,000.
4 Surviving probability
Reinforced temporal interactions can cause rumour spreading to terminate early or remain local, whereas memoryless dynamics remove the repetitive-interaction effect and allow global spreading. The model reproduces the corresponding pattern seen in data-driven simulations.
- Reinforced dynamics: Reinforced dynamics produce a rapidly decreasing surviving probability, indicating that rumours may die shortly after initiation and spread only locally.This effect is attributed to repetitive interactions and memory during the early spreading stage.
- Survival and global spreading: After surviving the initial stage, a rumour spreads globally and reaches a considerable fraction of the network.The surviving probability measures whether spreaders remain in the system at time t.
- Memoryless dynamics: Memoryless dynamics eliminate the repetitive-interaction effect in the initial regime, and the rumour spreads globally.The same qualitative behavior appears when event sequences are shuffled.
- Model–data comparison: The qualitative match between the reinforced model and data-driven simulations supports its representation of memory and reinforcement in spreading.The comparison is made using surviving-probability curves for synthetic and empirical temporal networks.
5 Effect of memory on spreading processes
Memory affects spreading through both static structural connectivity and temporal interaction ordering. Temporal contacts slow infection dramatically, while repeated interactions in reinforced dynamics slow it further.
- Structural growth: Memory also slows the growth of the largest connected component in RP networks, implying slower dynamics on time-varying networks with memory.The RP largest connected component grows considerably more slowly than in ML networks.
- Experimental setup: The study uses susceptible-infected processes on evolving ML and RP networks and on their static integrated structures.The SI process is treated as an extreme case of SIS, SIR, and rumour-spreading models under the chosen parameters.
- Static networks: On static integrated networks, infection reaches every node in a few iterations, with static RP spreading slightly slower because its connectivity is sparse.The static comparison isolates structural effects before temporal co-evolution is introduced.
- Temporal versus static networks: Temporal connectivity slows infection by three orders of magnitude compared with static integrated networks.In the temporal setting, infection can spread only through connections occurring during the evolving process.
- Reinforcement effects: In temporal reinforced dynamics, repeated interactions further reduce spreading speed and require approximately two times more iterations to infect every node.This comparison is made against the memoryless activity-driven temporal model.