Self-organized Segregation on the Grid

Abstract

We consider an agent-based model with exponentially distributed waiting times in which two types of agents interact locally over a graph, and based on this interaction and on the value of a common intolerance threshold $\tau$, decide whether to change their types. This is equivalent to a zero-temperature ising model with Glauber dynamics, an asynchronous cellular automaton with extended Moore neighborhoods, or a Schelling model of self-organized segregation in an open system, and has applications in the analysis of social and biological networks, and spin glasses systems. Some rigorous results were recently obtained in the theoretical computer science literature, and this work provides several extensions. We enlarge the intolerance interval leading to the expected formation of large segregated regions of agents of a single type from the known size $\epsilon >0$ to size $\approx 0.134$. Namely, we show that for $0.433< \tau < 1/2$ (and by symmetry $1/2<\tau <0.567$), the expected size of the largest segregated region containing an arbitrary agent is exponential in the size of the neighborhood. We further extend the interval leading to expected large segregated regions to size $\approx 0.312$ considering “almost segregated” regions, namely regions where the ratio of the number of agents of one type and the number of agents of the other type vanishes quickly as the size of the neighborhood grows. In this case, we show that for $0.344 < \tau \le 0.433$ (and by symmetry for $0.567 \le \tau <0.656$) the expected size of the largest almost segregated region containing an arbitrary agent is exponential in the size of the neighborhood. This behavior is reminiscent of supercritical percolation, where small clusters of empty sites can be observed within any sufficiently large region of the occupied percolation cluster. The exponential bounds that we provide also imply that complete segregation, where agents of a single type cover the whole grid, does not occur with high probability for $p=1/2$ and the range of intolerance considered.

Appendix

1.1 Concentration Bound on the Number of Agents in the Initial Configuration

Lemma 18

Let $\epsilon \in (0,1/2)$, and let $\mathcal {N}$ be an arbitrary neighborhood in the grid with ${N}$ agents. There exist $c,c' \in \mathbb {R}^+$, such that

$$\begin{aligned} P\left( |W - {N}/2| < c{N}^{1/2+\epsilon }\right) \ge 1-2e^{-c'{N}^{2\epsilon }}. \end{aligned}$$

(29)

Proof

Let $W_i$ be the random variable associated with the type of the i’th agent in $\mathcal {N}$ such that it is one whenever the type is (−1) and zero otherwise. Let $\mathcal {F}_i = \sigma (W_1,\ldots ,W_i)$. Then it is easy to see that $M_n = \mathbb {E}[W|\mathcal {F}_n]$ for $n=1,\ldots ,{N}$ is a martingale. It is also easy to see that $M_0 = \mathbb {E}[W] = {N}/2$, and $M_{{N}} = W$. We also have

$$\begin{aligned} |M_n - M_{n-1}|&= \left| \mathbb {E}\left( \sum _{i=1}^{{N}}W_i|\mathcal {F}_n\right) - \mathbb {E}\left( \sum _{i=1}^{{N}}W_i|\mathcal {F}_{n-1}\right) \right| \\&= \left| W_n+ ({N}-n)/2 - [{N}-(n-1)]/2\right| \\&\le \left| W_n - \frac{1}{2}\right| \le 1/2, \end{aligned}$$

for $n=1,2,\ldots ,{N}$. Now using Azuma’s inequality, there exist constants $c_1,c_2 \in \mathbb {R}^+$ such that

$$\begin{aligned} P\left( W - {N}/2 \ge c{N}^{1/2+\epsilon }\right) \le e^{-c_1{N}^{2\epsilon }}, \end{aligned}$$

and

$$\begin{aligned} P\left( W - {N}/2 \le -c'{N}^{1/2+\epsilon }\right) \le e^{-c_2{N}^{2\epsilon }}. \end{aligned}$$

It follows by an application of Boole’s inequality that there exists a constant $c\in \mathbb {R}^+$ such that (29) holds. $\square$

1.2 Preliminary Results for the Proof of Theorem 1

First, we give a bound on the probability of having an unhappy agent in the initial configuration, we then extend this bound for a radical region.

Lemma 19

Let $p_u$ be the probability of being unhappy for an arbitrary agent in the initial configuration. There exist positive constants $c_l$ and $c_u$ which depend only on $\tau$ such that

$$\begin{aligned} c_{l}\frac{2^{-[1-H(\tau ')]{N}}}{\sqrt{{N}}} \le p_u \le c_{u}\frac{2^{-[1-H(\tau ')]{N}}}{\sqrt{{N}}}. \end{aligned}$$

where $\tau ' = \frac{\tau {N} - 2}{{N}-1}$, and H is the binary entropy function.

Proof

We have

$$\begin{aligned} p_u = \frac{1}{2^{N}} \sum _{k = 0 }^{\tau {N} - 2}{{{N}-1}\atopwithdelims (){k}} + \frac{1}{2^{N}} \sum _{k = 0 }^{\tau {N} - 2}{{{N}-1}\atopwithdelims (){k}}, \end{aligned}$$

(30)

where the two unit reduction is to account for the strict inequality and the agent at the center of the neighborhood. Let $\tau ' = \frac{\tau {N} - 2}{{N}-1}$. After some algebra we have

$$\begin{aligned} {{N}-1 \atopwithdelims ()\tau ' ({N}-1)} \le \sum _{k = 0 }^{\tau ' ({N}-1)}{{{N}-1}\atopwithdelims (){k}} \le \frac{1-\tau '}{1-2\tau '}{{N}-1 \atopwithdelims ()\tau ' ({N}-1)}, \end{aligned}$$

and using Stirling’s formula, there exist constants $c,c' \in \mathbb {R}^+$ such that

$$\begin{aligned} {c\frac{2^{-[1-H(\tau ')]({N}-1)}}{\sqrt{({N}-1)\tau '(1-\tau ')}}} \le {{N}-1 \atopwithdelims ()\tau ' ({N}-1)} \le c'\frac{2^{-[1-H(\tau ')]({N}-1)}}{\sqrt{({N}-1)\tau '(1-\tau ')}}. \end{aligned}$$

The result follows by combining the above inequalities. $\square$

Lemma 20

There exist positive constants $c_l$ and $c_u$ which depend only on $\tau$ such that in the initial configuration, an arbitrary neighborhood with radius $(1+\epsilon ')w$ is a radical region with probability $p_{\epsilon '}$ where we have

$$\begin{aligned} c_{l}{2^{-[1-H(\tau '')](1+\epsilon ')^2{N}-o({N})} \le p_{\epsilon '} \le c_{u} 2^{-[1-H(\tau '')](1+\epsilon ')^2{N}+o({N})}}, \end{aligned}$$

where $\tau '' = (\lfloor \hat{\tau }(1+\epsilon ')^2{N} \rfloor - 1) / (1+\epsilon ')^2{N}$, $\hat{\tau } = (1-{1}/{(\tau {N}^{1/2-\epsilon })})\tau$, and H is the binary entropy function.

Proof

The proof follows the same lines as in the proof of Lemma 19. $\square$

Lemma 21

Let ${\rho } = 2^{[1-H(\tau ')]{N}/2}$ and

$$\begin{aligned} A = \left\{ \forall v\in \mathcal {N}_{\rho }, \; u^{+} \text{ would } \text{ be } \text{ happy } \text{ at } \text{ the } \text{ location } \text{ of } \text{ v } \text{ at } \text{ t=0 }\right\} . \end{aligned}$$

Then A occurs w.h.p.

Proof

Let $U_i$ for $i=1,2,\ldots ,|\mathcal {N}_{\rho }|$ be the event that agent $u^+$ would be happy at the location of i’th agent of $\mathcal {N}_{\rho }$. It is easy to see that $P(U_i)=p_u$ (see (30)). Hence we have

$$\begin{aligned} P(A)&= P\left( U_1^C \cap \cdots \cap U^C_{|\mathcal {N}_{\rho }|}\right) \\&= 1 - P\left( U_1 \cup \cdots \cup U_{|\mathcal {N}_{\rho }|}\right) \\&\ge 1 - |\mathcal {N}_{\rho }|\frac{2^{-[1-H(\tau ')]{N}}}{\sqrt{{N}}} \\&\ge 1 - \frac{5}{\sqrt{{N}}} \end{aligned}$$

which tends to one as ${N} \rightarrow \infty$. $\square$

The following lemma gives a simple lower bound for the probability of having a radical region inside a neighborhood which has radius $r= 2^{[1-H(\tau ')]{N}/2-o({N})}$. We call a radical region with radius $(1+\epsilon ')w$ an $\epsilon '$-radical region.

Lemma 22

Any arbitrary neighborhood $\mathcal {N}_r$ with radius $r= 2^{[1-H(\tau ')]{N}/2-o({N})}$ in the initial configuration has at least one $\epsilon '$-radical region in it with probability at least $2^{-[1-H(\tau ')](2\epsilon '+\epsilon '^2){N} -o({N})}$.

Proof

Divide the neighborhood into $2(1+\epsilon ')w$-blocks, and let ${N}_b$ denote the number of blocks in $\mathcal {N}_r$. Define the events

$$\begin{aligned} Q_i = \{ \text {The i-th block of } \mathcal {N}_r \text { is an } \epsilon '\text {-radical region} \}, \\ Q = \{\text {There is an } \epsilon '\text {-radical region in }\mathcal {N}_r \}. \end{aligned}$$

Using Lemma 20, it follows that

$$\begin{aligned} P(Q) \ge&\ P\left( Q_1 \cup \cdots \cup Q_{{N}_b}\right) \\ =&\ 1 - P\left( Q_1^C \cap \cdots \cap Q^C_{{N}_b}\right) \\ =&\ \frac{4r^2}{(1+\epsilon ')^2{N}}2^{-[1-H(\tau '')](1+\epsilon ')^2{N}-o({N})}\\ =&\ 2^{-[1-H(\tau ')][2\epsilon '+\epsilon '^2]{N} -[H(\tau ')-H(\tau '')](1+\epsilon ')^2{N}-o({N})} \\ =&\ 2^{-[1-H(\tau ')][2\epsilon '+\epsilon '^2]{N} -o({N})}. \end{aligned}$$

$\square$

1.3 FKG–Harris Inequality

The following is Theorem 4 in [28] which is originally by Harris [19]. Let $\sigma _t$ be the configuration of the agents on the grid at time t. Let $\mathbb {E}^{\sigma _0}[X]$ be the expected value of the random variable X, when the initial state of the system is $\sigma _0$. A probability distribution $\mu$ on $\{0,1\}^{\mathbb {Z}^d}$ is said to be positively associated if for all increasing f and g we have

$$\begin{aligned} \mathbb {E}[f(\sigma )g(\sigma )] \ge \mathbb {E}[f(\sigma )]\mathbb {E}[g(\sigma )]. \end{aligned}$$

Theorem 6

(Harris) Assume the process satisfies the following two properties:

• (a) Individual transitions affect the state at only one site.

• (b) For every continuous increasing function f and every $t>0$, the function $\sigma _0 \rightarrow \mathbb {E}^{\sigma _0}[f(\sigma _t)]$ is increasing. Then, if the initial distribution is positively associated, so is the distribution at all later times.

The following is a version of the FKG inequality [16] in our setting. The original inequality holds for a static setting and is extended here to our time-dynamic setting using Theorem 6.

Lemma 23

(FKG–Harris) Let A and B be two increasing events defined on our process on the grid. We have

$$\begin{aligned} P(A\cap B) \ge P(A)P(B). \end{aligned}$$

Proof

Assume A and B are increasing random variables which depend only on the states of the sites $v_1,v_2,\ldots ,v_k$ and first time step. We proceed by induction on k. First, let $k=1$. Let $\omega (v_1)$ be the realization of the site $v_1$. We also have

$$\begin{aligned} \left( 1_A(\omega _1)-1_A(\omega _2)\right) \left( 1_B(\omega _1)-1_B(\omega _2) \right) \ge 0, \end{aligned}$$

for all pairs of vectors $\omega _1$ and $\omega _2$ from the sample space. We have

$$\begin{aligned} 0&\le \sum _{\omega _1,\omega _2} \left( 1_A(\omega _1)-1_A(\omega _2)\right) \left( 1_B(\omega _1)-1_B(\omega _2)\right) P(\omega (v_1)=\omega _1)P(\omega (v_1)=\omega _2) \\&=2\left( P(A\cap B)-P(A)P(B)\right) , \end{aligned}$$

as required. Assume now that the result is valid for values of n satisfying $k<n$. Then

$$\begin{aligned} P(A\cap B)&= \mathbb {E}\left[ P\left( A\cap B \,\Big \vert \,\omega (v_1),\ldots ,\omega (v_{n-1})\right) \right] \\&\ge \mathbb {E}\left[ P\left( A\,\Big \vert \,\omega (v_1),\ldots ,\omega (v_{n-1})\right) P\left( B\,\Big \vert \,\omega (v_1),\ldots ,\omega (v_{n-1})\right) \right] , \end{aligned}$$

since, given $\omega (v_1),\ldots ,\omega (v_{n-1})$, $1_A$ and $1_B$ are increasing in the single variable $\omega (v_n)$. Now since $P\left( A|\omega (v_1),\ldots ,\omega (v_{n-1})\right)$ and $P\left( B|\omega (v_1),\ldots ,\omega (v_{n-1})\right)$ are increasing in the space of the $n-1$ sites, it follows from the induction hypothesis that

$$\begin{aligned} P(A\cap B)&\ge \mathbb {E}\left[ P\left( A\,\Big \vert \,\omega (v_1),\ldots ,\omega (v_{n-1})\right) \right] \mathbb {E}\left[ P\left( B\,\Big \vert \,\omega (v_1),\ldots ,\omega (v_{n-1})\right) \right] \nonumber \\&= P(A)P(B). \end{aligned}$$

(31)

Next, assume A and B are increasing random variables which depend only on the states of the sites in the first k time steps. We proceed by induction on $k<K$ such that K denotes the final time step over all the realizations. First, let $k=0$. Let $\omega (t_0)$ be the configuration of the graph at the first time step. We have

$$\begin{aligned} P(A\cap B) \ge P(A)P(B), \end{aligned}$$

by the above result. Assume now that the result is valid for all values of k satisfying $k<K$. Then, since our process satisfies the conditions of Theorem 6 and given $\omega (t_0),\ldots ,\omega (t_{K-1})$, $1_A$ and $1_B$ are increasing in $\omega (t_{K})$, we have

$$\begin{aligned} P(A\cap B)&= \mathbb {E}\left[ P\left( A\cap B \,\Big \vert \,\omega (t_0),\ldots ,\omega (t_{K-1})\right) \right] \\&\ge \mathbb {E}\left[ P\left( A\,\Big \vert \,\omega (t_0),\ldots ,\omega (t_{K-1})\right) P\left( B\,\Big \vert \,\omega (t_0),\ldots ,\omega (t_{K-1})\right) \right] . \end{aligned}$$

Now, since $P\left( A|\omega (t_0),\ldots ,\omega (t_{K-1})\right)$ and $P\left( B|\omega (t_0),\ldots ,\omega (t_{K-1})\right)$ are increasing in the space of the configurations of the graph in the first $K-1$ time steps, it follows from the induction hypothesis that

$$\begin{aligned} P(A\cap B)&\ge \mathbb {E}\left[ P\left( A\,\Big \vert \,\omega (t_0),\ldots ,\omega (t_{K-1})\right) \right] \mathbb {E}\left[ P\left( B\,\Big \vert \,\omega (t_0),\ldots ,\omega (t_{K-1})\right) \right] \\&= P(A)P(B). \end{aligned}$$

$\square$