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- 1. Graph Spectra through Network Complexity Measures Information Content of Eigenvalues Hector Zenil (joint work with Narsis Kiani and Jesper Tegn´er) Unit of Computational Medicine, Karolinska Institutet @ Department of Mathematics, Stockholm University Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 1 / 42
- 2. Outline: 1 Estimating Kolmogorov complexity 2 n-dimensional complexity 3 Graph Algorithmic Probability and Kolmogorov complexity of networks 4 Applications to complex networks and graph spectra Material mostly drawn from: 1 joint with Soler et al. Computability (2013). [1] 2 joint with Gauvrit et al. Behavior Research Methods (2013). [3] 3 Zenil et al. Physica A (2014). [4] 4 joint with Soler et al. PLoS ONE (2014). [6] 5 Zenil, Kiani and Tegn´er, LNCS 9044, (2015). [2] 6 Zenil and Tegn´er, Symmetry (forthcoming). 7 Zenil, Kiani and Tegn´er, Seminars in Cell and Developmental Biology (in revision). Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 2 / 42
- 3. Main goal Main goal throughout this talk: To study properties of graphs and networks with measures from information theory and algorithmic complexity. Table : Numerical calculations of (mostly) uncomputable functions: Busy Beaver problem upper semi-computable Kolmogorov-Chaitin complexity lower semi-computable Algorithmic Probability (Solomonoﬀ-Levin) upper semi-computable Bennett’s Logical Depth uncomputable Lower semi-computable: can be approximated from above. Upper semi-computable: can be approximated from below. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 3 / 42
- 4. The basic unit in Theoretical Computer Science The cell (the smallest unit of life) is to Biology what the Turing machine is to Theoretical Computer Science. Finite state diagram [A.M. Turing (1936)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 4 / 42
- 5. One machine for everything Computation (Turing-)universality (a) Turing proves that a M with input x can be encoded as an input M(x) for a machine U such that if M(x) = y then U(M(x)) = y for any Turing machine M. You do not need a computer for each diﬀerent task, only one! There is no distinction between software/hardware or data/program Together with Church’s thesis: Church-(Turing)’s thesis (b) Every eﬀectively computable function is computable by a Turing machine. Together the 2 suggest that: Anything can be programmed/simulated/emulated by a universal Turing machine. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 5 / 42
- 6. The undecidability of the Halting Problem The existence of the universal Turing machine U brings a fundamental G¨odel-type contradiction about the power of U (any universal machine): Let’s say we want to know whether a machine M will halt for input x. Assumption: We can program U in such a way that if M(x) halts then U(M(x)) = 0 otherwise U(M(x)) = 1. So U is a (halting) decider. Contradiction: Let M(x) = U(x), then U(U(x)) = 0, if and only if, U(x) = 1 and U(U(x)) = 1, if and only if, U(x) = 0. Therefore the assumption that we can know whether a Turing machine halts in general is not true. There is also a non-constructive proof using Cantor’s diagonalisation method. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 6 / 42
- 7. Computational irreducibility (1) Most fundamental irreducibility: If M halts for input x, you have to run either M(x) or U(M(x)) to know it, but if M does not halt, neither running M(x) or U(M(x)) will tell you that they do not halt. Most uncomputability results are of this type, you can know in one direction but not the other (e.g. when a string is random as we will see). (2) Secondary irreducibility (corollary): U(M(x)) can only produce time speedup on M(x) but not computation speed up (connected to time complexity, P = NP time results) in general, specially for (1). In other words, O(U(M(x))) ∼ O(M(x)), or O(U(M(x))) = c × O(M(x)), with c a constant. (2) is believed to be more pervasive than what (1) implies. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 7 / 42
- 8. Complexity and information content of strings Example (3 strings of length 40) a: 1111111111111111111111111111111111111111 b: 11001010110010010100111000101010100101011 c: 0101010101010101010101010101010101010101 According to Shannon (1948): (a) has minimum Entropy (only one micro-state). (b) has maximum Entropy (two micro-states with same frequency each). (c) has also maximum Entropy! (two micro-states with same frequency each). Shannon Entropy inherits from classical probability Shannon Entropy suﬀers of similar limitations: strings (b) and (c) have the same Shannon Entropy (same number of 0s and 1s) but they appear of very diﬀerent nature to us. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 8 / 42
- 9. Statistical v algorithmic Entropy rate can only ﬁx statistical regularities but not correlation Thue-Morse sequence: 01101001100101101001011001101001 Segment of π in binary: 0010010000111111011010101000100 Deﬁnition Kolmogorov(-Chaitin) complexity (1965,1966): KU(s) = min{|p|, U(p) = s} Algorithmic Randomness (also Martin L¨of and Schnorr) A string s is random if K(s) (in bits) ∼ |s|. Correlation versus causation Shannon Entropy is to correlation what Kolmogorov is to causation! Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 9 / 42
- 10. Example of an evaluation of K The string 01010101...01 can be produced by the following program: Program A: 1: n:= 0 2: Print n 3: n:= n+1 mod 2 4: Goto 2 The length of A (in bits) is an upper bound of K(010101...01) (+ the halting condition). Semi-computability of K Exhibiting a short version of a string is a suﬃcient test for non-randomness, but the lack of a short description (program) does not imply a suﬃcient test for randomness. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 10 / 42
- 11. The founding theorem of K complexity: Invariance to choice of U Do we measure K with programming language or universal TM U1 or U2? |KU1 (s) − KU2 (s)| < cU1,U2 It is not relevant in the limit, the diﬀerence is a constant that vanishes the longer the strings. Rate of convergence of K and the behaviour of c with respect to |s| The Invariance theorem in practice is a negative result The constant involved can be arbitrarily large, the theorem tells nothing about the convergence. Any estimating method of K is subject to it. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 11 / 42
- 12. Compression is Entropy rate not K Actual implementations of lossless compression have 2 main drawbacks and pitfalls: Lossless compression as entropy rate estimators Actual implementations of lossless compression algorithms (e.g. Lempev-Ziv, BZip2, PNG), seek for statistical regularities, repetitions in a sliding ﬁxed-length window of size w, hence entropy rate estimators up to block (micro-state) length w. Their success is only based on one side of the non-randomness test, i.e. low entropy = low K. Compressing short strings The compressor also adds the decompression instructions to the ﬁle. Any string shorter than say 100 bits is impossible to further compress or to get any meaningful ranking from compressing them (100 bps strings in structural molecular biology is long). Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 12 / 42
- 13. Alternative to lossless compression algorithms Figure : (originally Emile Borel’s inﬁnite monkey theorem): A monkey on a computer produces more structure by chance than a monkey on a typewriter. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 13 / 42
- 14. Algorithmic Probability (semi-measure, Levin’s Universal Distribution) Deﬁnition The classical probability of production of a bit string s among all 2n bit stings of size n (classical monkey theorem): Pr(s) = 1/2n (1) Deﬁnition Let U be a (preﬁx-free from Kraft’s inequality) universal Turing machine and p a program that produces s running on U, then m(s) = p:U(p)=s 1/2|p| < 1 (2) Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 14 / 42
- 15. The algorithmic Coding theorem Connection to K! The greatest contributor in the def. of m(s) is the shortest program p, i.e. K(s). The algorithmic Coding theorem describes the reverse connection between K(s) and m(s): Theorem K(s) = − log2(m(s)) + O(1) (3) Frequency and complexity are related If a string s is produced by many programs then there is also a short program that produces s (Thomas & Cover (1991)). [Solomonoﬀ (1964); Levin (1974); Chaitin (1976)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 15 / 42
- 16. The Coding Theorem Method (CTM) ﬂow chart Enumerate & run every TM ∈ (n, m) for increasing n and m (Busy Beaver values to determine halting time, otherwise informed runtime cutoﬀ value (see e.g. Calude & Stay, Most programs stop quickly or never halt, 2006). [Soler, Zenil et al, PLoS ONE (2014)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 16 / 42
- 17. Changes in computational formalism [H. Zenil and J-P. Delahaye, On the Algorithmic Nature of the World; 2010] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 17 / 42
- 18. Elementary Cellular Automata An elementary cellular automaton (ECA) is deﬁned by a local function f : {0, 1}3 → {0, 1}, Figure : Space-time evolution of a cellular automaton (ECA rule 30). f maps the state of a cell and its two immediate neighbours (range = 1) to a new cell state: ft : r−1, r0, r+1 → r0. Cells are updated synchronously according to f over all cells in a row. [Wolfram, (1994)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 18 / 42
- 19. Convergence in ECA classiﬁcation (CTM v Compress) Scatterplot of ECA classiﬁcation: CTM (x-axis) versus Compress (y-axis). [Soler-Toscano et al., Computability; 2013] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 19 / 42
- 20. Part II GRAPH ENTROPY, GRAPH ALGORITHMIC PROBABILITY AND GRAPH KOLMOGOROV COMPLEXITY Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 20 / 42
- 21. Graph Entropy deﬁnitions are not robust Several deﬁnitions (e.g. from molecular biology) if Graph Entropy have been proposed, e.g.: A complete graph has highest entropy H if deﬁned as containing all possible subgraphs up to the graph size, i.e. H(G) = − |G| i P(Gi ) log2 P(Gi ) where Gi is a subgraph of increasing size i in G. However, H(Adj(G)) = −P(Adj(G)) log2 P(Adj(G)) = 0 ! (and also all the adj matrices of all the subgraphs, so the sum would be 0 too !) Graph Entropy Complete and disconnected have then maximal and minimal entropy respectively. Alternative deﬁnitions include, for example, the number of bifurcations traversing the graph starting from any random node, etc. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 21 / 42
- 22. Graph Kolmogorov complexity (Physica A) Unlike Graph Entropy, Graph Kolmogorov complexity is very robust: complete graph: K ∼ log(|N|) E-R random graph: K ∼ |E| M. Gell-Mann (Nobel Prize 1969) thought that any reasonable measure of complexity of graphs should have both completely disconnected and completely connected graphs to have minimal complexity (The quark and the jaguar, 1994). Graph Kolmogorov complexity Complete and disconnected graphs with |N| nodes have low (algorithmic) information content. In a random graph every edge e ∈ E requires some information to be described. Both K(G) ∼ K(Adj(G)) ! Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 22 / 42
- 23. Numerical estimation of K(G) An labelled graph is uniquely represented by its adjacency matrix. So the question is What is the Kolmogorov complexity of an adjacency matrix? Figure : Two-dimensional Turing machines, also known as Turmites (Langton, Physica D, 1986). We will provide the deﬁnition of Kolmogorov complexity for unlabelled graphs later. [Zenil et al. Physica A, 2014] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 23 / 42
- 24. An Information-theoretic Divide-and-Conquer Algorithm! The Block Decomposition method uses the Coding Theorem method. Formally, we will say that an object c has (2D) Kolmogorov complexity: K2Dd×d (c) = (ru,nu)∈cd×d K2D(ru) + log2(nu) (4) where cd×d represents the set with elements (ru, nu), obtained from decomposing the object into (overlapping) blocks of d × d with boundary conditions. In each (ru, nu) pair, ru is one of such squares and nu its multiplicity. [Zenil et al., Two-Dimensional Kolmogorov Complexity and Validation of the Coding Theorem Method by Compressibility (2012)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 24 / 42
- 25. Classiﬁcation of ECA by BDM (= Km) and Compress Representative ECAs sorted by BDM (top row) and Compress (bottom row). [H. Zenil, F. Soler-Toscano, J.-P. Delahaye and N. Gauvrit, Two-Dimensional Kolmogorov Complexity and Validation of the Coding Theorem Method by Compressibility (2012)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 25 / 42
- 26. Complementary methods for diﬀerent object lengths The methods coexist and complement each other for diﬀerent string lengths (transitions are also smooth). method short strings long strings scalability time domain < 100 bits > 100 bits Lossless O(n) H compression × Coding Theorem O(exp) K method (CTM) × × CTM + Block Decomposition O(n) K → H method (BDM) Table : H stands for Shannon Entropy and K for Kolmogorov complexity. BDM can therefore be taken as an improvement to (Block) Entropy rate for a ﬁxed block size. For CTM: http://www.complexitycalculator.com Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 26 / 42
- 27. Graph algorithmic probability Works on directed and undirected graphs. Torus boundary conditions provide a solution to the boundaries problem. Overlapping sub matrices avoids the problem of not permutation invariance but leads to overﬁtting. The best option is to recursively divide into square matrices for which exact complexity estimations are known. [Zenil et al. Physica A (2014)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 27 / 42
- 28. K and graph automorphism group (Physica A) Figure : Left: An adjacency matrix is not a graph invariant yet isomorphic graphs have similar K. Right: Graphs with large automorphism group size (group symmetry) have lower K. This correlation suggests that the complexity of unlabelled graphs is captured by the complexity of their adjacency matrix (which is a labelled graph object). Indeed, in Zenil et al. LNCS we show that the complexity of a labelled graph is a good approximation to its unlabelled graph complexity. [Zenil et al. Physica A (2014)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 28 / 42
- 29. Unlabelled Graph Complexity The proof sketch of the labelled graph complexity ∼ unlabelled graph complexity uses the fact that there is an algorithm (e.g. brute force) of ﬁnite (small) size that produces any isomorphic graph from any other. Yet, one can deﬁne Graph unlabelled Kolmogorov complexity as follows: Deﬁnition Graph Unlabelled Kolmogorov Complexity: Let Adj(G) be the adjacency matrix of G and Aut(G) its automorphism group, then, K(G) = min{K(Adj(G))|Adj(G) ∈ A(Aut(G))} where A(Aut(G)) is the set of adjacency matrices of all G ∈ Aut(G). (The problem is believed to be in NP but not in NP-complete). [Zenil, Kiani and Tegn´er (forthcoming)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 29 / 42
- 30. Graph automorphisms and algorithmic complexity by BDM Classifying (and clustering) ∼ 250 graphs (no Aut(G) correction) with diﬀerent topological properties by K (BDM): [Zenil et al. Physica A (2014)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 30 / 42
- 31. Graph deﬁnitions Deﬁnition Dual graph: A dual graph of a plane graph G is a graph that has a vertex corresponding to each face of G, and an edge joining two neighboring faces for each edge in G. Deﬁnition Graph spectra: The set of graph eigenvalues of the adjacency matrix is called the spectrum of the graph. The Laplacian matrix of a graph is sometimes also known as the graph’s spectrum. Deﬁnition Cospectral graphs: Two graphs are called isospectral or cospectral if they have the same spectra. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 31 / 42
- 32. Testing compression and BDM on dual graphs [Zenil et al. Physica A (2014)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 32 / 42
- 33. H, compression and BDM on cospectral graphs Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 33 / 42
- 34. Quantifying Loss of Information in Network-based Dimensionality Reduction Techniques Figure : Flowchart of Quantifying Loss of Information in Network-based Dimensionality Reduction Techniques. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 34 / 42
- 35. Methods of (Algorithmic) Information Theory in network dimensionality reduction Figure : Information content of graph spectra and graph motif analysis. Information content of 16 graphs of diﬀerent types and the information content of their graph spectra approximated by Bzip2, Compress and BDM. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 35 / 42
- 36. Methods of (Algorithmic) Information Theory in network dimensionality reduction Figure : Information content progression of sparsiﬁcation. Information loss after keeping from 20 to 80% of the graph edges (100% corresponds to the information content of the original graph). Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 36 / 42
- 37. Methods of (Algorithmic) Information Theory in network dimensionality reduction Figure : Plot comparing all methods as applied to 4 artiﬁcial networks. The information content measured as normalized complexity with two diﬀerent lossless compression algorithms was used to assess the sparsiﬁcation, graph spectra and graph motif methods. The 6 networks from the Mendes DB are of the same size and each method displays diﬀerent phenomena. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 37 / 42
- 38. Eigenvalue information weight eﬀect on graph spectra In graph spectra either the largest eigenvalue (λ1) is only considered, or all eigenvalues (λ1...λn) are given the same weight. Yet eigenvalues capture diﬀerent properties and are sensitive to graph speciﬁty, e.g. in a complete graph λ1 provides the graph size. Figure : Graph spectra can be plotted in an n-dimensional space where n is the graph node size (and number of Eigenvalues). When a graph G evolves its spectra changes from Spec1(G) to Spec2(G ) as in the ﬁgure, but if not all eigenvalues are equally important hence the distance d(Spec1(G), Spec2(G )) is on a manifold and not on Euclidian space. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 38 / 42
- 39. Eigenvalues in Graph Spectra are not all the same Nor their magnitude is of any relevance (e.g. taking the largest one only): Figure : Statistics (ρ) and p-value plots between graph complexity (BDM) and largest, second largest and smallest Eigenvalues of 204 diﬀerent graph classes including 4913 graphs. Clearly the graph class complexity correlates in diﬀerent ways to diﬀerent Eigenvalues. [Source: Zenil, Kiani and Tegn´er LNCS (2015)] Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 39 / 42
- 40. Eigenvalues of evolving networks Most informative eigenvalues to characterize a family of networks and individuals in such a family: Figure : The complexity of graph versus the complexity of the list of eigenvalues per position (rows) provides information about the amount and kind of information stored in each eigenvalue, and the maximum entropy of rows also identiﬁes the eigenvalue that best characterize the changes in the evolving network that otherwise display very little topological changes. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 40 / 42
- 41. Entropy and complexity of Eigenvalue families Let n be the number of datapoints of an evolving graph (or a family of graphs to study), H the Shannon Entropy, K Kolmogorov complexity and KS the Kolmogorov-Sinai Entropy (∼ interval Shannon Entropy), then we are interested in: H(Spec(Gi )), K(Spec(Gi )), KS(Spec(Gi )) where i ∈ {1, . . . , n} to study the Eigenvalue behavior with respect to KBDM(Gi ), and KS(λ1 1, λ2 1, . . . , λn 1) . . . KS(λ2 2, λ2 2, . . . , λn 2) . . . maximizing the diﬀerences between Gi hence characterizing G in time. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 41 / 42
- 42. Part 2 Summary 1 We have a sound, robust and native 2-dimensional complexity measure applicable to graphs and networks. 2 The method is scalable, e.g. in 3 dimensions, I call CTM3D, the “3D printing complexity measure” because as you can see it only requires the Turing machine to operate in a 3D grid, eﬀectively the probability of a random computer program to print a 3D object! 3 The deﬁned graph complexity measure captures algebraic, topological (and, forthcoming, even physical properties) of graphs and networks. 4 There is a potential for applications in network and synthetic biology. 5 The method may prove to be very eﬀective at giving proper weight to eigenvalues and even shedding light on their meaning and information content. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 42 / 42
- 43. F. Soler-Toscano, H. Zenil, J.-P. Delahaye and N. Gauvrit, Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures, Computability, vol. 2, no. 2, pp. 125–140, 2013. H. Zenil, N.A. Kiani, J. Tegn´er, Numerical Investigation of Graph Spectra and Information Interpretability of Eigenvalues, IWBBIO 2015, LNCS 9044, pp. 395–405. Springer, 2015. N. Gauvrit, H. Zenil, F. Soler-Toscano and J.-P. Delahaye, Algorithmic complexity for short binary strings applied to psychology: a primer, Behavior Research Methods, vol. 46-3, pp 732-744, 2013. H. Zenil, F. Soler-Toscano, K. Dingle and A. Louis, Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks, Physica A: Statistical Mechanics and its Applications, vol. 404, pp. 341–358, 2014. J.-P. Delahaye and H. Zenil, Numerical Evaluation of the Complexity of Short Strings, Applied Mathematics and Computation, 2011. F. Soler-Toscano, H. Zenil, J.-P. Delahaye and N. Gauvrit, Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines, PLoS ONE, 9(5): e96223, 2014. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 42 / 42
- 44. J.P. Delahaye and H. Zenil, On the Kolmogorov-Chaitin complexity for short sequences, in Cristian Calude (eds), Complexity and Randomness: From Leibniz to Chaitin, World Scientiﬁc, 2007. G.J. Chaitin A Theory of Program Size Formally Identical to Information Theory, J. Assoc. Comput. Mach. 22, 329-340, 1975. R. Cilibrasi and P. Vit´anyi, Clustering by compression, IEEE Trans. on Information Theory, 51(4), 2005. A.N. Kolmogorov, Three approaches to the quantitative deﬁnition of information Problems of Information and Transmission, 1(1):1–7, 1965. L. Levin, Laws of information conservation (non-growth) and aspects of the foundation of probability theory, Problems of Information Transmission, 10(3):206–210, 1974. R.J. Solomonoﬀ. A formal theory of inductive inference: Parts 1 and 2, Information and Control, 7:1–22 and 224–254, 1964. S. Wolfram, A New Kind of Science, Wolfram Media, 2002. Zenil, Kiani, Tegn´er (Karolinska Institutet) Information Content of Eigenvalues May 27, 2015 42 / 42

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