"Riku is unpredictable, right?"
Yuki murmured.
"What? Is that a compliment?" Riku reacted.
"I don't know if it's a compliment, but your behavior is random."
Aoi showed interest. "A random walk."
"Random walk?"
"A process where each step proceeds in a random direction," Aoi explained. "Cannot predict where it goes next."
Riku laughed. "That's my life itself."
Yuki opened her notebook. "But there are types of randomness, right?"
"Sharp," Aoi nodded. "True random and patterned random."
"How are they different?"
"True random means past information doesn't help predict the future."
"But human behavior?"
"Has patterns," Aoi answered. "Not completely random. Has biases and tendencies."
Riku thought. "True, I do have preferences. Not completely whimsical."
"That's interesting," Aoi said. "Looks unpredictable, but statistically has regularity."
Yuki asked. "Does random walk arrive somewhere?"
"Depends on dimension," Aoi drew on the whiteboard. "In one and two dimensions, eventually returns to starting point. Three dimensions or more, doesn't return."
"Interesting property."
Riku asked another question. "Is friendship also a random walk?"
Aoi pondered. "Metaphorically, maybe."
"What do you mean?" Yuki was curious.
"Friendship is accumulation of unpredictable moments. But long-term, has direction."
Riku nodded. "Daily conversations are unpredictable. But gradually we get closer."
"That's drift," Aoi explained. "A slight tendency is added to random walk."
"Slight tendency?"
"Biased random walk. Not completely random, slightly skewed in a direction."
Yuki understood. "Friendship is a random walk biased toward intimacy."
"Beautiful expression," Aoi acknowledged.
Riku laughed. "But sometimes we move in reverse direction."
"Like arguments."
"Yes. But on average, we move forward. That's friendship's nature."
Aoi supplemented. "What matters is not individual steps, but overall tendency."
"Everything isn't decided by one conversation."
Yuki offered another perspective. "Then information-theoretically, how do we measure friendship?"
Aoi thought. "Mutual information, perhaps."
"Mutual information?"
"Amount of information two people share. Common experiences, knowledge, emotions."
"As it increases, friendship deepens," Riku understood.
"Yes. And with high mutual information, communication becomes more efficient."
Yuki wrote in her notebook. "Friendship = mutual information increase + biased random walk"
"Perfect," Aoi smiled.
Riku looked at the window. "But completely predictable friendship is boring."
"Agree," Aoi answered. "Randomness keeps relationships fresh."
"Interesting because there's surprise."
Yuki asked. "But if uncertainty is too high, doesn't it become unstable?"
"Good point," Aoi acknowledged. "Optimal is balance between predictability and surprise."
"Stability and freshness."
Riku laughed. "Could do relationship counseling with information theory."
"Maybe," Aoi answered seriously. "Actually, human relationships are information exchange."
Yuki summarized. "Random walk friendship is:
- Individual moments unpredictable
- Long-term has direction
- Mutual information increases
- Moderate surprise necessary"
Aoi nodded. "Mathematical model of human relationships. Imperfect, but gains insights."
Riku stood up. "Then let's go unpredictably today too."
"That's so like you," Yuki laughed.
Their friendship of three is surely also a random walk. But certainly deepening.
That was a statistical truth.