# Multiscale Information Theory and the Marginal Utility of Information

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Information

- Monotonicity: The information in a subset U that is contained in a subset V cannot have more information than V, that is, $U\subset V\Rightarrow H(U)\le H(V)$.
- Strong subadditivity: Given two subsets, the information contained in both cannot exceed the information in each of them separately minus the information in their intersection:$$H(U\cup V)\le H(U)+H(V)-H(U\cap V).$$

- Microcanonical or Hartley entropy: For a system with a finite number of joint states, ${H}_{0}(U)=logm$ is an information function, where m is the number of joint states available to the subset U of components. Here, information content measures the number of yes-or-no questions which must be answered to identify one joint state out of m possibilities.
- Shannon entropy: For a system characterized by a probability distribution over all possible joint states, $H(U)=-{\sum}_{i=1}^{m}{p}_{i}log{p}_{i}$ is an information function, where ${p}_{1},\dots ,{p}_{m}$ are the probabilities of the joint states available to the components in U [9]. Here, information content measures the number of yes-or-no questions which must be answered to identify one joint state out of all the joint states available to U, where more probable states can be identified more concisely.
- Tsallis entropy: The Tsallis entropy [51,52] is a generalization of the Shannon entropy with applications to nonextensive statistical mechanics. For the same setting as in Shannon entropy, Tsallis entropy is defined as ${H}_{q}(U)=-{\sum}_{i=1}^{m}{p}_{i}^{q}({p}_{i}^{1-q}-1)/(1-q)$ for some parameter $q\ge 0$. Shannon entropy is recovered in the limit $q\to 1$. Tsallis entropy is an information function for $q\ge 1$ (but not for $q<1$); this follows from Proposition 2.1 and Theorem 3.4 of [53].
- Logarithm of period: For a deterministic dynamic system with periodic behavior, an information function $L(U)$ can be defined as the logarithm of the period of a set U of components (i.e., the time it takes for the joint state of these components to return to an initial joint state) [54]. This information function measures the number of questions which one should expect to answer in order to locate the position of those components in their cycle.
- Vector space dimension: Consider a system of n components, each of whose state is described by a real number. Then the joint states of any subset U of $m\le n$ components can be described by points in some linear subspace of ${\mathbb{R}}^{m}$. The minimal dimension $d(U)$ of such a subspace is an information function, equal to the number of coordinates one must specify in order to identify the joint state of U.
- Matroid rank: A matroid consists of a set of elements called the ground set, together with a rank function that takes values on subsets of the ground set. Rank functions are defined to include the monotonicity and strong subadditivity properties [55], and generalize the notion of vector subspace dimension. Consequently, the rank function of a matroid is an information function, with the ground set identified as the set of system components.

## 3. Scale

## 4. Systems

- A finite set A of components,
- An information function ${H}_{\mathcal{A}}$, giving the information in each subset $U\subset A$,
- A scale function ${\sigma}_{\mathcal{A}}$, giving the intrinsic scale of each compoent $a\in A$.

- Example
**A**: Three independent components. Each component is equally likely to be in state 0 or state 1, and the system as a whole is equally likely to be in any of its eight possible states. - Example
**B**: Three completely interdependent components. Each component is equally likely to be in state 0 or state 1, but all three components are always in the same state. - Example
**C**: Independent blocks of dependent components. Each component is equally likely to take the value 0 or 1; however, the first two components always take the same value, while the third can take either value independently of the coupled pair. - Example
**D**: The $2+1$ parity bit system. The components can exist in the states 110, 101, 011, or 000 with equal probability. In each state, each component is equal to the parity (0 if even; 1 if odd) of the sum of the other two. Any two of the component are statistically independent of each other, but the three as a whole are constrained to have an even sum.

## 5. Multiscale Information Theory

#### 5.1. Dependencies

#### 5.2. Information Quantity in Dependencies

**D**, for which the tertiary shared information $I(a;b;c)$ is negative (Figure 2D). Such negative values appear to capture an important property of dependencies, but their interpretation is the subject of continuing discussion [34,36,37,62,63].

#### 5.3. Scale-Weighted Information

**C**(Figure 2C), the bit of information that applies to components a and c has scale 2, while the bit applying to component b has scale 1.

#### 5.4. Independence and Complete Interdependence

**C**, components a, b, and c are independent. This definition generalizes standard notions of independence in information theory, linear algebra, and matroid theory.

**C**, the subsystem comprised of components a and c is independent of the subsystem comprised of component b.

**C**, the set $\{a,c\}$ is completely interdependent, and can be replaced by a single component of scale two. We show in Appendix B that replacements of this kind preserve all relevant quantities of information and scale.

## 6. Complexity Profile

- Conservation law: The area under $C(y)$ is equal to the total scale-weighted information of the system, and is therefore independent of the way the components depend on each other [13]:$${\int}_{0}^{\infty}C(y)\phantom{\rule{0.277778em}{0ex}}dy=S({\mathfrak{D}}_{\mathcal{A}}).$$
- Total system information: At the lowest scale $y=0$, $C(y)$ corresponds to the overall joint information: $C(0)=H(A)$. For physical systems with the Shannon information function, this is the total entropy of the system, in units of information rather than the usual thermodynamic units.
- Additivity: If a system $\mathcal{A}$ is the union of two independent subsystems $\mathcal{B}$ and $\mathcal{C}$, the complexity profile of the full system is the sum of the profiles for the two subsystems, ${C}_{\mathcal{A}}(y)={C}_{\mathcal{B}}(y)+{C}_{\mathcal{C}}(y)$. We prove this property Appendix D.

## 7. Marginal Utility of Information

- (i)
- $0\le I(d;V)\le H(V)$ for all subsets $V\subset A$.
- (ii)
- For any pair of nested subsets $W\subset V\subset A$, $0\le I(d;V)-I(d;W)\le H(V)-H(W)$.
- (iii)
- For any pair of subsets $V,W\subset A$,$$I(d;V)+I(d;W)-I(d;V\cup W)-I(d;V\cap W)\le H(V)+H(W)-H(V\cup W)-H(V\cap W).$$

- 1.
- Conservation law: The total area under the curve $M(y)$ equals the total scale-weighted information of the system:$${\int}_{0}^{\infty}M(y)\phantom{\rule{0.166667em}{0ex}}dy=S({\mathfrak{D}}_{\mathcal{A}}).$$
- 2.
- Total system information: The marginal utility vanishes for information values larger than the total system information, $M(y)=0$ for $y>H(A)$, since, for higher values, the system has already been fully described.
- 3.
- Additivity: If $\mathcal{A}$ separates into independent subsystems $\mathcal{B}$ and $\mathcal{C}$, then$${U}_{\mathcal{A}}(y)=\underset{\begin{array}{c}{y}_{1}+{y}_{2}=y\\ {y}_{1},{y}_{2}\ge 0\end{array}}{max}\left({U}_{\mathcal{B}}({y}_{1})+{U}_{\mathcal{C}}({y}_{2})\right).$$$${M}_{\mathcal{A}}(y)=\underset{\begin{array}{c}{y}_{1}+{y}_{2}=y\\ {y}_{1},{y}_{2}\ge 0\end{array}}{min}max\left\{{M}_{\mathcal{B}}({y}_{1}),{M}_{\mathcal{C}}({y}_{2})\right\}.$$$$\tilde{f}(x)=max\{y:f(y)\le x\}.$$$${\tilde{M}}_{\mathcal{A}}(x)={\tilde{M}}_{\mathcal{B}}(x)+{\tilde{M}}_{\mathcal{C}}(x).$$

**D**, for which the marginal utility is

**D**, in contrast to the maximal independence and maximal interdependence in Examples

**A**and

**B**, respectively (Figure 5). For an N-component generalization of Example

**D**, in which each component acts as a parity bit for all others, we show in Appendix F that the MUI is given by

## 8. Reflection Principle for Systems of Independent Blocks

**C**is such a system, in that it can be partitioned into independent subsystems with component sets $\{a,c\}$ and $\{b\}$, and each of these sets is completely interdependent. We show in Appendix B that any set of completely interdependent components can be replaced by a single component, with scale equal to the sum of the scales of the replaced components, without altering the complexity profile or MUI. Thus, for systems of independent blocks, each block can be collapsed into a single component, whereupon Equation (29) applies and the reflection principle holds.

**D**, and, more generally, for a class of systems that exhibit negative information, as shown in Equation (26).

## 9. Application to Noisy Voter Model

- (i)
- $0\le {I}_{n}\le {H}_{n}$ for all $n\in \{1,\dots ,N\}$,
- (ii)
- $0\le {I}_{n}-{I}_{n-1}\le {H}_{n}-{H}_{n-1}$ for all $n\in \{1,\dots ,N\}$,
- (iii)
- ${I}_{n}+{I}_{m}-{I}_{n+m-\ell}-{I}_{\ell}\le {H}_{n}+{H}_{m}-{H}_{n+m-\ell}-{H}_{\ell}$ for all $n,m,\ell \in \{1,\dots ,N\}$.

## 10. Discussion

#### 10.1. Potential Applications

#### 10.2. Multivariate and Multiscale Information

#### 10.3. Relation of the MUI to Other Measures

#### 10.4. Multiscale Requisite Variety

#### 10.5. Mechanistic versus Informational Dependencies

## 11. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Total Scale-Weighted Information

**Theorem**

**A1.**

**Proof.**

**Theorem**

**A2.**

**Proof.**

## Appendix B. Complete Interdependence and Reduced Representations

**Lemma**

**A1.**

**Proof.**

**Theorem**

**A3.**

**Proof.**

## Appendix C. Properties of Independence

**Definition**

**A1.**

**Lemma**

**A2.**

**Proof.**

**Theorem**

**A4.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

**Theorem**

**A5.**

**Proof.**

## Appendix D. Additivity of the Complexity Profile

**Theorem**

**A6.**

**Proof.**

## Appendix E. Additivity of Marginal Utility of Information

- (i)
- $0\le I(d;V)\le H(V)$ for all $V\subset A$.
- (ii)
- For any $W\subset V\subset A$,$$0\le I(d;V)-I(d;W)\le H(V)-H(W).$$
- (iii)
- For any $V,W\subset A$,$$I(d;V)+I(d;W)-I(d;V\cup W)-I(d;V\cap W)\le H(V)+H(W)-H(V\cup W)-H(V\cap W).$$
- (iv)
- $I(d;A)\le y$.

**Lemma**

**A5.**

**Proof.**

**Lemma**

**A6.**

**Proof.**

**Theorem**

**A7.**

**Proof.**

- (1)
- $I(\widehat{d};V)>I(\widehat{d};V\cap B)+I(\widehat{d};V\cap C)+\u03f5,$
- (2)
- $I(\widehat{d};V)>I(\widehat{d};W)+\u03f5$, for all $W\subset V,W\notin S$.

- If either
- −
- none of V, W, $V\cup W$ and $V\cap W$ belong to S,
- −
- all of V, W, $V\cup W$ and $V\cap W$ belong to S,
- −
- V and $V\cup W$ belong to S while W and $V\cap W$ do not, or
- −
- W and $V\cup W$ belong to S while V and $V\cap W$ do not,

then the difference on the left-hand side of Constraint (iii) has the same value for $d=\widehat{d}$ and $d=\tilde{d}$—that is, the changes in each term cancel out in the difference. Thus Constraint (iii) is satisfied for $\tilde{d}$ since it is satisfied for $\widehat{d}$. - If V, W, and $V\cup W$ belong to S while $V\cap W$ does not, then$$\begin{array}{c}I(\tilde{d};V)+I(\tilde{d};W)-I(\tilde{d};V\cup W)-I(\tilde{d};V\cap W)\hfill \\ \phantom{\rule{4.em}{0ex}}\phantom{\rule{4.em}{0ex}}\phantom{\rule{4.em}{0ex}}\phantom{\rule{4.em}{0ex}}=I(\widehat{d};V)+I(\widehat{d};W)-I(\widehat{d};V\cup W)-I(\widehat{d};V\cap W)-\u03f5.\hfill \end{array}$$The left-hand side of Constraint (iii) therefore decreases when moving from $d=\widehat{d}$ to $d=\tilde{d}$. So Constraint (iii) is satisfied for $\tilde{d}$ since it is satisfied for $\widehat{d}$.
- The nontrivial case is that $V\cup W$ belongs to S while V, W and $V\cap W$ do not. Then$$\begin{array}{cc}I(\tilde{d};V)+I(\tilde{d};W)-I(\tilde{d};V\cup W)\hfill & -I(\tilde{d};V\cap W)\hfill \\ & =I(\widehat{d};V)+I(\widehat{d};W)-\left(I(\widehat{d};V\cup W)-\u03f5\right)-I(\widehat{d};V\cap W).\hfill \end{array}$$By the definition of S and condition (1) on $\u03f5$, we have$$\begin{array}{cc}\hfill I(\widehat{d};V\cup W)-\u03f5& >I\left(\widehat{d};(V\cup W)\cap B\right)+I\left(\widehat{d};(V\cup W)\cap C\right)\hfill \\ \hfill I(\widehat{d};V)& =I(\widehat{d};V\cap B)+I(\widehat{d};V\cap C)\hfill \\ \hfill I(\widehat{d};W)& =I(\widehat{d};W\cap B)+I(\widehat{d};W\cap C)\hfill \\ \hfill I(\widehat{d};V\cap W)& =I\left(\widehat{d};(V\cap W)\cap B\right)+I\left(\widehat{d};(V\cap W)\cap C\right).\hfill \end{array}$$Substituting into (A38) we have$$\begin{array}{c}I(\tilde{d};V)+I(\tilde{d};W)-I(\tilde{d};V\cup W)-I(\tilde{d};V\cap W)\hfill \\ \phantom{\rule{1.em}{0ex}}<I(\widehat{d};V\cap B)+I(\widehat{d};W\cap B)\hfill \\ \phantom{\rule{2.em}{0ex}}-I\left(\widehat{d};(V\cup W)\cap B\right)-I\left(\widehat{d};(V\cap W)\cap B\right)\hfill \\ \phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}+I(\widehat{d};V\cap C)+I(\widehat{d};W\cap C)\hfill \\ \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}-I\left(\widehat{d};(V\cup W)\cap C\right)-I\left(\widehat{d};(V\cap W)\cap C\right).\hfill \end{array}$$Applying Constraint (iii) on $\widehat{d}$ twice to the right-hand side above, we have$$\begin{array}{c}I(\tilde{d};V)+I(\tilde{d};W)-I(\tilde{d};V\cup W)-I(\tilde{d};V\cap W)\hfill \\ \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}<H(V\cap B)+H(W\cap B)-H\left((V\cup W)\cap B\right)-H\left((V\cap W)\cap B\right)\\ \hfill \phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}+H(V\cap C)+H(W\cap C)-H\left((V\cup W)\cap C\right)-H\left((V\cap W)\cap C\right).\end{array}$$But Lemma A2 implies that $H(Z\cap B)+H(Z\cap C)=H(Z)$ for any subset $Z\subset A$. We apply this to the sets V, W, $V\cup W$ and $V\cap W$ to simplify the right-hand side above, yielding$$I(\tilde{d};V)+I(\tilde{d};W)-I(\tilde{d};V\cup W)-I(\tilde{d};V\cap W)<H(V)+H(W)-H(V\cup W)-H(V\cap W),$$

**Theorem**

**A8.**

**Proof.**

- (v)
- $I(d;V)=I(d;V\cap B)+I(d;V\cap C)$,

**Theorem**

**A9.**

**Proof.**

**Case**

**A1.**

**Case**

**A2.**

**Case**

**A3.**

**Corollary**

**A1.**

**Proof.**

## Appendix F. Marginal Utility of Information for Parity Bit Systems

**D**as the case $N=3$. More generally, this family includes systems of $N-1$ independent random bits together with one parity bit.

- (i)
- $0\le {I}_{n}\le {H}_{n}$ for all $n\in \{1,\dots ,N\}$,
- (ii)
- $0\le {I}_{n}-{I}_{n-1}\le {H}_{n}-{H}_{n-1}$ for all $n\in \{1,\dots ,N\}$,
- (iii)
- ${I}_{n}+{I}_{m}-{I}_{n+m-\ell}-{I}_{\ell}\le {H}_{n}+{H}_{m}-{H}_{n+m-\ell}-{H}_{\ell}$ for all $n,m,\ell \in \{1,\dots ,N\}$,
- (iv)
- ${I}_{N}\le y$.

## References

- Bar-Yam, Y. Dynamics of Complex Systems; Westview Press: Boulder, CO, USA, 2003. [Google Scholar]
- Haken, H. Information and Self-Organization: A Macroscopic Approach to Complex Systems; Springer: New York, NY, USA, 2006. [Google Scholar]
- Miller, J.H.; Page, S.E. Complex Adaptive Systems: An Introduction to Computational Models of Social Life; Princeton University Press: Princeton, NJ, USA, 2007. [Google Scholar]
- Boccara, N. Modeling Complex Systems; Springer: New York, NY, USA, 2010. [Google Scholar]
- Newman, M.E.J. Complex Systems: A Survey. Am. J. Phys.
**2011**, 79, 800–810. [Google Scholar] [CrossRef] - Kwapień, J.; Drożdż, S. Physical approach to complex systems. Phys. Rep.
**2012**, 515, 115–226. [Google Scholar] [CrossRef] - Sayama, H. Introduction to the Modeling and Analysis of Complex Systems; Open SUNY: Bunghamton, NY, USA, 2015. [Google Scholar]
- Sethna, J.P. Statistical Mechanics: Entropy, Order Parameters, and Complexity; Oxford University Press: Oxford, UK, 2006. [Google Scholar]
- Shannon, C. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; Wiley: Hoboken, NJ, USA, 1991. [Google Scholar]
- Prokopenko, M.; Boschetti, F.; Ryan, A.J. An information-theoretic primer on complexity, self-organization, and emergence. Complexity
**2009**, 15, 11–28. [Google Scholar] [CrossRef] - Gallagher, R.G. Information Theory and Reliable Communication; Wiley: Hoboken, NJ, USA, 1968. [Google Scholar]
- Bar-Yam, Y. Multiscale complexity/entropy. Adv. Complex Syst.
**2004**, 7, 47–63. [Google Scholar] [CrossRef] - Bar-Yam, Y. Multiscale variety in complex systems. Complexity
**2004**, 9, 37–45. [Google Scholar] [CrossRef] - Bar-Yam, Y.; Harmon, D.; Bar-Yam, Y. Computationally tractable pairwise complexity profile. Complexity
**2013**, 18, 20–27. [Google Scholar] [CrossRef] - Metzler, R.; Bar-Yam, Y. Multiscale complexity of correlated Gaussians. Phys. Rev. E
**2005**, 71, 046114. [Google Scholar] [CrossRef] [PubMed] - Gheorghiu-Svirschevski, S.; Bar-Yam, Y. Multiscale analysis of information correlations in an infinite-range, ferromagnetic Ising system. Phys. Rev. E
**2004**, 70, 066115. [Google Scholar] [CrossRef] [PubMed] - Stacey, B.C.; Allen, B.; Bar-Yam, Y. Multiscale Information Theory for Complex Systems: Theory and Applications. In Information and Complexity; Burgin, M., Calude, C.S., Eds.; World Scientific: Singapore, 2017; pp. 176–199. [Google Scholar]
- Grassberger, P. Toward a quantitative theory of self-generated complexity. Int. J. Theor. Phys.
**1986**, 25, 907–938. [Google Scholar] [CrossRef] - Crutchfield, J.P.; Young, K. Inferring statistical complexity. Phys. Rev. Lett.
**1989**, 63, 105–108. [Google Scholar] [CrossRef] [PubMed] - Crutchfield, J.P. The calculi of emergence: Computation, dynamics and induction. Phys. D Nonlinear Phenom.
**1994**, 75, 11–54. [Google Scholar] [CrossRef] - Misra, V.; Lagi, M.; Bar-Yam, Y. Evidence of Market Manipulation in the Financial Crisis; Technical Report 2011-12-01; NECSI: Cambridge, MA, USA, 2011. [Google Scholar]
- Harmon, D.; Lagi, M.; de Aguiar, M.A.; Chinellato, D.D.; Braha, D.; Epstein, I.R.; Bar-Yam, Y. Anticipating Economic Market Crises Using Measures of Collective Panic. PLoS ONE
**2015**, 10, e0131871. [Google Scholar] [CrossRef] [PubMed] - Green, H. The Molecular Theory of Fluids; North–Holland: Amsterdam, The Netherlands, 1952. [Google Scholar]
- Nettleton, R.E.; Green, M.S. Expression in terms of molecular distribution functions for the entropy density in an infinite system. J. Chem. Phys.
**1958**, 29, 1365–1370. [Google Scholar] [CrossRef] - Wolf, D.R. Information and Correlation in Statistical Mechanical Systems. Ph.D. Thesis, University of Texas, Austin, TX, USA, 1996. [Google Scholar]
- Kardar, M. Statistical Physics of Particles; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Kadanoff, L.P. Scaling laws for Ising models near T
_{c}. Physics**1966**, 2, 263. [Google Scholar] - Wilson, K.G. The renormalization group: Critical phenomena and the Kondo problem. Rev. Mod. Phys.
**1975**, 47, 773. [Google Scholar] [CrossRef] - McGill, W.J. Multivariate information transmission. Psychometrika
**1954**, 46, 26–45. [Google Scholar] [CrossRef] - Han, T.S. Multiple mutual information and multiple interactions in frequency data. Inf. Control
**1980**, 46, 26–45. [Google Scholar] [CrossRef] - Yeung, R.W. A new outlook on Shannon’s information measures. IEEE Trans. Inf. Theory
**1991**, 37, 466–474. [Google Scholar] [CrossRef] - Jakulin, A.; Bratko, I. Quantifying and visualizing attribute interactions. arXiv, 2003; arXiv:cs.AI/0308002. [Google Scholar]
- Bell, A.J. The co-information lattice. In Proceedings of the Fifth International Workshop on Independent Component Analysis and Blind Signal Separation (ICA), Nara, Japan, 1–4 April 2003; Volume 2003. [Google Scholar]
- Bar-Yam, Y. A mathematical theory of strong emergence using multiscale variety. Complexity
**2004**, 9, 15–24. [Google Scholar] [CrossRef] - Krippendorff, K. Information of interactions in complex systems. Int. J. Gen. Syst.
**2009**, 38, 669–680. [Google Scholar] [CrossRef] - Leydesdorff, L. Redundancy in systems which entertain a model of themselves: Interaction information and the self-organization of anticipation. Entropy
**2010**, 12, 63–79. [Google Scholar] [CrossRef] - Kolchinsky, A.; Rocha, L.M. Prediction and modularity in dynamical systems. arXiv, 2011; arXiv:1106.3703. [Google Scholar]
- James, R.G.; Ellison, C.J.; Crutchfield, J.P. Anatomy of a bit: Information in a time series observation. Chaos Interdiscip. J. Nonlinear Sci.
**2011**, 21, 037109. [Google Scholar] [CrossRef] [PubMed] - Tononi, G.; Sporns, O.; Edelman, G.M. A measure for brain complexity: Relating functional segregation and integration in the nervous system. Proc. Natl. Acad. Sci. USA
**1994**, 91, 5033–5037. [Google Scholar] [CrossRef] [PubMed] - Ay, N.; Olbrich, E.; Bertschinger, N.; Jost, J. A unifying framework for complexity measures of finite systems. In Proceedings of the European Complex Systems Society (ECCS06), Oxford, UK, 25 September 2006. [Google Scholar]
- Bar-Yam, Y. Complexity of Military Conflict: Multiscale Complex Systems Analysis of Littoral Warfare; Technical Report; NECSI: Cambridge, MA, USA, 2003. [Google Scholar]
- Granovsky, B.L.; Madras, N. The noisy voter model. Stoch. Process. Appl.
**1995**, 55, 23–43. [Google Scholar] [CrossRef] - Faddeev, D.K. On the concept of entropy of a finite probabilistic scheme. Uspekhi Mat. Nauk
**1956**, 11, 227–231. [Google Scholar] - Khinchin, A.I. Mathematical Foundations of Information Theory; Dover: New York, NY, USA, 1957. [Google Scholar]
- Lee, P. On the axioms of information theory. Ann. Math. Stat.
**1964**, 35, 415–418. [Google Scholar] [CrossRef] - Rényi, A. Probability Theory; Akadémiai Kiadó: Budapest, Hungary, 1970. [Google Scholar]
- Daróczy, Z. Generalized information functions. Inf. Control
**1970**, 16, 36–51. [Google Scholar] [CrossRef] - Dos Santos, R.J. Generalization of Shannon’s theorem for Tsallis entropy. J. Math. Phys.
**1997**, 38, 4104–4107. [Google Scholar] [CrossRef] - Abe, S. Axioms and uniqueness theorem for Tsallis entropy. Phys. Lett. A
**2000**, 271, 74–79. [Google Scholar] [CrossRef] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Gell-Mann, M.; Tsallis, C. Nonextensive Entropy: Interdisciplinary Applications; Oxford University Press: Oxford, UK, 2004. [Google Scholar]
- Furuichi, S. Information theoretical properties of Tsallis entropies. J. Math. Phys.
**2006**, 47, 023302. [Google Scholar] [CrossRef] - Steudel, B.; Janzing, D.; Schölkopf, B. Causal Markov condition for submodular information measures. arXiv, 2010; arXiv:1002.4020. [Google Scholar]
- Dougherty, R.; Freiling, C.; Zeger, K. Networks, matroids, and non-Shannon information inequalities. IEEE Trans. Inf. Theory
**2007**, 53, 1949–1969. [Google Scholar] [CrossRef] - Li, M.; Vitányi, P. An Introduction to Kolmogorov Complexity and Its Applications; Springer Science & Business Media: New York, NY, USA, 2009. [Google Scholar]
- Chaitin, G.J. A theory of program size formally identical to information theory. J. ACM
**1975**, 22, 329–340. [Google Scholar] [CrossRef] - May, R.M.; Arinaminpathy, N. Systemic risk: The dynamics of model banking systems. J. R. Soc. Interface
**2010**, 7, 823–838. [Google Scholar] [CrossRef] [PubMed] - Haldane, A.G.; May, R.M. Systemic risk in banking ecosystems. Nature
**2011**, 469, 351–355. [Google Scholar] [CrossRef] [PubMed] - Beale, N.; Rand, D.G.; Battey, H.; Croxson, K.; May, R.M.; Nowak, M.A. Individual versus systemic risk and the Regulator’s Dilemma. Proc. Natl. Acad. Sci. USA
**2011**, 108, 12647–12652. [Google Scholar] [CrossRef] [PubMed] - Erickson, M.J. Introduction to Combinatorics; Wiley: Hoboken, NJ, USA, 1996. [Google Scholar]
- Williams, P.L.; Beer, R.D. Nonnegative decomposition of multivariate information. arXiv, 2010; arXiv:1004.2515. [Google Scholar]
- James, R.G.; Crutchfield, J.P. Multivariate Dependence Beyond Shannon Information. arXiv, 2016; arXiv:1609.01233. [Google Scholar]
- Perfect, H. Independence theory and matroids. Math. Gaz.
**1981**, 65, 103–111. [Google Scholar] [CrossRef] - Studenỳ, M.; Vejnarová, J. The multiinformation function as a tool for measuring stochastic dependence. In Learning in Graphical Models; Springer: Dodrecht, The Netherlands, 1998; pp. 261–297. [Google Scholar]
- Schneidman, E.; Still, S.; Berry, M.J.; Bialek, W. Network information and connected correlations. Phys. Rev. Lett.
**2003**, 91, 238701. [Google Scholar] [CrossRef] [PubMed] - Polani, D. Foundations and formalizations of self-organization. In Advances in Applied Self-Organizing Systems; Springer: New York, NY, USA, 2008; pp. 19–37. [Google Scholar]
- Wets, R.J.B. Programming Under Uncertainty: The Equivalent Convex Program. SIAM J. Appl. Math.
**1966**, 14, 89–105. [Google Scholar] [CrossRef] - James, R.G. Python Package for Information Theory. Zenodo
**2017**. [Google Scholar] [CrossRef] - Slonim, N.; Tishby, L. Agglomerative information bottleneck. Adv. Neural Inf. Process. Syst. NIPS
**1999**, 12, 617–623. [Google Scholar] - Shalizi, C.R.; Crutchfield, J.P. Information bottlenecks, causal states, and statistical relevance bases: How to represent relevant information in memoryless transduction. Adv. Complex Syst.
**2002**, 5, 91–95. [Google Scholar] [CrossRef] - Tishby, N.; Pereira, F.C.; Bialek, W. The information bottleneck method. arXiv, 2000; arXiv:physics/0004057. [Google Scholar]
- Ziv, E.; Middendorf, M.; Wiggins, C. An information-theoretic approach to network modularity. Phys. Rev. E
**2005**, 71, 046117. [Google Scholar] [CrossRef] [PubMed] - Peng, C.K.; Buldyrev, S.V.; Havlin, S.; Simons, M.; Stanley, H.E.; Goldberger, A.L. Mosaic organization of DNA nucleotides. Phys. Rev. E
**1994**, 49, 1685–1689. [Google Scholar] [CrossRef] - Gell-Mann, M.; Lloyd, S. Information measures, effective complexity, and total information. Complexity
**1996**, 2, 44–52. [Google Scholar] [CrossRef] - Hu, K.; Ivanov, P.; Chen, Z.; Carpena, P.; Stanley, H.E. Effect of Trends on Detrended Fluctuation Analysis. Phys. Rev. E
**2002**, 64, 011114. [Google Scholar] [CrossRef] [PubMed] - Vereshchagin, N.; Vitányi, P. Kolmogorov’s structure functions and model selection. IEEE Trans. Inf. Theory
**2004**, 50, 3265–3290. [Google Scholar] [CrossRef] - Grünwald, P.; Vitányi, P. Shannon information and Kolmogorov complexity. arXiv, 2004; arXiv:cs.IT/0410002. [Google Scholar]
- Vitányi, P. Meaningful information. IEEE Trans. Inf. Theory
**2006**, 52, 4617–4626. [Google Scholar] [CrossRef] - Moran, P.A.P. Random processes in genetics. Math. Proc. Camb. Philos. Soc.
**1958**, 54, 60–71. [Google Scholar] [CrossRef] - Harmon, D.; Stacey, B.C.; Bar-Yam, Y. Networks of Economic Market Independence and Systemic Risk; Technical Report 2009-03-01 (updated); NECSI: Cambridge, MA, USA, 2010. [Google Scholar]
- Stacey, B.C. Multiscale Structure in Eco-Evolutionary Dynamics. Ph.D. Thesis, Brandeis University, Waltham, MA, USA, 2015. [Google Scholar]
- Domb, C.; Green, M.S. (Eds.) Phase Transitions and Critical Phenomena; Academic Press: New York, NY, USA, 1972. [Google Scholar]
- Kardar, M. Statistical Physics of Fields; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Jacob, F.; Monod, J. Genetic regulatory mechanisms in the synthesis of proteins. J. Mol. Biol.
**1961**, 3, 318–356. [Google Scholar] [CrossRef] - Britten, R.J.; Davidson, E.H. Gene regulation for higher cells: A theory. Science
**1969**, 165, 349–357. [Google Scholar] [CrossRef] [PubMed] - Carey, M.; Smale, S. Transcriptional Regulation in Eukaryotes: Concepts, Strategies, and Techniques; Cold Spring Harbor Laboratory Press: Cold Spring Harbor, NY, USA, 2001. [Google Scholar]
- Elowitz, M.B.; Levine, A.J.; Siggia, E.D.; Swain, P.S. Stochastic gene expression in a single cell. Science
**2002**, 297, 1183–1186. [Google Scholar] [CrossRef] [PubMed] - Lee, T.I.; Rinaldi, N.J.; Robert, F.; Odom, D.T.; Bar-Joseph, Z.; Gerber, G.K.; Hannett, N.M.; Harbison, C.T.; Thompson, C.M.; Simon, I.; et al. Transcriptional Regulatory Networks in Saccharomyces cerevisiae. Science
**2002**, 298, 799–804. [Google Scholar] [CrossRef] [PubMed] - Boyer, L.A.; Lee, T.I.; Cole, M.F.; Johnstone, S.E.; Levine, S.S.; Zucker, J.P.; Guenther, M.G.; Kumar, R.M.; Murray, H.L.; Jenner, R.G.; et al. Core Transcriptional Regulatory Circuitry in Human Embryonic Stem Cells. Cell
**2005**, 122, 947–956. [Google Scholar] [CrossRef] [PubMed] - Chowdhury, S.; Lloyd-Price, J.; Smolander, O.P.; Baici, W.C.; Hughes, T.R.; Yli-Harja, O.; Chua, G.; Ribeiro, A.S. Information propagation within the Genetic Network of Saccharomyces cerevisiae. BMC Syst. Biol.
**2010**, 4, 143. [Google Scholar] [CrossRef] [PubMed] - Hopfield, J.J. Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA
**1982**, 79, 2554–2558. [Google Scholar] [CrossRef] [PubMed] - Rabinovich, M.I.; Varona, P.; Selverston, A.I.; Abarbanel, H.D.I. Dynamical principles in neuroscience. Rev. Mod. Phys.
**2006**, 78, 1213–1265. [Google Scholar] [CrossRef] - Schneidman, E.; Berry, M.J.; Segev, R.; Bialek, W. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature
**2006**, 440, 1007–1012. [Google Scholar] [CrossRef] [PubMed] - Bonabeau, E.; Dorigo, M.; Theraulaz, G. Swarm Intelligence: From Natural to Artificial Systems; Oxford University Press: Oxford, UK, 1999. [Google Scholar]
- Vicsek, T.; Zafeiris, A. Collective motion. Phys. Rep.
**2012**, 517, 71–140. [Google Scholar] [CrossRef] - Berdahl, A.; Torney, C.J.; Ioannou, C.C.; Faria, J.J.; Couzin, I.D. Emergent sensing of complex environments by mobile animal groups. Science
**2013**, 339, 574–576. [Google Scholar] [CrossRef] [PubMed] - Ohtsuki, H.; Hauert, C.; Lieberman, E.; Nowak, M.A. A simple rule for the evolution of cooperation on graphs and social networks. Nature
**2006**, 441, 502–505. [Google Scholar] [CrossRef] [PubMed] - Allen, B.; Lippner, G.; Chen, Y.T.; Fotouhi, B.; Momeni, N.; Yau, S.T.; Nowak, M.A. Evolutionary dynamics on any population structure. Nature
**2017**, 544, 227–230. [Google Scholar] [CrossRef] [PubMed] - Mandelbrot, B.; Taylor, H. On the distribution of stock price differences. Oper. Res.
**1967**, 15, 1057–1062. [Google Scholar] [CrossRef] - Mantegna, R.N. Hierarchical structure in financial markets. Eur. Phys. J. B Condens. Matter Complex Syst.
**1999**, 11, 193–197. [Google Scholar] [CrossRef] - Sornette, D. Why Stock Markets Crash: Critical Events in Complex Financial Systems; Princeton University Press: Princeton, NJ, USA, 2004. [Google Scholar]
- May, R.M.; Levin, S.A.; Sugihara, G. Complex systems: Ecology for bankers. Nature
**2008**, 451, 893–895. [Google Scholar] [CrossRef] [PubMed] - Schweitzer, F.; Fagiolo, G.; Sornette, D.; Vega-Redondo, F.; Vespignani, A.; White, D.R. Economic Networks: The New Challenges. Science
**2009**, 325, 422–425. [Google Scholar] [PubMed] - Harmon, D.; De Aguiar, M.; Chinellato, D.; Braha, D.; Epstein, I.; Bar-Yam, Y. Predicting economic market crises using measures of collective panic. arXiv, 2011; arXiv:1102.2620. [Google Scholar] [CrossRef]
- Schrödinger, E. What Is Life? The Physical Aspect of the Living Cell and Mind; Cambridge University Press: Cambridge, UK, 1944. [Google Scholar]
- Brillouin, L. The negentropy principle of information. J. Appl. Phys.
**1953**, 24, 1152–1163. [Google Scholar] [CrossRef] - Stacey, B.C. Multiscale Structure of More-than-Binary Variables. arXiv, 2017; arXiv:1705.03927. [Google Scholar]
- Ashby, W.R. An Introduction to Cybernetics; Chapman & Hall: London, UK, 1956. [Google Scholar]
- Stacey, B.C.; Bar-Yam, Y. Principles of Security: Human, Cyber, and Biological; Technical Report 2008-06-01; NECSI: Cambridge, MA, USA, 2008. [Google Scholar]
- Dorogovtsev, S.N. Lectures on Complex Networks; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Dantzig, G.B.; Wolfe, P. Decomposition principle for linear programs. Oper. Res.
**1960**, 8, 101–111. [Google Scholar] [CrossRef]

**Figure 1.**The dependency diagram of a system with three components, a, b and c, represented by the interiors of the three circles. The seven irreducible dependencies shown above correspond to the seven interior regions of the Venn diagram encompassed by the boundaries of the three circles. Irreducible dependencies are shaded according to their scale, assuming that each component has scale one. Reducible dependencies such as $a|b$ are not shown.

**Figure 2.**Dependency diagrams for our running example systems: (

**A**) three independent bits; (

**B**) three completely interdependent bits; (

**C**) independent blocks of dependent bits; and (

**D**) the $2+1$ parity bit system. Regions of information zero in (

**A**–

**C**) are not shown.

**Figure 3.**Schematic illustration of the (

**a**) complexity profile (CP) and (

**b**) marginal utility of information (MUI) for systems with varying degrees of interdependence among components. If the components are independent, all information applies at scale 1, so the complexity profile has $C(1)$ equal to the number of components and $C(x)=0$ for $x>1$. As the system becomes more interdependent, information applies at successively larger scales, resulting in a shallower decrease of $C(x)$. For the MUI, if components are independent, the optimal description scheme describes only a single component at a time, with marginal utility 1. As the system becomes more interdependent, information overlaps allow for more efficient descriptions that achieve greater marginal utility. For both the CP and MUI, the total area under the curve is equal to the total scale-weighted information $S(\mathfrak{D})$, which is preserved under reorganizations of the system. The CP and MUI are not reflections of each other in general, but they are for an important class of systems (see Section 8).

**Figure 4.**(

**A**–

**D**) Complexity profile $C(k)$ for Examples

**A**through

**D**. Note that the total (signed) area bounded by each curve equals $S({\mathfrak{D}}_{\mathcal{A}})={\sum}_{a\in A}H(a)=3.$ For Example

**D**(the parity bit), the information at scale 3 is negative.