2026-06-26 · paper

When the Gods Drift

Every network we have drawn so far has been a still photograph, bodies caught mid-gesture and held there. The arrows were fixed, the bias matrix was fixed, and the only randomness left was the coin flip that turned a probability into an edge. That was already enough to keep us busy for three posts.

Real networks rarely hold still. Desire kindles and cools and kindles again. Trade routes open and silt up. Gods rise through a pantheon and slide back down it. If we want to follow any of that, we have to let the thing we have been holding fixed, the hidden arrangement of arrows, start to move. That is where the slow walk goes next, and it is where my recent work actually lives.

Let’s begin, as we did the very first time, with two gods and a single edge: one of them leaning toward the other, wanting.

Two stories for a flickering edge

Suppose we run a short film of the network, a handful of frames spaced a little apart, and follow one edge across them. In the first frame Tefnut reaches toward her twin Shu, the moisture in her leaning into his dry air; in the next the reaching is gone, the two of them cooled and turned apart; a few frames later it is back. The edge flickered: there, gone, there again.

What happened?

There are two stories we could tell, and the whole of this post hangs on the difference between them.

The first story is that nothing happened. Back in part two we watched the same bias matrix produce different graphs each time we resampled it: the edge from Tefnut to Shu was never a certainty, only a probability, say $0.6$. A probability of $0.6$ that the edge appears is also a probability of $0.4$ that it does not. The underlying arrangement of arrows never budged; the coin came up heads, then tails, then heads, and the flicker is nothing but the coin blinking.

The second story is that something happened: the arrows themselves moved. Tefnut began to slip away from her family, the probability of that edge falling from $0.6$ toward $0.1$ as she went and climbing again as she returned. The edge did not blink at random; it followed her out and home, the warmth going out of her as she withdrew and flooding back as she returned to him.

What an ordinary on-and-off network leaves out is everything between present and absent: there an edge can only snap from $0$ to $1$, with nothing underneath it to drift. RDPG lays a continuous probability under each edge, so an edge can change by a hair, and a single tip can shift a node’s standing with the whole pantheon at once, not one tie at a time: move one arrow and every edge it owns answers together. That coupling, one small motion felt across all of a node’s edges, is the thread we pull on for the rest of the post, and write down exactly as $d\textcolor{#15803d}{P}/dt$ a little further on.

Telling luck from desire is the work the dynamics posts share. This post makes the first separation, and then uncovers a harder ambiguity waiting behind it.

Letting the tips move

First we need to say, carefully, what it means for the tips to move.

Recall the picture from part two. Every node carries two arrows from the origin, a yellow one in its role as giver and a blue one as receiver. The tip of each arrow, its far end, is the node’s latent position in that role: the giver tip is the part of it that reaches, the receiver tip the part that is reached for. Stack all the yellow tips and you get a cloud of points in yellow space; stack the blue tips, a second cloud in blue space. Until now those clouds were frozen.

Now we attach a clock. As time $t$ runs forward, each tip is allowed to slide along a path of its own. Tefnut’s tips swing out through both spaces and back; Bastet’s giver-arrow turns slowly in place; every node is free to follow its own path. The whole configuration of arrows, which we can gather into the two tables $\textcolor{#ca8a04}{Y}(t)$ and $\textcolor{#1d4ed8}{B}(t)$ from part two, is now a thing that depends on the moment you look at it.

Two words will be handy. A tip’s trajectory is the whole path it follows over time. Its velocity at a given instant is how fast it is sliding and in which direction. Drawn, the velocity is its own small arrow, perched at the tip and tangent to the path, a separate thing from the giver or receiver arrow that runs all the way from the origin. A tip parked at one spot has zero velocity; a tip racing across its space has a long one.

When the tips move, the bias matrix moves with them. At each instant we can multiply the current tables exactly as we did before,

$\textcolor{#15803d}{P}(t) \;=\; \textcolor{#ca8a04}{Y}(t)\,\textcolor{#1d4ed8}{B}(t)^{\top},$

and read off, in slot $(i, j)$, the current probability of an edge from node $i$ to node $j$. Because the tips are moving, $\textcolor{#15803d}{P}(t)$ becomes a movie rather than a single green table: a green table for every instant, its cells brightening and dimming as the bodies behind them turn toward one another and away.

Tefnut travels

Take the gods at their word about how they move. Part three told Tefnut’s beginning: Atum, alone in Heliopolis, took himself in hand and spat out the first twins, and Tefnut came into the world as moisture, born of desire. In the Demotic Myth of the Sun’s Eye, the Eye of Ra quarrels with her father and storms south to Nubia, and she carries her wetness away with her. The land she leaves cracks and parches and thirsts for her; Ra aches for his lost Eye, all that heat in him with nothing to fall on; and he sends Shu and Thoth to bring her back. Thoth, fable after patient fable, coaxes her home, and her return is the inundation itself, the Nile swelling, the dry fields drenched and opened under her, a whole country that had wanted her body slaked at last. In the geometry, that is a tip leaving the cloud. Her yellow and blue tips slide off toward an empty corner of their spaces, her dot product with every other god goes cold, her reaching finding no one, and one edge after another fades from the films. Then the tips travel back along nearly the same path, the overlaps warming again, her edges returning. A journey out and home again, a long loop traced through latent space. (Hold onto the fact that it is a loop. It will matter in the next post.)

The rate of an edge

We can even ask how fast a single edge is changing, how fast the wanting in it kindles or cools, a rate that turns out to be the engine of everything that follows.

Start with a single tip. Over a slice of time $\Delta t$ the giver tip $\textcolor{#ca8a04}{y}_i$ shifts by a discrete step $\Delta\textcolor{#ca8a04}{y}_i$, and the receiver by $\Delta\textcolor{#1d4ed8}{b}_j$. The ratio $\frac{\Delta\textcolor{#ca8a04}{y}_i}{\Delta t}$ is an average rate over the slice; only when the slice shrinks to nothing does it settle into the tip’s instantaneous velocity, which we then write $\frac{d\textcolor{#ca8a04}{y}_i}{dt}$. The $\Delta$ marks a finite step; the $d$ is kept for that limit.

Now climb to a single edge. Its bias is the dot product $\textcolor{#15803d}{P}_{ij} = \textcolor{#ca8a04}{y}_i \cdot \textcolor{#1d4ed8}{b}_j$, which part one drew as the overlap of the two arrows, the area their shadows sweep out where the bodies meet: full when they line up and lie along each other, slight when they turn away. How fast does that overlap change as the arrows drift?

Take the two motions one at a time. Hold the receiver $\textcolor{#1d4ed8}{b}_j$ still and let the giver shift by $\Delta\textcolor{#ca8a04}{y}_i$; its shadow on the receiver slides with it, and the overlap grows by $\Delta\textcolor{#ca8a04}{y}_i\cdot\textcolor{#1d4ed8}{b}_j$, the giver’s step taken against the receiver. Hold the giver still instead, and it grows by $\textcolor{#ca8a04}{y}_i\cdot\Delta\textcolor{#1d4ed8}{b}_j$.

When both shift at once, the change in the overlap over the slice opens into three pieces:

$\Delta\textcolor{#15803d}{P}_{ij} \;=\; (\textcolor{#ca8a04}{y}_i + \Delta\textcolor{#ca8a04}{y}_i)\cdot(\textcolor{#1d4ed8}{b}_j + \Delta\textcolor{#1d4ed8}{b}_j) - \textcolor{#ca8a04}{y}_i\cdot\textcolor{#1d4ed8}{b}_j \;=\; \Delta\textcolor{#ca8a04}{y}_i\cdot\textcolor{#1d4ed8}{b}_j \;+\; \textcolor{#ca8a04}{y}_i\cdot\Delta\textcolor{#1d4ed8}{b}_j \;+\; \Delta\textcolor{#ca8a04}{y}_i\cdot\Delta\textcolor{#1d4ed8}{b}_j.$

The first two are the single motions we just found. The third, $\Delta\textcolor{#ca8a04}{y}_i\cdot\Delta\textcolor{#1d4ed8}{b}_j$, is the overlap of the two steps with each other, the grey corner in the figure below: a step times another step. It is second order, and as the slice shrinks toward zero it dwindles far faster than the other two and leaves no trace. Divide by $\Delta t$ and pass to the limit as the slice closes to nothing, where each ratio $\frac{\Delta}{\Delta t}$ becomes its $\frac{d}{dt}$ and the second-order corner falls away; the rate of a single edge is the two pieces that survive:

$\frac{d\textcolor{#15803d}{P}_{ij}}{dt} \;=\; \frac{d\textcolor{#ca8a04}{y}_i}{dt} \cdot \textcolor{#1d4ed8}{b}_j \;+\; \textcolor{#ca8a04}{y}_i \cdot \frac{d\textcolor{#1d4ed8}{b}_j}{dt}.$

The bias $\textcolor{#15803d}{P}_{ij} = \textcolor{#ca8a04}{y}_i \cdot \textcolor{#1d4ed8}{b}_j$ has two equal-area rectangle views, the shadow rectangles from part one. Pull the velocity $\frac{d\textcolor{#ca8a04}{y}_i}{dt}$ out of $\textcolor{#ca8a04}{y}$‘s tip and $\frac{d\textcolor{#1d4ed8}{b}_j}{dt}$ out of $\textcolor{#1d4ed8}{b}$‘s tip. Because each term is a dot product, only the part of a velocity along the other arrow counts: that part drives the gold strip $\frac{d\textcolor{#ca8a04}{y}_i}{dt} \cdot \textcolor{#1d4ed8}{b}_j$ and the blue strip $\textcolor{#ca8a04}{y}_i \cdot \frac{d\textcolor{#1d4ed8}{b}_j}{dt}$ on both views. A velocity pulled across the other arrow leaves its term zero; pulled backward, away from it, the term turns negative and the overlap shrinks, shown by the red dashed strip cut from inside. The grey corner is the both-move piece, second order, dwindling far faster than the strips as you pull the velocities in. See if you can pull the two velocities so the rate holds at zero, the overlap unmoved even though neither velocity is.

Two terms, one for each arrow that can move: the giver’s motion as the receiver sees it, plus the receiver’s motion as the giver sees it. This is the product rule from calculus, the one for the derivative of a product of two functions, $(fg)' = f'g + fg'$, read straight off the dot product: a dot product is a sum of such products, and inherits it term by term.

In words: an edge’s probability changes at a rate that adds the giver moving, taken against where the receiver sits, to the receiver moving, taken against where the giver sits. If neither tip moves the edge holds still; if the two turn from each other, the wanting cools fast.

Now climb to the whole network. Stack that single-edge rule over every pair $(i, j)$ and the same product rule reappears for the entire table at once, the givers’ motion and the receivers’ motion each contributing a term:

$\frac{d\textcolor{#15803d}{P}}{dt} \;=\; \frac{d\textcolor{#ca8a04}{Y}}{dt}\,\textcolor{#1d4ed8}{B}^{\top} \;+\; \textcolor{#ca8a04}{Y}\,\frac{d\textcolor{#1d4ed8}{B}^{\top}}{dt}.$

This is the coupling we promised at the start. A single tip never moves in private: shift one giver, a single row of $\textcolor{#ca8a04}{Y}$, and a whole row of $\textcolor{#15803d}{P}$ stirs at once, every edge that node sends out, each one its moving tip taken against a different partner. Some of those biases rise while others fall, by how the motion lines up with each receiver; how tight that trade-off is depends on the dimension, a roomier latent space leaving more ways to strengthen some ties without weakening the rest.

And here is something quietly strange, worth pausing on. We have just taken the derivative of a network. Stretching the word only a little, $d\textcolor{#15803d}{P}/dt$ is the velocity of the entire web of connection probabilities, an infinitesimal nudge to the whole network at once. A classic on-and-off network admits no such object: there an edge is present or absent, and the smallest move available is to flip one edge from $0$ to $1$, a discrete jump with nothing in between, so there is no “slightly more of an edge”, and no rate, and no derivative. The continuous probabilities of RDPG are what lay a smooth surface beneath the discrete graph, and a smooth surface is the kind of thing calculus can grip. The whole machinery of motion we are leaning on exists only because we slipped a geometry underneath the network.

And from each frame of that movie we can draw a graph, the same way we always have: flip a coin for every cell, with that cell’s current value as its bias. Run the clock and the network flickers, frame after frame, each one a noisy snapshot of the smoothly moving probabilities underneath.

Bastet turns

Her motion is slower, and of a different kind. Across the first millennium BCE she changes from a fierce lioness, one of the solar daughters all appetite and claw, into the cat of Bubastis, goddess of music and pleasure and the warm dark of the house. Her tip barely changes its distance from the origin; she stays a major goddess the whole way through. What changes is the direction her giving arrow points. Early on it aims toward the company she keeps as a lioness; over centuries it swings around to aim at a quite different quarter of the network, the festival and the bed: the barges going down to Bubastis loud with flutes and clappers, the women calling out and lifting their skirts to the men on the banks, and more wine drunk along the way, Herodotus says, than in all the rest of the year. Same length, new heading: where Tefnut’s tip changed its position, Bastet’s changes its bearing.

And some gods barely move at all. Ra sits near the centre the whole time, the sun the others turn around, his arrows close to fixed.

One last thing, and the gods themselves insist on it. Tefnut, Hathor, Sekhmet, Bastet are not always held apart; the texts call each of them the Eye of Ra, the same wild solar daughter, all heat and appetite, poured into different vessels. A model that only watches how the network reaches toward them has no way to prise such interchangeable figures apart, and it sets their tips almost on top of one another, exactly as part three found Hathor and Tefnut nearly coinciding in blue space. The mythology blurs them for the same reason the geometry does: seen from the outside, they play the same role.

Drag the time slider, or let it play. Tefnut (ringed in red) journeys out toward the empty origin and home again; her row and column in the bias matrix $\textcolor{#15803d}{P}(t)$ fade as she leaves, and her edges flicker out of the realised graph along with them. Bastet (violet) holds her giving arrow’s length while its heading turns, and her outgoing edges rewire from the fierce goddesses toward Hathor, the goddess of music and love. Because givers and receivers differ, $\textcolor{#15803d}{P}(t)$ is not symmetric: Ra’s column glows, the sun the whole pantheon reaches toward, while his row stays dim. Tefnut and Sekhmet, two faces of the same fierce Eye, begin almost on top of one another.

Chance or motion?

Now put yourself back on the observer’s side, where the arrows and the green movie are hidden and all you hold is the flickering film of graphs.

Watch a single edge across a few frames and you learn almost nothing. An edge that blinks off for a frame and on again could be a tip on the move, or it could be the same steady probability landing tails once. Chance and motion produce the same flicker when you only have one edge to look at.

Company rescues us here, exactly as it did in part two, and now in two directions at once. There we saw that a single node’s arrow becomes knowable because the rest of the network acts as a crowd of probes, each edge a noisy measurement against a different partner. Time adds a second crowd. Tefnut’s departure shows up as the same probability sliding the same way across many of her edges and across many successive frames, all of them dimming together. Coin-flip jitter has no such consistency: it scatters, frame to frame and edge to edge, with nothing to line it up. Average over enough partners and enough moments, and the steady drift of real desire pulls clear of the random speckle, the Law of Large Numbers working now across time as well as across the population.

So the honest answer to “did the world change, or did the coin?” is: from one edge and two frames, you cannot say; from many edges and many frames, you very often can. The continuous geometry under the graph is what makes the question answerable at all. A bare on-and-off network gives you nothing to average toward.

A second veil

We have let the cloud move, turned the bias matrix into a movie, and met the first thing standing between a changing graph and a changing world. Call it the veil of chance: the gap between an edge flickering because a tip moved and an edge flickering because a coin fell the other way. The Egyptians would not have given it a god; they kept no deity of pure chance, and read the gods’ own will in the fall of the dice, with Thoth to oversee the throw. This veil is ours, a modern and statistical thing, and we have seen how to thin it, by gathering enough edges and enough frames that real motion separates from noise.

Lift the veil of chance and convince yourself the cloud genuinely moved. A second veil waits behind it, stranger than the first. Even when the motion is real and measured as cleanly as the data allow, the thing we observe cannot see all of it. Some of the motion happens in a direction the network is blind to. To find that direction, we leave the Nile for a moment and stand on a different shore.

The camel and the shark

Picture a camel paddling at the surface of the Red Sea, and a shark cruising roughly beneath her; the two are old acquaintances by now, having met like this many times, each the other’s neighbour in a small watery web. The shark moves freely in three dimensions: this way or that, and up toward the surface or down into the dark. The camel, floating on top, takes in only the flat sheet of water around her. What reaches her of the shark is its path pressed flat onto the surface, the horizontal wandering with the depth squeezed out.

Now the shark dives. On the surface almost nothing happens: the flattened mark holds its place while the animal drops straight down beneath her. From the camel’s vantage, surfacing and diving look the same, because depth is exactly the direction the surface cannot record. Many different journeys through the water, rising here, plunging there, leave the very same trail on top. The camel cannot undo the flattening; the information is not in what she sees.

Our situation is the camel’s. The moving latent cloud, $\textcolor{#ca8a04}{Y}(t)$ and $\textcolor{#1d4ed8}{B}(t)$, is the shark, free in its full space. What we observe, the bias matrix $\textcolor{#15803d}{P}(t)$ and the graphs drawn from it, is the surface trace. And there is a whole direction of motion that leaves no mark on it.

That direction is rotation, and only the kind that turns the whole cloud together, every yellow tip and every blue tip through the same angle. Let even a handful of nodes hold still while the rest turn, and the spin would drag those few visibly out of place, enough to serve as probes that betray the motion; that fixed handful is the foothold the next post uses to pin the turning frame down. Spin the whole cloud together, though, and every coordinate changes while the bias matrix does not move at all. We saw why in part two: an edge bias is a dot product, and a dot product survives rotation. Turn the giver and the receiver through the same angle and their dot product holds, so

$\textcolor{#15803d}{P} = \textcolor{#ca8a04}{Y}\,\textcolor{#1d4ed8}{B}^{\top} = (\textcolor{#ca8a04}{Y}R)\,(\textcolor{#1d4ed8}{B}R)^{\top}$

for any rotation $R$. This is the thing the rotate-together toggle in part one was quietly showing you, and that part two made precise.

Rotation is the shark’s depth: a direction of change the surface cannot see. The analogy is loose; the lesson is exact. The cloud can spin however it likes and the bias matrix we observe holds perfectly still. So when the latent cloud appears to swirl over time, we are left with a real ambiguity: did the gods move, or is the whole frame turning under them while they sit where they are? A slow spin of the entire cloud and a true migration of its tips can produce the very same film of graphs. From the observed side, nothing tells them apart.

You have already watched each half on its own: in part two a shared rotation turned every arrow while the bias matrix held perfectly still, and in the moving cloud above a real migration of tips lit the matrix up and let it dim. The gauge is the gap between those two films, the spins of the first kind hiding inside the motion of the second.

The whole-network rule states this exactly. A rigid rotation gives the givers the velocity $\frac{d\textcolor{#ca8a04}{Y}}{dt} = \textcolor{#ca8a04}{Y}\Omega$ and the receivers $\frac{d\textcolor{#1d4ed8}{B}}{dt} = \textcolor{#1d4ed8}{B}\Omega$, both turned by the same infinitesimal twist $\Omega$, a $d \times d$ matrix, with $d$ the number of coordinates each tip carries (two, in our pictures). A pure rotation neither stretches nor shrinks, which forces $\Omega$ to be antisymmetric: that is, $\Omega^{\top} = -\Omega$, its transpose is its own negative. Feed those velocities into the rule and the two terms fold into one that vanishes:

$\frac{d\textcolor{#15803d}{P}}{dt} \;=\; (\textcolor{#ca8a04}{Y}\Omega)\,\textcolor{#1d4ed8}{B}^{\top} + \textcolor{#ca8a04}{Y}\,(\textcolor{#1d4ed8}{B}\Omega)^{\top} \;=\; \textcolor{#ca8a04}{Y}\,(\Omega + \Omega^{\top})\,\textcolor{#1d4ed8}{B}^{\top} \;=\; 0,$

because $\Omega + \Omega^{\top} = 0$. The cloud is genuinely turning and not one bias moves. So we can name the blind spot precisely: it is the set of cloud-velocities that leave every bias fixed, the motions with $d\textcolor{#15803d}{P}/dt = 0$ throughout, and rigid rotation, the antisymmetric $\Omega$, is the inhabitant of it we keep meeting.

How roomy is that blind spot? Exactly as roomy as the antisymmetric twists $\Omega$ allow, one for every pair of axes that can be turned into each other. In the flat two-dimensional world of our drawings that is a single angle to spin through, which is why the rotate-together toggle had just one knob. Add a third dimension and there are three independent spins; in $d$ dimensions there are $d(d-1)/2$ in all. The shark had a single hidden depth. A latent space of any real size hides a whole room of invisible motions, and that room grows with the square of the dimension.

Above every observation, then, sits a whole family of clouds, all rotations of one another, every one giving the identical bias matrix. Mathematicians call that family a fibre, and we can call this second obstacle the fibre veil. This one does have a god of its own: Amun, whose name means the hidden, for which cloud of the fibre is the true one is exactly what cannot be seen. To recover the motion of the gods is to choose, at each instant, one cloud out of its fibre, and to choose consistently enough that the motion we read off belongs to the gods and not to our turning frame.

Where we go next

Two veils stand between the graphs we see and the motion we want. The veil of chance, the modern and statistical one, we can thin with enough data. The fibre veil, Amun’s, is sterner, built into the model itself, and no amount of data removes it. What we can still do is choose our frame with care, lining up each moment’s cloud against the last so that what we read as movement belongs to the gods and not to our own shifting vantage.

That careful lining-up, and the strange thing that happens when a god’s path closes into a loop and the frame fails to come home with her, is where the slow walk goes next. Tefnut came back to him. The question we will have to face is whether the world she returned to had quietly turned beneath her while she was away, the bed not quite where she left it.