The dual problems of coordination and anti-coordination on random bipartite graphs

In this paper, the individuals located at each vertex will operate using a simple set of update rules. These rules can incorporate random behavior, but the update decisions depend only on the color of an individual's neighbors. Consider the relationship between update rules for anti-coordination and coordination games. We will see that any update rule for an individual playing an anti-coordination game can be adapted to an update rule for playing a coordination game and vice versa. At its most basic, an anti-coordination rule aims to minimize the number of neighbors with the same color, and the goal of a coordination rule is to maximize the number of neighbors with the same color. Therefore, we can turn an anti-coordination update rule into a coordination update rule just by picking the opposite color every time.

Suppose we have an individual vertex with a neighbors playing color A and b neighbors playing color B, like in figure 1(a). When we define an anti-coordination rule where the central individual will select color A with probability p(a, b) and color B with probability 1 − p(a, b), we can make the corresponding coordination rule as follows: choose A with probability 1 − p(a, b) and B with probability p(a, b).

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Consider an update rule (anti-coordination or coordination) that has a function p(a, b) that gives the probability of choosing color A. If we switch the colors of all neighbors, the probability of choosing A is now p(b, a) because now b neighbors are playing A and a neighbors are playing B. There is a natural restriction to impose on the possible update rules. If we switch the color of every neighbor, moving from figures 1(a) and (b), the probabilities of the central vertex choosing color A, p(a, b), and color B, 1 − p(a, b), should switch as well. This restriction gives us the following complementary condition, by setting the probability of choosing A equal to the probability of choosing B after switching all the neighbors' colors:

$$\begin{equation}p(a,b)=1-p(b,a).\end{equation} \tag{ 1 }$$

For any anti-coordination update rule, a vertex with aA neighbors and bB neighbors will choose A with some probability p(a, b). If we switch the colors of all the neighbors, the vertex will choose A with probability p(b, a) = 1 − p(a, b), but this is equal to the probability of a coordination player choosing A. Therefore, an anti-coordination algorithm can be converted into its dual algorithm for a coordination game by temporarily switching the colors of all the neighbors, using the anti-coordination update rule, and switching the neighbors' colors back. As an example, an anti-coordination update rule on figure 1(a) will have the same behavior as a coordination update rule on figure 1(b).

The same process can be used to convert a coordination algorithm to an anti-coordination algorithm.

To put the above individual choice function p(a, b) in context, it is worthwhile to introduce a few intuitive anti-coordination update rules. The first update rule, called randomness-first, involves making a random choice with probability r, and otherwise with probability 1 − r makes a color choice that minimizes color conflicts. This update rule can be expressed as:

$$\begin{equation}p(a,b)=\begin{cases}1-\frac{1}{2}r\quad & a< b\\ \frac{1}{2}\quad & a=b\\ \frac{1}{2}r\quad & b< a\end{cases}.\end{equation} \tag{ 2 }$$

Under the second update rule, called memory-0, individuals first attempt to choose any color that eliminates all color conflicts. If that is not possible, the individual chooses randomly with probability r and otherwise with probability 1 − r chooses the color minimizing conflicts with neighbors. In our terms, this algorithm is

$$\begin{equation}p(a,b)=\begin{cases}1\quad & a=0\\ 1-\frac{1}{2}r\quad & 0< a< b\\ \frac{1}{2}\quad & a=b\\ \frac{1}{2}r\quad & 0< b< a\\ 0\quad & b=0\end{cases}.\end{equation} \tag{ 3 }$$

The third main update rule, called memory-1, is like the memory-0 rule except that the agent only makes a random choice if no neighbors have changed color in the last round of updates. Since this is not a memory-less update rule, it does not have a corresponding p(a, b) function, and the following proof would need to be slightly modified, particularly by significantly enlarging the state space of the Markov chains to include the last N colorings of the graph, to prove the equivalence for update rules with finite memory. While we do not go over all the details of proving that a finite-memory update rule also satisfies the isomorphism, we do show results of computer simulations to demonstrate that the duality of coordination and anti-coordination holds in section 3.

This is only a small selection of all possible update rules. Any function that satisfies equation (1) and returns values between 0 and 1 could be an update rule, although many would be very ineffective. The three update rules described above are all intuitively reasonable and simple to express, which made them excellent candidates for study in prior work on network graph coloring problems [11]. However, there are other natural update rules that we do not explicitly describe here. For example, an individual may wish to choose each color proportional to the number of neighbors playing that color.

In what follows, we demonstrate that an anti-coordination update rule is exactly as effective at finding a two-coloring as the corresponding coordination update rule is at finding a uniform color for the whole bipartite network.

For a connected, bipartite graph G of size N, let col(G) be the set of all possible labelings of the graph G. Note that here we refer to all ways of labeling the vertices of G with either color A or color B, not just two-colorings in which no neighbors share the same color.

The system will update as follows: the graph is initialized by randomly assigning each vertex a color. An update order is created that describes the order in which the labelled vertices will update their color. The update order is represented as a list of the numbers 1 through N, which is just a permutation of N elements. The set of all permutations of N elements, called the symmetric group on N elements, is denoted S_N . The vertices continually update their colors in this order, one at a time, until the desired coloring (either a two-coloring or uniform coloring) is found.

Now we can define our Markov chains. Let {X_i } be a Markov chain using an anti-coordination update rule, and let {Y_i } be the Markov chain using the associated coordination update rule, as described above. The state space Ω of both chains is the set of ordered triples (G*, σ, m) where G* ∈ col(G), σ ∈ S_n , and m ∈ {1, 2, ..., n}. Unsurprisingly, G* represents the colors of the vertices of the graph at some time i. σ is the order in which the vertices update, and m is the current position in the update step.

The state space is quite large, but for each state, there are exactly two states to which the Markov chains can move with non-zero probability, shown in figure 2.

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To begin, we initialize both Markov chains (anti-coordination and coordination) by sampling from the uniform distribution Π over Ω, so each starting coloring is equally likely.

Without loss of generality, let X_j = Y_j = (G*, σ, m). Here, σ(m) is the vertex that is about to update. Let ${G}_{A}^{\ast }$ be the colored graph that is the same as G* except possibly σ(m) which has color A, and ${G}_{B}^{\ast }$ the same but for color B. In each step of the Markov chains, σ(m) selects one of two colors and the position in the update cycle increases by one, resetting to 1 if necessary. The update order σ remains unchanged. Thus, if σ(m) has a color A neighbors and b color B neighbors,

$$\begin{equation}P({X}_{j+1}=({G}_{A}^{\ast },\sigma ,m\enspace \text{mod}(n)+1))=p(a,b)\end{equation} \tag{ 4 }$$

$$\begin{equation}P({X}_{j+1}=({G}_{B}^{\ast },\sigma ,m\enspace \text{mod}(n)+1))=1-p(a,b)=p(b,a)\end{equation} \tag{ 5 }$$

$$\begin{equation}P({Y}_{j+1}=({G}_{A}^{\ast },\sigma ,m\enspace \text{mod}(n)+1))=1-p(a,b=p(b,a))\end{equation} \tag{ 6 }$$

$$\begin{equation}P({Y}_{j+1}=({G}_{B}^{\ast },\sigma ,m\enspace \text{mod}(n)+1))=p(a,b).\end{equation} \tag{ 7 }$$

For bipartite graphs, we claim that these Markov chains {X_i } and {Y_i } are isomorphic. First, because G is a connected, bipartite graph, the vertices can be divided into two groups. In a two-coloring, all the vertices in the same group will be the same color, and all vertices in different groups will be different colors. Let S be the set of vertices of one of these groups. Because we are working with two-colorings of bipartite graphs, we can define a function ϕ : col(G) → col(G) by switching the color of every vertex in S, and define ψ_S : Ω → Ω as the extension of ϕ in the natural way. We claim that this is a Markov chain isomorphism between X_i and Y_i . This requires proving two conditions hold. First, ψ_S must be bijective. Second, ψ_S must commute with the transition matrices of X_i and Y_i , i.e. the probability of X_i moving from x to y is the same as Y_i moving from ψ_S (x) to ψ_S (y). More formally, for all x, y ∈ Ω,

$$\begin{equation}P({X}_{i+1}=y\vert {X}_{i}=x)=P({Y}_{i+1}={\psi }_{S}(y)\vert {Y}_{i}={\psi }_{S}(x)).\end{equation} \tag{ 8 }$$

If equation (8) holds, the two Markov chains are equivalent in that after relabelling the states in Ω (according to ψ_S ), the Markov chains are identical.

That ψ_S is bijective is fairly obvious. For any colored graph G*, because we are only working with two-colorings on bipartite graphs, ϕ(G*) is well-defined, and only ϕ(G*) maps to G*, so it is both one-to-one and onto, and therefore ψ is as well.

Now we will prove equation (8). Since we are considering Markov chains moving from x to y (or ψ_S (x) to ψ_S (y)), let x = (G*, σ, m). Let a and b be the number of color A and color B neighbors of σ(m) in G*, respectively.

We begin with the conditional statement X_i = x = (G*, σ, m). Equations (4) and (5) give the only two possibles states of X_i+1 and their transition probabilities:

$$\begin{equation}P\left({X}_{i+1}=({G}_{A}^{\ast },\sigma ,m\enspace \text{mod}(n)+1)\right)=p(a,b)\end{equation} \tag{ 9 }$$

$$\begin{equation}P\left({X}_{i+1}=({G}_{B}^{\ast },\sigma ,m\enspace \text{mod}(n)+1)\right)=1-p(a,b).\end{equation} \tag{ 10 }$$

Once again, ${G}_{A}^{\ast }$ and ${G}_{B}^{\ast }$ are the same as G* except σ(m) which has color A or B, respectively.

Now we consider Y_i+1 given that Y_i = ψ(x) = ψ((G*, σ, m)) = (ϕ(G*), σ, m). σ(m) is the next vertex to update, and either it is in the subset S or it is not. These two cases must be handled separately.

If σ(m) ∈ S, none of σ(m)'s neighbors are in S, so σ(m) still has a color A neighbors and b color B neighbors. Because we are now in the coordination Markov chain {Y_i }, σ(m) chooses its color according to equations (6) and (7).

With probability p(a, b), σ(m) chooses color B. Because σ(m) ∈ S, ϕ(G*) becomes $\phi ({G}_{A}^{\ast })$ when σ(m) chooses B. Thus, ${Y}_{i+1}=(\phi ({G}_{A}^{\ast }),\sigma ,m\enspace \text{mod}(n)+1)=\psi ({G}_{A}^{\ast },\sigma ,m\enspace \text{mod}(n)+1)$.

With probability 1 − p(a, b), σ(m) chooses color A, and ${Y}_{i+1}=(\phi ({G}_{B}^{\ast }),\sigma ,m\enspace \text{mod}(n)+1)=\psi ({G}_{B}^{\ast },\sigma ,m\enspace \text{mod}(n)+1)$.

Thus, when σ ∈ S, equation (8) holds (figure 3).

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If σ(m) ∉ S, then all of its neighbors are. So in ϕ(G*), σ(m) has b color A neighbors and a color B neighbors.

With probability 1 − p(b, a) = p(a, b), σ(m) chooses color A, and ${Y}_{i+1}=(\phi ({G}_{A}^{\ast }),\sigma ,m\enspace \text{mod}(n)+1)$.

With probability p(b, a) = 1 − p(a, b), σ(m) chooses B, and ${Y}_{i+1}=(\phi ({G}_{A}^{\ast }),\sigma ,m\enspace \text{mod}(n)+1)$.

So equation (8) holds when σ(m) ∉ S (figure 4). Therefore, ψ is a Markov chain isomorphism.

Now we are prepared to state and defend the main claim of this work: when using local information, the anti-coordination and coordination problems are equivalent. Any result regarding the efficacy of an update rule p(a, b) for an anti-coordination game can also be applied to a coordination game, and vice versa.

Since the initial distribution Π is the uniform distribution and ψ is bijective, ψ(Π) = Π and both Markov chains begin from the same distribution. Furthermore, because ψ switches the color of the set S, for any state X_i that is a valid two-coloring, ψ(X_i ) = Y_i is a uniform coloring. For all times i, applying equation (8) i times tells us that moving the anti-coordination chain from a state X₀ to state X_i happens with the same probability as moving the coordination chain from Y₀ = ψ(X₀) to Y_i = ψ(X_i ). Because Π is the uniform distribution, for all x ∈ Ω and for all times i:

$$\begin{equation}P({X}_{i}=x\vert {X}_{0}\sim {\Pi})=P({Y}_{i}=\psi (x)\vert {Y}_{0}\sim \psi ({\Pi})={\Pi}).\end{equation} \tag{ 11 }$$

Critically, this says that the probability of solving the anti-coordination problem in i steps is the same as solving the coordination problem in i steps, for all i. Additionally, the process is linked at each step, so the expected number of player color changes will be the same, for example.

This result also holds for any update rules with finite memory. Any stochastic process whose transition probabilities only depend on a finite number of previous states can be reexpressed as a Markov chain by defining the new state space to be lists of elements from the previous state space, and this works here with any update rule that considers the last n update steps.