The distribution of violent event and interevent times in conflicts

Despite a mathematical theorem predicting that fine-grained data should reveal lognormal distributions for violent event intervals, empirical analysis shows that the commonly observed power law distribution remains valid, indicating that violent events are energy-intensive processes.

The Gist

The Big Question: How Long Do We Wait Between Fights?

Imagine you are watching a chaotic street fight. It's not a non-stop brawl; it's a series of punches, kicks, and shoves separated by moments of breathing, backing away, or stumbling.

Scientists have long studied these "pauses" (called interevent times) in big wars and conflicts. They found that the time between fights follows a specific pattern called a Power Law. Think of a Power Law like a "long tail" on a graph: most pauses are short, but there are a few massive pauses that go on for a very long time. This pattern suggests that violence is somewhat unpredictable and "scale-free."

However, there was a problem with this old data. It was "coarse-grained," meaning researchers only looked at data in chunks of one day. If a fight stopped for 23 hours and started again 1 hour later, the data just said "1 day." It was too blurry to see the tiny details.

The New Idea: Zooming In with a Microscope

Jeroen Bruggeman, the author of this paper, decided to zoom in. Instead of looking at wars over days, he looked at street fights recorded on mobile phones, analyzing them down to the second (or even fractions of a second).

He had a hunch based on a mathematical theorem:

• If you look at violence in tiny, fine-grained details, the pauses shouldn't be a Power Law.
• Instead, they should be Lognormal.

The Analogy: The Multiplicative Chain
Imagine you are trying to time how long it takes to get ready to throw the next punch.

1. You have a baseline time (say, 2 seconds).
2. To get ready for the next punch, you have to solve a new problem: dodge a chair, catch your breath, or wait for a friend to move.
3. Each of these problems multiplies your time. Maybe the chair adds 1.5x the time. Maybe catching your breath adds 2x.

If you keep multiplying these random factors together over and over, math says the result creates a Lognormal distribution. It's like a chain reaction of small delays.

What Did the Data Show?

Bruggeman analyzed 59 video clips of small group fights. He looked at two things:

1. The Pauses (Interevent Times)

• The Expectation: He thought the pauses would be Lognormal because they are made of many small, multiplying delays.
• The Reality: Surprisingly, the data looked equally good for both the Power Law and the Lognormal.
• Why? The author suggests that the "Power Law" might still be hiding there. Why? Because bystanders filming fights get bored. If a fight stops for 10 minutes, the person filming might put the phone down. This means the "long pauses" are missing from the data. If you miss the long tail, the data looks like a Power Law even if it's actually something else.

2. The Fights Themselves (Event Times)

• The Expectation: Bruggeman argued that actual fighting is different from waiting. Fighting is exhausting. You can't punch for 10 minutes straight; you run out of energy.
• The Reality: The data showed that the duration of the actual violence fits the Lognormal pattern perfectly.
• The Metaphor: Think of a sprinter. They can run fast for a short burst, but they can't keep it up forever. The distribution of how long they sprint before collapsing follows a Lognormal curve. Violence is the same: it burns energy, so it has a natural "short tail." It doesn't have the "infinite long tail" of a Power Law.

The Bottom Line

1. Common Wisdom Holds Up (Sort of): The idea that violence follows a Power Law isn't completely wrong, but it might be an illusion caused by our inability to film long pauses.
2. Violence is Tiring: The actual act of fighting is clearly Lognormal. It's a burst of energy that runs out quickly, rather than an endless, scale-free process.
3. The Takeaway: When we look at violence with high-definition, second-by-second precision, we see that the pauses are complex and messy (hard to categorize), but the fighting itself is a finite, energy-draining event that follows a predictable, bell-curve-like pattern (Lognormal).

In short: Waiting for the next punch is a chaotic game of chance that looks like a Power Law (or maybe isn't). But the punch itself? That's just a burst of energy that follows a Lognormal curve because nobody can fight forever.

Technical Summary

1. Problem Statement

The paper addresses a discrepancy in the statistical modeling of violent conflicts:

• Existing Consensus: Previous studies on enduring violent conflicts (e.g., warfare, terrorism) have established that interevent times (the time intervals between violent acts) follow a power law distribution. However, these findings are based on coarse-grained data with a temporal resolution of at least one day.
• Theoretical Contradiction: A mathematical theorem by Sornette and Cont suggests that if interevent times can be arbitrarily close to zero (fine-grained data), the distribution should theoretically be lognormal, not power law, provided the process is multiplicative.
• Research Gap: There is a lack of empirical evidence using fine-grained data (resolution of seconds or less) to test whether the power law holds at high temporal resolutions or if it reverts to a lognormal distribution as predicted by theory.

2. Methodology

The author employs a novel approach combining theoretical modeling with high-resolution empirical data:

• Theoretical Framework:

  • Conflict is modeled as a multiplicative process.
  • Let $\tau_1$ be the first interevent time. Subsequent times are defined as $\tau_{t+1} = v_t \cdot v_{t-1} \dots v_1 \cdot \tau_1$, where $v$ represents random, independent variables with finite mean and variance.
  • According to the Central Limit Theorem applied to the logarithm of these variables ($\log(\tau)$), the distribution should converge to a lognormal distribution.
  • The author posits that if the minimum time ($\tau_{min}$) is not arbitrarily close to zero (as in daily data), the distribution appears as a power law. If $\tau_{min} \to 0$ (fine-grained), it should be lognormal.

• Data Collection:

  • Source: 59 video recordings of street fights involving small groups ($2 \le n < 10$; mean $n=3.6$) collected from public platforms (LiveLeak, YouTube, WorldStarHipHop).
  • Resolution: High temporal resolution (1 second or shorter).
  • Sample Size:
    • Interevent times: 287 intervals (ranging from 0.002 to 95 seconds).
    • Event times: 375 durations (ranging from 0.342 to 36 seconds).
  • Coding: Violence was defined as force against a body or forceful movement of a body. "Interevent times" are periods where no violence occurs; "event times" are the duration of the violent act itself.

• Statistical Analysis:

  • Used Maximum Likelihood Estimation (MLE) via R's poweRlaw package.
  • Fitted both lognormal and power law distributions to the data.
  • Applied Vuong's test to compare the goodness-of-fit between the two models.

3. Key Results

The study yielded distinct results for interevent times versus event times:

• Interevent Times (Intervals between violence):

  • Finding: The data did not support the hypothesis that fine-grained interevent times are lognormally distributed.
  • Statistical Evidence: Both the lognormal and power law models fit the data approximately equally well.
  • Vuong's Test: Two-sided $p = 0.403$; one-sided test for lognormal superiority $p = 0.201$.
  • Conclusion: The "common wisdom" that interevent times follow a power law was not refuted, even at second-level resolution. The author notes a potential sampling bias: bystanders may stop filming during long lulls, leading to undersampling of the distribution's tail, which could artificially favor power law fits.

• Event Times (Duration of violence):

  • Finding: Violent events are clearly lognormally distributed.
  • Statistical Evidence: The lognormal fit was significantly better than the power law fit ($p = 0.00077$).
  • Theoretical Alignment: This aligns with the conjecture that violent events consume significant energy and are therefore shorter than interevent times, resulting in a distribution with a short tail.
  • Note on Artifacts: A "jump" in the cumulative distribution of event times was observed due to coding limitations (single punches/kicks were often coded as exactly 1 second). However, the author argues this does not affect the Maximum Likelihood Estimates, as distributions are fitted to the tails rather than the shortest intervals.

4. Key Contributions

• Novel Data Resolution: Provides one of the first empirical tests of conflict dynamics using fine-grained data (seconds) rather than the standard coarse-grained data (days).
• Multiplicative Process Representation: Introduces a novel representation of conflict data as a multiplicative process, linking the theoretical work of Sornette and Cont directly to empirical conflict data.
• Differentiation of Metrics: Distinguishes between the statistical nature of interevent times (intervals) and event times (duration), showing they follow different distributions (Power Law vs. Lognormal) even within the same dataset.
• Validation of Theory (Partial): Confirms the lognormal distribution for event times but challenges the assumption that fine-grained interevent times automatically shift from power law to lognormal.

5. Significance and Implications

• Challenging Common Wisdom: The paper suggests that the power law distribution of interevent times is robust, persisting even when data resolution increases from days to seconds. This implies that the underlying mechanisms of conflict generation may be more complex than simple multiplicative processes with a lower bound approaching zero.
• Methodological Caution: Highlights the importance of data resolution and sampling bias. The potential for "undersampling" long pauses in video data (due to bystander behavior) suggests that observed distributions might be artifacts of data collection rather than pure physical laws.
• Energy Dynamics: The confirmation of lognormal distribution for event times supports the physical intuition that violence is an energy-intensive process with natural limits on duration, distinct from the waiting times between events.
• Future Research: As high-resolution data (CCTV, mobile phones) becomes more abundant, this study sets a precedent for re-evaluating established statistical laws in social physics and conflict studies.

In summary, while the study confirms that violent event durations are lognormally distributed, it fails to refute the power law distribution of interevent times, suggesting that the transition from power law to lognormal in multiplicative processes may depend on factors beyond just temporal resolution, such as sampling bias or the specific nature of the "minimum time" constraint in human conflict.