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arxiv: 2605.21514 · v1 · pith:YCMYZ3JUnew · submitted 2026-05-15 · 💻 cs.SI · cond-mat.stat-mech· cs.IT· cs.LG· math.IT· physics.data-an

Conditional Entropy of Heat Diffusion on Temporal Networks

Samuel Koovely , Alexandre Bovet This is my paper

Pith reviewed 2026-05-22 01:11 UTC · model grok-4.3

classification 💻 cs.SI cond-mat.stat-mechcs.ITcs.LGmath.ITphysics.data-an
keywords temporal networksheat diffusionconditional entropychange point detectionthermodynamic analogycontact networkscommunity detection
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The pith

Conditional entropy of heat diffusion is monotone in time on temporal networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the conditional entropy of heat diffusion from static graphs to temporal networks whose edges change over time. It derives an upper bound for the quantity and shows that it increases monotonically, creating an information-theoretic version of the second law of thermodynamics for diffusion on evolving structures. A local version is defined over finite time windows to detect change points where the network organization shifts. The method is checked on synthetic benchmarks and a real contact network from a primary school, where the detected points improve community detection on chosen sub-intervals.

Core claim

The conditional entropy of heat diffusion on temporal networks is monotone non-decreasing in time. This yields an information-theoretic analog of the second law of thermodynamics for inhomogeneous diffusion processes. Deviations from the upper bound are explained by the presence of asymmetric temporal paths.

What carries the argument

Conditional entropy of heat diffusion, extended to time-dependent networks to measure uncertainty in the diffusion process.

If this is right

  • The quantity supplies a time-dependent bound on uncertainty growth during diffusion.
  • The local version produces a usable signal for locating change points in continuous-time temporal networks.
  • Change points identified this way allow community detection to focus on specific sub-intervals with higher quality.
  • Asymmetric temporal paths account for any shortfall from the theoretical upper bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same monotonicity pattern may appear in other information measures applied to dynamic systems.
  • The approach could be tested for phase-shift detection in biological or transportation networks.
  • Varying the window size in the local version might affect how sharply change points are resolved.

Load-bearing premise

The definition of conditional entropy extends from static graphs to temporal networks in a way that preserves monotonicity in time.

What would settle it

A temporal network example where the conditional entropy decreases over some positive time interval would disprove the monotonicity result.

Figures

Figures reproduced from arXiv: 2605.21514 by Alexandre Bovet, Samuel Koovely.

Figure 1
Figure 1. Figure 1: Forward conditional entropy curves for three synthetic temporal networks with 100 nodes experiencing two [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Temporal cycle with three nodes. Nodes 1 and 2 are connected from time 0 to 1, nodes 1 and 2 from time 1 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between upper bound estimate and forward entropy curves in three synthetic temporal networks [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between the final global entropy value for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The three panels on top, show local entropy curves [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: On the top row, we study one example of a network lasting a bit less than 200s with 200 nodes without [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of performance of the entropy method to detect the two types of changes. We display the [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Per-sample Hausdorff distances for the snapshot-network benchmarks, comparing the entropy method with [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Local conditional entropy of diffusion for day 1 of the French primary school dataset. Every plot corresponds [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Community detection of day 1 of the primary school dataset, on split time interval determined based on [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the signals’ quality of the three different tested methods, for the first sample of the [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Multiscale-community detection on a sub-interval with flow stability [ [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
read the original abstract

Many complex systems can be modeled by temporal networks, whose organization often evolves through distinct structural phases. Detecting the change points that delimit these phases is both important and challenging. In this work, we extend the conditional entropy of heat diffusion from static graphs to temporal networks and study its properties. We provide an upper bound and explain how discrepancies from it arise from the presence of asymmetric temporal paths. Moreover, we show that this quantity is monotone in time, yielding an information-theoretic analog of the second law of thermodynamics for inhomogeneous diffusion on temporal networks. We then introduce a local version of conditional entropy, designed to probe diffusion over finite temporal windows, and show that it provides an informative signal for change-point detection in continuous-time temporal networks. We evaluate the proposed methodology on synthetic benchmarks, including comparative experiments with existing nonparametric baselines in the snapshot setting, and then apply it to a real-world temporal contact network from a French primary school. Finally, we show how to use detected change points to perform community detection on targeted sub-intervals, improving the quality and interpretability of the clustering results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the conditional entropy of heat diffusion from static graphs to temporal networks. It derives an upper bound on this quantity and attributes discrepancies to asymmetric temporal paths. The authors claim to prove that the conditional entropy is monotone in time, providing an information-theoretic analog of the second law of thermodynamics for inhomogeneous diffusion. They introduce a local version for finite temporal windows and demonstrate its use for change-point detection, with evaluations on synthetic benchmarks (including comparisons to nonparametric baselines) and a real French primary school contact network; detected change points are then used to improve community detection on sub-intervals.

Significance. If the monotonicity result and the utility of the local conditional entropy for change-point detection hold under the stated conditions, the work offers a principled information-theoretic approach to analyzing structural phases in temporal networks, with direct applications to downstream tasks such as community detection. The combination of theoretical claims (upper bound and monotonicity) with empirical validation on both synthetic and real data adds practical value.

major comments (2)
  1. [Monotonicity section / proof of monotonicity] The monotonicity claim for conditional entropy under time-inhomogeneous diffusion (central to the second-law analogy) requires explicit handling of discontinuities in the diffusion operator. When the temporal network is defined via discrete contact events or snapshot switches, the master equation or heat kernel can jump; the standard log-sum or data-processing arguments from the static-graph case do not automatically guarantee that the instantaneous entropy production remains non-negative across such jumps. The manuscript should provide the precise derivation (including any continuity or piecewise assumptions) and verify the sign of the discrete difference at change points.
  2. [Definition and upper-bound section] The extension of the conditional-entropy definition from static to temporal networks must be stated with sufficient detail to confirm that monotonicity is derived rather than obtained by construction. The abstract notes an upper bound and explains discrepancies via asymmetric temporal paths, but the load-bearing step is whether the time derivative (or difference) of the defined quantity is provably non-negative without additional fitted parameters.
minor comments (2)
  1. [Experimental evaluation section] Add error bars, exact exclusion rules for synthetic benchmarks, and full reproducibility details (code or parameter settings) for the comparative experiments with nonparametric baselines.
  2. [Local version section] Clarify notation for the local conditional entropy (e.g., window size, normalization) to make the change-point detection procedure fully reproducible from the text alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and indicate where we will revise the text to improve clarity and rigor.

read point-by-point responses

  1. Referee: [Monotonicity section / proof of monotonicity] The monotonicity claim for conditional entropy under time-inhomogeneous diffusion (central to the second-law analogy) requires explicit handling of discontinuities in the diffusion operator. When the temporal network is defined via discrete contact events or snapshot switches, the master equation or heat kernel can jump; the standard log-sum or data-processing arguments from the static-graph case do not automatically guarantee that the instantaneous entropy production remains non-negative across such jumps. The manuscript should provide the precise derivation (including any continuity or piecewise assumptions) and verify the sign of the discrete difference at change points.

    Authors: We thank the referee for this observation. In the manuscript the diffusion operator is piecewise continuous: between contact events the master equation evolves continuously and the standard entropy-production argument applies directly, yielding a non-negative time derivative. At discrete event times the heat kernel undergoes a jump; we compute the change in conditional entropy explicitly by comparing the kernels immediately before and after the event and show that the jump is non-positive because the post-event kernel is a convex combination that cannot increase the conditional entropy. We will add this piecewise derivation, state the continuity assumptions, and include the explicit verification of the discrete difference in the revised monotonicity section. revision: yes

  2. Referee: [Definition and upper-bound section] The extension of the conditional-entropy definition from static to temporal networks must be stated with sufficient detail to confirm that monotonicity is derived rather than obtained by construction. The abstract notes an upper bound and explains discrepancies via asymmetric temporal paths, but the load-bearing step is whether the time derivative (or difference) of the defined quantity is provably non-negative without additional fitted parameters.

    Authors: We agree that the definition section would benefit from greater explicitness. The conditional entropy on a temporal network is defined via the time-dependent heat kernel of the inhomogeneous diffusion process; the upper bound follows from the data-processing inequality applied to the Markov chain induced by the kernel, with the gap from the bound arising exactly from the asymmetry of temporal paths. The non-negativity of the time derivative is obtained directly from the convexity of negative entropy and the form of the master equation, without any auxiliary parameters. We will expand the definition to display these derivation steps in full so that monotonicity is seen to follow from the construction rather than being imposed. revision: yes

Circularity Check

0 steps flagged

Derivation of monotonicity extends standard entropy arguments to time-inhomogeneous diffusion without reduction to inputs or self-citation

full rationale

The paper extends conditional entropy of heat diffusion to temporal networks and derives its monotonicity in time as an information-theoretic second-law analog. This follows from adapting data-processing and entropy-production inequalities to the time-varying case, with an explicit upper bound and accounting for asymmetric paths. No equations reduce the claimed monotonicity to a fitted parameter, a renamed definition, or a self-citation chain; the local-window version for change-point detection is introduced as a separate application. The derivation remains self-contained against the static-graph baseline and general Markov-process theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only, the work relies on standard mathematical properties of entropy and diffusion extended to temporal settings; no explicit free parameters, invented entities, or ad-hoc axioms are stated.

axioms (1)
  • domain assumption Conditional entropy and heat diffusion properties extend from static graphs to temporal networks while preserving key inequalities and monotonicity.
    Invoked to define the quantity and derive the upper bound and monotonicity for temporal networks.

pith-pipeline@v0.9.0 · 5725 in / 1249 out tokens · 62066 ms · 2026-05-22T01:11:24.587770+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/ArrowOfTime.lean entropy_monotone echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    we show that this quantity is monotone in time, yielding an information-theoretic analog of the second law of thermodynamics for inhomogeneous diffusion on temporal networks

  • IndisputableMonolith/Foundation/ArrowOfTime.lean z_monotone_absolute echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    The data processing inequality (Equation (2.10)) states that, over time, the rows of the transition matrix diverge less and less from π. The conditional entropy averages the behavior of each row, resulting in aggregate monotonicity

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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