The rich landscape of iterated simplicity

The title of the thesis, over a landscape in the style of its cover. It is the result of a five-year joint PhD at BionamiX (Faculty of Bioscience Engineering, Ghent University, under Jan M. Baetens) and the Scientific Computing Group (São Carlos Institute of Physics, University of São Paulo, under Odemir M. Bruno). More on the research page and on ORCID.

The picture is not explained yet: within ten minutes the room will know what it is, and it comes back on the last slide.

The talk starts cold, with a small experiment in participatory science: nothing is explained yet. Everyone picks two people near them and keeps them for the whole game. Each round you look only at their hands: if exactly one of them has a hand up, yours goes up; if none or both do, yours goes down. Your own hand does not count, and neither does which of the two is which: only how many are up.

Everybody first looks and decides, and only moves on the signal. That keeps the whole room in step.

The start is random, but in a way everyone can check for themselves: hands up if you were born between January and June.

Two practice rounds, then back to the start and five real rounds. Each click starts a three-second countdown, which the room says out loud: three, two, one, switch. The rule stays on screen the whole time. Nothing on screen shows the pattern the room makes: the room itself is the computer.

That was the parity rule: your hand goes up when an odd number of your neighbours have theirs up. This is what a row of cells following it does, drawn round by round: space runs to the right, time runs down, each round one row lower. From a random start, like tonight’s, it makes a jumble of triangles; from a single raised hand in the middle it draws a Sierpiński triangle. The reset button draws a new random start.

Nobody saw the whole pattern; everyone only followed a simple, local rule. That is a cellular automaton, and that was the game: from here on, the cells are squares on a screen.

The parity rule is one of many. A cell looks at itself and its two neighbours, which makes eight possible situations; a rule says, for each of them, whether the cell is on or off next. Read those eight answers as a binary number and you get the rule’s name: 01011010 is 90, so the parity rule is rule 90. Eight yes-or-no answers make 256 rules in all, the elementary cellular automata. Click an answer in the table to flip it and see which rule you get.

Four of them, started one by one from the same random row: rule 168 dies out, rule 1 repeats itself, blinking on and off, rule 30 looks random, and rule 110 builds structures that move and collide. Same recipe, very different outcomes. Each reset draws a new random row.

Moving to two spatial dimensions: John Conway’s Game of Life, from 1970. Every cell now has eight neighbours, the Moore neighbourhood, and counts how many of them are alive. A dead cell with exactly three live neighbours is born; a live cell with two or three survives; with fewer it dies of loneliness, with more of overpopulation.

From a random soup the grid churns for a while and then calms down into small blocks, blinkers and the odd traveller; a reset draws a new soup. These four rules are enough to build machines: this is Bill Gosper’s glider gun, which fires a glider every thirty steps, forever, into an eater that swallows each one.

Some patterns are left exactly as they are by the rule: every live cell has two or three live neighbours, and no dead cell has exactly three. These are still lifes. The five here are tiles of the mosaics of the cover, of five sizes (levels 2 to 6).

Fit many of them together and you can make a picture in which every cell obeys the Game of Life and nothing ever changes. The cover of the thesis is made this way. The colours are only decoration: drain them away and every cell is simply alive or dead, and still nothing changes. Take away one cell and the picture starts to unravel. A reset packs a new landscape of tiles; on this page you can also click or drag on the picture to disturb it yourself.

Just as rule 90 is one of 256 elementary rules, the Game of Life is one of many Life-like rules: a dead cell is born with some numbers of live neighbours (B), a live cell survives with others (S). The Game of Life is B3/S23; there are $2^{18} = 262\,144$ such rules, and again they behave very differently.

Three of them, each started with its own button: Maze (B3/S12345) freezes into corridors within a few steps, the Game of Life keeps something going that is hard to name, and Replicator (B1357/S1357), the parity rule on eight neighbours, looks like noise. Each reset draws a new random start.

What if the room played the game differently? First, from here on the people become nodes, and a hand up becomes a node that is on.

In the game you looked at your two neighbours, so the room was really one long row: link everybody to the person before and after them, snaking up from the back of the room to the front and round again, and everyone has two neighbours. That is a one-dimensional cellular automaton, an elementary one, like the rows of cells we just saw.

Now link everyone to the eight people around them: a grid, like the one the Game of Life and the other Life-like cellular automata live on. The room wraps round at its edges, so the people at a wall have eight too; the stubs at the edges are links to the far side.

Then take some of those links and swing them across the room to anyone at all. A few long links like these make a small-world network: everyone is suddenly only a few steps from everyone else, and running the same Life-like rule for everyone on it gives a Life-like network automaton. On this page, drag the slider: at 0 the grid comes back, at 100% the links are completely random.

But now some have four contacts and some thirteen, so a rule like “exactly three neighbours” no longer makes sense for everybody.

A node with ten neighbours and a node with three cannot follow a rule that counts neighbours. So count the share instead: the fraction of a node’s neighbours that are on, which runs round each ring from 0 at the top, clockwise, to 1, from light to dark. Cut it into intervals, five here. A node that is off (left) is born when the share falls in one of its chosen intervals; a node that is on (right) survives when the share falls in one of its chosen intervals. The chosen intervals are the outlined ones. An end drawn solid belongs to the interval, a dotted one does not: in $\phi^5_{6,11}$ a node that is off is born from $\tfrac15$ up to and including $\tfrac35$. On this page, click an interval to choose it or not. The rule’s name reads the chosen intervals as two binary numbers, as rule 90 did on the elementary table: born is 6, the second and third intervals, and survive is 11, the first, second and fourth.

Each click cuts the share more finely, into 7, 9 and then 11 intervals: the resolution $r$. The rule keeps the regions it had, but its name changes, and every finer resolution can write rules that a coarser one cannot: there is no end to them. At $r = 9$, on a grid where everyone has eight neighbours, each interval holds exactly one number of live neighbours, and the Game of Life is $\phi^9_{8,12}$. The slider on this page sets any resolution from 2 to 13.

Defining these rules properly, for any network and at any resolution, is the first contribution of the thesis (Chapter 5, Physica D 2025), and the point of the definition is symmetry. Which ends each interval includes is chosen so that every rule has a cousin that behaves exactly the same with on and off swapped: the same patterns, in the other colours. The button on this page swaps a rule for its cousin: born becomes the complement of survive read backwards, and survive the complement of born read backwards; some rules are their own cousin. At an odd resolution the intervals below $\tfrac12$ include their left end, those above it their right end, and the middle one both. An even resolution has no middle interval, so there the two rings give $\tfrac12$ to different sides: the born ring to the interval just left of it, the survive ring to the one just right of it (Appendix C of the thesis).

Life-like network automata can run any rule on any network, and the network matters as much as the rule. Here one rule, $\phi^9_{168,486}$, on the three networks of the rest of the talk, 150 nodes each, from a random start with three in ten on; the click starts all three. On the random network it never settles; on the small-world ring it fills up and freezes; on the scale-free network it all but dies out. Why? On the scale-free network 80 of the 150 nodes, more than half, have exactly four links, and a node with four links only ever sees none, a quarter, half, three quarters or all of its neighbours on: shares in which this rule never lets a node be born. So they can never switch on. On the ring every node has about eight neighbours, and on the random network anything from two to thirteen. Same rule, three different behaviours: a great many combinations to study. (On the page, ↺ draws new starts.)

A word on the names. The random network is an Erdős–Rényi network: every pair of nodes is linked with the same small probability. The ring is a Watts–Strogatz network (WSr in the thesis): a ring of 150 nodes, each linked to its four nearest neighbours on either side, with a tenth of the links rewired at random. “Small-world” is a shorthand: depending on how much is rewired, such a network is close to the lattice, a small world, or close to a random network, and most of the rewired networks in the thesis lean towards random. The third is grown by nonlinear preferential attachment (NPA): every newcomer links to earlier nodes with a probability that grows with their degree $k$, as $k^\alpha + 1$. With $\alpha = 1$ it resembles a Barabási–Albert scale-free network without being quite one; here $\alpha = 1.8$, so one hub is linked to 106 of the other nodes and the next ones to about fifty, on the way to a star ($\alpha = 0$ gives no hubs at all, a large $\alpha$ one giant hub).

Nature is full of the same recipe: simple local rules, on a network of who reacts to whom. Every bird in a flock follows a few local rules: align with your neighbours, stay close, don’t collide. Two birds are easy to understand; the flock as a whole, turning and splitting and closing up again around a threat, is not. The flock on the left is made of boids, the simulated birds of Craig Reynolds (1987), each following just those rules; the click starts it together with the film beside it. Move the mouse over them: they scatter from it and close up behind it. The same happens in the sky over Rome: a starling flock splits around a peregrine falcon (the small dark bird near the gap) and closes up again behind it. Then, in one long shot, the falcon chases the flock, and dark waves run through it as the birds swerve away, one after the other. Nobody in the flock sees the whole; each bird reacts only to the birds nearest to it. (Video: C. Hemelrijk et al. 2019, from the videos to “Damping of waves of agitation in starling flocks”, CC BY 4.0; two clips joined and cropped.)

A single neuron is well understood: it fires, or not, depending on its inputs. How billions of them together experience reality and steer through it, and even dream, is one of the hardest questions there is. The brain on the left is only an illustration: a neuron fires when enough of its inputs fired, then rests for a moment; the click starts it together with the film beside it. On the right, an octopus fast asleep. Every so often a sleeping octopus slips into a short phase of active sleep, and its skin comes alive: it darkens, breaks into blotches and pales again, running through the same patterns it uses when it is awake, to hide or to signal. That skin is a screen of tiny colour cells, each opened and closed by nerves from the brain, so what we are watching is the brain at work, with nothing outside to react to. The researchers found that its brain activity in these moments looks like waking activity, and compare it to our own REM sleep, the sleep in which we dream. Nobody knows what an octopus experiences; but simple cells, wired together, produce something that looks remarkably like a dream. (Video: an octopus in active sleep, A. Pophale et al., Nature 2023, CC BY 4.0; cropped and trimmed.)

Borrowing two words from biology (after S. A. Kauffman, 1990): the rule is the genotype, local and simple, what each node does; the behaviour is the phenotype, global and rich, what the whole does. Here the rule is Replicator, $\phi^{9}_{170,170}$, on the random network of 150 nodes from before, started with a click: it never settles. The rule fits in two rings, and yet you cannot read what the network does off those rings. On this page, click the rings to change the rule while it runs.

That gap is what the thesis is about, and a Life-like network automaton is a good place to study it: the rule is as simple as a rule gets, the network can be anything, and every step can be watched. Its research question has two halves: what links a simple local rule to the rich behaviour it produces, and can we put that link to use? The rule never tells the whole story, but it tells more than nothing, and the behaviour in turn tells us something about the rule and about the network it runs on.

The thesis has four parts. Part I sets the stage: a taxonomy of the many kinds of cellular automata (A comprehensive taxonomy of cellular automata, CNSNS 2025) and the networks they can live on. Part II is about rules and their metrics: the Lyapunov spectrum, exact growth rates of defects for the simplest rules (under review, and too technical for tonight), and the metrics and the synchronisers of Essential metrics for Life on graphs (Physica D 2025). Part III, skipped tonight, is about cellular automata and neural networks (ACRI 2024 and a Springer book chapter, 2025). Part IV uses rules as instruments in network science: network classification (under review in Information Sciences, first author Lucas C. de Oliveira), and impact analysis, which node matters (unpublished, work in progress).

The two halves of the research question fall on two parts: Part II asks what links a rule and its behaviour, Part IV puts that link to use.

Four chapters make up the rest of the talk, two for each half: getting to know the rules (metrics, synchronisation) and applying them (network classification, impact analysis). They drop to the bottom of the screen and stay there, with the current one filled, so you can always see where we are.

Can we say something about a rule before we run it? Start with one number. Take a node and its eight neighbours, each on or off at random, as in a network that is half on: 512 possible inputs. The rule’s two rings say which of them turn the node on: born, round a node that is off, and survive, round one that is on. The share of the 512 inputs that turn it on is the Hamming weight, $\mathrm{HW}_8 = \frac{1}{2^9}\sum_{s\in\{0,1\}}\sum_{q=0}^{8}\binom{8}{q}\,\phi(s, q/8)$: each slice counts as often as there are neighbourhoods in it, so the slices at the bottom of the rings, around half on, count the most. It is the average next density from half on. For the Game of Life it is 0.27: from half on, about a quarter of the nodes is on one timestep later. (From any other start the same question gives a curve, the average next density or mean-field curve, which Chapter 5 works out too; the talk shows those curves only briefly, in the race of the next chapter.) On this page the buttons switch between the Game of Life and a consensus-seeking rule; the two slides of this chapter share the rule, and its name is under the buttons.

One click brings in the consensus-seeking rule, the kind of rule the next chapter goes looking for: 0.50. It treats on and off alike, so from half on it stays at half on, on average.

The second number is about differences. Take the same node and its eight neighbours, and flip one of its nine inputs. Flip a neighbour, and the share of neighbours on moves one slice along the ring, one way or the other: the red arrows. Flip the node itself, and it moves to the other ring. Does the node’s next state change? Count the inputs whose flip changes it, averaged over all 512 inputs: that is the Boolean sensitivity, $\mathrm{BS}_8 = \frac{1}{2^9}\sum_{s\in\{0,1\}}\sum_{q=0}^{8}\binom{8}{q}\big(\mathrm{IS}(s,q,8)+\mathrm{NS}(s,q,8)\big)$, split into the node itself (IS) and its neighbours (NS), at most 9. For the Game of Life it is 3.17: of a node’s nine inputs, about three matter. It is also how many nodes a single difference reaches in one timestep, on average: above one, a difference spreads. (Chapter 5 follows a difference of any size too, the average next defect or Derrida curve; its slope at the origin is this number.)

The consensus-seeking rule: 6.40, twice as sensitive as the Game of Life. A single difference spreads fast under this rule, and that will matter in the next chapter.

Our result: we worked out these numbers, and the two curves behind them, for Life-like network automata, so that they can be read off any of the 262 144 rules before running it. They are what the filter of the next slide is made of.

Can we really use what a rule looks like to predict what it will do? A test, in the form of a mission: looking for consensus. From a random start, after a while every node should have the same colour, all on or all off. The obvious candidates are threshold rules: do what most of your neighbours do. On a network they get stuck, in patches that each agree with themselves but never with each other.

There are 262 144 rules at resolution 9: every way of colouring the two rings, here the Game of Life. First filter: we do not care which colour wins, only that both are equally likely, so the rule must treat on and off alike: symmetry, it must be its own cousin, like the simple threshold rule shown here (the button says so). That leaves 512. Second filter, on the average next density: from half on and half off, the network must move away, towards all on or all off. That leaves 27, about one rule in ten thousand, found in a fraction of a second without a single simulation. Here each of them stands where it will run, with the two curves behind the numbers of before, just for a glance (the thesis has the details): in blue the average next density from any start, in red the average next defect, each with its tangent. Every blue curve leaves one half more steeply than the diagonal, so from half and half the network is pushed away; the red tangent is the Boolean sensitivity.

Now we can afford to run every one of them. One click puts them in place, the next runs them: all 27 on the same grid of 900 nodes, a fifth of its links rewired, from the same random start, for at most 100 timesteps, as in the thesis. Most of them freeze in patches within a few rounds, or keep churning, and fade; the simple threshold rule is stuck from timestep 10. Three reach agreement, and they are the only ones: of all 512 rules that treat on and off alike, no other reaches a lasting agreement in this race. A strong filter: the metrics inform the behaviour, and a run decides which rule is best. Run over hundreds of networks, one of them clearly is: $phi^9_{488,464}$, which brought more than 96% of the rewired grids of the thesis to agreement. It is among the most sensitive of the 27, but sensitivity alone is not enough (Chapter 6, Physica D 2025).

The winner at work, on the grid of the race (900 nodes, eight neighbours each, a fifth of the links rewired), from the same random start with half of the nodes on. Patches form and fight, and from timestep 31 every node is on. The curve on the right is the share of nodes that are on, timestep by timestep. The buttons switch to the simple threshold rule, on the same network from the same start: within a dozen timesteps it freezes into patches, each content with its own majority, a few nodes blinking at the borders.

The slider under the grid rewires it: none at the left end, then on a logarithmic scale from one link in a thousand to all of them; back at 20% it is the race’s grid again. Moved while the rule runs, it only swaps the links: every node keeps its colour and the run goes on (one that was stuck may come loose). The talk stays there: on rewired grids like this one, the winner brought more than 96% of the networks of the thesis to agreement at a fifth rewired, but only 21% of the grids without rewiring and 37% of the fully rewired ones. On this grid and from this start it agrees within the thesis’s 100 timesteps from about 2% to 50% rewired, and not below or above. The thesis ran 900 samples for every setting, on rewired rings, rewired grids and scale-free networks; these grids are a small illustration.

Our result: with the metrics as a filter, we find rules that do what we want before running a single one. (To say out loud, not on the slide: which raises the question the other way round. Can we start from rules that already do something we want, such as telling networks apart, and find out what makes them work?)

The first application turns the question around: instead of asking what a rule does on a network, ask what it tells us about the network. Here one rule runs on the three networks of before, 150 nodes each, from a random start with three nodes in ten on: first the consensus-seeking rule of the mission. Under each network, three empty lines. The click starts it, for at most 100 timesteps, as in the thesis.

Under each network, three histograms sum up what its nodes did over the run: how evenly each node split its time between on and off, how irregular its row of ons and offs was, and how many of its neighbours were on (the Shannon entropy and the Lempel–Ziv complexity of Miranda et al., and the neighbourhood density of Chapter 9). Together they are the network’s fingerprint, drawn over them as a real one, and a simple classifier can learn to tell such fingerprints apart. Under the consensus-seeking rule every network agrees on all off within about ten timesteps, and from then on every node does the same: three fingerprints alike, and the run stops a second later. A bad detective.

The next click switches to a good detective, $\phi^{9}_{170,48}$, from a new start, with the lines empty again; the click after runs it. Now the histograms differ, and so do the fingerprints: a whorl, a loop and a tented arch. On the ring every node keeps switching, irregularly, on about half of the time; on the random network about one node in five goes off for good (a node with two or four links can never be born under this rule); on the scale-free network more than half of the nodes never switch on again, and the rest switch in a more regular rhythm. So we look for rules whose fingerprints are clearly different. (G. H. B. Miranda and colleagues started classifying networks this way in 2016; the thesis did it on 23 datasets, with histograms split by degree and a support-vector machine. On the page the buttons switch the rule and ↺ draws a new start; about one start in fifteen leaves the ring stuck under the consensus-seeking rule.)

What makes the difference can be read off the rule itself. Back to the rings, and to the red arrow of the sensitivity slide: one neighbour more or fewer moves a node one slice along its ring. The next click draws that arrow wherever it changes the answer, between every two neighbouring slices where the rule says something else. The consensus-seeking rule changes its mind at 6 of the 16 steps, the good detective at 10: a rule that changes its mind often is a sensitive rule, since a small change in the neighbourhood often changes what a node does.

This is our contribution to network classification, Chapter 9, work led by Lucas C. de Oliveira (with M. Rollier, J. M. Baetens and O. M. Bruno, under review). Over thousands of rules and 23 datasets, the best detectives change their mind often, but not always: at between half and nine in ten of the steps. That makes sense: such rules are sensitive enough to pick up how networks differ, but not so sensitive that it drowns in chaos; they behave in a complex way (a reading the thesis proposes but does not test). Our detective, at 10 of 16, sits inside that band. On 13 of the 23 datasets, more often tends to classify better (correlations between about 0.2 and 0.46), and searching among the rules that change their mind often, but not always, finds a rule within 5% of the best while trying only 10 to 20% of the rules. On social and bioinformatics networks there is no such pattern, but the method itself is mediocre there as well. (The count is what the thesis calls the jaggedness: the neighbour part of the Boolean sensitivity with every slice weighed the same, whatever the network. The Boolean sensitivity itself shows the same pattern, slightly weaker.)

What could this be good for, when nobody has the network? Think of a contact-tracing app such as Coronalert, launched in Belgium six years ago this week: phones sense which phones come near and, for privacy, nobody ever holds the network of who met whom. If every phone ran one simple rule with the phones around it and reported only its own ons and offs, their fingerprint could still say what kind of network we live in, for instance whether our contacts run through a few superspreaders. We cannot try thousands of rules on real people, so the rule would be chosen in advance, from simulations alone: one that changes its mind often, but not always. (A thought experiment; a backup slide shows it.) And the consensus-seeking rules, such bad detectives, turn out to be good at something else: finding who matters.

The second application uses the consensus-seeking rules that were such bad detectives. Here a toy village of 100 people, about eight contacts each: which individuals steer the group?

First, find a converging system: everyone follows a consensus-seeking rule, from a start with half of them on. The click runs it: with nobody stubborn, the village ends with every node on.

Then force a node to be stubborn: it stays off, no matter what. Some are not strong enough: when a quiet person at the edge is stubborn (ringed), nobody else follows. Others convince the entire group: when an important person is stubborn, the whole village flips to all off. Holding one node fixed like this is called clamping, or pinning: it is an intervention, not just an observation. (An illustration, from the same start every time.)

Now make every node stubborn in turn, one at a time, and record how much of the village ends up following it: that is each node’s dynamical importance score, the darker, the more (in the thesis, holding a node fixed is called clamping). In this village most people sway only a small part of it, and the most important about half of it, on average. (Chapter 10, work in progress; an illustration over 100 starts that reach agreement, with each node stubborn in each.)

Why would we want to know who matters? Take an example: a health campaign with a small budget. Marketing campaigns and political propaganda ask the same question; the health campaign is the friendlier example. In 32 villages in Honduras, researchers started a water-purification and a vitamin programme with a few villagers each, to see how far it would spread, and compared ways of picking those few (Kim and colleagues, The Lancet, 2015). So which three villagers should be our ambassadors? We do not know how villagers really persuade each other, so we use a stand-in: the consensus-seeking rule, not to model the villagers, but to measure them. The three most important, ringed in red, become our ambassadors: an ambassador is a stubborn node, who keeps promoting the programme, no matter what. The dashed rings are the three villagers with the most contacts, 21 each: not the same people. Importance goes with the number of contacts (a rank correlation of 0.82 here), but not one-to-one: the many contacts of those three also listen to the centre of the village, and follow it. In the Honduran trial too, picking the villagers with the most friends did no better than picking at random, while picking friends that villagers named did better, at least for the vitamins. (An example, not a campaign I studied.)

Our result: a ranking of who steers the group that needs no knowledge of the real rule, only a stand-in rule and a simulator. Whether the stand-in picks the same people as the real rule would is the most important next step of Chapter 10: comparing it with, say, an epidemic model on the same networks. Taking the top three one by one also ignores that they may reach the same people; choosing them together is a next step. This part of the thesis is unpublished and still in progress.

To sum up, along the two halves of the research question, and the chapters along the bottom of the slides.

Getting to know the rules. Metrics: we defined Life-like network automata so that every rule has its cousin, and worked out numbers and curves that quantify a rule before it runs, such as the Hamming weight and the Boolean sensitivity, here for one rule after another (Chapter 5).

Synchronisation: with those metrics as a filter, we find rules that do what we want, here a consensus-seeking rule that brings a rewired grid to agreement, from one random start after another (Chapter 6).

Applying them. Network classification: we understand which rules give networks distinct fingerprints, good detectives that change their mind often, but not always, here on the three networks (Chapter 9).

And impact analysis: making every node stubborn in turn gives each a dynamical importance score. Here the best-connected villager, the most important one, a quiet one and another important one, in turn: twice the village follows, twice it does not, each time what that person does in most starts (Chapter 10, work in progress).

Where does this go next? First, at BionamiX: spatiotemporal modelling, where cellular automata model processes in space and time. Those need the extensions from the taxonomy of Chapter 2: rules that differ from cell to cell, more states, larger neighbourhoods.

Groundwater level: the MACCA-GW model, a cellular automaton for groundwater flow, applied to De Wilde Landen in the Netherlands, is accepted for the ACRI 2026 proceedings (with L. Landuyt and J. M. Baetens). It is not part of the thesis: a model in space, not a network automaton. Every cell holds the height of the water table; water flows to it from its four neighbours, rain falls, the land cover draws some back, and a line of wells pumps, more in summer. The map is a simplified version on a made-up reserve, not the calibrated model, started with this click: a year in ten seconds, the water table highest in spring and lowest in late summer, deepest along the wells. Next to established models such as MODFLOW it offers simplicity, and many runs at once for ensembles; in the paper, pumping more in winter and less in summer raises the spring water table by about a third of a metre, at no cost.

And wildfires, close to home: in August a fire in the Hautes Fagnes burnt some 3,000 hectares, one of the largest in Belgian history, and villages had to be evacuated. The picture is a wildfire as a simple cellular automaton, an illustration and not the BionamiX model, started with this click: green trees, red fire, charcoal where it has passed, a wind from the left; a burning tree burns for several timesteps and can set each of its eight neighbours alight, most easily downwind. Models like this ask where a fire goes next, and where a firebreak would help most.

Second: influence maximisation. I wish I had started on node importance earlier: it holds many open questions and many relevant applications. Which few people, chosen together, reach the most? Does a stand-in rule pick the same people as the real processes on a network, an epidemic, an opinion, a product spreading? The claims of the impact chapter still have to be checked against such processes, and on real networks. The picture, started with this click, is the village of before with our three ambassadors (ringed) holding on to the programme: from one start after another, the village follows them, nearly always.

And next for me: what goes on inside artificial minds. A neural network is also simple operations, iterated: every node adds up what comes in, each input with its weight, adds a bias, and passes the sum on only if it is positive (a ReLU). Here a small one with random weights and biases, started with this click, processes one input after another: each link lights up with what it passes on, a node fills as it fires, and the largest output is ringed. Nobody writes down what a trained network does: the same gap between rule and behaviour, with very high stakes. The results of this thesis do not carry over directly, but the way of thinking, from the parts to the whole and back, can help.

The title once more, in two halves: the rich landscape is what we see, the behaviour; iterated simplicity is what we write down, the rule. This thesis brought those two worlds a little closer together.

Thank you. The colours drain out of the landscape, as on the still-life slide: every cell simply alive or dead, and nothing changes. The QR code leads to this site, with every demo of tonight and the thesis chapter by chapter.

The last click takes away one cell, and the still life breaks down, as on the still-life slide: the debris spreads over the landscape and keeps going while the slide is up. Where it comes near the words and the logos it is faded, so they stay readable.

What could this be good for, when nobody has the network? Six years ago this week, on 30 September 2020, Belgium launched its contact-tracing app, Coronalert: every phone sensed over Bluetooth which phones came near it, and to protect our privacy nobody ever held the network of who met whom. Now imagine every phone running one simple rule with the phones around it, and reporting only its own ons and offs, never whom it met. The next click starts them: a small network of 48 phones, hidden.

From those rows of ons and offs alone, the fingerprint says what kind of network we live in: whether our contacts run through a few hubs, superspreaders, or are well mixed. Here the hidden network is scale-free, with one phone linked to 34 others (its links appear on this click), and the phones run the good detective, so the fingerprint is the scale-free one of before. A thought experiment, of course: real contacts change from hour to hour.

But which rule do we load into millions of phones? We cannot try thousands of rules on real people. This is where knowing what makes a good detective pays off: a rule that changes its mind often, but not always, chosen in advance from simulations alone. And the consensus-seeking rules, such bad detectives, turn out to be good at something else: finding who matters.