The iPod Model

We introduce a Voter Model variant, inspired by social evolution of musical preferences. In our model, agents have preferences over a set of songs and upon meeting update their own preferences incrementally towards those of the other agents they meet. Using the spectral gap of an associated Markov chain, we give a geometry dependent result on the asymptotic consensus time of the model.


Introduction
The terminology of Finite Markov Information Exchange (FMIE) models has been introduced [1] [2] as a catch-all for the interpretation of Interacting Particle Systems (IPS) models as stochastic social dynamics. Many important and classical models fit under this two-level framework; the bottom level a meeting model among agents, and the top level an information exchange algorithm performed at each meeting.
For classic IPS models, such as the Voter Model, with a simple meeting algorithm the FMIE perspective is perhaps unnecessary. In this paper however, we will introduce and study a (much) generalized Voter Model -inspired by the evolution of musical preferences among a group of friends -as an FMIE process.

The iPod Model
Here we introduce the iPod FMIE model. The underlying framework of the stochastic process is a weighted graphs G on N vertices. We will refer to each vertex as an agent and occasionally to our vertex set as I. Associated to the edges are symmetric meeting rates ν i,j for 1 ≤ i = j ≤ N . We assume that all meeting rates are normalized, i.e. j ν i,j = 1 for all agents i.
Each agent i is equipped at each time t with a probability measure X t (i) on {1, 2, . . . , σ} which we will reference by its distribution X k t (i) for 1 ≤ k ≤ σ. We consider σ as a fixed number of songs and X k t (i) the preference of agent i at time t for song k. The stochastic process X t updates over time as follows. Between every pair of agents i, j we associate a Poisson process with rate ν i,j whose times we refer to as meetings between i and j. At a meeting time t between agents i and j, each agent picks a song σ i and σ j independently and distributed according to X t− (i) and X t− (j).
We interpret this as each agent choosing a song to play to the other agent based on their preferences. After agent i hears the song j chose he updates his preferences according to X σj t (i) = (1 − η)X σj t− (i) + η and X k t (i) = (1 − η)X k t− (i) for all other k = σ j . Here 0 < η < 1 is a fixed interaction parameter. Agent j updates her preferences similarly. It is immediate that if X t− (i) is a probability measure than so is X t (i). Note that we are implicitly working with cadlag paths.
Analogous to results on the consensus time of the Voter Model -for instance [4] or more generally [8] -in this paper we will estimate the fixation time (to be defined) of the iPod process. Interestingly, again similar to the Voter Model our proof will explore a connection between this process and the Wright-Fisher diffusion [4].
A special feature of the model (Proposition 2.4) is that the average (over agents) preference for a given song evolves as a martingale, analogous to the total proportion of agents with a given opinion on the voter model. This distinguishes the iPod model from many other variants of the voter model that have been studied [3].

Fixation Time
We will be focused on estimating the fixation time T fix of the iPod process. Every time two agents meet at least one distinct song is played between them and so at least one of the σ songs is played infinitely often. Given that only one song is played infinitely often, we define T fix to be the last time any other song is played.
We note that T fix is not a stopping time and a priori could be infinite, i.e. if more than one song is played infinitely often. However, we will show that this is not the case and in fact T fix has finite expectation, the bounding of which will be our primary goal. Theorem 1.1. There exists a constant C(η) so that from any initial configuration of σ songs, the fixation time T fix has expectation where λ is the spectral gap of G.
The spectral gap λ of reversible Markov chain is interpreted as its asymptotic rate of convergence to its stationary distribution, and can be defined by the second eigenvalue of the chain's transition matrix [7]. In our setting, we define the spectral gap λ in terms of the edge weights ν i,j . First, for any function f : I → R we define the Dirichlet form ε(f, f ) by The spectral gap λ is then defined in our context by the extremal characterization There is extensive literature [7] giving order of magnitude bounds on the N → ∞ asymptotic behaviour of λ N for particular families of N -vertex graphs. For such families, Theorem 1.1 gives an order of magnitude upper bound on the asymptotic fixation time, for fixed η and σ. We will show (Theorem 5.1) the tightness of this bound in the case of particular special family of graphs.

Projection on a Single Song
Our main technique will be focusing on the projection of our system to a single song. For some fixed (arbitrary) song k we will consider only X k (i) which we will write simply as x(i) dropping the k. When two agents i, j meet, each independently chooses to either play song k or not; with probability x(i) and x(j) respectively. Writing Ber(x(i)) and Ber(x(j)) for independent Bernoulli variables with given success parameters, we see that if i and j meet at time t that with x(j) updating similarly. At such a meeting, for all other agents k = i, j, x(k) remains unchanged. This implies that the evolution of any given song can be considered separately from the others -though not independently. We will therefore focus first on the FMIE system {x t (i)} i∈I,t≥0 evolving as above and then later return to the original multi-song model. The primary object of study in our one song model will be the average preference for the song, written Our goal in this section will be bounding how long it takes M t to approach the boundary {0, 1}. Specifically we will prove a bound on the stopping time We will use the shorthand x t = {x t (i) : 1 ≤ i ≤ N } for the configuration at time t. In particular, we will often use x 0 for an arbitrary initial configuration. By comparison, we will use X t (respectively X 0 ) for a configuration of the multisong model. To state our bound, we introduce the function φ(x) given by Theorem 2.1. There exists a constant A(η) so that from any initial configu- where λ is the spectral gap of G.
To prove our theorem, we first estimate how long it takes M t to exit small intervals. Then, we use embedding to compare M t to the Wright-Fisher Diffusion.

Derived Quantities
We will begin by analysing a few quantities derived from x t . For ease of notation we will occasionally drop t. Our primary object of study will be the (L 1 ) average of the preferences x(i), denoted M t which is introduced above. We will repeatedly make use of the following lemma on the step sizes of M t .

Lemma 2.2. If t is a meeting time then
Proof. If agent i is involved in a meeting at t, then either As only two agents are involved in any meeting, our bound follows easily.
As a warm-up for the more complicated quantities to appear later, we begin by showing that M t evolves as a continuous time martingale. We here implicitly use the filtration F t generated by {x t (i)} i∈I,t≥0 . Also, note that we may clearly assume that almost surely meeting times between agents are unique and that the set of meeting times has no accumulation point.
We will make use of the process dynamics notation E (dA t |F t− ) = (resp. ≥, ≤)B t dt to mean that is a martingale (respectively submartingale, supermartingale). Clearly this notation is compatible with arithmetic operations. To calculate a process's dynamics, we make repeated use of the following lemma, the proof of which is straightforward.
In particular, for the average preference M t we have the following dynamics. Proof. To begin we note that since E Ber(x t (j)) = x t (j) we have that and similarly for x t (j). Summing both we find that As only x(i) and x(j) change at such a time t, this gives us that i.e. M t is a martingale.
We next look at the process dynamics of M 2 t . To do so we introduce the quantity Q t given by N .
In particular we use Lemma 2.3 to calculate the following.
Proof. As before, we begin by calculating that for k = i, j, since x(k) does not change after a meeting between i and j that: Next we calculate that , and similarly for x 2 (j). Finally we have that Putting this all together we find that Using Lemma 2.3, summing over i, j and normalizing by N 2 we find that Instead of M 2 , we will often be more concerned with M t (1 − M t ). As M t is a martingale, from Proposition 2.5 we easily have that A central tool for the study of the underlying Markov Chain on G is the Dirichlet form ε. We recall that the Dirchilet form ε(f, f ) for a function f : We will write ε(x t , x t ) for the Dirichlet form of the function i → x t (i).
The main fact that we will need about the Dirichlet form is its relationship to the spectral gap. We recall the definition of the spectral gap of a Markov Chain is given by where Var(f ) is the variance of the function f (i) with respect to the uniform measure on I. A simple but important fact we make repeated use of is that 0 < λ ≤ 1. Following Lemma 2.3 we can calculate dQ.
as well as Proof. We begin by noting that Q t = M t − i We have from Proposition 2.5 that . When agents i and j meet, only x(i) and x(j) change and so Summing over i and j our first equation for dQ t is done. The second is an immediate consequence of the first using the identity

Escaping an ǫ Neighbourhood
Next we focus our attention on how long it takes M t to escape from the neighbourhood (M 0 − ǫ, M 0 + ǫ) for some small (unspecified for now) ǫ. Let τ be the escape time, i.e.
For ease of notation in this section we will often write E for E x0 -that is the expectation starting from some initial condition x 0 , perhaps with some (to be specified) condition on M 0 .
Our main goal in this section will be to show the following bound.
Proposition 2.7. There exists a positive constant A(η) so that for any M 0 and ǫ satisfying η

Lower Bound for
First we look for a bound on the heterozygosity M t (1 − M t ). We will make repeated use of the following calculus exercise.
Using our process dynamics calculations we may now begin to bound τ .
Lemma 2.9. There exist positive constants C(η), D(η) so that Proof. First we recall that from Proposition 2.6 we have a submartingale Next, we note that since λ ≤ 1 Substituting this in and rearranging the inequality 4η−2η 2 . Using this we are ready for our lower bound.
Proof. Proposition 2.5 shows is a martingale. The Optional Stopping Theorem and Lemma 2.9 show which finishes our proof.

Upper Bound for
We will now impose the condition on our ǫ neighbourhoods that

Bounding E τ
We can now prove Proposition 2.7.
Proof. Combining Lemma 2.10 and Lemma 2.11 in opposite ways, we have Rearranging this and redefining the constants A(η) and B(η) we have that

Noting that by assumption
always and so combining these two facts which completes our proof.

Embedding in the Wright-Fisher Diffusion
We will now analyse the martingale M t , starting from some initial configuration x 0 , by discretizing it and embedding that into a Brownian motion. We begin by defining a series of stopping times τ k for the martingale M t . Let τ 0 = 0 and for k ≥ 1 define τ k inductively as that is the first time after τ k−1 that M t exits the ball of size Using Proposition 2.7, we have the following bound on the expectation of the increments of our stopping times.

The Wright-Fisher Diffusion
Here we introduce the Wright-Fisher diffusion W t . We recall that W t is the continuous martingale in [0, 1] solving the stochastic differential equation We will frequently think of W t as a diffusion process and so record here that W t has instantaneous drift µ(x) = 0 and variance σ 2 (x) = x(1 − x).
then the first escape τ from the ǫ-ball about w 0 satisfies Applying a standard argument for diffusion processes we find that u(x) satisfies the equation subject to the boundary conditions u(w − ) = u(w + ) = 0. Integrating, we find that for we have the solution for some constants A, B. Applying the boundary conditions we get and Therefore, for w 0 we have To simplify this we apply the Taylor approximation to f at w 0 which plugging into u(w 0 ) we find To approximate the error, we use Taylor's remainder theorem. First, On the interval [w 0 − ǫ, w 0 + ǫ], applying a bound we used for M t (1 − M t ) above, we have Therefore the third derivative is bounded on the same interval by Therefore the error R(w 0 + x) for |x| ≤ ǫ of the 2nd Taylor approximation is bounded by So, for w ± , using that ǫ ≤ w0(1−w0) 2 we have Applying this to our calculation of u(w 0 ) -once to each of f (w 0 ± ǫ) we have from which our bounds on u(w 0 ) follow easily.
We will also need the well known bound for the absorption time T abs of W t at the boundary {0, 1}. To state this bound, recall the function Lemma 2.14. Let T abs be the stopping time Proof. This is easy using the same techniques as the previous proof. Letting

Embedding M τ k in W t
Let W t be a Wright-Fisher diffusion started at W 0 = M 0 . As |M t | ≤ 1 the discrete martingale {M τ k } k≥0 is clearly square integrable and so [5] we can find a sequence of stopping timesτ k for W t so that We will use this embedding to bound the first escape time S by comparison with the absorption time of the Wright-Fisher diffusion.

The Comparison Calculation
We will focus on the first time that M t exits the interval ( η 2N , 1 − η 2N ). Recall the stopping time S defined by and let K = inf{k ≥ 0 : τ k ≥ S}.
Proof. For any 0 ≤ x ≤ 1 2 we have and so if M 0 ≤ 1 2 , then for any k ≥ ln( η N ) ln(3/4) we have M τ k ≤ η 2N with positive probability. Fix such a k 0 . In fact, as {M τn } n≥0 is a martingale, we have A similar argument holds for an initial configuration with 1 2 ≤ M 0 ≤ 1 going above the level 1 − η 2N . Therefore, for any initial M 0 , there is a uniform lower bound on the probability that K < k 0 . Thus, by a standard Strong Markov argument, we must have K < ∞ almost surely.
Next, we define equivalent stopping times for W t . LetK be the index and recall the absorption time T abs given by For M t we have clearly that S ≤ τ K .
Furthermore, as M τK ∈ (0, 1) so must be WτK by the equivalence in distribution (and thus support). Therefore, τK ≤ T abs , as for t ≥ T abs , W t is constant and in {0, 1}.
First we need to bound theτ k .
Lemma 2.16. The hitting timesτ k satisfies Proof. For this, we note that starting at w 0 = Wτ k−1 , the timeτ k can only occur after W t leaves the interval (Wτ k − w0(1−w0) as Wτ k is already outside this interval and W t is continuous. Write τ for the first exit time of this interval. Applying the Strong Markov Property we see that completing our proof.
We are now ready to prove T heorem 2.1.
Proof. We recall by Equation (5), E S ≤ E τ K and so we will focus on bounding E τ K . As K < ∞ almost surely by Lemma 2.15 we have that for some constant A(η) depending only on η. Therefore we can calculate using the Strong Markov property that Therefore we see that Because Equation (5) S ≤ τ K and using Lemma 2.14 to bound E T abs we can conclude that from which our conclusion follows.

The General Model
We are now ready to prove our bound on the fixation time of the general iPod model with σ songs. We recall that for each agent i, we write their preference for song k by X k t (i). For each song k, we write the average preference for that song as M k t , given by We have shown in Proposition 2.4 that for each 1 ≤ k ≤ σ, M k t is a martingale.

Approaching the Boundary
We begin by using Theorem 2.1 to bound the time it takes for one of the M k t to approach the boundary 1. Specifically, we will analyse the stopping time Let S k be the first time that M k t approaches either boundary, that is and set S max = max 1≤k≤σ S k . We will bound τ in two steps, first by bounding E S max and second by showing that E τ is on the same order of magnitude as E S max .
Proposition 3.1. There is a constant C(η) so that from any initial configuration X 0 we have Proof. To begin we recall that by Theorem 2.1 there is a constant C(η) so that Recalling that the M k 0 satisfy the constraint σ k=1 φ(M k 0 ) = 1, a simple calculus exercise in Lagrange multipliers shows that completing the proof.
Next we will show that E τ is on the same order of magnitude (w.r.t N ) as E S max .
that is the event that song k approaches the boundary 1 at time S k (as opposed to the boundary 0). Clearly A k ⊂ {τ ≤ S max } and so We claim that the A k are almost surely disjoint. Assuming otherwise, given A k ∩ A j one of S k , S j must occur first -since both M k and M j can't be greater than 1 − η 2N at the same time. Assume without loss of generality that S k < S j . At time S k , M k ≥ 1 − η 2N and so we must have M j ≤ η 2N , meaning that S j has already occurred, a contradiction. Therefore the events A k , 1 ≤ k ≤ σ are almost surely disjoint.
Next, as M k t is a martingale -with step size bounded by ∆ = 2η N -we have and so for N >> 0 Thus for any initial configuration X 0 Applying the Strong Markov Property we have

Fixation Time
Next we will estimate the fixation time given that the preference M k t for some (fixed but arbitrary) song k has approached the boundary 1. Specifically, we will consider starting from an initial configuration X 0 with When M k is near 1, the fixation time T fix can only be the last time any song other than k plays. Of course this need not occur. Projecting on k, this is the last time one of the Bernoulli trials for k has failed. We begin by showing that from such an initial configuration, T fix has likely already occurred.
Proof. We will consider the stopping time R, the first time any song other than k plays. Before R, each X k (i) can only increase. Therefore at time R -without loss of generality, a meeting of i and j -if another song is played by only one of i, j then If both agents play a different song, then X k (i) + X k (j) is even smaller at R. This then implies that on {R < ∞} Now, applying the Optional Stopping Theorem to R ∧ t, we find that Solving for P (t < R) we find that As this is true for arbitrary t, we have P (R = ∞) ≥ 1 2 from which our result follows.
Next we need to consider what happens when M k t approaches 1, but the song k fails to play at a meeting.

From any initial configuration
Proof. Let T n , 1 ≤ n < ∞ be the n-th meeting time. We first definẽ R = inf{n ≥ 0 : some song other than k plays at T n }.
We will calculate how M k t changes after the first meeting time, given that song k is played by both agents at the meeting time T 1 .
If agents i and j meet and both play k at T 1 then X k T1 (i) = (1 − η)X k 0 (i) + η and similarly for X k (j). So given that i and j meet and play k Summing over pairs of agents we find that By the same calculation we find that Continuing the same easy inductive calculation we find that Next, we need to know the chance of some song other than k being played at time T n given M k Tn−1 . We will need the identity for x, y ≤ 1 -which follows easily from 1 + (1 − x)(1 − y) ≥ 1. Using that, and that the probability of at least one of i, j not playing k is 1 − X k (i)X k (j), we have P A song other than k is played at T n |M k . Therefore we have that P R = n|F 0 = P R > n − 1, Another song is played at T n |F 0 ≤ P Another song is played at T n |R > n − 1, F 0 For the first inequality here we used the simple bound This allows us to calculate that using our assumption that M k 0 ≥ 1 − η 2N and the Taylor series expansion for |x| < 1.
Our result then follows since meetings occur independently at rate 1 N and so We are finally prepared to prove Theorem 1.1.
Proof. We will calculate here an upper bound for i.e. the upper bound over all initial configurations X 0 . Let τ be the stopping time from Equation (6), i.e. the first time that some song k has M k t ≥ 1 − η 2N and let K be that song. Note that this defines K uniquely as 1 − η 2N ≥ 1 2 . Let R be stopping time (as above) defined by R = inf{t ≥ τ | some song other than K is played}.
We first recall from Proposition 3.3 that at time τ , we have Also, at time τ , if T fix has not yet occurred, then some song other than K will play again and so R < ∞. Combining Proposition 3.1 and Proposition 3.2 we have that there exists a constant C(η) so that from any initial configuration X 0 We then have for any initial X 0 : Here the 1 8η is clearly dominated by the first term. Therefore, we have that and so E X0 T fix ≤ 4C(η) ln(σ)N λ from which our conclusion follows.

The Interaction Parameter η
Our goal here is find the asymptotic of our bound with respect to η. Tracing through the steps of our proof of Theorem 2.1, we may actually prove the following improved bound.
Proposition 4.1. There exists a constant C so that from any initial configuration x 0 , the first escape time S satisfies Then, repeating the arguments in Section 3, we may improve our bound in Theorem 1.1 on the expectation of the fixation time T fix . Theorem 4.2. There exists a constant C so that from any initial X 0 the fixation time T fix satisfies We conjecture that this can actually be improved to depend on η as 1 η(1−η) .

The Complete Graph Case
As an example of a geometry in which more can be said than Theorem 1.1, we look at the complete graph K N on N vertices. Specifically, we have uniform meeting rates between agents, that is ν ij = 1 N −1 for all pairs of agents i, j. It is standard fact that the spectral gap λ KN = 1 and so Theorem 1.1 shows that the fixation time has A simple argument will show that this order of magnitude bound is in fact tight.

A Lower Bound
Throughout this section we assume that there are at least two songs, i.e. σ ≥ 2. To achieve any reasonable lower bound, we need to ignore starting conditions that are likely already at fixation by time t = 0. We call an initial configuration non-trivial if there exists at least one song k with and will consider only non-trivial initial configurations. The choice of the factor of 1 2 here is of course arbitrary. Theorem 5.1. There exists a constant C(η, σ) such that for K N started from any non-trivial initial configuration, the fixation time T fix has Proof. Recalling Theorem 2.1, first consider any one song and consider its average preference M t , t ≥ 0. From the proof of Proposition 2.5 which combined with Q ≤ 1 4 gives that is a sub-martingale. By assumption, there exists at least one song k with M k 0 (1 − M k 0 ) ≥ 1 4σ . Let be the first time that M k t leaves the interval 1 8σ , 1 − 1 8σ . Then applying the Optional Stopping Theorem At time T 2 , we have have and so we can conclude that To complete the proof, we need only show that the fixation time T fix is with high probability the same order of magnitude as T 2 .
Consider the first meeting after time T 2 , between some agents i and j. If two different songs are played at that meeting, then by definition T fix must not have yet occurred. The probability that at a meeting at time t that agent i plays song k and j does not, or vis-versa, is Therefore, on the complete graph, the probability that two different songs play at a meeting at time t is Recalling Lemma 2.2, at time T 2 we still have and so at time T 2 we have Thus, the probability at time T 2 that fixation has occurred is bounded by Applying the Strong Markov property, we can conclude that finishing the proof.

Further Directions
We conclude by presenting a few possible further directions for research on the iPod model.

Improve the Fixation Time Bound
Heuristically, from any initial configuration the processes X k t mixes on a time scale of the order of the relaxation time λ −1 . Then, for any song k, when x t (i) ≈ M t we have Q t ≈ M t (1 − M t ) and so Following through the same embedding and comparison arguments, we then find a fixation time of O(N ). Therefore we conjecture that for any initial configuration E T fix = O(λ −1 + N ) = O(max(λ −1 , N ).

Remove the Dependence on σ
When the processes X k t are well mixed, i.e. when x k (i) ≈ M k again we have Q t ≈ M t (1 − M t ). Then, the σ-dimensional process {M k t } 1≤k≤σ,t≥0 has a comparable covariation structure to the σ-allele Wright Fisher Diffusion.
By a well known calculation [6] the σ-allele process has an expected absorption time of O(1), i.e. independent of σ. Therefore, we conjecture that by a similar embedding and comparison argument, the iPod process fixates in a time scale independent of the number of songs σ.
Combining this with our other conjectured improvements to Theorem 1.1, we conclude with the following conjectured bound for the fixation time of the iPod model.

Conjecture 1.
There exists a constant C so that for any graph G on N vertices, the fixation time T fix of the iPod model on G with σ songs, started from any initial configuration, satisfies where λ is the spectral gap of G.