Source-linked AI summary
Scaling laws of human interaction activity
Diego Rybski, Sergey V. Buldyrev, Shlomo Havlin, Fredrik Liljeros, Hernan A. Makse
TL;DR
The paper addresses limited understanding of the statistical laws underlying human communication activity. It analyzes message dynamics in two Internet communities and reports a generalized Gibrat-like scaling relation associated with long-term correlated activity, with implications for communication-resource allocation and related social systems.
Problem
The paper asks how statistical laws underlying human communication activity arise and how they relate to broader growth patterns.
Method
The authors analyze message-sending dynamics in two Internet communities and develop a mathematical framework relating growth behavior to long-term correlations.
Results
The study identifies long-term persistence in message-sending activity and relates it mathematically to a generalized Gibrat-like growth law.
Takeaways & Limitations
The findings may support estimating future message probabilities by activity level and improving communication-resource allocation.
Takeaways & Limitations
The paper flags the assumption that the studied human activity can be treated as a large number of independent random events.
Abstract
from arXiv · showhide
Even though people in our contemporary, technological society are depending on communication, our understanding of the underlying laws of human communicational behavior continues to be poorly understood. Here we investigate the communication patterns in two social Internet communities in search of statistical laws in human interaction activity. This research reveals that human communication networks dynamically follow scaling laws that may also explain the observed trends in economic growth. Specifically, we identify a generalized version of Gibrat's law of social activity expressed as a scaling law between the fluctuations in the number of messages sent by members and their level of activity. Gibrat's law has been essential in understanding economic growth patterns, yet without an underlying general principle for its origin. We attribute this scaling law to long-term correlation patterns in human activity, which surprisingly span from days to the entire period of the available data of more than one year. Further, we provide a mathematical framework that relates the generalized version of Gibrat's law to the long-term correlated dynamics, which suggests that the same underlying mechanism could be the source of Gibrat's law in economics, ranging from large firms, research and development expenditures, gross domestic product of countries, to city population growth. These findings are also of importance for designing communication networks and for the understanding of the dynamics of social systems in which communication plays a role, such as economic markets and political systems.
I. INTRODUCTION
The paper investigates whether collective social activity follows emergent statistical laws, focusing on communication patterns in two Internet communities. Using message timing and sender–receiver identifiers, it examines activity dynamics while preserving anonymity.
- The study asks whether unforeseen outcomes of social activity follow emergent statistical laws, whose origins remain an open question.
- The authors analyze message-sending dynamics in two growing Internet communities to search for statistical laws of human communication activity.
- OC1 includes over 80,000 members and more than 12.5 million messages collected over 63 days, while OC2 covers 492 days, more than 500,000 messages, and almost 30,000 members.
- The anonymous data contain only message times and sender and receiver identification numbers, not message content.
- Message sending is treated as an intentional social action whose collective emergent properties are unintended.
- Members show larger temporal activity fluctuations than shuffled messages, with one OC1 member sending many messages early and far fewer late in data acquisition.
II. RESULTS
Across two online communities, message activity follows a generalized Gibrat-like scaling law: average growth is nearly activity-independent, while growth fluctuations decay as a power law. Long-term temporal correlations distinguish the observed dynamics from randomized activity and connect fluctuation scaling with growth across social networks and other systems.
- Growth in message activity: The conditional average message-growth rate is fairly independent of members’ initial activity, whereas its standard deviation decreases as a power law.The analysis conditions growth on the initial number of messages m0 and compares both online communities.
- Growth in message activity: βOC1 = 0.22 ± 0.01 and βOC2 = 0.17 ± 0.03, with similar exponents despite different member populations.The fitted values deviate slightly at large m0 because of low statistics.
- Growth in message activity: The approximate agreement with growth exponents reported for firms, countries, universities, scientific output, and cities generalizes Gibrat’s law across social and human systems.The paper suggests that mechanisms behind growth properties in different systems may originate in human activity.
- Temporal correlations: In both communities, shuffled records give Hrnd = 1/2, whereas H > 1/2 indicates bursts persisting from days to years and a self-similar activity distribution.The randomized result confirms that the observed correlations arise from temporal structure.
- Growth–correlation relation: Equation (5), β = 1 − H, formalizes the link between growth fluctuations and long-term correlations and agrees approximately with direct measurements.The predicted values are HOC1 ≈ 0.78 versus measured H = 0.75 ± 0.05, and HOC2 ≈ 0.83 versus measured H = 0.88 ± 0.03.
III. DISCUSSION
The discussion interprets persistent communication activity as a possible source of the observed scaling relationship, while leaving its precise origin unresolved. It also considers implications for prediction and modeling in communication-based systems.
- Open questions: The origin of the persistence pattern remains unresolved and is identified as a focus for future research.The paper states that careful statistical analysis is needed to determine whether the behavior is Levy-like or reflects pure memory.
- Origin of persistence: Two possible origins are considered: Levy-type inter-event intervals or pure long-term memory in activity.Distinguishing these scenarios requires analyzing both inter-event intervals and their correlations.
- Origin of persistence: Long-term correlations may explain the scaling relationship between members’ activity and its fluctuations.The discussion links this persistence to possible mechanisms involving heavy-tailed inter-message intervals or correlated activity.
- Implications: Persistent interactions may support prediction of future activity, including the probability that members at a given activity level exceed a message threshold.The authors connect this possibility to improved resource allocation in communication-based systems.
- Implications: Mean-field models based on independent Poisson events may produce faulty predictions for this type of human activity.The discussion instead relates the observed dynamics to an underlying long-term correlated process and individual states driven by stimuli.
- Broader connections: The persistence found in individual messaging activity could underlie long-term persistence in aggregated records such as traffic and stock markets.The paper presents this as a possible connection rather than an established causal explanation.
Calculations of ⟨r(m0)⟩, σ(m0) and optimal times t0 and t1
This section defines how growth rates and their fluctuations are calculated from members’ message counts and explains the observation-window choices. The analysis uses the full record while selecting a midpoint start time to balance usable activity and follow-up duration.
- Calculation setup: The growth-rate calculation uses initial and final message counts, with t1 chosen at the end of the observation period.Using t1 = T maximizes the available data for estimating growth.
- Calculation setup: Choosing t0 too early excludes members with zero initial activity, whereas choosing it too late leaves insufficient time to observe growth.The window therefore requires a balance between retaining members and allowing follow-up.
- Optimal times: t0 = T/2 is the selected optimal start time in both online communities.The criterion uses members active at t0 who show subsequent activity through t1 = T.
- Data representation: The raw data records each message’s time, sender identifier, and receiver identifier.These entries provide the event-level basis for constructing members’ activity records.
- Surrogate construction: The randomized surrogate preserves the total message count and message times while shuffling their temporal associations.Randomly swapping event times destroys temporal correlations without changing the set of message instants.
SUPPORTING INFORMATION (SI)
The supporting information accompanies the paper on scaling laws of human interaction activity and lists its authors.
- The supporting information belongs to “Scaling laws of human interaction activity.”
- The paper lists Diego Rybski, Sergey V. Buldyrev, and Shlomo Havlin among its authors.
- The author list also includes Fredrik Liljeros and Hernán A. Makse.
I. NOTATION
The notation defines message-event times, cumulative and interval counts, growth rates, and analogous degree-growth quantities. It also specifies how activity levels and observation times are selected.
- Each member j sends M_j messages at event times t_j(n), with n ranging from 1 to M_j.
- The message-count sequence records the number of messages in non-overlapping periods δt, with daily resolution given by δt = 1 day.
- The cumulative count records how many messages a member has sent up to time t, and its normalized cumulative sum defines a random-walk displacement.
- Members are grouped by total activity M, while t0 is varied and t1 fixed at the observation-period endpoint to maximize eligible growth-rate cases.Eligibility requires at least one message by t0 and at least one new message between t0 and t1.
- Message and degree growth use logarithmic quantities r = ln m1 and r_k = ln k1, with corresponding preferential-attachment growth quantities.
II. OPTIMAL TIMES t0 AND t1
The analysis selects observation times for calculating growth rates in the two online communities. Figure 5 reports these optimal times separately for OC1 and OC2.
- Figure 5 displays the optimal times t0 and t1 used to calculate growth rates for OC1 and OC2.
USING DETRENDED FLUCTUATION ANALYSIS
Detrended Fluctuation Analysis quantifies long-term correlations in message activity while removing local trends. The procedure estimates fluctuation and correlation exponents across member activity levels.
- DFA quantifies long-term correlations in a record µ(t) through its cumulative profile, segmentation, polynomial detrending, segment variances, and fluctuation function.
- Long-term correlations produce power-law fluctuation growth F(∆t), allowing estimation of the fluctuation exponent H and correlation exponent γ.
- Uncorrelated dynamics correspond to H_rnd = 1/2, whereas long-term correlations imply 1/2 < H < 1 and 0 < γ < 1.
- Because individual fluctuation functions are noisy, members are grouped into logarithmic bins by total message activity M before averaging.The bins include activity ranges such as 1–2, 3–7, and 8–20 messages.
- The resulting fluctuation exponents are compared across activity groups, with error bars obtained from standard deviations across subdivisions of each group.
IV. GROWTH IN THE DEGREE
The preferential-attachment analysis examines degree-growth rates as functions of initial degree and compares the model’s fluctuations with theoretical expectations. Under uncorrelated increments, the model yields the standard exponent β_PA = 1/2.
- The preferential-attachment model is analyzed through average growth rates and growth-rate fluctuations conditional on initial degree k0.
- The model has a constant average growth rate and a standard deviation that decreases as a power law with exponent β_PA = 1/2.
- In the standard preferential-attachment model, node degree increases according to a growth law with dynamics exponent b = 1/2.
- β_PA = 1/2 is obtained for uncorrelated increments, matching the numerical result.
- The model assumes new links follow a Poisson process, making link-arrival intervals exponentially distributed.
- Fitness extensions introduce heterogeneous node attractiveness, while random fluctuations are characterized by the exponent β.