Complexity Science

Complexity science is the study of how large systems made up of many interacting parts give rise to emergent behaviors — patterns, structures, or dynamics that cannot be understood by analyzing the parts in isolation. It seeks to understand how order, adaptation, and organization emerge in natural, social, and technological systems — from ant colonies and ecosystems to economies, cities, and human societies. Complexity science studies systems that are more than the sum of their parts.

Why Big Animals Eat Less (Per Kilo) And Other Weird-But-True Body Rules

~ Manuel Castañón-Puga

Ever wonder why a mouse seems to snack nonstop while an elephant can take its time? West, Brown, and Enquist explain that many things in living creatures follow simple size rules. As animals grow larger, their total energy use increases, but not at the same rate as their body mass. That’s why a small pet burns way more energy per kilogram than you do. Heartbeats and breaths follow the same pattern: smaller bodies tick faster, while bigger bodies tick slower. Even though long things, such as circulating blood or growing, tend to stretch with size.

The reason sits inside our plumbing. Life runs on networks of branching tubes that reach everywhere: your arteries and airways, the veins in a leaf, even the tiny pipes in an insect. These networks spread out like a fractal tree, splitting and splitting until they reach the tiniest units that actually do the job—capillaries in us, alveoli in lungs, fine vessels in plants. According to the authors, the end units remain approximately the same size across species, and the entire network is designed to utilize as little energy as possible. From those simple ideas, the size rules just… fall out.

This helps make sense of everyday stuff. A hamster needs frequent meals because its fast-beating heart and speedy breathing burn fuel quickly, while a cow can graze slowly because its network moves resources more calmly. Blood pressure stays about the same across sizes, but total blood volume scales with body size, so bigger animals carry more fuel on board. Lungs fit the story too: as bodies grow, the number of tiny air sacs increases a lot, but each sac’s size changes only a little. The result is a huge total surface for gas exchange without wasting energy pushing air around.

Once you notice these patterns, you see them everywhere. Tree trunks and aortas expand in size in a similar manner. Tiny creatures live life in fast-forward, big ones in slow motion, because their delivery networks set the pace. You don’t need equations to use this idea: it’s enough to remember that branching networks plus smart energy use create smooth scaling rules. That’s why animals of all sizes—from shrews to whales—play by the same hidden design, just at different speeds.

Reference:
West, G. B., Brown, J. H., & Enquist, B. J. (1997). A General Model for the Origin of Allometric Scaling Laws in Biology. Science, 276(5309), 122–126. https://doi.org/10.1126/science.276.5309.122

Why There’s No “Best” Strategy (and How to Pick One That Fits You)

~ Manuel Castañón-Puga

We all love shortcuts. The perfect study hack. The ultimate workout plan. The “best” way to search for answers online. Wolpert and Macready show that this dream has a catch: there’s no single method that wins across every kind of problem. When you average over all possible situations, every strategy performs the same. If one approach excels in some tasks, it must falter in others. Even a random strategy can look just as good—on average—if you judge it across every problem out there.

So what actually works? Match the method to the kind of problem you face. The authors explain this as “alignment.” Think of it like playlists. A gym playlist pumps you up, but it’s awful for falling asleep. In the same way, an algorithm—or any plan—needs to fit the pattern of the task. If you know something about your problems (for example, your homework tends to be practice-with-small-twists, not total curveballs), build your approach around that. Without using what you know, you’re basically picking at random and hoping for luck.

Life also changes while you’re working. Projects shift, goals change, and new information arrives midway. The same “no free lunch” idea still bites in these time-varying situations: after the very first step, no fixed approach dominates across all possible ways things can change. What helps is paying attention to how your world usually shifts. If your schedule becomes busy near exams, use strategies that adapt—such as quick checkpoints and backups—rather than rigid plans that assume nothing will change.

One more practical warning: don’t overhype wins from tiny tests. The authors demonstrate that outperforming another method on a small set of examples doesn’t prove much; it only indicates that you were better in those specific cases. Instead, track results over the kinds of tasks you actually face, and compare to simple baselines. If your fancy routine isn’t clearly better than a plain, honest approach, rethink it. In short, there’s no universal champion. But by learning the shape of your own problems and choosing tactics that match that shape, you turn “no free lunch” into a recipe that works for your everyday life.

Reference:
Wolpert, D. H., & Macready, W. G. (1997). No Free Lunch Theorems for Optimization. In IEEE Transactions on Evolutionary Computation (Vol. 1, Issue 1). doi: 10.1109/4235.585893

Learn Faster with the “Natural” Gradient

~ Manuel Castañón-Puga

When you’re learning something new, you don’t just step randomly—you look for the path that gets you downhill fastest. Amari explains that many machine-learning models live on curved spaces, so the usual gradient doesn’t actually point straight “down.” The fix is the natural gradient, which adjusts each step to the true shape of the space so updates follow the steepest descent where it really matters. In simple terms, the algorithm stops slipping sideways and starts moving directly toward better settings. This idea originates from information geometry and applies to perceptrons, mixing-matrix problems such as blind source separation, and even linear dynamical systems used for deconvolution, not just toy examples.

Why care? Because using the natural gradient in online learning (updating as each new example arrives) can be as accurate, in the long run, as training with all data at once. Statistically, Amari shows this reaches “Fisher efficiency,” which means the online method eventually matches the gold-standard batch estimator instead of settling for second best. For everyday intuition, think of studying a little every day and still getting the same score as if you’d crammed with the full textbook—provided you study in the smartest direction.

This smarter direction can also dodge the annoying “plateaus” that slow standard backprop training, where progress feels stuck even though you’re doing everything “right.” By respecting the curvature of the model’s parameter space, natural-gradient steps help the learner escape these flat regions more readily, speeding up practical training of neural networks. Amari highlights this benefit while positioning the method across common tasks, from multilayer perceptrons to separating mixed signals, such as voices in a room or unmixing time-smeared audio.

There’s also a tip for tuning your learning rate without guesswork. The paper proposes an adaptive rule that makes big steps when you’re far from the goal and smaller steps as you get close, helping you converge quickly without overshooting. It’s like running on open ground but slowing near the finish line to avoid slipping past it. This adaptive schedule aligns naturally with the natural gradient, offering a practical approach that can be applied in real-world training loops.

Reference:
Amari, S. (1998). Natural Gradient Works Efficiently in Learning. Neural Computation, 10(2), 251–276. https://doi.org/10.1162/089976698300017746

How Markets Quietly Shape What We Want

~ Manuel Castañón-Puga

We often think our tastes and values are purely personal, but Bowles argues that everyday systems—like shops, apps, schools, and jobs—nudge them all the time. Markets don’t just set prices; they set the scene. Paying taxes for a service feels different from buying the same thing yourself. One frames you as a citizen with rights; the other as a customer engaging in a transaction. That frame alters how fair something appears and how generous we perceive it to be. In lab games, people offered less when the situation was described like a market “exchange” and more when it felt like “splitting a pie.” Even money itself can be a powerful simplifier. In older communities studied by Bohannan, certain items weren’t traded across categories. As money spread, more items became comparable, and that changed what felt OK to swap—and what a “good life” looked like.

Motivation shifts, too. When we do things for a reward, we often start liking the activity less. Psychologists such as Deci and Ryan have demonstrated that paying or punishing can crowd out pride, curiosity, and the sense of choice. Bowles reviews real-world hints of this: when people were offered cash to accept an unpopular facility in their town, support fell; paying for blood donation sometimes made willing donors less likely to give. The takeaway isn’t “money bad.” It’s subtler: clear quid-pro-quo deals push us to focus on the payoff, while choice and autonomy keep our inner drive alive. In your daily life, that might mean mixing paid gigs with passion projects, or keeping some hobbies reward-free so they stay fun.

Norms and reputations also depend on the setting. In tight communities or teams where you’ll meet again, being trustworthy pays off. In fast, anonymous markets, identity matters less, so it’s harder for reputations to grow—and easier to act only for yourself. But market life isn’t destiny. Simple tweaks—such as talking face-to-face, showing names, or building group identity—can increase cooperation. Consider how you buy and sell online: profiles, reviews, and repeat interactions make kindness and reliability more prevalent, as your behavior now follows you later.

Finally, we learn what to value from the people around us. Bowles demonstrates that culture spreads vertically (from parents), obliquely (through teachers and creators), and horizontally (among friends). Conformity isn’t always mindless; it can be a smart shortcut when learning is costly. That’s why “what everyone does” is so sticky. Markets can shift who we see, what gets praised, and which paths look successful—so the role models change, and so do we. For everyday life, the message is empowering: choose your frames and your crowds. Decide which activities you’ll keep intrinsic. Build circles where your future self will meet you again. Small design choices—how you pay, how you participate, who you follow—quietly train your preferences. Use them on purpose.

Reference:
Bowles, S. (1998). Endogenous Preferences: The Cultural Consequences of Markets and Other Economic Institutions. Journal of Economic Literature, 36(1), 75–111. http://www.jstor.org/stable/2564952

The “Middle Way” of Nature: Where Big Patterns Come from Small Parts

~ Manuel Castañón-Puga

We’re often told that if you know the tiny rules, you can predict everything. But that’s not how the world always works. Laughlin et al. argue that when lots of pieces come together, new patterns appear that don’t depend on every little detail. They call these “protected” behaviors—stable, reliable patterns, such as magnetism or superfluid flow — that appear repeatedly across different materials. You can’t easily compute them by tracking every atom; instead, you spot them by looking for the collective rules that emerge when many parts interact. In other words, between the world of atoms and the everyday world, there’s a “middle” zone where organization pops up on its own.

This middle zone shows up in living things and in everyday materials. Proteins don’t just sit there; they fold into useful shapes, pick out the right molecules to work with, and even team up to build structures—like a tiny construction crew that knows the plan. Glasses—think phone screens or plastic—can “age,” remember, and change slowly even when their atoms don’t look very different under x-rays. Soap-like molecules naturally form into bubbles and membranes, much like cell walls. And in some electronic materials, charges arrange themselves into stripes or domains that behave in surprising ways. All of this says the same thing: when enough pieces interact, they organize, and the result can be smarter than any single part.

How does this organization happen? One helpful picture is the “energy landscape.” Imagine a map of hills and valleys that indicates which shapes or arrangements are easier or harder for a system to reach. Some landscapes are rugged, with many similar valleys, so the system can get stuck in different long-lived states. Others are like a funnel that guides you toward one best shape—this is how many natural proteins fold quickly and reliably, because their landscapes gently steer them to the right place. Another big idea is “frustration,” which occurs when different forces want different things at the same time, such as in a group project with conflicting goals. That conflict can create patterns—such as domains, droplets, or stripes—as a compromise. Seeing problems this way helps you expect slow changes, memory effects, and multiple outcomes, rather than a single perfect answer.

Why should you care? These ideas make everyday choices easier. When a task involves many moving parts—planning an event, training a team, organizing your study schedule—don’t expect one tiny tweak to predict everything. Look for the protected patterns: routines that stay stable even when details change. Build funnels: set up habits and checklists that guide you toward the good outcome from many starting points. Expect frustration: competing goals will create trade-offs, so design simple “rules of thumb” everyone can follow. And be patient with “aging” and “memory”: systems often change slowly, keep history, and need time to reorganize. Scientists are developing new tools to see this middle world more clearly—like advanced microscopes and smart scattering methods—because understanding these rules won’t just explain cells and materials; it could help us design artificial systems that adapt and learn. That mindset can help you build projects, communities, and even personal routines that work with complexity instead of fighting it.

Reference:
Laughlin, R. B., Pines, D., Schmalian, J., Stojkovic, B. P., & Wolynes, P. (1999). The middle way. Proceedings of the National Academy of Sciences (PNAS), 97(1), 32–37. https://doi.org/10.1073/pnas.97.1.32