Network Theory I

 

Here’s a video of a talk I gave last Tuesday—part of a series. You can see the slides here:

• Network Theory I: electrical circuits and signal-flow graphs.

Click on items in blue, or pictures, for more information.

One reason I’m glad I gave this talk is because afterwards Jamie Vicary pointed out some very interesting consequences of the relations among signal-flow diagrams listed in my talk. It turns out they imply equations familiar from the theory of complementarity in categorical quantum mechanics!

This is the kind of mathematical surprise that makes life worthwhile for me. It seemed utterly shocking at first, but I think I’ve figured out why it happens. Now is not the time to explain… but I’ll have to do it soon, both here and in the paper that Jason Eberle are writing about control theory.

For now, besides the slides, the best place to read more about this program is here:

• Brendan Fong, A compositional approach to control theory.

Network Theory Overview

 

Here’s a video of a talk I gave yesterday, made by Brendan Fong. You can see the slides here—and then click the items in blue, and the pictures, for more information!

The idea: nature and the world of human technology are full of networks! People like to draw diagrams of networks. Mathematical physicists know that in principle these diagrams can be understood using category theory. But why should physicists have all the fun? This is the century of understanding living systems and adapting to life on a finite planet. Math isn’t the main thing we need for this, but it’s got to be part of the solution… so one thing we should do is develop a unified and powerful theory of networks.

We are still far from doing this. In this overview, I briefly described three parts of the jigsaw puzzle, and invited everyone to help fit them together:

• electrical circuits and signal-flow graphs.

• stochastic Petri nets, chemical reaction networks and Feynman diagrams.

• Bayesian networks, information and entropy.

In my talks coming up, I’ll go into more detail on each of these. With luck, you’ll be able to see videos here.

But if you’re near Oxford, you might as well actually attend! You can see dates, times, locations, my slides, and the talks themselves as they show up by going here.

Network Theory Talks at Oxford

I’m giving some talks at Oxford:

Network Theory

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. After an overview, we will look at three portions of the jigsaw puzzle in three separate talks:

I. Electrical circuits and signal-flow graphs.

II. Stochastic Petri nets, chemical reaction networks and Feynman diagrams.

III. Bayesian networks, information and entropy.

If you’re nearby I hope you can come! All these talks will take place in Lecture Theatre B in the Computer Science Department—see the map below. Here are the times:

• Friday 21 February 2014, 2 pm: Network Theory: overview. See the slides or watch a video.

• Tuesday 25 February, 3:30 pm: Network Theory I: electrical circuits and signal-flow graphs. See the slides or watch a video.

• Tuesday 4 March, 3:30 pm: Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams. See the slides or watch a video.

• Tuesday 11 March, 3:30 pm: Network Theory III: Bayesian networks, information and entropy. See the slides or watch a video

The first talk will be part of the OASIS series, meaning the “Oxford Advanced Seminar on Informatic Structures”.

I thank Samson Abramsky, Bob Coecke and Jamie Vicary of the Computer Science Department for inviting me, and Ulrike Tillmann and Minhyong Kim of the Mathematical Institute for helping me get set up. I also thank all the people who helped do the work I’ll be talking about, most notably Jacob Biamonte, Jason Erbele, Brendan Fong, Tobias Fritz, Tom Leinster, Tu Pham, and Franciscus Rebro.

Ulrike Tillmann has also kindly invited me to give a topology seminar:

Operads and the Tree of Life

Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists use increasingly sophisticated mathematical techniques to reconstruct the most likely “phylogenetic tree” describing how these species evolved from earlier ones. In their work on this subject, they have encountered an interesting example of an operad, which is obtained by applying a variant of the Boardmann–Vogt “W construction” to the operad for commutative monoids. The operations in this operad are labelled trees of a certain sort, and it plays a universal role in the study of stochastic processes that involve branching. It also shows up in tropical algebra. This talk is based on work in progress with Nina Otter.

I’m not sure exactly where this will take place, but surely somewhere in the Mathematical Institute building:

• Monday 24 February, 3:30 pm, Operads and the Tree of Life. See the slides.

The Computer Science Department is shown in the map here:

 

The Mathematical Institute is a bit to the west:

Lyapunov Functions for Complex-Balanced Systems

guest post by Manoj Gopalkrishnan

A few weeks back, I promised to tell you more about a long-standing open problem in reaction networks, the ‘global attractor conjecture’. I am not going to quite get there today, but we shall take one step in that direction.

Today’s plan is to help you make friends with a very useful function we will call the ‘free energy’ which comes up all the time in the study of chemical reaction networks. We will see that for complex-balanced systems, the free energy function decreases along trajectories of the rate equation. I’m going to explain this statement, and give you most of the proof!

The point of doing all this work is that we will then be able to invoke Lyapunov’s theorem which implies stability of the dynamics. In Greek mythology, Sisyphus was cursed to roll a boulder up a hill only to have it roll down again, so that he had to keep repeating the task for all eternity. When I think of an unstable equilibrium, I imagine a boulder delicately balanced on top of a hill, which will fall off if given the slightest push:

or, more abstractly:

On the other hand, I picture a stable equilibrium as a pebble at the very bottom of a hill. Whichever way a perturbation takes it is up, so it will roll down again to the bottom:

Lyapunov’s theorem guarantees stability provided we can exhibit a nice enough function that decreases along trajectories. ‘Nice enough’ means that, viewing as a height function for the hill, the equilibrium configuration should be at the bottom, and every direction from there should be up. If Sisyphus had dug a pit at the top of the hill for the boulder to rest in, Lyapunov’s theorem would have applied, and he could have gone home to rest. The moral of the story is that it pays to learn dynamical systems theory!

Because of the connection to Lyapunov’s theorem, such functions that decrease along trajectories are also called Lyapunov functions. A similar situation is seen in Boltzmann’s H-theorem, and hence such functions are sometimes called H-functions by physicists.

Another reason for me to talk about these ideas now is that I have posted a new article on the arXiv:

• Manoj Gopalkrishnan, On the Lyapunov function for complex-balanced mass-action systems.

The free energy function in chemical reaction networks goes back at least to 1972, to this paper:

• Friedrich Horn and Roy Jackson, General mass action kinetics, Arch. Rational Mech. Analysis 49 (1972), 81–116.

Many of us credit Horn and Jackson’s paper with starting the mathematical study of reaction networks. My paper is an exposition of the main result of Horn and Jackson, with a shorter and simpler proof. The gain comes because Horn and Jackson proved all their results from scratch, whereas I’m using some easy results from graph theory, and the log-sum inequality.

We shall be talking about reaction networks. Remember the idea from the network theory series. We have a set whose elements are called species, for example

A complex is a vector of natural numbers saying how many items of each species we have. For example, we could have a complex But chemists would usually write this as

A reaction network is a set of species and a set of transitions or reactions, where each transition goes from some complex to some complex For example, we could have a transition with

and

In this situation chemists usually write

but we want names like for our transitions, so we might write

or

As John explained in Part 3 of the network theory series, chemists like to work with a vector of nonnegative real numbers saying the concentration of each species at time If we know a rate constant for each transition we can write down an equation saying how these concentrations change with time:

This is called the rate equation. It’s really a system of ODEs describing how the concentration of each species change with time. Here an expression like is shorthand for the monomial

John and Brendan talked about complex balance in Part 9. I’m going to recall this definition, from a slightly different point of view that will be helpful for the result we are trying to prove.

We can draw a reaction network as a graph! The vertices of this graph are all the complexes where The edges are all the transitions We think of each edge as directed, going from to

We will call the map that sends each transition to the positive real number the flow on this graph. The rate equation can be rewritten very simply in terms of this flow as:

where the right-hand side is now a linear expression in the flow

Flows of water, or electric current, obey a version of Kirchhoff’s current law. Such flows are called conservative flows. The following two lemmas from graph theory are immediate for conservative flows:

Lemma 1. If f is a conservative flow then the net flow across every cut is zero.

A cut is a way of chopping the graph in two, like this:

It’s easy to prove Lemma 1 by induction, moving one vertex across the cut at a time.

Lemma 2. If a conservative flow exists then every edge is part of a directed cycle.

Why is Lemma 2 true? Suppose there exists an edge that is not part of any directed cycle. We will exhibit a cut with non-zero net flow. By Lemma 1, this will imply that the flow is not conservative.

One side of the cut will consist of all vertices from which is reachable by a directed path in the reaction network. The other side of the cut contains at least since is not reachable from by the assumption that is not part of a directed cycle. There is flow going from left to right of the cut, across the transition Since there can be no flow coming back, this cut has nonzero net flow, and we’re done.     ▮

Now, back to the rate equation! We can ask if the flow is conservative. That is, we can ask if, for every complex :

In words, we are asking if the sum of the flow through all transitions coming in to equals the sum of the flow through all transitions going out of If this condition is satisfied at a vector of concentrations so that the flow is conservative, then we call a point of complex balance. If in addition, every component of is strictly positive, then we say that the system is complex balanced.

Clearly if is a point of complex balance, it’s an equilibrium solution of the rate equation. In other words, is a solution of the rate equation, where never changes.

I’m using ‘equilibrium’ the way mathematicians do. But I should warn you that chemists use ‘equilibrium’ to mean something more than merely a solution that doesn’t change with time. They often also mean it’s a point of complex balance, or even more. People actually get into arguments about this at conferences.

Complex balance implies more than mere equilibrium. For starters, if a reaction network is such that every edge belongs to a directed cycle, then one says that the reaction network is weakly reversible. So Lemmas 1 and 2 establish that complex-balanced systems must be weakly reversible!

From here on, we fix a complex-balanced system, with a strictly positive point of complex balance.

Definition. The free energy function is the function

where the sum is over all species in

The whole point of defining the function this way is because it is the unique function, up to an additive constant, whose partial derivative with respect to is This is important enough that we write it as a lemma. To state it in a pithy way, it is helpful to introduce vector notation for division and logarithms. If and are two vectors, we will understand to mean the vector such that coordinate-wise. Similarly is defined in a coordinate-wise sense as the vector with coordinates

Lemma 3. The gradient of equals

We’re ready to state our main theorem!

Theorem. Fix a trajectory of the rate equation. Then is a decreasing function of time Further, it is strictly decreasing unless is an equilibrium solution of the rate equation.

I find precise mathematical statements reassuring. You can often make up your mind about the truth value from a few examples. Very often, though not always, a few well-chosen examples are all you need to get the general idea for the proof. Such is the case for the above theorem. There are three key examples: the two-cycle, the three-cycle, and the figure-eight.

The two-cycle. The two-cycle is this reaction network:

It has two complexes and and two transitions and with rates and respectively.

Fix a solution of the rate equation. Then the flow from to equals and the backward flow equals The condition for to be a conservative flow requires that This is one binomial equation in at least one variable, and clearly has a solution in the positive reals. We have just shown that every two-cycle is complex balanced.

The derivative can now be computed by the chain rule, using Lemma 3. It works out to times

This is never positive, and it’s zero if and only if

Why is this? Simply because the logarithm of something greater than 1 is positive, while the log of something less than 1 is negative, so that the sign of is always opposite the sign of We have verified our theorem for this example.

(Note that occurs when but also at other points: in this example, there is a whole hypersurface consisting of points of complex balance.)

In fact, this simple calculation achieves much more.

Definition. A reaction network is reversible if for every transition there is a transition going back, called the reverse of Suppose we have a reversible reaction network and a vector of concentrations such that the flow along each edge equals that along the edge going back:

whenever is the reverse Then we say the reaction network is detailed balanced, and is a point of detailed balance.

For a detailed-balanced system, the time derivative of is a sum over the contributions of pairs consisting of an edge and its reverse. Hence, the two-cycle calculation shows that the theorem holds for all detailed balanced systems!

This linearity trick is going to prove very valuable. It will allow us to treat the general case of complex balanced systems one cycle at a time. The proof for a single cycle is essentially contained in the example of a three-cycle, which we treat next:

The three-cycle. The three-cycle is this reaction network:

We assume that the system is complex balanced, so that

Let us call this nonnegative number A small calculation employing the chain rule shows that equals times

We need to think about the sign of this quantity:

Lemma 3. Let be positive numbers. Then is less than or equal to zero, with equality precisely when

The proof is a direct application of the log sum inequality. In fact, this holds not just for three numbers, but for any finite list of numbers. Indeed, that is precisely how one obtains the proof for cycles of arbitrary length. Even the two-cycle proof is a special case! If you are wondering how the log sum inequality is proved, it is an application of Jensen’s inequality, that workhorse of convex analysis.

The three-cycle calculation extends to a proof for the theorem so long as there is no directed edge that is shared between two directed cycles. When there are such edges, we need to argue that the flows and can be split between the cycles sharing that edge in a consistent manner, so that the cycles can be analyzed independently. We will need the following simple lemma about conservative flows from graph theory. We will apply this lemma to the flow

Lemma 4. Let be a conservative flow on a graph Then there exist directed cycles in and nonnegative real ‘flows’ such that for each directed edge in the flow equals the sum of over such the cycle contains the edge

Intuitively, this lemma says that conservative flows come from constant flows on the directed cycles of the graph. How does one show this lemma? I’m sure there are several proofs, and I hope some of you can share some of the really neat ones with me. The one I employed was algorithmic. The idea is to pick a cycle, any cycle, and subtract the maximum constant flow that this cycle allows, and repeat. This is most easily understood by looking at the example of the figure-eight:

The figure-eight. This reaction network consists of two three-cycles sharing an edge:

Here’s the proof for Lemma 4. Let be a conservative flow on this graph. We want to exhibit cycles and flows on this graph according to Lemma 4. We arbitrarily pick any cycle in the graph. For example, in the figure-eight, suppose we pick the cycle We pick an edge in this cycle on which the flow is minimum. In this case, is the minimum. We define a remainder flow by subtracting from this constant flow which was restricted to one cycle. So the remainder flow is the same as on edges that don’t belong to the picked cycle. For edges that belong to the cycle, the remainder flow is minus the minimum of on this cycle. We observe that this remainder flow satisfies the conditions of Lemma 4 on a graph with strictly fewer edges. Continuing in this way, since the lemma is trivially true for the empty graph, we are done by infinite descent.

Now that we know how to split the flow across cycles, we can figure out how to split the rates across the different cycles. This will tell us how to split the flow across cycles. Again, this is best illustrated by an example.

The figure-eight. Again, this reaction network looks like

Suppose as in Lemma 4, we obtain the cycles

with constant flow

and

with constant flow such that

Here’s the picture:

Then we obtain rates and by solving the equations

Using these rates, we can define non-constant flows on and on by the usual formulas:

and similarly for In particular, this gives us

and similarly for

Using this, we obtain the proof of the Theorem! The time derivative of along a trajectory has a contribution from each cycle as in Lemma 4, where each cycle is treated as a separate system with the new rates and the new flows and So, we’ve reduced the problem to the case of a cycle, which we’ve already done.

Let’s review what happened. The time derivative of the function has a very nice form, which is linear in the flow The reaction network can be broken up into cycles. Th e conservative flow for a complex balanced system can be split into conservative flows on cycles by Lemma 4. This informs us how to split the non-conservative flow across cycles. By linearity of the time derivative, we can separately treat the case for every cycle. For each cycle, we get an expression to which the log sum inequality applies, giving us the final result that decreases along trajectories of the rate equation.

Now that we have a Lyapunov function, we will put it to use to obtain some nice theorems about the dynamics, and finally state the global attractor conjecture. All that and more, in the next blog post!

Autocatalysis in Reaction Networks

guest post by Manoj Gopalkrishnan

Since this is my first time writing a blog post here, let me start with a word of introduction. I am a computer scientist at the Tata Institute of Fundamental Research, broadly interested in connections between Biology and Computer Science, with a particular interest in reaction networks. I first started thinking about them during my Ph.D. at the Laboratory for Molecular Science. My fascination with them has been predominantly mathematical. As a graduate student, I encountered an area with rich connections between combinatorics and dynamics, and surprisingly easy-to-state and compelling unsolved conjectures, and got hooked.

There is a story about Richard Feynman that he used to take bets with mathematicians. If any mathematician could make Feynman understand a mathematical statement, then Feynman would guess whether or not the statement was true. Of course, Feynman was in a habit of winning these bets, which allowed him to make the boast that mathematics, especially in its obsession for proof, was essentially irrelevant, since a relative novice like himself could after a moment’s thought guess at the truth of these mathematical statements. I have always felt Feynman’s claim to be unjust, but have often wondered what mathematical statement I would put to him so that his chances of winning were no better than random.

Today I want to tell you of a result about reaction networks that I have recently discovered with Abhishek Deshpande. The statement seems like a fine candidate to throw at Feynman because until we proved it, I would not have bet either way about its truth. Even after we obtained a short and elementary proof, I do not completely ‘see’ why it must be true. I am hoping some of you will be able to demystify it for me. So, I’m just going to introduce enough terms to be able to make the statement of our result, and let you think about how to prove it.

John and his colleagues have been talking about reaction networks as Petri nets in the network theory series on this blog. As discussed in part 2 of that series, a Petri net is a diagram like this:

Following John’s terminology, I will call the aqua squares ‘transitions’ and the yellow circles ‘species’. If we have some number #rabbit of rabbits and some number #wolf of wolves, we draw #rabbit many black dots called ‘tokens’ inside the yellow circle for rabbit, and #wolf tokens inside the yellow circle for wolf, like this:

Here #rabbit = 4 and #wolf = 3. The predation transition consumes one ‘rabbit’ token and one ‘wolf’ token, and produces two ‘wolf’ tokens, taking us here:

John explained in parts 2 and 3 how one can put rates on different transitions. For today I am only going to be concerned with ‘reachability:’ what token states are reachable from what other token states. John talked about this idea in part 25.

By a complex I will mean a population vector: a snapshot of the number of tokens in each species. In the example above, (#rabbit, #wolf) is a complex. If are two complexes, then we write

if we can get from to by a single transition in our Petri net. For example, we just saw that

via the predation transition.

Reachability, denoted , is the transitive closure of the relation . So (read is reachable from ) iff there are complexes

such that

For example, here by repeated predation.

I am very interested in switches. After all, a computer is essentially a box of switches! You can build computers by connecting switches together. In fact, that’s how early computers like the Z3 were built. The CMOS gates at the heart of modern computers are essentially switches. By analogy, the study of switches in reaction networks may help us understand biochemical circuits.

A siphon is a set of species that is ‘switch-offable’. That is, if there are no tokens in the siphon states, then they will remain absent in future. Equivalently, the only reactions that can produce tokens in the siphon states are those that require tokens from the siphon states before they can fire. Note that no matter how many rabbits there are, if there are no wolves, there will continue to be no wolves. So {wolf} is a siphon. Similarly, {rabbit} is a siphon, as is the union {rabbit, wolf}. However, when Hydrogen and Oxygen form Water, {Water} is not a siphon.

For another example, consider this Petri net:

The set {HCl, NaCl} is a siphon. However, there is a conservation law: whenever an HCl token is destroyed, an NaCl token is created, so that #HCl + #NaCl is invariant. If both HCl and NaCl were present to begin with, the complexes where both are absent are not reachable. In this sense, this siphon is not ‘really’ switch-offable. As a first pass at capturing this idea, we will introduce the notion of ‘critical set’.

A conservation law is a linear expression involving numbers of tokens that is invariant under every transition in the Petri net. A conservation law is positive if all the coefficients are non-negative. A critical set of states is a set that does not contain the support of a positive conservation law.

For example, the support of the positive conservation law #HCl + #NaCl is {HCl, NaCl}, and hence no set containing this set is critical. Thus {HCl, NaCl} is a siphon, but not critical. On the other hand, the set {NaCl} is critical but not a siphon. {HCl} is a critical siphon. And in our other example, {Wolf, Rabbit} is a critical siphon.

Of particular interest to us will be minimal critical siphons, the minimal sets among critical siphons. Consider this example:

Here we have two transitions:

and

The set is a critical siphon. But so is the smaller set So, is not minimal.

We define a self-replicable set to be a set of species such that there exist complexes and with such that for all we have

So, there are transitions that accomplish the job of creating more tokens for all the species in In other words: these species can ‘replicate themselves’.

We define a drainable set by changing the to a . So, there are transitions that accomplish the job of reducing the number of tokens for all the species in These species can ‘drain away’.

Now here comes the statement:

Every minimal critical siphon is either drainable or self-replicable!

We prove it in this paper:

• Abhishek Deshpande and Manoj Gopalkrishnan, Autocatalysis in reaction networks.

But first note that the statement becomes false if the critical siphon is not minimal. Look at this example again:

The set is a critical siphon. However is neither self-replicable (since every reaction destroys ) nor drainable (since every reaction produces ). But we’ve already seen that is not minimal. It has a critical subsiphon, namely This one is minimal—and it obeys our theorem, because it is drainable.

Checking these statements is a good way to make sure you understand the concepts! I know I’ve introduced a lot of terminology here, and it takes a while to absorb.

Anyway: our proof that every minimal critical siphon is either drainable or self-replicable makes use of a fun result about matrices. Consider a real square matrix with a sign pattern like this:

If the matrix is full-rank then there is a positive linear combination of the rows of the matrix so that all the entries are nonzero and have the same sign. In fact, we prove something stronger in Theorem 5.9 of our paper. At first, we thought this statement about matrices should be equivalent to one of the many well-known alternative statements of Farkas’ lemma, like Gordan’s theorem.

However, we could not find a way to make this work, so we ended up proving it by a different technique. Later, my colleague Jaikumar Radhakrishnan came up with a clever proof that uses Farkas’ lemma twice. However, so far we have not obtained the stronger result in Theorem 5.9 with this proof technique.

My interest in the result that every minimal critical siphon is either drainable or self-replicable is not purely aesthetic (though aesthetics is a big part of it). There is a research community of folks who are thinking of reaction networks as a programming language, and synthesizing molecular systems that exhibit sophisticated dynamical behavior as per specification:

• International Conference on DNA Computing and Molecular Programming.

• Foundations of Nanoscience: Self-Assembled Architectures and Devices.

• Molecular Programming Architectures, Abstractions, Algorithms and Applications.

Networks that exhibit some kind of catalytic behavior are a recurring theme among such systems, and even more so in biochemical circuits.

Here is an example of catalytic behavior:

The ‘catalyst’ helps transform to In the absence of the reaction is turned off. Hence, catalysts are switches in chemical circuits! From this point of view, it is hardly surprising that they are required for the synthesis of complex behaviors.

In information processing, one needs amplification to make sure that a signal can propagate through a circuit without being overwhelmed by errors. Here is a chemical counterpart to such amplification:

Here the catalyst catalyzes its own production: it is an ‘autocatalyst’, or a self-replicating species. By analogy, autocatalysis is key for scaling synthetic molecular systems.

Our work deals with these notions on a network level. We generalize the notion of catalysis in two ways. First, we allow a catalyst to be a set of species instead of a single species; second, its absence can turn off a reaction pathway instead of a single reaction. We propose the notion of self-replicable siphons as a generalization of the notion of autocatalysis. In particular, ‘weakly reversible’ networks have critical siphons precisely when they exhibit autocatalytic behavior. I was led to this work when I noticed the manifestation of this last statement in many examples.

Another hope I have is that perhaps one can study the dynamics of each minimal critical siphon of a reaction network separately, and then somehow be able to answer interesting questions about the dynamics of the entire network, by stitching together what we know for each minimal critical siphon. On the synthesis side, perhaps this could lead to a programming language to synthesize a reaction network that will achieve a specified dynamics. If any of this works out, it would be really cool! I think of how abelian group theory (and more broadly, the theory of abelian categories, which includes categories of vector bundles) benefits from a fundamental theorem that lets you break a finite abelian group into parts that are easy to study—or how number theory benefits from a special case, the fundamental theorem of arithmetic. John has also pointed out that reaction networks are really presentations of symmetric monoidal categories, so perhaps this could point the way to a Fundamental Theorem for Symmetric Monoidal Categories.

And then there is the Global Attractor Conjecture, a
long-standing open problem concerning the long-term behavior of solutions to the rate equations. Now that is a whole story by itself, and will have to wait for another day.

Quantum Network Theory (Part 2)

guest post by Tomi Johnson

Last time I told you how a random walk called the ‘uniform escape walk’ could be used to analyze a network. In particular, Google uses it to rank nodes. For the case of an undirected network, the steady state of this random walk tells us the degrees of the nodes—that is, how many edges come out of each node.

Now I’m going to prove this to you. I’ll also exploit the connection between this random walk and a quantum walk, also introduced last time. In particular, I’ll connect the properties of this quantum walk to the degrees of a network by exploiting its relationship with the random walk.

This is pretty useful, considering how tricky these quantum walks can be. As the parts of the world that we model using quantum mechanics get bigger and have more complicated structures, like biological network, we need all the help in understanding quantum walks that we can get. So I’d better start!

Flashback

Starting with any (simple, connected) graph, we can get an old-fashioned ‘stochastic’ random walk on this graph, but also a quantum walk. The first is the uniform escape stochastic walk, where the walker has an equal probability per time of walking along any edge leaving the node they are standing at. The second is the related quantum walk we’re going to study now. These two walks are generated by two matrices, which we called and The good thing is that these matrices are similar, in the technical sense.

We studied this last time, and everything we learned is summarized here:

where:

• is a simple graph that specifies

• the adjacency matrix (the generator of a quantum walk) with elements equal to unity if nodes and are connected, and zero otherwise (), which subtracted from

• the diagonal matrix of degrees gives

• the symmetric Laplacian (generator of stochastic and quantum walks), which when normalized by returns both

• the generator of the uniform escape stochastic walk and

• the quantum walk generator to which it is similar!

Now I hope you remember where we are. Next I’ll talk you through the mathematics of the uniform escape stochastic walk and how it connects to the degrees of the nodes in the large-time limit. Then I’ll show you how this helps us solve aspects of the quantum walk generated by

Stochastic walk

The uniform escape stochastic walk generated by is popular because it has a really useful stationary state.

To recap from Part 20 of the network theory series, a stationary state of a stochastic walk is one that does not change in time. By the master equation

the stationary state must be an eigenvector of with eigenvalue

A fantastic pair of theorems hold:

• There is always a unique (up to multiplication by a positive number) positive eigenvector of with eigenvalue That is, there is a unique stationary state

• Regardless of the initial state any solution of the master equation approaches this stationary state in the large-time limit:

To find this unique stationary state, consider the Laplacian which is both infinitesimal stochastic and symmetric. Among other things, this means the rows of sum to zero:

Thus, the ‘all ones’ vector is an eigenvector of with zero eigenvalue:

Inserting the identity into this equation we then find is a zero eigenvector of :

Therefore we just need to normalize this to get the large-time stationary state of the walk:

If we write for the basis vector that is zero except at the ith node of our graph, and 1 at that node, the inner product is large-time probability of finding a walker at that node. The equation above implies this is proportional to the degree of node

We can check this for the following graph:

We find that is

which implies large-time probability for nodes and and for nodes and Comparing this to the original graph, this exactly reflects the arrangement of degrees, as we knew it must.

Math works!

The quantum walk

Next up is the quantum walk generated by Not a lot is known about quantum walks on networks of arbitrary geometry, but below we’ll see some analytical results are obtained by exploiting the similarity of and

Where to start? Well, let’s start at the bottom, what quantum physicists call the ground state. In contrast to stochastic walks, for a quantum walk every eigenvector of is a stationary state of the quantum walk. (In Puzzle 5, at the bottom of this page, I ask you to prove this). The stationary state is of particular interest physically and mathematically. Physically, since eigenvectors of the correspond to states of well-defined energy equal to the associated eigenvalue, is the state of lowest energy, energy zero, hence the name ‘ground state’. (In Puzzle 3, I ask you to prove that all eigenvalues of are non-negative, so zero really does correspond to the ground state.)

Mathematically, the relationship between eigenvectors implied by the similarity of and means

So in the ground state, the probability of our quantum walker being found at node is

Amazingly, this probability is proportional to the degree and so is exactly the same as the probability in the stationary state of the stochastic walk!

In short: a zero energy quantum walk leads to exactly the same distribution of the walker over the nodes as in the large-time limit of the uniform escape stochastic walk The classically important notion of degree distribution also plays a role in quantum walks!

This is already pretty exciting. What else can we say? If you are someone who feels faint at the sight of quantum mechanics, well done for getting this far, but watch out for what’s coming next.

What if the walker starts in some other initial state? Is there some quantum walk analogue of the unique large-time state of a stochastic walk?

In fact, the quantum walk in general does not converge to a stationary state. But there is a probability distribution that can be thought to characterize the quantum walk in the same way as the large-time state characterizes the stochastic walk. It’s the large-time average probability vector

If you didn’t know the time that had passed since the beginning of a quantum walk, then the best estimate for the probability of your measuring the walker to be at node would be the large-time average probability

There’s a bit that we can do to simplify this expression. As usual in quantum mechanics, let’s start with the trick of diagonalizing This amounts to writing

where are projectors onto the eigenvectors of and are the corresponding eigenvalues of If we insert this equation into

we get

and thus

Due to the integral over all time, the interference between terms corresponding to different eigenvalues averages to zero, leaving:

The large-time average probability is then the sum of terms contributed by the projections of the initial state onto each eigenspace.

So we have a distribution that characterizes a quantum walk for a general initial state, but it’s a complicated beast. What can we say about it?

Our best hope of understanding the large-time average probability is through the term associated with the zero energy eigenspace, since we know everything about this space.

For example, we know the zero energy eigenspace is one-dimensional and spanned by the eigenvector This means that the projector is just the usual outer product

where we have normalized according to the inner product (If you’re wondering why I’m using all these angled brackets, well, they’re a notation named after Dirac that is adored by quantum physicists.)

The zero eigenspace contribution to the large-time average probability then breaks nicely into two:

This is just the product of two probabilities:

• first, the probability for a quantum state in the zero energy eigenspace to be at node as we found above,

and

• second, the probability of being in this eigenspace to begin with. (Remember, in quantum mechanics the probability of measuring the system to have an energy is the modulus squared of the projection of the state onto the associated eigenspace, which for the one-dimensional zero energy eigenspace means just the inner product with the ground state.)

This is all we need to say something interesting about the large-time average probability for all states. We’ve basically shown that we can break the large-time probability vector into a sum of two normalized probability vectors:

the first being the stochastic stationary state associated with the zero energy eigenspace, and the second $\Omega$ associated with the higher energy eigenspaces, with

The weight of each term is governed by the parameter

which you could think of as the quantumness of the result. This is one minus the probability of the walker being in the zero energy eigenspace, or equivalently the probability of the walker being outside the zero energy eigenspace.

So even if we don’t know we know its importance is controlled by a parameter that governs how close the large-time average distribution of the quantum walk is to the corresponding stochastic stationary distribution

What do we mean by ‘close’? Find out for yourself:

Puzzle 1. Show, using a triangle inequality, that the trace distance between the two characteristic stochastic and quantum distributions and is upper-bounded by

Can we say anything physical about when the quantumness is big or small?

Because the eigenvalues of have a physical interpretation in terms of energy, the answer is yes. The quantumness is the probability of being outside the zero energy state. Call the next lowest eigenvalue the energy gap. If the quantum walk is not in the zero energy eigenspace then it must be in an eigenspace of energy greater or equal to Therefore the expected energy of the quantum walker must bound the quantumness

This tells us that a quantum walk with a low energy is similar to a stochastic walk in the large-time limit. We already knew this was exactly true in the zero energy limit, but this result goes further.

So little is known about quantum walks on networks of arbitrary geometry that we were very pleased to find this result. It says there is a special case in which the walk is characterized by the degree distribution of the network, and a clear physical parameter that bounds how far the walk is from this special case.

Also, in finding it we learned that the difficulties of the initial state dependence, enhanced by the lack of convergence to a stationary state, could be overcome for a quantum walk, and that the relationships between quantum and stochastic walks extend beyond those with shared generators.

What next?

That’s all for the latest bit of idea sharing at the interface between stochastic and quantum systems.

I hope I’ve piqued your interest about quantum walks. There’s so much still left to work out about this topic, and your help is needed!

Other questions we have include: What holds analytically about the form of the quantum correction? Numerically it is known that the so-called quantum correction tends to enhance the probability of being found on nodes of low degree compared to Can someone explain why? What happens if a small amount of stochastic noise is added to a quantum walk? Or a lot of noise?

It’s difficult to know who is best placed to answer these questions: experts in quantum physics, graph theory, complex networks or stochastic processes? I suspect it’ll take a bit of help from everyone.

Background reading

A couple of textbooks with comprehensive sections on non-negative matrices and continuous-time stochastic processes are:

• Peter Lancaster and Miron Tismenetsky, The Theory of Matrices: with Applications, 2nd edition, Academic Press, San Diego, 1985.

• James R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1997.

There is, of course, the book that arose from the Azimuth network theory series, which considers several relationships between quantum and stochastic processes on networks:

• John Baez and Jacob Biamonte, A Course on Quantum Techniques for Stochastic Mechanics, 2012.

Another couple of books on complex networks are:

• Mark Newman, Networks: An Introduction, Oxford University Press, Oxford, 2010.

• Ernesto Estrada, The Structure of Complex Networks: Theory and Applications, Oxford University Press, Oxford, 2011. Note that the first chapter is available free online.

There are plenty more useful references in our article on this topic:

• Mauro Faccin, Tomi Johnson, Jacob Biamonte, Sabre Kais and Piotr Migdał, Degree distribution in quantum walks on complex networks.

Puzzles for the enthusiastic

Sadly I didn’t have space to show proofs of all the theorems I used. So here are a few puzzles that guide you to doing the proofs for yourself:

Stochastic walks and stationary states

Puzzle 2. (For the hard core.) Prove there is always a unique positive eigenvector for a stochastic walk generated by You’ll need the assumption that the graph is connected. It’s not simple, and you’ll probably need help from a book, perhaps one of those above by Lancaster and Tismenetsky, and Norris.

Puzzle 3. Show that the eigenvalues of (and therefore ) are non-negative. A good way to start this proof is to apply the Perron-Frobenius theorem to the non-negative matrix This implies that has a positive eigenvalue equal to its spectral radius

where are the eigenvalues of and the associated eigenvector is positive. Since it follows that shares the eigenvectors of and the associated eigenvalues are related by inverted translation:

Puzzle 4. Prove that regardless of the initial state the zero eigenvector is obtained in the large-time limit of the walk generated by This breaks down into two parts:

(a) Using the approach from Puzzle 5, to show that the positivity of and the infinitesimal stochastic property imply that and thus is actually the unique zero eigenvector and stationary state of (its uniqueness follows from puzzle 4, you don’t need to re-prove it).

(b) By inserting the decomposition into and using the result of puzzle 5, complete the proof.

(Though I ask you to use the diagonalizability of the main results still hold if the generator is irreducible but not diagonalizable.)

Quantum walks

Here are a couple of extra puzzles for those of you interested in quantum mechanics:

Puzzle 5. In quantum mechanics, probabilities are given by the moduli squared of amplitudes, so multiplying a state by a number of modulus unity has no physical effect. By inserting

into the quantum time evolution matrix show that if

then

hence is a stationary state in the quantum sense, as probabilities don’t change in time.

Puzzle 6. By expanding the initial state in terms of the complete orthogonal basis vectors show that for a quantum walk never converges to a stationary state unless it began in one.

Quantum Network Theory (Part 1)

guest post by Tomi Johnson

If you were to randomly click a hyperlink on this web page and keep doing so on each page that followed, where would you end up?

As an esteemed user of Azimuth, I’d like to think you browse more intelligently, but the above is the question Google asks when deciding how to rank the world’s web pages.

Recently, together with the team (Mauro Faccin, Jacob Biamonte and Piotr Migdał) at the ISI Foundation in Turin, we attended a workshop in which several of the attendees were asking a similar question with a twist. “What if you, the web surfer, behaved quantum mechanically?”

Now don’t panic! I have no reason to think you might enter a superposition of locations or tunnel through a wall. This merely forms part of a recent drive towards understanding the role that network science can play in quantum physics.

As we’ll find, playing with quantum networks is fun. It could also become a necessity. The size of natural systems in which quantum effects have been identified has grown steadily over the past few years. For example, attention has recently turned to explaining the remarkable efficiency of light-harvesting complexes, comprising tens of molecules and thousands of atoms, using quantum mechanics. If this expansion continues, perhaps quantum physicists will have to embrace the concepts of complex networks.

To begin studying quantum complex networks, we found a revealing toy model. Let me tell you about it. Like all good stories, it has a beginning, a middle and an end. In this part, I’ll tell you the beginning and the middle. I’ll introduce the stochastic walk describing the randomly clicking web surfer mentioned above and a corresponding quantum walk. In part 2 the story ends with the bounding of the difference between the two walks in terms of the energy of the walker.

But for now I’ll start by introducing you to a graph, this time representing the internet!

If this taster gets you interested, there are more details available here:

• Mauro Faccin, Tomi Johnson, Jacob Biamonte, Sabre Kais and Piotr Migdał, Degree distribution in quantum walks on complex networks, arXiv:1305.6078 (2013).

What does the internet look like from above?

As we all know, the idea of the internet is to connect computers to each other. What do these connections look like when abstracted as a network, with each computer a node and each connection an edge?

The internet on a local scale, such as in your house or office, might look something like this:

 

with several devices connected to a central hub. Each hub connects to other hubs, and so the internet on a slightly larger scale might look something like this:

 

What about the full global, not local, structure of the internet? To answer this question, researchers have developed representations of the whole internet, such as this one:

 

While such representations might be awe inspiring, how can we make any sense of them? Or are they merely excellent desktop wallpapers and new-age artworks?

In terms of complex network theory, there’s actually a lot that can be said that is not immediately obvious from the above representation.

For example, we find something very interesting if we plot the number of web pages with different incoming links (called degree) on a log-log axis. What is found for the African web is the following:

This shows that very few pages are linked to by a very large number others, while a very large number of pages receive very few links. More precisely, what this shows is a power law distribution, the signature of which is a straight line on a log-log axis.

In fact, power law distributions arise in a diverse number of real world networks, human-built networks such as the internet and naturally occurring networks. It is often discussed alongside the concept of the preferential attachment; highly connected nodes seem to accumulate connections more quickly. We all know of a successful blog whose success had led to an increased presence and more success. That’s an example of preferential attachment.

It’s clear then that degree is an important concept in network theory, and its distribution across the nodes a useful characteristic of a network. Degree gives one indication of how important a node is in a network.

And this is where stochastic walks come in. Google, who are in the business of ranking the importance of nodes (web pages) in a network (the web), use (up to a small modification) the idealized model of a stochastic walker (web surfer) who randomly hops to connected nodes (follows one of the links on a page). This is called the uniform escape model, since the total rate of leaving any node is set to be the same for all nodes. Leaving the walker to wander for a long while, Google then takes the probability of the walker being on a node to rank the importance of that node. In the case that the network is undirected (all links are reciprocated) this long-time probability, and therefore the rank of the node, is proportional to the degree of the node.

So node degrees and the uniform escape model play an important role in the fields of complex networks and stochastic walks. But can they tell us anything about the much more poorly understood topics of quantum networks and quantum walks? In fact, yes, and demonstrating that to you is the purpose of this pair of articles.

Before we move on to the interesting bit, the math, it’s worth just listing a few properties of quantum walks that make them hard to analyze, and explaining why they are poorly understood. These are the difficulties we will show how to overcome below.

• No convergence. In a stochastic walk, if you leave the walker to wander for a long time, eventually the probability of finding a walker at a node converges to a constant value. In a quantum walk, this doesn’t happen, so the walk can’t be characterized so easily by its long-time properties.

• Dependence on initial states. In some stochastic walks the long-time properties of the walk are independent of the initial state. It is possible to characterize the stochastic walk without referring to the initialization of the walker. Such a characterization is not so easy in quantum walks, since their evolution always depends on the initialization of the walker. Is it even possible then to say something useful that applies to all initializations?

• Stochastic and quantum generators differ. Those of you familiar with the network theory series know that some generators produce both stochastic and quantum walks (see part 16 for more details). However, most stochastic walk generators, including that for the uniform escape model, do not generate quantum walks and vice versa. How do we then compare stochastic and quantum walks when their generators differ?

With the task outlined, let’s get started!

Graphs and walks

In the next couple of sections I’m going to explain the diagram below to you. If you’ve been following the network theory series, in particular part 20, you’ll find parts of it familiar. But as it’s been a while since the last post covering this topic, let’s start with the basics.

A simple graph can be used to define both stochastic and quantum walks. A simple graph is something like this:

where there is at most one edge between any two nodes, there are no edges from a node to itself and all edges are undirected. To avoid complications, let’s stick to simple graphs with a finite number of nodes. Let’s also assume you can get from every node to every other node via some combination of edges i.e. the graph is connected.

In the particular example above the graph represents a network of nodes, where nodes 3 and 4 have degree (number of edges) 3, and nodes 1, 2 and 5 have degree 2.

Every simple graph defines a matrix called the adjacency matrix. For a network with nodes, this matrix is of size and each element is unity if there is an edge between nodes and , and zero otherwise (let’s use this basis for the rest of this post). For the graph drawn above the adjacency matrix is

By construction, every adjacency matrix is symmetric:

(the means the transposition of the elements in the node basis) and further, because each is real, it is self-adjoint:

(the means conjugate transpose).

This is nice, since (as seen in parts 16 and 20) a self-adjoint matrix generates a continuous-time quantum walk.

To recap from the series, a quantum walk is an evolution arising from a quantum walker moving on a network.

A state of a quantum walk is represented by a size complex column vector . Each element of this vector is the so-called amplitude associated with node and the probability of the walker being found on that node (if measured) is the modulus of the amplitude squared Here is the standard basis vector with a single non-zero th entry equal to unity, and is the usual inner product.

A quantum walk evolves in time according to the Schrödinger equation

where is called the Hamiltonian. If the initial state is then the solution is written as

The probabilities are guaranteed to be correctly normalized when the Hamiltonian is self-adjoint.

There are other matrices that are defined by the graph. Perhaps the most familiar is the Laplacian, which has recently been a topic on this blog (see parts 15, 16 and 20 of the series, and this recent post).

The Laplacian is the matrix

where the degree matrix is an diagonal matrix with elements given by the degrees

For the graph drawn above, the degree matrix and Laplacian are:

The Laplacian is self-adjoint and generates a quantum walk.

The Laplacian has another property; it is infinitesimal stochastic. This means that its off diagonal elements are non-positive and its columns sum to zero. This is interesting because an infinitesimal stochastic matrix generates a continuous-time stochastic walk.

To recap from the series, a stochastic walk is an evolution arising from a stochastic walker moving on a network.

A state of a stochastic walk is represented by a size non-negative column vector . Each element of this vector is the probability of the walker being found on node

A stochastic walk evolves in time according to the master equation

where is called the stochastic Hamiltonian. If the initial state is then the solution is written

The probabilities are guaranteed to be non-negative and correctly normalized when the stochastic Hamiltonian is infinitesimal stochastic.

So far, I have just presented what has been covered on Azimuth previously. However, to analyze the important uniform escape model we need to go beyond the class of (Dirichlet) generators that produce both quantum and stochastic walks. Further, we have to somehow find a related quantum walk. We’ll see below that both tasks are achieved by considering the normalized Laplacians: one generating the uniform escape stochastic walk and the other a related quantum walk.

Normalized Laplacians

The two normalized Laplacians are:

• the asymmetric normalized Laplacian (that generates the uniform escape Stochastic walk) and

• the symmetric normalized Laplacian (that generates a Quantum walk).

For the graph drawn above the asymmetric normalized Laplacian is

The identical diagonal elements indicates that the total rates of leaving each node are identical, and the equality within each column of the other non-zero elements indicates that the walker is equally likely to hop to any node connected to its current node. This is the uniform escape model!

For the same graph the symmetric normalized Laplacian is

That the diagonal elements are identical in the quantum case indicates that all nodes are of equal energy, this is type of quantum walk usually considered.

Puzzle 1. Show that in general is infinitesimal stochastic but not self-adjoint.

Puzzle 2. Show that in general is self-adjoint but not infinitesimal stochastic.

So a graph defines two matrices: one that generates a stochastic walk, and one that generates a quantum walk. The natural question to ask is whether these walks are related. The answer is that they are!

Underpinning this relationship is the mathematical property that and are similar. They are related by the following similarity transformation

which means that any eigenvector of associated to eigenvalue gives a vector

that is an eigenvector of with the same eigenvalue! To show this, insert the identity into

and multiply from the left with to obtain

The same works in the opposite direction. Any eigenvector of gives an eigenvector

of with the same eigenvalue

The mathematics is particularly nice because is self-adjoint. A self-adjoint matrix is diagonalizable, and has real eigenvalues and orthogonal eigenvectors.

As a result, the symmetric normalized Laplacian can be decomposed as

where is real and are orthogonal projectors. Each acts as the identity only on vectors in the space spanned by and as zero on all others, such that

Multiplying from the left by and the right by results in a similar decomposition for :

with orthogonal projectors

I promised above that I would explain the following diagram:

Let’s summarize what it represents now:

• is a simple graph that specifies

• the adjacency matrix (generator of a quantum walk), which subtracted from

• the diagonal matrix of the degrees gives

• the symmetric Laplacian (generator of stochastic and quantum walks), which when normalized by returns both

• the generator of the uniform escape stochastic walk and

• the quantum walk generator to which it is similar!

What next?

Sadly, this is where we’ll finish for now.

We have all the ingredients necessary to study the walks generated by the normalized Laplacians and exploit the relationship between them.

Next time, in part 2, I’ll talk you through the mathematics of the uniform escape stochastic walk and how it connects to the degrees of the nodes in the long-time limit. Then I’ll show you how this helps us solve aspects of the quantum walk generated by

In other news

Before I leave you, let me tell you about a workshop the ISI team recently attended (in fact helped organize) at the Institute of Quantum Computing, on the topic of quantum computation and complex networks. Needless to say, there were talks on papers related to quantum mechanics and networks!

Some researchers at the workshop gave exciting talks based on numerical examinations of what happens if a quantum walk is used instead of a stochastic walk to rank the nodes of a network:

• Giuseppe Davide Paparo and Miguel Angel Martín-Delgado, Google in a quantum network, Sci. Rep. 2 (2012), 444.

• Eduardo Sánchez-Burillo, Jordi Duch, Jesús Gómez-Gardenes and David Zueco, Quantum navigation and ranking in complex networks, Sci. Rep. 2 (2012), 605.

Others attending the workshop have numerically examined what happens when using quantum computers to represent the stationary state of a stochastic process:

• Silvano Garnerone, Paolo Zanardi and Daniel A. Lidar, Adiabatic quantum algorithm for search engine ranking, Phys. Rev. Lett. 108 (2012), 230506.

It was a fun workshop and we plan to organize/attend more in the future!