PHYSICAL REVIEW E 92, 042149 (2015)

Stationary properties of maximum-entropy random walks

Purushottam D. Dixit*

Department of Systems Biology, Columbia University, New York, New York 10032, United States(Received 22 June 2015; published 23 October 2015)

Maximum-entropy (ME) inference of state probabilities using state-dependent constraints is popular in thestudy of complex systems. In stochastic systems, how state space topology and path-dependent constraintsaffect ME-inferred state probabilities remains unknown. To that end, we derive the transition probabilities andthe stationary distribution of a maximum path entropy Markov process subject to state- and path-dependentconstraints. A main finding is that the stationary distribution over states differs significantly from the Boltzmanndistribution and reflects a competition between path multiplicity and imposed constraints. We illustrate our resultswith particle diffusion on a two-dimensional landscape. Connections with the path integral approach to diffusionare discussed.

DOI: 10.1103/PhysRevE.92.042149 PACS number(s): 05.30.−d, 02.50.Tt, 05.40.Fb, 89.70.Cf

I. INTRODUCTION

On the one hand, in biological, sociological, and techno-logical systems, our ability to collect large amounts of data hasbeen improving continuously. Examples include measurementof expression levels of thousands of mRNAs in individual cells[1,2], estimation of species abundance in different ecologies[3,4], connectivity, and dynamics of socio-technological plat-forms such as FaceBook and Twitter [5,6]. On the other hand,in many such cases our intuition to build prescriptive modelshas not yet fully developed. Indeed, probabilistic descriptiveapproaches have become popular. A popular example is theprinciple of maximum entropy (ME) [7–10]. Intuitively, MEpicks the “least informative” distribution while requiring itto reproduce certain low-dimensional aspects of the data, forexample, lower order moments. The result is the Boltzmannexponential distribution in constrained quantities. ME has beenemployed to study a diverse set of systems, for example,neuronal dynamics [11], bird flocks [12], ecological speciesdistribution [13], gene expression noise [14], protein sequencevariability [15,16], and behavior [17].

In many cases [11–14], but not always [15–17], theexperimental data are a realization of a stochastic process.Here, in order to estimate the distribution over states, onemay impose path-dependent constraints in addition to state-dependent constraints in the inference procedure. Also, thedynamical neighborhood of any state—states reachable in asingle transition—is usually finite, which defines the statespace topology. How these three factors affect the maximumpath entropy distribution over states and how different thisdistribution is from the Boltzmann exponential distributionremains currently unknown. We solve this problem forMarkovian dynamics. We derive transition probabilities andthe stationary distribution over states of the maximum pathentropy Markov process subject to state- and path-dependentconstraints. We show that the stationary distribution is the outerproduct of the left and the right Perron-Frobenius eigenvectorsof a transfer matrix and depends nontrivially on the topologyand imposed constraints and is almost always substantially

*pd2447@c2b2.columbia.edu

different than the Boltzmann distribution. We illustrate ourresults with a particle diffusing on a two-dimensional lattice.

This article is organized as follows. First, we derive themaximum path entropy Markovian random walk subject tostate- and path-dependent constraints. Next, we illustrateour results with a particle diffusing on a two dimensionalgrid. Finally, we speculate connections with path integralformulation of diffusion processes.

II. THEORY

We begin with an observation. Discrete state stochastic sys-tems can be modeled as a random walk in higher dimensions.For example, the Glauber dynamics [18] of the Ising modelwith N spins is a random walk in 2N dimensions. In Glauberdynamics, every state is connected to only N − 1 other statesout of the 2N , defining the state space topology. To that end, weconsider an irreducible and aperiodic discrete time Markovianrandom walk on a directed graph G with nodes V and edgesE. We denote the unique stationary distribution over the statesby {pa} and the transition probabilities p(a → b) by kab. Weassume that kab �= 0 iff (a,b) ∈ E and pa > 0∀a.

A. State- and path-dependent constraints

The appropriate ensemble to impose state- and path-dependent constraints is the ensemble {�} of stationary statetrajectories � ≡ · · · → a → b → · · · of fixed duration T �1. The entropy of the ensemble, normalized by T , is given by[19–22]

S = − 1

T

∑�

p(�) log p(�) ≈ −∑a,b

pakab log kab. (1)

The second approximation holds when T � 1 for Markoviandynamics and is equivalent to constraining two point correla-tions in a more general formulation [23–26]. In Eq. (1) andfrom here onwards, all summations involving quantities withtwo indices are restricted on the edges of G unless statedotherwise.

We constrain path ensemble average of a state- (or path-)dependent quantity rab. Generalization to multiple constraintsis straightforward (see below). State-dependent quantities suchas energy and particle number depend only on the initial state

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PURUSHOTTAM D. DIXIT PHYSICAL REVIEW E 92, 042149 (2015)

a or the final state b. Path-dependent quantities such as fluxesand rates depend on both states. The path ensemble averagesare given by [19–21]

〈r〉 =∑a,b

pakabrab. (2)

Additionally, {pa} and {kab} are not mutually independent. Wehave∑

b

pakab = pa,∑

a

pakab = pb,∑a,b

pakab = 1. (3)

B. Maximizing path entropy

We maximize S in Eq. (1) with respect to unknownstationary distribution pa and transition probabilities kab whileimposing constraints in Eqs. (2) and (3). Using Lagrange mul-tipliers to impose the constraints, we write the unconstrainedLagrange function, sometimes called the Caliber [10,27,28],

C = −∑a,b

pakab log kab +∑

a

ma

(∑b

pakab − pa

)

+∑

b

nb

(∑a

pakab − pb

)+ ζ

(∑a,b

pakab − 1

)

− γ

(∑a,b

pakabrab − 〈r〉)

. (4)

In Eq. (4), we have worked with a single path-dependentconstraint for notational simplicity. Generalization to multipleconstraints introduces additional Lagrange multipliers but isconceptually straightforward. Next, we maximize the Caliberwith respect to kab and pa . Differentiating the Caliber withrespect to kab, we have

pa(log kab + 1) = pa(ma + nb + ζ − γ rab)

⇒ kab = ema+nb+ζ−1−γ rab . (5)

Differentiating the Caliber with respect to pa , we have

0 = −∑

b

kab log kab + ma

∑b

kab − ma +∑

b

nbkab − na

+ ζ∑

b

kab − γ∑

b

kabrab. (6)

Substituting kab from Eq. (5) and using∑

b kab = 1, we get

ma + na = 1. (7)

Substituting in Eq. (5), we find that the maximum path entropytransition probabilities are given by

kab = φb

ηφa

Wab, (8)

when (a,b) ∈ E and zero otherwise. Here, Wab = e−γ rab when(a,b) ∈ E and zero otherwise, φa = e−ma , and η = e−ζ . Inorder to determine φa , we impose

∑b kab = 1. We have∑

b

Wabφb = ηφa. (9)

Thus, φ = {. . . ,φa, . . . ,φb . . . } is the right eigenvector of Wwith eigenvalue η.

How do we guarantee that kab is positive for all a andb connected by an edge and that the inferred Markov chainhas the maximum path entropy? Since the graph representingthe network of states is assumed to be aperiodic and ergodic,the matrix W is non-negative. The Perron-Frobenius theoremguarantees that the maximum eigenvalue of W is positive andthe corresponding right and left eigenvectors have positiveelements. Moreover, the path entropy maximization problemis convex and has a unique maximum. Thus, if we choose η

to be the maximum eigenvalue and φ the corresponding righteigenvector, transition probabilities kab are guaranteed to bepositive for all a and b and the corresponding Markov chain isguaranteed to have the maximum path entropy under imposedconstraints. From here onwards, we work with this choice.

C. Stationary distribution

The stationary distribution over states {pa} is determinedby solving

∑a

pakab = pb ⇒∑

a

pa

φa

Wab = ηpb

φb

. (10)

Thus, if ψ is the left Perron-Frobenius eigenvector and φ isthe right Perron-Frobenius eigenvector of W with the sameeigenvalue η, the stationary distribution is their outer product,

pa = ψaφa. (11)

Equation (11) is remniscent of quantum mechanics. W is atransfer operator, φ is the wave function, and ψ is its conjugate.The stationary probability pa is the outer product of φa andψa (see below). The Perron-Frobenius eigenvectors and thusthe stationary distribution depend on the topology and theimposed constraints in a nontrivial fashion. In other words,the Boltzmann distribution, the maximum-entropy distributionover states, is no longer guaranteed when topological anddynamical information is introduced.

D. Detailed balance

Is the inferred Markov process detailed balanced? Let uscalculate its entropy production rate s [29],

s =∑a,b

pakab logkab

kba

= −γ 〈rab − rba〉. (12)

In Eq. (12), the antisymmetric part of constraint rab contributesto entropy production. If all constraints are symmetric, theentropy production is zero and the Markov process is automat-ically detailed balanced. In fact, if detailed balance pakab =pbkba is explicitly imposed, the inference problem is equiva-lent to constraining symmetrized quantities r

†ab = 1

2 (rab + rba)(see Appendix for details). In this case, the transfer matrixW is symmetric and the left and the right Perron-Frobeniuseigenvectors coincide. The stationary distribution is simplythe outer product of this eigenvector with itself.

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E. Path probability

Finally, we write down the probability of an arbitrarystationary state path � ≡ a1 → a2 → · · · → an,

p(�) = pa1 · ka1a2 · · · kan−1an

= ψa1φa1

φa2

ηφa1

Wa1a2 · · ·

= 1

ηn−1e−A(�), (13)

where the “action” A(�) associated with the path � is

A(�) = γ

n−1∑t=1

rat at+1 − log ψa1 − log φan. (14)

The construction of the maximum path entropy Markovprocess is complete. Recently, similar results were developedfrom a large deviation point of view which provides aprobabilistic rationale for maximizing path entropy [30,31].

III. NUMERICAL ILLUSTRATION

Equation (11) gives a recipe to calculate the stationarydistribution from constraints but does not give us an intuitiveunderstanding of how it depends on them. To that end, wehighlight three important features: (1) path entropy-enthalpycompensation, (2) state space topology, and (3) currents witha numerical example. We study the stationary distribution ofa particle diffusing on a two-dimensional [−N,N ] × [−N,N ]square lattice to illustrate these features. We assume that theparticle jumps to one of its nearest neighbors in a singletransition.

A. Path entropy-enthalpy compensation

First, consider an aperiodic square lattice with boundaries.What is the stationary distribution when no state- and path-dependent constraints are imposed? We start with the transfermatrix W which here is the adjacency matrix of G [32]. Thestationary distribution is the square of its Perron-Frobeniuseigenvector φ. For a = (x,y), pa is given by

pa = 1

(N + 1)2sin2 (2x + 2N + 1)π

4N + 4sin2 (2y + 2N + 1)π

4N + 4.

(15)

Here, x,y ∈ [−N,N ]. Equation (15) is recognized as theprobability distribution corresponding to the lowest energyeigenfunction of the discretized form of the Schrodingerequation for a particle in a square box (see Appendix fordetails). The stationary distribution localizes near the centerof the lattice (see Fig. 1), a striking departure from the micro-canonical ME distribution where all states are equiprobable.The entropic localization can be understood as follows. Allpaths of a fixed duration T have equal probability [see Eq.(13)]. But, there are many more such paths near the centercompared to the boundaries. Thus, the states in the center arefrequented often compared to the boundaries which leads totheir higher probability [32].

How does the stationary distribution change when state-dependent quantities are constrained? To illustrate this, we

FIG. 1. (Color online) The stationary distribution of a freelydiffusing particle on an aperiodic square lattice.

define a potential εa at every point a = (x,y),

εa = A

x2 + y2 + B. (16)

We fix A = 11 and B = 10. The radially symmetric potentialhas a peak in the middle of the lattice and takes its lowest valuesin the four corners (see Appendix for a graph). Let us obtainthe stationary distribution by constraining the average poten-tial and imposing detailed balance. The symmetric transfermatrix is

Wab = exp

[−γ

(εa + εb

2

)], (17)

when a and b are nearest neighbors on the lattice and zerootherwise. γ is the Lagrange multiplier associated with theaverage potential. In Fig. 2 we show the stationary distributionfor γ = 0.01 and 0.1. When constrained by average potential,the particle balances path multiplicity with energetic unfavora-bility of states. This balance is reminiscent of entropy-enthalpycompensation [33] well known in chemistry. We note thatthe values of parameters used in this study are strictly forillustrative purposes and do not carry any special meaning.

FIG. 2. (Color online) The stationary distribution of particlewhen average potential is constrained.

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PURUSHOTTAM D. DIXIT PHYSICAL REVIEW E 92, 042149 (2015)

FIG. 3. (Color online) The maximum path entropy stationarydistribution pa and the Boltzmann distribution qa when averagedpotential is constrained.

B. Topology

Thus, defects and asymmetries in state space topologyhave a prominent impact on the stationary distribution. Ona topologically symmetric state space, for example, a periodiclattice, a freely diffusing particle indeed has a flat stationarydistribution (not shown). Nonetheless, the symmetry can bebroken by a combination of finite dynamical neighborhoodof states and the potential landscape. In Fig. 3 we plotthe stationary distribution pa [Eq. (11)] and the Boltzmanndistribution qa ∝ e−βεa after constraining the average potentialon a periodic lattice. pa is calculated as above with aslight modification that the underlying graph of connectivityrepresents a periodic lattice. γ in Eq. (17) is fixed at 0.1.Inverse temperature β of the maximum-entropy Boltzmanndistribution qa is adjusted to match the numerical value of theaverage potential of the maximum path entropy distribution pa ,which allows a direct comparison. Indeed pa is significantlydifferent than qa .

How do we understand this difference? On the one hand, theME distribution qa depends solely on the state potential εa . Onthe other hand, Eq. (13) shows that the paths that frequent bothhigh and low potential states have a non-negligible probabilitythereby comparatively increasing the stationary probability ofhigh potential states that are in the dynamic neighborhood oflow potential states. As the dynamical neighborhood of theparticle is expanded, for example, when the particle is allowedto jump to nth(n > 1) nearest neighbor, pa and qa are expectedto agree more and more. Indeed, if the particle can jump fromany state to any other state in a single transition, the ME andthe maximum path entropy predictions are trivially identicalto each other [21].

C. Currents

In addition to state-dependent quantities, one may constrainpath-dependent quantities such as rates and currents. How dopath-dependent constraints change the stationary distribution?On the periodic lattice, we constrain the average potentialand a current along the positive Y axis (Fig. 4). To obtain thestationary distribution, we first identify the asymmetric transfermatrix,

Wab = exp

[−γ

(εa + εb

2

)− αJab

]. (18)

FIG. 4. (Color online) The stationary distribution in the presenceof nonequilibrium current. As α increases from 0.1 to 0.5, the netcurrent increases and modulates the stationary distribution to a greaterextent. γ is fixed at 0.1.

As above, γ is the Lagrange multiplier associated withpotential and α is associated with current. The current in thepositive Y direction between states a = (x,y) and b = (z,w) isdefined as Jab = ±1 if w = y ± 1 with appropriate correctionsat y,w = 1,N . Jab is zero for sideways movement. Jab isantisymmetric and contributes to entropy production, makingthe Markov process a nonequilibrium stationary state. We findthe left and the right Perron-Frobenius eigenvectors ψ and φ

of W. The stationary distribution is the outer product of thesetwo vectors, pa = ψaφa .

Figure 4 shows pa at α = 0.1 and α = 0.5 with γ fixedat γ = 0.1. We see that net current modulates the stationarydistribution, a fact well known in statistical physics [34]. Thiseffect can be understood by looking at path probabilities. FromEq. (13), we know that paths that traverse through regionsof high potential have a low probability. But, this may bealleviated if they simultaneously carry a net favorable current.This leads to a higher probability for energetically unfavorablestates that are represented frequently in current carrying paths.

IV. DISCUSSION

The maximum-entropy (ME) principle was first introducedby Gibbs in statistical physics more than a century ago [8] andwas later established as an inference principle in its own rightby Shore and Johnson [9]. As discussed in the introduction,ME has been applied to a variety of problems across fields.Additionally, ME is often invoked as the information theoreticbasis of statistical physics [7].

The majority of the applications of ME have concentratedon inferring distributions over states of a system. Yet, ME canalso be applied to study dynamics where the probabilistic “statespace” is the collective of all possible paths. In this work, wederived the maximum path entropy distribution over the “statespace” of stationary paths of a Markov chain subject to state-and path- dependent constraints. We derived the analyticalform of the corresponding transition probability matrix and thestationary distribution over states. While the maximum pathentropy distribution has a Boltzmann-like exponential formfor path probabilities [see Eq. (13)], the stationary distributionover states is profoundly affected by asymmetry in state spacetopology, finite dynamical reach of states, and path-dependent

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STATIONARY PROPERTIES OF MAXIMUM-ENTROPY . . . PHYSICAL REVIEW E 92, 042149 (2015)

constraints. Notably, even in the absence of path-dependentconstraints, state space topology affects the maximum pathentropy distribution projected over states, which deviatesconsiderably from the Boltzmann exponential distribution.These effects will be magnified in higher dimensions and areclearly very relevant in many discrete state systems wherestate-based ME has previously been employed [11–14]. Apartfrom their theoretical importance, the results developed hereshould allow better numerical predictions in ME modeling ofdynamical probability distributions.

In this work, we focused on probability distributionsover states. But, we have access to path probabilities aswell [see Eq. (13)]. What is the relevance of the inferredMarkovian dynamics to the study of diffusive random walksin general? We provide a speculation. The two mathematicalframeworks that describe random walks, the local Fokker-Planck formulation and the nonlocal path-integral formulationare often equivalent. For example, the local assertion thatall nearest neighbor jumps on an infinite regular lattice areequiprobable is equivalent to the nonlocal assertion that allpaths of equal duration are equiprobable. But, confinement andlattice irregularities lead to prominent localization away fromthe boundary, a striking difference between the two approaches[32]. This nonenergetic localization has to be explainedby invoking fictitious entropic forces in the Fokker-Planckapproach. We believe that path-based approaches may be betterdescriptors of stochastic dynamics especially for discrete statefinite systems such as spin systems and chemical reactionnetworks. We leave this for future theoretical and experimentalstudies.

ACKNOWLEDGMENTS

We thank Dr. Sumedh Risbud, Dr. Manas Raach, ProfessorThierry Mora, and Dr. Karthik Shekhar for valuable discus-sions and comments on the manuscript.

APPENDIX

1. Imposing detailed balance

As above, we consider the Caliber,

C = −∑a,b

pakab log kab +∑

a

ma

(∑b

pakab − pa

)

+∑

b

nb

(∑a

pakab − pb

)+ ζ

(∑pakab − 1

)

+∑a,b

εab(pakab − pbkba) − γ

(∑ab

pakabrab − 〈r〉)

.

(A1)

We have introduced Lagrange multipliers εab to enforcedetailed balance. As above, all summations involving twoindices are restricted to edges of the graph.

Differentiating the Caliber with respect to kab, we have

pa(log kab + 1) = pama + panb + paζ + pa(εab − εba)

−paγ rab (A2)

⇒ kab = e(ma+nb+ζ−1−γ rab+εab−εba ). (A3)

Differentiating the Caliber with respect to pa , we have

0 = −∑

b

kab log kab + ma

∑b

kab − ma +∑

b

nbkab − na

+ ζ∑

b

kab +∑

b

kab(εab − εba) − γ∑

b

rabkab. (A4)

Substituting kab from Eq. (A3), we get

ma + na = 1. (A5)

Substituting in Eq. (A3), we get

kab = αb

ηαa

e−γ rabκab. (A6)

Here, αa = e−ma , η = e−ζ , and κab = eεab−εba . Notice thatκabκba = 1.

To determine κab, we impose detailed balance,

kab

kba

= pb

pa

= α2b

α2a

e−γ rab+γ rba κ2ab (A7)

⇒ κab =√

pb

pa

αa

αb

e12 γ (rab−rba ). (A8)

Thus, the transition probabilities are

kab =√

pb

pa

αa

αb

eγ

2 (rab−rba ) αb

ηαa

e−γ rab (A9)

= 1

η

√pb

pa

e− 12 γ (rab+rba ). (A10)

Let φa = √pa and Wab = e− 1

2 γ (rab+rba ) when (a,b) ∈ E

and zero otherwise. Using∑

b kab = 1, we have∑b

Wabφb = ηφa. (A11)

Thus, φ, the vector of square roots of probabilities is theeigenvector of W with eigenvalue η. Thus, imposing detailedbalance is equivalent to constraining a symmetrized form ofthe constraints.

2. Probability distribution of a free particle

To gain intuition, consider a free particle on a one-dimensional aperiodic lattice between x = −N to x = N .Assume that at every step, the particle jumps to one ofits nearest neighbors. Let us construct the maximum pathentropy random walk for this particle without any state- orpath-dependent constraints. The transfer matrix W has thefollowing form.

W =

⎡⎢⎢⎢⎢⎣

0 1 0 0 . . . 01 0 1 0 . . . 00 1 0 1 . . . 0...

......

.... . .

...0 0 0 . . . 1 0

⎤⎥⎥⎥⎥⎦. (A12)

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PURUSHOTTAM D. DIXIT PHYSICAL REVIEW E 92, 042149 (2015)

FIG. 5. (Color online) Potential landscape on a [−N,N ] ×[−N,N ] square lattice with N = 20. The potential is highest at thecenter of the lattice and decreases as the reciprocal of the squareddistance from the center [see Eq. (16)]. We have chosen A = 11 andB = 10.

If φ is an eigenvector of W corresponding to an eigenvalue η,we have

φa+1 + φa−1 − φa = (η − 1)φa, (A13)

if a ∈ [−N + 1,N − 1] and

φ2 = ηφ1,φN−1 = ηφN. (A14)

Recognize that Eqs. (A13) and (A14) are the discretizedversion of the time-independent Schrodinger equation for afree particle,

− ∂2φ

∂x2= (1 − η)φ. (A15)

The Perron-Frobenius eigenvector is then the lowest energystate of the particle in a box [35],

φa =√

1

n + 1sin

aπ

2n + 2. (A16)

The stationary distribution is simply the square of thiseigenvector. From Eq. (A16), it is straightforward to writedown the stationary distribution for a two-dimensional case[see Eq. (15)].

3. Potential landscape

Figure 5 has a graphical representation of the potentiallandscape.

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