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The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy^{ †}

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## Abstract

**:**

Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it doesn’t bother you any more.Arnold Sommerfeld [1].

## 1. Classical Thermodynamics and Related Work

- A measure of statistical disorder
- Some quantity or property that increases but never decreases
- A process that defines the direction of time
- A measure of statistical information

## 2. Notation and Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

#### Graph Entropy

## 3. Algorithmic Information Dynamics

#### 3.1. Approximations to Algorithmic Complexity

#### 3.2. Block Decomposition Method

#### 3.3. Normalised BDM

#### 3.4. A Reprogrammability Calculus

**Definition**

**10.**

**Definition**

**11.**

## 4. The Thermodynamics of Computer Programs

#### Graphs as Computer Programs

## 5. Principle of Maximum Algorithmic Randomness (MAR)

#### 5.1. Maximal Algorithmic Randomness Preferential Attachment (MARPA) Algorithm

**Definition**

**12.**

#### 5.2. Supremacy of Algorithmic Maxent

**Theorem**

**1.**

**Theorem**

**2.**

#### 5.3. Numerical Examples

- Start with an empty or complete graph G.
- Perform all possible perturbations for a single operation (e.g., edge deletion or addition) on G to produce ${G}^{\prime}$.
- Keep only the perturbation that maximised algorithmic randomness, i.e., $C\left({G}^{\prime}\right)\ge C\left(G\right)$.
- Set $G:={G}^{\prime}$
- Repeat 1 until final target size is reached or $C\left(G\right)>C\left({G}^{\prime}\right)$.

#### Time Complexity and Terminating Criterion

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The thermodynamics of computer programming. (

**Top**) The programs producing simple versus random data have different reprogrammability properties. If repurposed to generate programs to print blocks of 0s, we only need a single intervention in the generative program of (

**1**), changing 1 to 0 inside the Print instruction indicating that 200 0s be printed instead of 200 1s. In contrast, asking a program that prints a random binary string s to print only 0s will require on average $\left|s\right|/2$ interventions to manually change every bit 1 to 0. Random perturbations can be seen as the exploration of the possible paths through which an object may evolve over time.

**Figure 2.**Graphs as produced by programs. All real-world networks lie between the extreme cases of being as simple as a complete graph whose algorithmic complexity C is minimal and grows by only $log\left|V\right(k\left)\right|$, and a random (also statistically random and thus E-R) graph whose algorithmic complexity is maximal and grows by its number of edges $\left|E\right(\text{E-R}\left)\right|$. If we ask what it takes to change the program producing k to produce $\text{E-R}$ and vice versa, in a random graph, any single algorithmic-random node or edge removal does not entail a major change in the program-size of its generating program, which is similar in size to the random graph itself, i.e., $\left|E\right(G\left)\right|$. The curve shows how, without loss of generality, the reprogramming capability of networks, as produced by computer programs, produces an asymmetry imposed by algorithmic complexity and reminiscent of traditional thermodynamics as based on classical probability. A maximally random network has only positive (blue) elements (Figure 3 and Figure 4) because there exists no perturbation that can increase the randomness of the network either by removing a node or an edge, as it is already random (and thus non-deterministic). Thus, changing its (near) minimal program-size length by edge or node removal is slow. However, a simple graph may have elements that incline its program-size length toward randomness. In each extreme case (simple vs. random), the distribution of sorted elements capable of shifting in each direction is shown in the form of what we call “signatures”, both for algorithmically random edge and node removal. The highest reprogrammability point is the place where a graph has as many elements to steer it in one direction as in the other.

**Figure 3.**Algorithmic Maxent in the generation of graphs (without loss of generality, no other restrictions are imposed). The paths to statistical and algorithmic randomness are different, and they determine different principles for different purposes. Algorithmic Maxent tells apart recursive cases from non-recursive (thus algorithmic random) ones. Classical Maxent quantifies statistical randomness but its algorithmic refinement quantifies both statistical and algorithmic randomness. This opens up the range of possibilities for moving toward and reaching a random graph, by not only considering whether it is random-looking but also whether it is actually algorithmically random. Because the worse case for algorithmic Maxent is not to distinguish recursive cases (due to, e.g., semi-computability) its worse performance is to retrieve the same Gibbs measure and produce the same ensemble as classical Maxent would do.

**Figure 4.**Reduction of the Gibbs/Boltzmann distribution. While objects may appear maximally statistically random in all other properties but the ones constrained by a problem of interest for which Maxent is relevant, there may actually be of different nature and some may actually be algorithmically random while others recursively generated hence they should not be placed in the same distribution. The original graph G has a shortest description $minG$ and perturbations to either the original and compressed version can lead to randomised graphs for different purposes. Some of them will have maximum entropy denoted by $max{G}_{n}^{\prime}$ but among them only some will also be of $maxK\left(G\right)$.

**Figure 5.**(

**Top left**) One can produce a MAR graph starting from an empty graph and adding one edge at a time (see Figure 6) or one can start from a complete graph and start deleting edge by edge keeping only those that maximise the algorithmic randomness of the resulting graph. (

**Bottom left**) Following this process, MAR graphs top E-R graphs meaning BDM effectively separate low algorithmic complexity (algorithmic randomness) from high entropy (statistical randomness), where entropy would simply be blind collapsing all recursive and non-recursive cases. (

**Right**) degree distribution comparison between E-R and MAR graphs.

**Figure 6.**(

**Top**) A MAR graph constructed by adding one by one every possible edge and keeping only those that maximise the uncompressibility/algorithmic complexity of the graph according to BDM. Shown are only eight steps from 10 edges to 150. (

**Bottom**) MAR graphs versus ensemble of pseudo-randomly generated E-R graphs (with edge density 0.5) versus the ZK graph [15] designed to have minimal adjacency matrix entropy but maximal degree sequence entropy.

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**MDPI and ACS Style**

Zenil, H.; Kiani, N.A.; Tegnér, J. The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy. *Entropy* **2019**, *21*, 560.
https://doi.org/10.3390/e21060560

**AMA Style**

Zenil H, Kiani NA, Tegnér J. The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy. *Entropy*. 2019; 21(6):560.
https://doi.org/10.3390/e21060560

**Chicago/Turabian Style**

Zenil, Hector, Narsis A. Kiani, and Jesper Tegnér. 2019. "The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy" *Entropy* 21, no. 6: 560.
https://doi.org/10.3390/e21060560