Contribution
Two information-theoretic accounts of emergence, Hoel’s coarse-graining account and the integrated-information-decomposition (ΦID) account of Mediano, Rosas and colleagues, both score a system by an expected value of the mutual information between its past and its future. The paper takes those scores apart transition by transition. It shows, in one toy system for each framework and in one human connectome, that a system emergent on average can have many particular transitions whose local value runs the other way. It names this “flickering emergence” and argues that any measure built on time-delayed mutual information admits it.
Key insight
Mutual information is never negative, but its pointwise value is, whenever a realised pair (x, y) is less likely under the coupling than under independence. An emergence score that is an expectation of such values can therefore be positive while some of the moments it averages are negative. Then “the system is emergent” holds of the ensemble of transitions and fails at those moments, which the paper glosses as the system briefly being “more like a collection of ‘parts’”.
Assumptions
- Discrete, stationary, Markov dynamics. Excess entropy is cut to one step, E(X) = I(X_{t−1}; X_t), and the system is given by a transition probability matrix.
- Coarse-graining account: effective information is computed with a maximum-entropy (uniform) distribution imposed on past states, as in Hoel et al.; the coarse-graining is fixed by the analyst (here a logical AND over pairs of nodes).
- ΦID account: a localisable double-redundancy function is assumed, Varley’s I_τsx; the decomposition and its lattice of sixteen atoms (for two variables) are taken from Mediano et al. and not re-derived.
- Network application: a random walker on a structural connectivity matrix, with Infomap communities as macro-nodes.
- The general claim (“any approach based on excess entropy”) is argued from localisability, not proved for each framework.
Key results
- Local mutual information (Eqs. 4–5). i(x; y) = log P(x,y)/(P(x)P(y)) = h(x) − h(x|y). It is negative when P(x,y) < P(x)P(y), “when a system is transiently breaking from its own average, long-term behaviour”. Holds: for any joint distribution; I(X;Y) = E[i(x;y)] ≥ 0.
- Local excess entropy (Eq. 8). e(x) = i(x_{t−1}; x_t), negative when P(x_t | x_{t−1}) < P(x_t): “The past transiently ceases to inform of the future and instead misinforms.”
- Coarse-graining emergence (Eqs. 9–12). Effectiveness F̄(X) = I(X_{t−1}^{Hmax}; X_t)/log N; emergence when log(F̄(X̃)/F̄(X)) > 0. The toy four-node Boolean network coarse-grained by AND over pairs gives log(F̄(X̃)/F̄(X)) ≈ 0.202.
- Incongruous dynamics, coarse-graining (§3.1). Defined as e(x̃) < 0 while e(x) > 0. “≈52.73% of informative micro-scale edges map to misinformative macro-scale edges”; these are mostly low-probability transitions, which is why the average stays emergent. A random walk on the micro chain shows sign flips of e(x̃_t)/e(x_t) along the trajectory. The paper notes the normalisation by log N does not carry over locally, since local values are unbounded, so only signs are compared.
- Network application (§3.1). A Human Connectome Project structural matrix, Infomap macro-nodes: effectiveness ≈0.37 bit micro against ≈0.38 bit macro; ≈22.47% of informative micro edges map to misinformative macro edges. Informative edges were about as likely within communities (≈52.3%) as between (≈47.7%); misinformative edges were mostly between communities (≈80.4% against ≈19.6%). Replicated with spinglass and leading-eigenvector communities (supplement, not seen). Holds: one subject’s matrix.
- ΦID emergence (§§2.4.2, 3.2). Temporal synergy is the atom {X1,X2}→{X1,X2}, “that portion of E(X) that is not disclosed by any simpler combination of elements”. The toy macro system has Φ∂ ≈ 0.031 bit on average; 31.25% of its sixteen transitions have a negative local value of that atom. Two transitions ending in (0,0) have opposite-signed synergy, double-redundancy and “downward causation” atoms.
- The general claim (§§1, 5). Flickering is “an unavoidable consequence of any formal approach based on time-delayed mutual information” in the sense that the possibility is; the Conclusions say such approaches are “likely to display” it. Barnett and Seth’s dynamical independence and Chang et al.’s information closure are expected to show it “barring an unexpected mathematical result”, not shown.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Pointwise past–future mutual information can be negative while its expectation is positive | strong | standard identity, Eqs. 4–8 |
| C2 | A coarse-graining emergent on average can map most informative micro transitions to misinformative macro ones | moderate | one constructed toy (§3.1, Fig. 3) |
| C3 | The ΦID synergy atom can be positive on average and negative at particular transitions | moderate | the same toy at macro scale (§3.2, Fig. 5) |
| C4 | In a connectome, misinformative macro edges lie mostly between communities | weak | one subject, one walker model (§3.1, Fig. 4) |
| C5 | Every excess-entropy-based account of emergence admits flickering | weak | argued from localisability; two frameworks shown, two others conjectured |
| C6 | If integrated emergence is identified with a sense of self, the sense of self should flicker reportably | weak | a conditional raised in Discussion, not tested |
Method
Take a transition probability matrix, compute local excess entropy for every transition at micro and macro scale, and count sign mismatches. For ΦID, solve the double-redundancy lattice locally for each transition with I_τsx and Möbius inversion (Eq. 15), and read the sign of the synergy atom. Run a random walk to show the sign changes over time.
Concepts
- Local (pointwise) mutual information — i(x; y), the realised-pair term whose expectation is I(X; Y).
- Misinformative transition — one with e(x) < 0: knowing the past makes the realised future less expected than chance.
- Incongruous dynamics — a transition informative at one scale and misinformative at another (coarse-graining), or a negative local value of an atom whose expectation is positive (ΦID).
- Flickering emergence — the transient loss of an average emergent property at particular transitions, “like a candle sputtering”.
- Temporal synergy (Mediano et al.’s “causal decoupling”) — the {X1,X2}→{X1,X2} atom of ΦID.
Connections
It builds on Hoel, Albantakis and Tononi’s effective information (LIT-1038), on the ΦID programme reviewed in LIT-025, and on Lizier’s local information dynamics. Varley and Hoel’s “emergence as the conversion of information” (LIT-027) is cited as the observer-relative reading under which flickering is only model failure. Klein and Hoel’s network coarse-graining supplies the connectome analysis. The Discussion argues a “pragmatic pluralist” view, that the two frameworks are distinct valid kinds of emergence.
Bearing on the record
- QUESTION-035 (how long a combined form lasts). The seed claim in LIT-953 is confirmed: the paper does show emergence holding on average and failing at particular configurations. What it adds for the question is that a criterion of unity computed as an average over the dynamics gives no verdict about a particular moment. “How long does it last” then needs a window and a rule for how many misinformative moments end the form, which none of these measures supplies.
- CLAIM-157 (the block criterion and its duration clause). “It lasts as long as that block persists” is stated over joint dynamics, which are estimated over an interval. Varley shows that for information measures the interval verdict and the moment verdict can differ in sign, so the duration clause inherits the same question of window. The network result also bears on the boundary clause: misinformative transitions concentrate on edges between communities, so the boundary of a block is where the macro description fails most often.
- CLAIM-161 (persistence as conserved coherence). If the coherence measure is an information quantity, “approximately conserved along its own trajectory” must tolerate negative local values that average out. The paper does not discuss persistence, so this is the record’s application, not the paper’s claim.
- THEORY-222 (causal emergence). Complements it: the average effective-information gain THEORY-222 holds can coexist with most informative micro transitions becoming misinformative at the macro scale. Nothing in THEORY-222 cites this paper, and nothing there needs correcting.
- THEORY-036. Cites Varley and Hoel, not this paper; no bearing.
- New THEORY. THEORY-268 holds the claim no existing THEORY holds: emergence measured as an expectation is a property of the ensemble of transitions, not of each moment.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- One four-node toy, chosen to display the effect, carries both framework results; one structural connectome carries the network result.
- Whether a negative local value marks a change in the system or a fluctuation is left open: “it does not necessarily define why a particular moment breaks from the long-term behaviour”.
- The local ΦID result depends on the choice of redundancy function (I_τsx); other redundancy functions are not tried.
- The general claim for every excess-entropy measure is argued, not proved, and two named frameworks are not computed.
- For a deterministic, invertible dynamics with a uniform state distribution every realised transition has e(x) = log N > 0, so flickering is possible in these frameworks, not guaranteed for every system; the paper does not state the conditions under which it must occur.
Open questions
- What fraction of moments may be misinformative before a form is no longer one system, and is there a principled window?
- Do misinformative transitions cluster in time (a breakdown) or scatter (noise)? A test on a system with a known regime change, such as the flock that scatters or the fibrillating heart the paper names, would separate the two.
- Do the coarse-graining and ΦID measures flicker at the same moments on a shared system?