Azimuth

Resource Convertibility (Part 3)

13 April, 2015

guest post by Tobias Fritz

In Part 1 and Part 2, we learnt about ordered commutative monoids and how they formalize theories of resource convertibility and combinability. In this post, I would like to say a bit about the applications that have been explored so far. First, the study of resource theories has become a popular subject in quantum information theory, and many of the ideas in my paper actually originate there. I’ll list some references at the end. So I hope that the toolbox of ordered commutative monoids will turn out to be useful for this. But here I would like to talk about an example application that is much easier to understand, but no less difficult to analyze: graph theory and the resource theory of zero-error communication.

A graph consists of a bunch of nodes connected by a bunch of edges, for example like this:

This particular graph is the pentagon graph or 5-cycle. To give it some resource-theoretic interpretation, think of it as the distinguishability graph of a communication channel, where the nodes are the symbols that can be sent across the channel, and two symbols share an edge if and only if they can be unambiguously decoded. For example, the pentagon graph roughly corresponds to the distinguishability graph of my handwriting, when restricted to five letters only:

So my ‘w’ is distinguishable from my ‘u’, but it may be confused for my ‘m’. In order to communicate unambiguously, it looks like I should restrict myself to using only two of those letters in writing, since any third of them may be mistaken for one of the other three. But alternatively, I could use a block code to create context around each letter which allows for perfect disambiguation. This is what happens in practice: I write in natural language, where an entire word is usually not ambiguous.

One can now also consider graph homomorphisms, which are maps like this:

The numbers on the nodes indicate where each node on the left gets mapped to. Formally, a graph homomorphism is a function taking nodes to nodes such that adjacent nodes get mapped to adjacent nodes. If a homomorphism $G\to H$ exists between graphs $G$ and $H,$ then we also write $H\geq G$ ; in terms of communication channels, we can interpret this as saying that $H$ simulates $G,$ since the homomorphism provides a map between the symbols which preserves distinguishability. A ‘code’ for a communication channel is then just a homomorphism from the complete graph in which all nodes share an edge to the graph which describes the channel. With this ordering structure, the collection of all finite graphs forms an ordered set. This ordered set has an intricate structure which is intimately related to some big open problems in graph theory.

We can also combine two communication channels to form a compound one. Going back to the handwriting example, we can consider the new channel in which the symbols are pairs of letters. Two such pairs are distinguishable if and only if either the first letters of each pair are distinguishable or the second letters are,

$(a,b) \sim (a',b') \:\Leftrightarrow\: a\sim a' \:\lor\: b\sim b'$

When generalized to arbitrary graphs, this yields the definition of disjunctive product of graphs. It is not hard to show that this equips the ordered set of graphs with a binary operation compatible with the ordering, so that we obtain an ordered commutative monoid denoted Grph. It mathematically formalizes the resource theory of zero-error communication.

Using the toolbox of ordered commutative monoids combined with some concrete computations on graphs, one can show that Grph is not cancellative: if $K_{11}$ is the complete graph on 11 nodes, then $3C_5\not\geq K_{11},$ but there exists a graph $G$ such that

$3 C_5 + G \geq K_{11} + G$

The graph $G$ turns out to have 136 nodes. This result seems to be new. But if you happen to have seen something like this before, please let me know!

Last time, we also talked about rates of conversion. In Grph, it turns out that some of these correspond to famous graph invariants! For example, the rate of conversion from a graph $G$ to the single-edge graph $K_2$ is Shannon capacity $\Theta(\overline{G}),$ where $\overline{G}$ is the complement graph. This is of no surprise since $\Theta$ was originally defined by Shannon with precisely this rate in mind, although he did not use the language of ordered commutative monoids. In any case, the Shannon capacity $\Theta(\overline{G})$ is a graph invariant notorious for its complexity: it is not known whether there exists an algorithm to compute it! But an application of the Rate Theorem from Part 2 gives us a formula for the Shannon capacity:

$\Theta(\overline{G}) = \inf_f f(G)$

where $f$ ranges over all graph invariants which are monotone under graph homomorphisms, multiplicative under disjunctive product, and normalized such that $f(K_2) = 2.$ Unfortunately, this formula still not produce an algorithm for computing $\Theta.$ But it nonconstructively proves the existence of many new graph invariants $f$ which approximate the Shannon capacity from above.

Although my story ends here, I also feel that the whole project has barely started. There are lots of directions to explore! For example, it would be great to fit Shannon’s noisy channel coding theorem into this framework, but this has turned out be technically challenging. If you happen to be familiar with rate-distortion theory and you want to help out, please get in touch!

References

Here is a haphazard selection of references on resource theories in quantum information theory and related fields:

• Igor Devetak, Aram Harrow and Andreas Winter, A resource framework for quantum Shannon theory.

• Gilad Gour, Markus P. Müller, Varun Narasimhachar, Robert W. Spekkens and Nicole Yunger Halpern, The resource theory of informational nonequilibrium in thermodynamics.

• Fernando G.S.L. Brandão, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim and Stephanie Wehner, The second laws of quantum thermodynamics.

• Iman Marvian and Robert W. Spekkens, The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations.

• Elliott H. Lieb and Jakob Yngvason, The physics and mathematics of the second law of thermodynamics.

14 Comments | chemistry, mathematics, networks, physics | Permalink
Posted by John Baez

Resource Convertibility (Part 2)

10 April, 2015

guest post by Tobias Fritz

In Part 1, I introduced ordered commutative monoids as a mathematical formalization of resources and their convertibility. Today I’m going to say something about what to do with this formalization. Let’s start with a quick recap!

Definition: An ordered commutative monoid is a set $A$ equipped with a binary relation $\geq,$ a binary operation $+,$ and a distinguished element $0$ such that the following hold:

• $+$ and $0$ equip $A$ with the structure of a commutative monoid;

• $\geq$ equips $A$ with the structure of a partially ordered set;

• addition is monotone: if $x\geq y,$ then also $x + z \geq y + z.$

Recall also that we think of the $x,y\in A$ as resource objects such that $x+y$ represents the object consisting of $x$ and $y$ together, and $x\geq y$ means that the resource object $x$ can be converted into $y.$

When confronted with an abstract definition like this, many people ask: so what is it useful for? The answer to this is twofold: first, it provides a language which we can use to guide our thoughts in any application context. Second, the definition itself is just the very start: we can now also prove theorems about ordered commutative monoids, which can be instantiated in any particular application context. So the theory of ordered commutative monoids will provide a useful toolbox for talking about concrete resource theories and studying them. In the remainder of this post, I’d like to say a bit about what this toolbox contains. For more, you’ll have to read the paper!

To start, let’s consider catalysis as one of the resource-theoretic phenomena neatly captured by ordered commutative monoids. Catalysis is the phenomenon that certain conversions become possible only due to the presence of a catalyst, which is an additional resource object which does not get consumed in the process of the conversion. For example, we have

$\text{timber + nails}\not\geq \text{table},$

$\text{timber + nails + saw + hammer} \geq \text{table + saw + hammer}$

because making a table from timber and nails requires a saw and a hammer as tools. So in this example, ‘saw $+$ hammer’ is a catalyst for the conversion of ‘timber $+$ nails’ into ‘table’. In mathematical language, catalysis occurs precisely when the ordered commutative monoid is not cancellative, which means that $x + z\geq y + z$ sometimes holds even though $x\geq y$ does not. So, the notion of catalysis perfectly matches up with a very natural and familiar notion from algebra.

One can continue along these lines and study those ordered commutative monoids which are cancellative. It turns out that every ordered commutative monoid can be made cancellative in a universal way; in the resource-theoretic interpretation, this boils down to replacing the convertibility relation by catalytic convertibility, in which $x$ is declared to be convertible into $y$ as soon as there exists a catalyst which achieves this conversion. Making an ordered commutative monoid cancellative like this is a kind of ‘regularization’: it leads to a mathematically more well-behaved structure. As it turns out, there are several additional steps of regularization that can be performed, and all of these are both mathematically natural and have an appealing resource-theoretic interpretation. These regularizations successively take us from the world of ordered commutative monoids to the realm of linear algebra and functional analysis, where powerful theorems are available. For now, let me not go into the details, but only try to summarize one of the consequences of this development. This requires a bit of preparation.

In many situations, it is not just of interest to convert a single copy of some resource object $x$ into a single copy of some $y;$ instead, one may be interested in converting many copies of $x$ into many copies of $y$ all together, and thereby maximizing (or minimizing) the ratio of the resulting number of $y$ ‘s compared to the number of $x$ ‘s that get consumed. This ratio is measured by the maximal rate:

$\displaystyle{ R_{\mathrm{max}}(x\to y) = \sup \left\{ \frac{m}{n} \:|\: nx \geq my \right\} }$

Here, $m$ and $n$ are natural numbers, and $nx$ stands for the $n$ -fold sum $x+\cdots+x,$ and similarly for $my.$ So this maximal rate quantifies how many $y$ ’ s we can get out of one copy of $x,$ when working in a ‘mass production’ setting. There is also a notion of regularized rate, which has a slightly more complicated definition that I don’t want to spell out here, but is similar in spirit. The toolbox of ordered commutative monoids now provides the following result:

Rate Theorem: If $x\geq 0$ and $y\geq 0$ in an ordered commutative monoid $A$ which satisfies a mild technical assumption, then the maximal regularized rate from $x$ to $y$ can be computed like this:

$\displaystyle{ R_{\mathrm{max}}^{\mathrm{reg}}(x\to y) = \inf_f \frac{f(y)}{f(x)} }$

where $f$ ranges over all functionals on $A$ with $f(y)\neq 0.$

Wait a minute, what’s a ‘functional’? It’s defined to be a map $f:A\to\mathbb{R}$ which is monotone,

$x\geq y \:\Rightarrow\: f(x)\geq f(y)$

and additive,

$f(x+y) = f(x) + f(y)$

In economic terms, we can think of a functional as a consistent assignment of prices to all resource objects. If $x$ is at least as useful as $y,$ then the price of $x$ should be at least as high as the price of $y$ ; and the price of two objects together should be the sum of their individual prices. So the $f$ in the rate formula above ranges over all ‘markets’ on which resource objects can be ‘traded’ at consistent prices. The term ‘functional’ is supposed to hint at a relation to functional analysis. In fact, the proof of the theorem crucially relies on the Hahn–Banach Theorem.

The mild technical mentioned in the Rate Theorem is that the ordered commutative monoid needs to have a generating pair. This turns out to hold in the applications that I have considered so far, and I hope that it will turn out to hold in most others as well. For the full gory details, see the paper.

So this provides some idea of what kinds of gadgets one can find in the toolbox of ordered commutative monoids. Next time, I’ll show some applications to graph theory and zero-error communication and say a bit about where this project might be going next.

• Resource convertibility: part 3.

4 Comments | chemistry, mathematics, networks, physics | Permalink
Posted by John Baez

Resource Convertibility (Part 1)

7 April, 2015

guest post by Tobias Fritz

Hi! I am Tobias Fritz, a mathematician at the Perimeter Institute for Theoretical Physics in Waterloo, Canada. I like to work on all sorts of mathematical structures which pop up in probability theory, information theory, and other sorts of applied math. Today I would like to tell you about my latest paper:

• The mathematical structure of theories of resource convertibility, I.

It should be of interest to Azimuth readers as it forms part of what John likes to call ‘green mathematics’. So let’s get started!

Resources and their management are an essential part of our everyday life. We deal with the management of time or money pretty much every day. We also consume natural resources in order to afford food and amenities for (some of) the 7 billion people on our planet. Many of the objects that we deal with in science and engineering can be considered as resources. For example, a communication channel is a resource for sending information from one party to another. But for now, let’s stick with a toy example: timber and nails constitute a resource for making a table. In mathematical notation, this looks like so:

$\mathrm{timber} + \mathrm{nails} \geq \mathrm{table}$

We interpret this inequality as saying that “given timber and nails, we can make a table”. I like to write it as an inequality like this, which I think of as stating that having timber and nails is at least as good as having a table, because the timber and nails can always be turned into a table whenever one needs a table.

To be more precise, we should also take into account that making the table requires some tools. These tools do not get consumed in the process, so we also get them back out:

$\text{timber} + \text{nails} + \text{saw} + \text{hammer} \geq \text{table} + \text{hammer} + \text{saw}$

Notice that this kind of equation is analogous to a chemical reaction equation like this:

$2 \mathrm{H}_2 + \mathrm{O}_2 \geq \mathrm{H}_2\mathrm{O}$

So given a hydrogen molecules and an oxygen molecule, we can let them react such as to form a molecule of water. In chemistry, this kind of equation would usually be written with an arrow ‘ $\rightarrow$ ’ instead of an ordering symbol ‘ $\geq$ ’ , but here we interpret the equation slightly differently. As with the timber and the nails and nails above, the inequality says that if we have two hydrogen atoms and an oxygen atom, then we can let them react to a molecule of water, but we don’t have to. In this sense, having two hydrogen atoms and an oxygen atom is at least as good as having a molecule of water.

So what’s going on here, mathematically? In all of the above equations, we have a bunch of stuff on each side and an inequality ‘ $\geq$ ’ in between. The stuff on each side consists of a bunch of objects tacked together via ‘ $+$ ’ . With respect to these two pieces of structure, the collection of all our resource objects forms an ordered commutative monoid:

Definition: An ordered commutative monoid is a set $A$ equipped with a binary relation $\geq,$ a binary operation $+,$ and a distinguished element $0$ such that the following hold:

• $+$ and $0$ equip $A$ with the structure of a commutative monoid;

• $\geq$ equips $A$ with the structure of a partially ordered set;

• addition is monotone: if $x\geq y,$ then also $x + z \geq y + z.$

Here, the third axiom is the most important, since it tells us how the additive structure interacts with the ordering structure.

Ordered commutative monoids are the mathematical formalization of resource convertibility and combinability as follows. The elements $x,y\in A$ are the resource objects, corresponding to the ‘collections of stuff’ in our earlier examples, such as $x = \text{timber} + \text{nails}$ or $y = 2 \text{H}_2 + \text{O}_2.$ Then the addition operation simply joins up collections of stuff into bigger collections of stuff. The ordering relation $\geq$ is what formalizes resource convertibility, as in the examples above. The third axiom states that if we can convert $x$ into $y,$ then we can also convert $x$ together with $z$ into $y$ together with $z$ for any $z,$ for example by doing nothing to $z.$

A mathematically minded reader might object that requiring $A$ to form a partially ordered set under $\geq$ is too strong a requirement, since it requires two resource objects to be equal as soon as they are mutually interconvertible: $x \geq y$ and $y \geq x$ implies $x = y.$ However, I think that this is not an essential restriction, because we can regard this implication as the definition of equality: ‘ $x = y$ ’ is just a shorthand notation for ‘ $x\geq y$ and $y\geq x$ ’ which formalizes the perfect interconvertibility of resource objects.

We could now go back to the original examples and try to model carpentry and chemistry in terms of ordered commutative monoids. But as a mathematician, I needed to start out with something mathematically precise and rigorous as a testing ground for the formalism. This helps ensure that the mathematics is sensible and useful before diving into real-world applications. So, the main example in my paper is the ordered commutative monoid of graphs, which has a resource-theoretic interpretation in terms of zero-error information theory. As graph theory is a difficult and traditional subject, this application constitutes the perfect training camp for the mathematics of ordered commutative monoids. I will get to this in Part 3.

In Part 2, I will say something about what one can do with ordered commutative monoids. In the meantime, I’d be curious to know what you think about what I’ve said so far!

• Resource convertibility: part 2.

• Resource convertibility: part 3.

28 Comments | chemistry, mathematics, networks, physics | Permalink
Posted by John Baez

Information and Entropy in Biological Systems (Part 3)

6 April, 2015

I think you can watch live streaming video of our workshop on Information and Entropy in Biological Systems, which runs Wednesday April 8th to Friday April 10th. Later, videos will be made available in a permanent location.

To watch the workshop live, go here. Go down to where it says

Investigative Workshop: Information and Entropy in Biological Systems

Then click where it says live link. There’s nothing there now, but I’m hoping there will be when the show starts!

Below you can see the schedule of talks and a list of participants. The hours are in Eastern Daylight Time: add 4 hours to get Greenwich Mean Time. The talks start at 10 am EDT, which is 2 pm GMT.

Schedule

There will be 1½ hours of talks in the morning and 1½ hours in the afternoon for each of the 3 days, Wednesday April 8th to Friday April 10th. The rest of the time will be for discussions on different topics. We’ll break up into groups, based on what people want to discuss.

Each invited speaker will give a 30-minute talk summarizing the key ideas in some area, not their latest research so much as what everyone should know to start interesting conversations. After that, 15 minutes for questions and/or coffee.

Here’s the schedule. You can already see slides or other material for the talks with links!

Wednesday April 8

• 9:45-10:00 — the usual introductory fussing around.
• 10:00-10:30 — John Baez, Information and entropy in biological systems.
• 10:30-11:00 — questions, coffee.
• 11:00-11:30 — Chris Lee, Empirical information, potential information and disinformation.
• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — John Harte, Maximum entropy as a foundation for theory building in ecology.
• 2:00-2:15 — questions, coffee.
• 2:15-2:45 — Annette Ostling, The neutral theory of biodiversity and other competitors to the principle of maximum entropy.
• 2:45-3:00 — questions, coffee.
• 3:00-5:30 — break up into groups for discussions.

• 5:30 — reception.

Thursday April 9

• 10:00-10:30 — David Wolpert, The Landauer limit and thermodynamics of biological organisms.
• 10:30-11:00 — questions, coffee.
• 11:00-11:30 — Susanne Still, Efficient computation and data modeling.
• 11:30-11:45 — questions.

• 11:45-1:30 — group photo, lunch, conversations.

• 1:30-2:00 — Matina Donaldson-Matasci, The fitness value of information in an uncertain environment.
• 2:00-2:15 — questions, coffee.
• 2:15-2:45 — Roderick Dewar, Maximum entropy and maximum entropy production in biological systems: survival of the likeliest?
• 2:45-3:00 — questions, coffee.
• 3:00-6:00 — break up into groups for discussions.

Friday April 10

• 10:00-10:30 — Marc Harper, Information transport and evolutionary dynamics.
• 10:30-11:00 — questions, coffee.
• 11:00-11:30 — Tobias Fritz, Characterizations of Shannon and Rényi entropy.
• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Christina Cobbold, Biodiversity measures and the role of species similarity.
• 2:00-2:15 — questions, coffee.
• 2:15-2:45 — Tom Leinster, Maximizing biological diversity.
• 2:45-3:00 — questions, coffee.
• 3:00-6:00 — break up into groups for discussions.

Participants

Here are the confirmed participants. This list may change a little bit:

• John Baez – mathematical physicist.

• Romain Brasselet – postdoc in cognitive neuroscience knowledgeable about information-theoretic methods and methods of estimating entropy from samples of probability distributions.

• Katharina Brinck – grad student at Centre for Complexity Science at Imperial College; did masters at John Harte’s lab, where she extended his Maximum Entropy Theory of Ecology (METE) to trophic food webs, to study how entropy maximization on the macro scale together with MEP on the scale of individuals drive the structural development of model ecosystems.

• Christina Cobbold – mathematical biologist, has studied the role of species similarity in measuring biodiversity.

• Troy Day – mathematical biologist, works with population dynamics, host-parasite dynamics, etc.; influential and could help move population dynamics to a more information-theoretic foundation.

• Roderick Dewar – physicist who studies the principle of maximal entropy production.

• Barrett Deris – MIT postdoc studying the studying the factors that influence evolvability of drug resistance in bacteria.

• Charlotte de Vries – a biology master’s student who studied particle physics to the master’s level at Oxford and the Perimeter Institute. Interested in information theory.

• Matina Donaldson-Matasci – a biologist who studies information, uncertainty and collective behavior.

• Chris Ellison – a postdoc who worked with James Crutchfield on “information-theoretic measures of structure and memory in stationary, stochastic systems – primarily, finite state hidden Markov models”. He coauthored “Intersection Information based on Common Randomness”, http://arxiv.org/abs/1310.1538. The idea: “The introduction of the partial information decomposition generated a flurry of proposals for defining an intersection information that quantifies how much of “the same information” two or more random variables specify about a target random variable. As of yet, none is wholly satisfactory.” Works on mutual information between organisms and environment (along with David Krakauer and Jessica Flack), and also entropy rates.

• Cameron Freer – MIT postdoc in Brain and Cognitive Sciences working on maximum entropy production principles, algorithmic entropy etc.

• Tobias Fritz – a physicist who has worked on “resource theories” and haracterizations of Shannon and Rényi entropy and on resource theories.

• Dashiell Fryer – works with Marc Harper on information geometry and evolutionary game theory.

• Michael Gilchrist – an evolutionary biologist studying how errors and costs of protein translation affect the codon usage observed within a genome. Works at NIMBioS.

• Manoj Gopalkrishnan – an expert on chemical reaction networks who understands entropy-like Lyapunov functions for these systems.

• Marc Harper – works on evolutionary game theory using ideas from information theory, information geometry, etc.

• John Harte – an ecologist who uses the maximum entropy method to predict the structure of ecosystems.

• Ellen Hines – studies habitat modeling and mapping for marine endangered species and ecosystems, sea level change scenarios, documenting of human use and values. Her lab has used MaxEnt methods.

• Elizabeth Hobson – behavior ecology postdoc developing methods to quantify social complexity in animals. Works at NIMBioS.

• John Jungk – works on graph theory and biology.

• Chris Lee – in bioinformatics and genomics; applies information theory to experiment design and evolutionary biology.

• Maria Leites – works on dynamics, bifurcations and applications of coupled systems of non-linear ordinary differential equations with applications to ecology, epidemiology, and transcriptional regulatory networks. Interested in information theory.

• Tom Leinster – a mathematician who applies category theory to study various concepts of ‘magnitude’, including biodiversity and entropy.

• Timothy Lezon – a systems biologist in the Drug Discovery Institute at Pitt, who has used entropy to characterize phenotypic heterogeneity in populations of cultured cells.

• Maria Ortiz Mancera – statistician working at CONABIO, the National Commission for Knowledge and Use of Biodiversity, in Mexico.

• Yajun Mei – statistician who uses Kullback-Leibler divergence and how to efficiently compute entropy for the two-state hidden Markov models.

• Robert Molzon – mathematical economist who has studied deterministic approximation of stochastic evolutionary dynamics.

• David Murrugarra – works on discrete models in mathematical biology; interested in learning about information theory.

• Annette Ostling – studies community ecology, focusing on the influence of interspecific competition on community structure, and what insights patterns of community structure might provide about the mechanisms by which competing species coexist.

• Connie Phong – grad student at Chicago’s Institute of Genomics and System biology, working on how “certain biochemical network motifs are more attuned than others at maintaining strong input to output relationships under fluctuating conditions.”

• Petr Plechak – works on information-theoretic tools for estimating and minimizing errors in coarse-graining stochastic systems. Wrote “Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems”.

• Blake Polllard – physics grad student working with John Baez on various generalizations of Shannon and Renyi entropy, and how these entropies change with time in Markov processes and open Markov processes.

• Timothee Poisot – works on species interaction networks; developed a “new suite of tools for probabilistic interaction networks”.

• Richard Reeve – works on biodiversity studies and the spread of antibiotic resistance. Ran a program on entropy-based biodiversity measures at a mathematics institute in Barcelona.

• Rob Shaw – works on entropy and information in biotic and pre-biotic systems.

• Matteo Smerlak – postdoc working on nonequilibrium thermodynamics and its applications to biology, especially population biology and cell replication.

• Susanne Still – a computer scientist who studies the role of thermodynamics and information theory in prediction.

• Alexander Wissner-Gross – Institute Fellow at the Harvard University Institute for Applied Computational Science and Research Affiliate at the MIT Media Laboratory, interested in lots of things.

• David Wolpert – works at the Santa Fe Institute on i) information theory and game theory, ii) the second law of thermodynamics and dynamics of complexity, iii) multi-information source optimization, iv) the mathematical underpinnings of reality, v) evolution of organizations.

• Matthew Zefferman – works on evolutionary game theory, institutional economics and models of gene-culture co-evolution. No work on information, but a postdoc at NIMBioS.

3 Comments | conferences, information and entropy | Permalink
Posted by John Baez

Categorical Foundations of Network Theory

4 April, 2015

Jacob Biamonte got a grant from the Foundational Questions Institute to run a small meeting on network theory:

• The categorical foundations of network theory.

It’s being held 25-28 May 2015 in Turin, Italy, at the ISI Foundation. We’ll make slides and/or videos available, but the main goal is to bring a few people together, exchange ideas, and push the subject forward.

The idea

Network theory is a diverse subject which developed independently in several disciplines. It uses graphs with additional structure to model everything from complex systems to theories of fundamental physics.

This event aims to further our understanding of the mathematical theory underlying the relations between seemingly different networked systems. It’s part of the Azimuth network theory project.

Timetable

With the exception of the first day (Monday May 25th) we will kick things off with a morning talk, with plenty of time for questions and interaction. We will then break for lunch at 1:00 p.m. and return for an afternoon work session. People are encouraged to give informal talks and to present their ideas in the afternoon sessions.

Monday May 25th, 10:30 a.m.

• Jacob Biamonte: opening remarks.

For Jacob’s work on quantum networks visit www.thequantumnetwork.org.

• John Baez: network theory.

For my stuff see the Azimuth Project network theory page.

Tuesday May 26th, 10:30 a.m.

• David Spivak: operadic network design.

Operads are a formalism for sticking small networks together to form bigger ones. David has a 3-part series of articles sketching his ideas on networks.

Wednesday May 27th, 10:30 a.m.

• Eugene Lerman: continuous time open systems and monoidal double categories.

Eugene is especially interested in classical mechanics and networked dynamical systems, and he wrote an introductory article about them here on the Azimuth blog.

Thursday May 28th, 10:30 a.m.

• Tobias Fritz: ordered commutative monoids and theories of resource convertibility.

Tobias has a new paper on this subject, and a 3-part expository series here on the Azimuth blog!

Location and contact

ISI Foundation
Via Alassio 11/c
10126 Torino — Italy

Phone: +39 011 6603090
Email: isi@isi.it
Theory group details: www.TheQuantumNetwork.org

23 Comments | conferences, networks | Permalink
Posted by John Baez

A Networked World (Part 3)

2 April, 2015

guest post by David Spivak

• Part 1: The problem.

• Part 2: Creating a knowledge network.

From parts to wholes

Remember where we were. Ologs, linguistically-enhanced sketches, just weren’t doing justice to the idea that each step in a recipe is itself a recipe. But the idea seemed ripe for mathematical formulation.

Thus, I returned to a question I’d wondered about in the very beginning: how is macro-understanding built from micro-understanding? How can multiple individual humans come together, like cells in a multicellular organism, to make a whole that is itself a surviving decision-maker?

There were, and continue to be, a lot of “open-to-Spivak” questions one can ask: How are stories about events built from sub-stories about sub-events? How is macro-economics built from micro-economics? Are large-scale phenomena always based on, and relatable to, interactions between smaller-scale phenomena? For example, I still want to understand, in very basic terms, how classical (large-scale) phenomena are a manifestation of quantum phenomena.

Neuroscience professor Michael Gazzaniga has a similar question: How does cognition arise from the interaction of tiny event-noticers, and how does society emerge and effect individual brains? As put it in the last paragraph of his book Who’s In Charge, we are in need of a language by which to understand the interfaces of “our layered hierarchical existence”, because doing so “holds the answer to our quest for understanding mind/brain relationships.” He goes on:

Understanding how to develop a vocabulary for those layered interactions, for me, constitutes the scientific problem of this century.

I tend to be infatuated with this same kind of idea: cognition emerging from interactions between sub-cognitive pieces. This is what got me interested in what I now call “operadic modularity”. Luckily again, my Office of Naval Research hero (now at the Air Force Office of Scientific Research) granted me a chance to study it.

The idea is this: modularity is about arranging many modules into a single whole, which is another module, usable as part of a larger system. Each system of modules is a module of its own, and we see the nesting phenomenon. Operads can model the language of nestable interface arrangements, and their algebras can model how interfaces are filled in with the required sorts of modules.

Here, by operad, I mean symmetric colored operad, or symmetric multicategory. Operads are like categories—they have objects, morphisms, identities, and a unital and associative composition formula—the difference is that the domain of a morphism is a finite set of objects (in an operad) rather than a single object (as in a category). So morphisms in an operad are written like $\varphi\colon X_1,\ldots,X_n\to Y;$ we call such a morphism n-ary.

An early example, formulated operadically by Peter May (the inventor of operads) is called the little 2-cubes operad, denoted $E_2.$ It has only one object, say a square ⬜, and an n-ary morphism

⬜ ,…, ⬜ $\longrightarrow$ ⬜

is any arrangement of $n$ non-overlapping squares in a larger square. These arrangements clearly display a nesting property.

Another source of examples comes from the fact that every monoidal category $(C,\otimes)$ has an underlying operad $\mathcal{O}_C,$ with

$\mathrm{Hom}_{\mathcal{O}_C}(X_1,\ldots,X_n;Y):=\mathrm{Hom}_{C}(X_1\otimes\cdots\otimes X_n,Y)$

(Either $C$ was symmetric monoidal to begin with or you can add in symmetries, roughly by multiplying each hom-set by $n!.$ ) The operad $\mathbf{Set}$ underlying the cartesian monoidal category $(\mathbf{Set},\times,\{1\})$ of sets is an example I’ll use later.

If you want to think about operads as modeling modularity—building one thing out of many—the first trick is to imagine the codomain object as the exterior and all the domain objects as sitting inside it, on the interior. May’s little 2-cubes operad gives the idea: squares in a square. From now on, if I speak of many little objects arranged inside one big object, I always mean it this way: the interior objects constitute the domain, the exterior object is the codomain, and the arrangement itself is the morphism. These arrangements can be nested inside one another, corresponding to composition in the operad.

What are other types of nested phenomena, which we might be able to think about operadically? How about circles wired together in a larger circle? An object in this operad is a circle with some number of wires sticking out; let’s call it a ported-circle. A morphism from n-many ported-circles to one ported-circle is any connection pattern involving—i.e., wiring together of—the ports. This description can be interpreted in a few different ways; I usually mean the underlying operad of the monoidal category of “sets and cospans under disjoint union”, but the “spaghetti and meatballs operad” of circular planar arc diagrams is another good interpretation.

Once you have an operad $\mathcal{O},$ you have a kind of calculus for nestable arrangements. As I’ve been saying, I often think of the morphisms in an operad in terms of pictures, such as wiring diagrams or squares-in-a-square. If you say you want these pictures to “mean something”, you’re probably looking for an algebra $F:\mathcal{O}\to\textbf{Sets}.$ This operad functor $F,$ which acts like a lax functor between monoidal categories, would tell you the set $F(X)$ of fillers or fills that can be placed into each object $X\in\mathcal{O}$ in the picture.

I often think of the operad $\mathcal{O}$ as a picture language, and the $\mathcal{O}$ -algebra $F$ its intended semantics. Not only does such a set-valued functor on $\mathcal{O}$ give you a set of fills for each object $X\in\mathcal{O}$ , it would also give you a formula for producing a large-scale fill (element of $F(Y)$ ) from any arrangement $\varphi\colon X_1,\ldots,X_n\to Y$ of small-scale fills (element of $F(X_1)\times\cdots\times F(X_n)$ ).

For example, given a pointed space $A$ , you can ask for the set of based 2-spheres

$L_A($ ⬜ $)=\{\ell\colon S^2\to A\}$

in it. Here, $\ell$ is any element of $L_A($ ⬜ $).$ Think of a based sphere in $A$ as a continuous map from the filled-in square to $A$ that sends the boundary of the square to the basepoint of $A.$ Given n spheres $\ell_1,\ldots, \ell_n$ in $A,$ an arrangement $\varphi$ of non-overlapping squares in a square prescribes a new based sphere $L_A(\varphi)(\ell_1,\ldots, \ell_n)\in L_A($ ⬜ $).$ The idea is that you send all the unused space in the big exterior square to the basepoint of $A$ , and follow the instructions $\ell_i$ when you get to the $i$ th little square inside. Thus any “2-fold loop space” gives an algebra of May’s little 2-cubes operad.

So recently, I’ve been thinking a lot about operadic modularity, i.e., cases in which a thing can be built out of a bunch of simpler things. Note that not all cases of “nesting” have such a clear picture language. For example, context-free grammars are modular: you build [postal-address] out of [name-part], [street-address] and [zip-part], you build each of these, e.g., [name-part], in any of several ways (there is an optional suffix part and the option to abbreviate your first name using an initial). The point is, you build things out of smaller parts, nested inside still smaller parts. Seeing context-free grammars as free operads is one of the things Hermida, Makkai, and Power explained in their paper on higher dimensional multigraphs.

The operadic notion of modularity can also be applied to building hierarchical protein materials. Like context-free grammars, the operad of such materials doesn’t come with a nice picture language. However, it can be formalized as an operad nonetheless. That is, there is a grammar of actions that one can apply to a bunch of polypeptides, actions such as “attach”, “overlay”, “rigidMotion”, “helix”, “makeArray”. From these you can build proteins that are quite complex from a simple vocabulary of 20 amino acids. I’ve joined forces with Tristan Giesa and Ravi Jagadeesan to make such a program. The software package, called Matriarch, for “Materi-als Arch-itecture”, should be available soon as an open source Python library.

There are lots of operads whose morphisms look like string diagrams of various sorts. These operads, which generalize a set-theoretic version of May’s topological little 2-cubes, have clear picture languages. The algebras on such “visualizable” operads can model things like databases and dynamical systems. Over the past year or so, I’ve been writing a series of “worked example” papers, such as those above, in which I explain various picture languages and semantics for them.

I find that operads provide a nice language in which to discuss string diagrams of various sorts. String-diagrammatic languages exist for many different “doctrines”, such as categories, categories without identities, monoidal categories, cartesian monoidal categories, traced monoidal categories, operads, etc. For example, Dylan Rupel and I realized that traced monoidal categories are (well, if you have enough equipment and an expert like Patrick Schultz around) algebras on the operad of oriented 1-cobordisms. It seems to me that the other doctrines above are similarly associated to operads that are “nearby” Cob, e.g., sub-operads of Cob, operads under Cob, etc. Maps between these various operads should induce known adjunctions between the corresponding doctrines.

That brings us to present day. There will be a workshop in Turin in a couple of months, and I think it’ll be a lot of fun:

• Categorical Foundations of Network Theory, May 25-28, ISI Foundation, Turin, Italy.

I’m looking forward to hearing from John Baez, Jacob Biamonte, Eugene Lerman, Tobias Fritz and others, about what they’ve been thinking about recently. I think we’ll find interesting common ground. If there’s interest, I’d be happy to talk about categorical models for how information is communicated throughout a network, and whether this gives any insight that can lead to better decision-making by the larger whole.

12 Comments | mathematics, networks | Permalink
Posted by John Baez

A Networked World (Part 2)

30 March, 2015

guest post by David Spivak

• Part 1: The problem.

Creating a knowledge network

In 2007, I asked myself: as mathematically as possible, what can formally ground meaningful information, including both its successful communication and its role in decision-making? I believed that category theory could be useful in formalizing the type of object that we call information, and the type of relationship that we call communication.

Over the next few years, I worked on this project. I tried to understand what information is, how it is stored, and how it can be transferred between entities that think differently. Since databases store information, I wanted to understand databases category-theoretically. I eventually decided that databases are basically just categories $\mathcal{C},$ corresponding to a collection of meaningful concepts and connections between them, and that these categories are equipped with functors $\mathcal{C}\to\mathbf{Set}.$ Such a functor assigns to each meaningful concept a set of examples and connects them as dictated by the morphisms of $\mathbf{Set}.$ I later found out that this “databases as categories” idea was not original; it is due to Rosebrugh and others. My view on the subject has matured a bit since then, but I still like this basic conception of databases.

If we model a person’s knowledge as a database (interconnected tables of examples of things and relationships), then the network of knowledgeable humans could be conceptualized as a simplicial complex equipped with a sheaf of databases. Here, a vertex represents an individual, with her database of knowledge. An edge represents a pair of individuals and a common ground database relating their individual databases. For example, you and your brother have a database of concepts and examples from your history. The common-ground database is like the intersection of the two databases, but it could be smaller (if the people don’t yet know they agree on something). In a simplicial complex, there are not only vertices and edges, but also triangles (and so on). These would represent databases held in common between three or more people.

I wanted “regular people” to actually make such a knowledge network, i.e., to share their ideas in the form of categories and link them together with functors. Of course, most people don’t know categories and functors, so I thought I’d make things easier for them by equipping categories with linguistic structures: text boxes for objects, labeled arrows for morphisms. For example, “a person has a mother” would be a morphism from the “person” object, to the “mother” object. I called such a linguistic category an olog, playing on the word blog. The idea (originally inspired during a conversation with my friend Ralph Hutchison) was that I wanted people, especially scientists, to blog their ontologies, i.e., to write “onto-logs” like others make web-logs.

Ologs codify knowledge. They are like concept webs, except with more rules that allow them to simultaneously serve as database schemas. By introducing ologs, I hoped I could get real people to upload their ideas into what is now called the cloud, and make the necessary knowledge network. I tried to write my papers to engage an audience of intelligent lay-people rather than for an audience of mathematicians. It was a risk, but to me it was the only honest approach to the larger endeavor.

(For students who might want to try going out on a limb like this, you should know that I was offered zero jobs after my first postdoc at University of Oregon. The risk was indeed risky, and one has to be ok with that. I personally happened to be the beneficiary of good luck and was offered a grant, out of the clear blue sky, by a former PhD in algebraic geometry, who worked at the Office of Naval Research at the time. That, plus the helping hands of Haynes Miller and many other brilliant and wonderful people, can explain how I lived to tell the tale.)

So here’s how the simplicial complex of ologs would ideally help humanity steer. Suppose we say that in order for one person to learn from another, the two need to find a common language and align some ideas. This kind of (usually tacit) agreement on, or alignment of, an initial common-ground vocabulary and concept-set is important to get their communication onto a proper footing.

For two vertices in such a simplicial network, the richer their common-ground olog (i.e., the database corresponding to the edge between them) is, the more quickly and accurately the vertices can share new ideas. As ideas are shared over a simplex, all participating databases can be updated, hence making the communication between them richer. In around 2010, Mathieu Anel and I worked out a formal way this might occur; however, we have not yet written it up. The basic idea can be found here.

In this setup, the simplicial complex of human knowledge should grow organically. Scientists, business people, and other people might find benefit in ologging their ideas and conceptions, and using them to learn from their peers. I imagined a network organizing itself, where simplices of like-minded people could share information with neighboring groups across common faces.

I later wrote a book called Category Theory for the Sciences, available free online, to help scientists learn how category theory could apply to familiar situations like taxonomies, graphs, and symmetries. Category theory, simply explained, becomes a wonderful key to the whole world of pure mathematics. It’s the closest thing we have to a universal language of thought, and therefore an appropriate language for forming connections.

My working hypothesis for the knowledge network was this. The information held by people whose worldview is more true—more accurate—would have better predictive power, i.e., better results. This is by definition: I define ones knowledge to be accurate as the extent to which, when he uses this knowledge to direct his actions, he has good luck handling his worldly affairs. As Louis Pasteur said, “Luck favors the prepared mind.” It follows that if someone has a track record of success, others will value finding broad connections into his olog. However, to link up with someone you must find a part of your olog that aligns with his—a functorial connection—and you can only receive meaningful information from him to the extent that you’ve found such common ground.

Thus, people who like to live in fiction worlds would find it difficult to connect, except to other like-minded “Obama’s a Christian”-type people. To the extent you are imbedded in a fictional—less accurate, less predictive—part of the network, you will find it difficult to establish functorial connections to regions of more accurate knowledge, and therefore you can’t benefit from the predictive and conceptual value of this knowledge.

In other words, people would be naturally inclined to try to align their understanding with people that are better informed. I felt hope that this kind of idea could lead to a system in which honesty and accuracy were naturally rewarded. At the very least, those who used it could share information much more effectively than they do now. This was my plan; I just had to make it real.

I had a fun idea for publicizing ologs. The year was in 2008, and I remember thinking it would be fantastic if I could olog the political platform and worldview of Barack Obama and of Sarah Palin. I wished I could sit down with them and other politicians and help them write ologs about what they believed and wanted for the country. I imagined that some politicians might have ologs that look like a bunch of disconnected text boxes—like a brain with neurons but no synapses—a collection of talking points but no real substantive ideas.

Anyway, there I was, trying to understand everything this way: all information was categories (or perhaps sketches) and presheaves. I would work with interested people from any academic discipline, such as materials science, to make ologs about whatever information they wanted to record category-theoretically. Ologs weren’t a theory of everything, but instead, as Jack Morava put it, a theory of anything.

One day I was working on a categorical sketch to model processes within processes, but somehow it really wasn’t working properly. The idea was simple: each step in a recipe is a mini-recipe of its own. Like chopping the carrots means getting out a knife and cutting board, putting a carrot on there, and bringing the knife down successively along it. You can keep zooming into any of these and see it as its own process. So there is some kind of nested, fractal-like behavior here. The olog I made could model the idea of steps in a recipe, but I found it difficult to encode the fact that each step was itself a recipe.

This nesting thing seemed like an idea that mathematics should treat beautifully, and ologs weren’t doing it justice. It was then when I finally admitted that there might be other fish in the mathematical sea.

• Part 3: From parts to wholes.

10 Comments | mathematics, networks | Permalink
Posted by John Baez

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Azimuth

Resource Convertibility (Part 3)

References

Resource Convertibility (Part 2)

Resource Convertibility (Part 1)

Information and Entropy in Biological Systems (Part 3)

Schedule

Wednesday April 8

Thursday April 9

Friday April 10

Participants

Categorical Foundations of Network Theory

The idea

Timetable

Monday May 25th, 10:30 a.m.

Tuesday May 26th, 10:30 a.m.

Wednesday May 27th, 10:30 a.m.

Thursday May 28th, 10:30 a.m.

Location and contact

A Networked World (Part 3)

From parts to wholes

A Networked World (Part 2)

Creating a knowledge network

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