The Flip Schelling Process on Random Geometric and Erdös-Rényi Graphs
Thomas Bläsius, Tobias Friedrich, Martin S. Krejca, Louise Molitor
TThe Flip Schelling Process on Random Geometricand Erdős–Rényi Graphs
Thomas Bläsius ! ˇ Karlsruhe Institute of Technology, Karslruhe, Germany
Tobias Friedrich ! ˇ Hasso Plattner Institute, University of Potsdam, Potsdam, Germany
Martin S. Krejca ! Sorbonne Université, CNRS, LIP6, France
Louise Molitor ! ˇ Hasso Plattner Institute, University of Potsdam, Potsdam, Germany
Abstract
Schelling’s classical segregation model gives a coherent explanation for the wide-spread phenomenonof residential segregation. We consider an agent-based saturated open-city variant, the Flip-Schelling-Process (FSP), in which agents, placed on a graph, have one out of two types and, based on thepredominant type in their neighborhood, decide whether to changes their types; similar to a newagent arriving as soon as another agent leaves the vertex.We investigate the probability that an edge { u, v } is monochrome, i.e., that both vertices u and v have the same type in the FSP, and we provide a general framework for analyzing the influence ofthe underlying graph topology on residential segregation. In particular, for two adjacent vertices,we show that a highly decisive common neighborhood, i.e., a common neighborhood where theabsolute value of the difference between the number of vertices with different types is high, supportssegregation and moreover, that large common neighborhoods are more decisive.As an application, we study the expected behavior of the FSP on two common random graphmodels with and without geometry: (1) For random geometric graphs, we show that the existence ofan edge { u, v } makes a highly decisive common neighborhood for u and v more likely. Based onthis, we prove the existence of a constant c > / c . (2) For Erdős–Rényi graphs we show that large commonneighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP isat most 1 / Theory of computation → Network formation; Theory of compu-tation → Random network models
Keywords and phrases
Agent-based Model, Schelling Segregation, Spin System
Funding
Martin S. Krejca : This work was supported by the Paris Île-de-France Region.
Acknowledgements
We want to thank Thomas Sauerwald for the discussions on random walks. a r X i v : . [ m a t h . P R ] F e b The Flip Schelling Process on Random Graphs
Residential segregation is a well-known sociological phenomenon [43] where different groupsof people tend to separate into largely homogeneous neighborhoods. Studies, e.g. [16], showthat individual preferences are the driving force behind present residential patterns and bearmuch to the explanatory weight. Local choices therefore lead to a global phenomenon [41].A simple model for analyzing residential segregation was introduced by Schelling [40, 41] inthe 1970s. In his model, two types of agents, placed on a grid, act according to the followingthreshold behavior, with τ ∈ (0 ,
1) as the intolerance threshold : agents are content withtheir current position on the grid if at least a τ -fraction of neighbors is of their own type.Otherwise they are discontent and want to move, either via swapping with another randomdiscontent agent or via jumping to a vacant position. Schelling demonstrated by experimentsthat, starting from a uniform random distribution, the described process drifts towardsstrong segregation, even if agents are tolerant and agree to live in mixed neighborhoods, i.e.,if τ ≤ . Many empirical studies have been conducted to investigate the influence of variousparameters on the obtained segregation, see [7, 8, 23, 36, 39].On the theoretical side, Schelling’s model started recently gaining traction within thealgorithmic game theory and artificial intelligence communities [1, 10, 14, 15, 19, 20, 30],with focus on core game theoretic questions, where agents strategically select locations.Henry et al. [28] described a simple model of graph clustering motivated by Schellingwhere they showed that segregated graphs always emerge. Variants of the random Schellingsegregation process were analyzed by a line of work that showed that residential segregationoccurs with high probability [4, 6, 9, 11, 29, 45].We consider an agent-based model, called the Flip-Schelling-Process (FSP) , which canbe understood as the Schelling model in a saturated open city . In contrast to closedcities [6, 11, 29, 45], which require fixed populations, open cities [3, 4, 9, 25] allow residentsto move away. In saturated city models, also known as voter models [18, 31, 32], verticesare not allowed to be unoccupied, hence, a new agent enters as soon as one agent vacatesa vertex. In general, in voter models two types of agents are placed on a graph. An agentexamines their neighbors and, if a certain threshold is of another type, they change theirtype. Also in this model segregation is visible. There is a line of work, mainly in physics,that studies the voting dynamics on several types of graphs [2, 12, 33, 38, 44].In the FSP, agents have binary types. An agent is content if the fraction of agents intheir neighborhood with the same type is larger . Otherwise, if the fraction is smaller , anagent is discontent and is willing to flip their type to become content. If the fraction of sametype agents in their neighborhood is exactly , an agent flips their type with probability .Starting from an initial configuration where the type of each agent is chosen uniformly atrandom, we investigate a simultaneous-move, one-shot process and bound the number ofmonochrome edges, which is a popular measurement for segregation strength [17, 24].Close to our model is the work by Omidvar and Franceschetti [34, 35], who initiated ananalysis of the size of monochrome regions in the so called Schelling Spin Systems . Agents oftwo different types are placed on a grid [34] and a geometric graph [35], respectively. Thenindependent and identical Poisson clocks are assigned to all agents and, every time a clockrings, the state of the corresponding agent is flipped if and only if the agent is discontent w.r.t.a certain intolerance threshold τ regarding the neighborhood size. The model corresponds tothe Ising model with zero temperature with Glauber dynamics [13, 42].The commonly used underlying topology for modeling the residential areas are (toroidal)grid graphs [10, 29, 34], regular graphs [10, 15, 19], paths [10, 30], cycles [3, 5, 6, 11, 45] . Bläsius, T. Friedrich, M. S. Krejca and L. Molitor 3 Erdős-Rényi graphs random geometric graphs5 k 10 k 15 k 20 k 25 k 5 k 10 k 15 k 20 k 25 k0.500.550.600.650.70 number of vertices n f r a c t i o n o f m o n o c h r o m ee d g e s averagedegree Figure 1
The fraction of monochrome edges after the Flip-Schelling-Process (FSP) in Erdős–Rényigraphs and random geometric graphs for different graph sizes (number of vertices n ) and differentexpected average degrees. Each data point shows the average over 1000 generated graphs with onesimulation of the FSP per graph. The error bars show the interquartile range, i.e., 50 % of themeasurements lie between the top and bottom end of the error bar. and trees [1, 10, 20, 30]. Considering the influence of the given topology that models theresidential area regarding, e.g., the existence of stable states and convergence behaviorleads to phenomena like non-existence of stable states [19, 20], non-convergence to stablestates [10, 15, 19], and high-mixing times [9, 26].To avoid such undesirable characteristics, we suggest to investigate random geometricgraphs [37], like in [35]. Random geometric graphs demonstrate, in contrast to other randomgraphs without geometry, such as Erdős–Rényi graphs [21, 27], community structures, i.e.,densely connected clusters of vertices. An effect observed by simulating the FSP is that thefraction of monochrome edges is significantly higher in random geometric graphs comparedto Erdős–Rényi graphs, where the fraction stays almost stable around , cf. Fig 1.We set out for rigorously proving this phenomenon. In particular, we prove for randomgeometric graphs that there exists a constant c such that, given an edge { u, v } , the probabilitythat { u, v } is monochrome is lower-bounded by + c , cf. Theorem 6. In contrast, we showfor Erdős–Rényi graphs that segregation is not likely to occur and the probability that { u, v } is monochrome is upper-bounded by + o (1), cf. Theorem 17.We introduce a general framework to deepen the understanding of the influence of theunderlying given topology on residential segregation. To this end, we first show that ahighly decisive common neighborhood supports segregation, cf. Section 3.1. In particular,we provide a lower bound that an edge { u, v } is monochrome based on the probability thatthe difference between the majority and the minority regarding both types in the commonneighborhood, i.e., the number of agents which are adjacent to u and v , is larger compared totheir exclusive neighborhoods, i.e., the number of agents which are adjacent to either u or v .Next, we show that large sets are more decisive, cf. Section 3.2. This implies that a largecommon neighborhood, compared to the exclusive neighborhood, is likely to be more decisive,i.e., makes it more likely that the absolute value of the difference between the number ofdifferent types in the common neighborhood is larger than in the exclusive neighborhoods.These considerations hold for arbitrary graphs. Hence, we reduce the question concerning alower bound for the fraction of monochrome edges in the FSP to the probability that, given { u, v } , the common neighborhood of u and v is larger than the exclusive neighborhoods of u and v , respectively. The Flip Schelling Process on Random Graphs
For random geometric graphs, we prove that a large geometric region, i.e., the intersectingregion that are formed by intersecting disks, leads to a large vertex set, cf. Section 3.3, andthat random geometric graphs have enough edges that have sufficiently large intersectingregions, cf. Section 3.4, such that segregation is likely to occur. In contrast, for Erdős–Rényigraphs, we show that the common neighborhood between two vertices u and v is with highprobability empty and therefore segregation is not likely to occur, cf. Section 4.Overall, we shed light on the influence of the structure of the underlying graph anddiscovered the significant impact of the community structure as an important factor on theobtained segregation strength. We reveal for random geometric graphs that already afterone round a provable tendency is apparent and a strong segregation occurs. Let G = ( V, E ) be an unweighted and undirected graph, with vertex set V and edge set E . Forany vertex v ∈ V , we denote the neighborhood of v in G by N v = { u ∈ V : { u, v } ∈ E } andthe degree of v in G by δ v = | N v | . We consider random geometric graphs and Erdős–Rényigraphs with a total of n ∈ N + vertices and an expected average degree δ > r ∈ R + , a random geometric graph G ∼ G ( n, r ) is obtained by distributing n vertices uniformly at random in some geometric ground space and connecting vertices u and v if and only if dist( u, v ) ≤ r . We use a two-dimensional toroidal Euclidean space withtotal area 1 as ground space. More formally, each vertex v is assigned a point ( v , v ) ∈ [0 , and the distance between u = ( u , u ) and v is dist( u, v ) = p | u − v | ◦ + | u − v | ◦ for | u i − v i | ◦ = min {| u i − v i | , − | u i − v i |} . We note that using a torus instead of, e.g., a unitsquare, has the advantage that we do not have to consider edge cases, for vertices that areclose to the boundary. In fact, a disk of radius r around any point has the same area π r .As every vertex v is connected to all vertices in the disk of radius r around it, its expectedaverage degree is δ = ( n − π r .For a given p ∈ [0 , G ( n, p ) denote an Erdős–Rényi graph. Each edge { u, v } isincluded with probability p , independently from every other edge. It holds that δ = ( n − p .Consider two different vertices u and v . Let N u ∩ v := | N u ∩ N v | be the number of verticesin the common neighborhood , let N u \ v := | N u \ N v | be the number of vertices in the exclusiveneighborhood of u , and let N v \ u := | N v \ N u | be the number of vertices in the exclusiveneighborhood of v . Furthermore, with N u ∪ v := | V \ ( N u ∪ N v ) | , we denote the number ofvertices that are neither adjacent to u nor to v .Let G be a graph where each vertex represents an agent of type t + or t − . The type ofeach agent is chosen independently and uniformly at random. An edge { u, v } monochrome ifand only if u and v are of the same type. The Flip-Schelling-Process (FSP) is defined asfollows: an agent v whose type is aligned with the type of more than δ v / δ v / v changestheir type. In case of a tie, i.e., if exactly δ v / v changestheir type with probability . FSP is a simultaneous-move, one-shot process, i.e., all agentsmake their decision at the same time and, moreover, only once.For x, y ∈ N , we define [ x..y ] = [ x, y ] ∩ N and for x ∈ N , we define [ x ] = [1 ..x ]. In this section, we state several lemmas that we will use in order to prove our results in thenext sections. . Bläsius, T. Friedrich, M. S. Krejca and L. Molitor 5 ▶ Lemma 1.
Let X ∼ Bin( n, p ) and Y ∼ Bin( n, q ) with p ≥ q . Then Pr [ X ≥ Y ] ≥ . Proof.
Let Y , . . . , Y n be the individual Bernoulli trials for Y , i.e., Y = P i ∈ [ n ] Y i . Define newrandom variables Y ′ , . . . , Y ′ n such that Y i = 1 implies Y ′ i = 1 and if Y i = 0, then Y ′ i = 1 withprobability ( p − q ) / (1 − q ) and Y ′ i = 0 otherwise. Note that for each individual Y ′ i , we have Y ′ i = 1 with probability p , i.e., Y ′ = P i ∈ [ n ] Y ′ i ∼ Bin( n, p ). Moreover, as Y ′ ≥ Y for everyoutcome, we have Pr [ X ≥ Y ] ≥ Pr [ X ≥ Y ′ ]. It remains to show that Pr [ X ≥ Y ′ ] ≥ .As X and Y ′ are equally distributed, we have Pr [ X ≥ Y ′ ] = Pr [ X ≤ Y ′ ]. Moreover, asone of the two inequalities holds in any event, we get Pr [ X ≥ Y ′ ] + Pr [ X ≤ Y ′ ] ≥
1, andthus equivalently 2Pr [ X ≥ Y ′ ] ≥
1, which proves the claim. ◀▶ Lemma 2.
Let n ∈ N + , p ∈ [0 , , and let X ∼ Bin( n, p ) . Then, for all i ∈ [0 ..n ] , it holdsthat Pr [ X = i ] ≤ Pr [ X = ⌊ p ( n + 1) ⌋ ] . Proof.
We interpret the distribution of X as a finite series and consider the sign of thedifferences of two neighboring terms. A maximum of the distribution of X is located atthe position at which the difference switches from positive to negative. To this end, let b : [0 , n − → [ − ,
1] be defined such that, for all i ∈ [0 , n − ∩ N , it holds that b ( d ) = (cid:18) nd + 1 (cid:19) p d +1 (1 − p ) n − d − − (cid:18) nd (cid:19) p d (1 − p ) n − d . We are interested in the sign of b . In more detail, for any d ∈ [0 , n − ∩ N , if sgn (cid:0) b ( d ) (cid:1) ≥ (cid:0) b ( d + 1) (cid:1) ≤
0, then d + 1 is a local maximum. If the sign is always negative, thenthere is a global maximum in the distribution of X at position 0.In order to determine the sign of b , for all i ∈ [0 ..n − b ( i ) = n ! i !( n − i − p i (1 − p ) n − i − pi + 1 − n ! i !( n − i − p d (1 − p ) n − i − − pn − i = n ! i !( n − i − p i (1 − p ) n − i − (cid:18) pi + 1 − − pn − i (cid:19) . Since the term n ! p i (1 − p ) n − i − is always non-negative, the sign of b ( i ) is determined by thesign of p/ ( i + 1) − (1 − p ) / ( n − i ). Solving for i , we get that pi + 1 − − pn − i ≥ ⇔ i ≤ p ( n + 1) − . Note that p ( n + 1) − X isunimodal, as the sign of b changes at most once. Thus, each local maximum is also a globalmaximum. As discussed above, the largest value d ∈ [0 , n − ∩ N such that sgn (cid:0) b ( d ) (cid:1) ≥ (cid:0) b ( d + 1) (cid:1) ≤ d + 1. Since d needs tobe integer, the largest value that satisfies this constraint is ⌊ p ( n + 1) − ⌋ . If the sign of b is always negative ( p ≤ / ( n + 1)), then the distribution of X has a global maximum at 0,which is also satisfied by ⌊ p ( n + 1) − ⌋ + 1, which concludes the proof. ◀▶ Theorem 3 (Stirling’s Formula [22, page 54]) . For all n ∈ N + , it holds that √ π n n +1 / e − n · e (12 n +1) − < n ! < √ π n n +1 / e − n · e (12 n ) − . ▶ Corollary 4.
For all n ≥ with n ∈ N , it holds that n ! > √ π n n +1 / e − n and (1) n ! < e n n +1 / e − n . (2) The Flip Schelling Process on Random Graphs
Proof.
For both inequalities, we aim at using Theorem 3.Equation (1): Note that e (12 n +1) − >
1, since n +1 >
0. Hence, √ π n n +1 / e − n < √ π n n +1 / e − n · e (12 n +1) − . Equation (2): We prove this case by showing that √ π e (12 n ) − < e . (3)Note, that e (12 n ) − is strictly decreasing. Hence, we only have to check whether Equation (3)holds for n = 2. √ π e (12 n ) − ≤ √ π e < . < e . ◀▶ Lemma 5.
Let A , B , and C be random variables such that Pr [
A > C ∧ B > C ] > and Pr [
A > C ∧ B ≤ C ] > . Then Pr [
A > B ∧ A > C ] ≥ Pr [
A > B ] · Pr [
A > C ] . Proof.
Using the definition of conditional probability, we obtainPr [
A > B ∧ A > C ] = Pr [
A > B | A > C ] · Pr [
A > C ] . Hence, we are left with bounding Pr [
A > B | A > C ] ≥ Pr [
A > B ]. Partitioning the samplespace into the two events
B > C and B ≤ C and using the law of total probability, we obtainPr [ A > B | A > C ] = Pr [
B > C | A > C ] · Pr [
A > B | A > C ∧ B > C ]+ Pr [ B ≤ C | A > C ] · Pr [
A > B | A > C ∧ B ≤ C ] . Note that the condition
A > C ∧ B ≤ C already implies A > B and thus the last probabilityequals to 1. Moreover, using the definition of conditional probability, we obtainPr [
A > B | A > C ] = Pr [
B > C | A > C ] · Pr [
A > B ∧ A > C ∧ B > C ]Pr [
A > C ∧ B > C ]+ Pr [ B ≤ C | A > C ] . Using that Pr [
B > C | A > C ] ≥ Pr [
A > C ∧ B > C ], that
A > B ∧ B > C alreadyimplies
A > C , that Pr [ B ≤ C | A > C ] ≥ Pr [
A > B ∧ B ≤ C ], and finally the law of totalprobability, we obtainPr [ A > B | A > C ] ≥ Pr [
A > B ∧ A > C ∧ B > C ] + Pr [ B ≤ C | A > C ]= Pr [
A > B ∧ B > C ] + Pr [ B ≤ C | A > C ] ≥ Pr [
A > B ∧ B > C ] + Pr [
A > B ∧ B ≤ C ]= Pr [ A > B ] . ◀ In this section, we prove the following main theorem. ▶ Theorem 6.
Let G ∼ G ( n, r ) be a random geometric graph with expected average degree δ = o ( √ n ) . The expected fraction of monochrome edges after the FSP is at least
12 + 9800 · − q π ⌊ δ/ ⌋ · (cid:18) − e − δ/ (cid:18) δ (cid:19)(cid:19) · (1 − o (1)) . . Bläsius, T. Friedrich, M. S. Krejca and L. Molitor 7 Note that the bound in Theorem 6 is bounded away from for all δ ≥
2. Moreover, the twofactors depending on δ go to and 1, respectively, for a growing δ .Given an edge { u, v } , we prove the above lower bound on the probability that { u, v } is monochrome in the following four steps. For a vertex set, we introduce the conceptof decisiveness that measures how much the majority is ahead of the minority in the FSP.With this, we give a lower bound on the probability that { u, v } is monochrome based on theprobability that the common neighborhood of u and v is more decisive than their exclusiveneighborhoods. We show that large neighborhoods are likely to be more decisive than smallneighborhoods. To achieve this, we give bounds on the likelihood that two similar randomwalks behave differently. This step reduces the question of whether the common neighborhoodis more decisive than the exclusive neighborhoods to whether the former is larger than thelatter. Turning to geometric random graphs, we show that the common neighborhoodis sufficiently likely to be larger than the exclusive neighborhoods if the geometric regioncorresponding to the former is sufficiently large. We do this by first showing that the actualdistribution of the neighborhood sizes is well approximated by independent random variablesthat follow binomial distributions. Afterwards, we give the desired bounds for these randomvariables. We show that the existence of the edge { u, v } in the geometric random graphmakes it sufficiently likely that the geometric region hosting the common neighborhood of u and v is sufficiently large. Let { u, v } be an edge of a given graph. To formally define the above mentioned decisiveness,let N + u ∩ v and N − u ∩ v be the number of vertices in the common neighborhood of u and v thatare occupied by agents of type t + and t − , respectively. Then D u ∩ v := | N + u ∩ v − N − u ∩ v | is the decisiveness of the common neighborhood of u and v . Analogously, we define D u \ v and D v \ u for the exclusive neighborhoods of u and v , respectively.The following theorem bounds the probability for { u, v } to be monochrome based onthe probability that the common neighborhood is more decisive than each of the exclusiveneighborhoods. ▶ Theorem 7.
In the FSP, let { u, v } ∈ E be an edge and let D be the event { D u ∩ v >D u \ v ∧ D u ∩ v > D v \ u } . Then { u, v } is monochrome with probability at least / D ] / . Proof. If D occurs, then the types of u and v after the FSP coincide with the predominanttype before the FSP in the shared neighborhood. Thus, { u, v } is monochrome.Otherwise, assuming D , w.l.o.g., let D u ∩ v ≤ D u \ v and assume the type of v has alreadybeen determined. If D u ∩ v = D u \ v , then u chooses a type uniformly at random, whichcoincides with the type of v with probability . Otherwise, D u ∩ v < D u \ v and thus u takes the type that is predominant in u ’s exclusive neighborhood, which is t + and t − withprobability , each. Moreover, this is independent from the type of v as v ’s neighborhood isdisjoint to u ’s exclusive neighborhood.Thus, for the event M that { u, v } is monochrome, we get Pr [ M | D ] = 1 and Pr (cid:2) M | D (cid:3) = . Hence, we get Pr [ M ] > Pr [ D ] + (1 − Pr [ D ]) = + Pr [ D ] / ◀ The goal of this section is to reduce the question of how decisive a neighborhood is to thequestion of how large it is. To be more precise, assume we have a set of vertices of size a and give each vertex the type t + and t − , each with probability . Let A i for i ∈ [ a ] be the The Flip Schelling Process on Random Graphs random variable that takes the value +1 and − i -th vertex in this set has type t + and t − , respectively. Then, for A = P i ∈ [ a ] A i , the decisiveness of the vertex set is | A | . Inthe following, we study the decisiveness | A | depending on the size a of the set. Note thatthis can be viewed as a random walk on the integer line: Starting at 0, in each step, it movesone unit either to the left or to the right with equal probabilities. We are interested in thedistance from 0 after a steps.Assume for the vertices u and v that we know that b vertices lie in the common neigh-borhood and a vertices lie in the exclusive neighborhood of u . Moreover, let A and B bethe positions of the above random walk after a and b steps, respectively. Then the event D u ∩ v > D u \ v is equivalent to | B | > | A | . Motivated by this, we study the probability of | B | > | A | , assuming b ≥ a . The core difficulty here comes from the fact that we require | B | to be strictly larger than | A | . Also note that a + b corresponds to the degree of u in thegraph. Thus, we have to study the random walks also for small numbers of a and b . We notethat all results in this section are independent from the specific application to the FSP, andthus might be of independent interest.Before we give a lower bound on the probability that | B | > | A | , we need the followingtechnical lemma. It states that doing more steps in the random walk only makes it morelikely to deviate further from the starting position. ▶ Lemma 8.
For i ∈ [ a ] and j ∈ [ b ] with ≤ a ≤ b , let A i and B j be independent randomvariables that are − and each with probability . Let A = P i ∈ [ a ] A i and B = P j ∈ [ b ] B j .Then Pr [ | A | < | B | ] ≥ Pr [ | A | > | B | ] . Proof.
Let ∆ k be the event that | B | − | A | = k . First note thatPr [ | A | < | B | ] = X k ∈ [ b ] Pr [∆ k ] and Pr [ | A | > | B | ] = X k ∈ [ a ] Pr [∆ − k ] . To prove the statement of the lemma, it thus suffices to prove the following claim. ▷ Claim 9.
For k ≥
0, Pr [∆ k ] ≥ Pr [∆ − k ].We prove this claim via induction on b − a . For the base case a = b , A and B are equallydistributed and thus Claim 9 clearly holds.For the induction step, let B + be the random variable that takes the values B + 1 and B − each. Note that B + represents the same type of random walk as A and B but with b + 1 steps. Moreover B + is coupled with B to make the same decisions inthe first b steps. Let ∆ + k be the event that | B + | − | A | = k . It remains to show that Claim 9holds for these ∆ + k . For this, first note that the claim trivially holds for k = 0. For k ≥ + k and the induction hypothesis to obtainPr (cid:2) ∆ + k (cid:3) = Pr [∆ k − ]2 + Pr [∆ k +1 ]2 ≥ Pr [∆ − k +1 ]2 + Pr [∆ − k − ]2 = Pr (cid:2) ∆ + − k (cid:3) . ◀ Using Lemma 8, we can now prove the following general bound for the probability that | A | < | B | , depending on certain probabilities for binomially distributed variables. ▶ Lemma 10.
For i ∈ [ a ] and j ∈ [ b ] with ≤ a ≤ b , let A i and B j be independent randomvariables that are − and each with probability . Let A = P i ∈ [ a ] A i and B = P j ∈ [ b ] B j .Moreover, let X ∼ Bin( a, ) , Y ∼ Bin( b, ) , and Z ∼ Bin( a + b, ) . Then Pr [ | A | < | B | ] ≥ − Pr (cid:20) Z = a + b (cid:21) + Pr (cid:2) X = a (cid:3) · Pr (cid:2) Y = b (cid:3) . . Bläsius, T. Friedrich, M. S. Krejca and L. Molitor 9 Proof.
Using that Pr [ | A | < | B | ] ≥ Pr [ | A | > | B | ] (see Lemma 8), we obtainPr [ | A | < | B | ] + Pr [ | A | > | B | ] + Pr [ | A | = | B | ] = 1 ⇒ | A | < | B | ] + Pr [ | A | = | B | ] ≥ ⇔ Pr [ | A | < | B | ] ≥ − Pr [ | A | = | B | ]2 . (4)Thus, it remains to give an upper bound for Pr [ | A | = | B | ].Using the inclusion–exclusion principle and the fact that B is symmetric around 0, i.e.,Pr [ B = x ] = Pr [ B = − x ] for any x , we obtainPr [ | A | = | B | ] = Pr [ A = B ∨ A = − B ]= Pr [ A = B ] + Pr [ A = − B ] − Pr [ A = B = 0]= 2Pr [ A = − B ] − Pr [ A = B = 0] . (5)We estimate Pr [ A = − B ] and Pr [ A = B = 0] using bounds for binomially distributed vari-ables. To this end, define new random variables X i = A i +12 for i ∈ [ a ] and let X = P i ∈ [ a ] X i .Note that the X i are independent and take values 0 and 1, each with probability . Thus, X ∼ Bin( a, ). Moreover, A = 2 X − a . Analogously, we define Y with Y ∼ Bin( b, ) and B = 2 Y − b . Note that X and Y are independent and thus Z = X + Y ∼ Bin( a + b, ).With this, we getPr [ A = − B ] = Pr [2 X − a = − Y + b ] = Pr (cid:20) Z = a + b (cid:21) , andPr [ A = B = 0] = Pr [ A = 0] · Pr [ B = 0] = Pr h X = a i · Pr (cid:20) Y = b (cid:21) . This, together with Equations (4) and (5) yield the claim. ◀ The bound in Lemma 10 becomes worse for smaller values of a and b . Considering this worstcase, we obtain the following specific bound. ▶ Theorem 11.
For i ∈ [ a ] and j ∈ [ b ] with ≤ a ≤ b , let A i and B j be independent randomvariables that are − and each with probability . Let A = P i ∈ [ a ] A i and B = P j ∈ [ b ] B j .If a = b = 0 or a = b = 1 , then Pr [ | A | < | B | ] = 0 . Otherwise Pr [ | A | < | B | ] ≥ . Proof.
Clearly, if a = b = 0, then A = B = 0 and thus Pr [ | A | < | B | ] = 0. Similarly, if a = b = 1, then | A | = | B | = 1 and thus Pr [ | A | < | B | ] = 0. For the remainder, assume thatneither a = b = 0 nor a = b = 1, and let X , Y , and Z be defined as in Lemma 10, i.e., X ∼ Bin( a, ), Y ∼ Bin( b, ), and Z ∼ Bin( a + b, ).If a + b is odd, then Pr (cid:2) Z = a + b (cid:3) = 0. Thus, by Lemma 10, we have Pr [ | A | < | B | ] ≥ .If a + b is even and a + b ≥
6, thenPr (cid:20) Z = a + b (cid:21) = (cid:18) a + b a + b (cid:19) (cid:18) (cid:19) a + b ≤ (cid:18) (cid:19) (cid:18) (cid:19) = 516 . Hence, by Lemma 10, we have Pr [ | A | < | B | ] ≥ − = .If a + b < a + b even), there are four cases: a = 0 , b = 2; a = 0 , b = 4; a = 1 , b = 3; a = 2 , b = 2. If a = 0 and b = 2, then A = 0 with probability 1 and | B | = 2 withprobability . Thus, Pr [ | A | < | B | ] = . If a = 0 and b = 4, then | A | < | B | unless B = 0.As Pr [ B = 0] = (cid:0) (cid:1) · ( ) = , we get Pr [ | A | < | B | ] = 1 − = . If a = 1 and b = 3, then | A | = 1 with probability 1 and | B | = 3 with probability (either B = B = B = 1or B = B = B = − | A | < | B | ] = . If a = b = 2, then | A | = 0 withprobability and | B | = 2 with probability . Thus Pr [ | A | < | B | ] = .We note that the bound of Pr [ | A | < | B | ] = is tight for a = b = 3. ◀ To be able to apply Theorem 11 to an edge { u, v } , we need to make sure that the size oftheir common neighborhood (corresponding to b in the corollary) is at least the size of theexclusive neighborhoods (corresponding to a in the corollary). In the following, we givebounds for the probability that this happens. Note that this is the first time we actuallytake the graph into account. Thus, all above considerations hold for arbitrary graphs.Recall that we consider random geometric graphs G ( n, r ) and u and v are each connectedto all vertices that lie within a disk of radius r around them. As u and v are adjacent, theirdisks intersect, which separates the ground space into four regions; cf. Fig 2a. Let R u ∩ v bethe intersection of the two disks. Let R u \ v be the set of points that lie in the disk of u butnot in the disk of v , and analogously, let R v \ u be the disk of v minus the disk of u . Finally,let R u ∪ v the set of points outside both disks. Then, each of the n − p = µ ( R u ∩ v ) and q = µ ( R u \ v ) = µ ( R v \ u ) be the probabilities for the common and exclusiveregions, respectively. The probability for R u ∪ v is then 1 − p − q .We are now interested in the sizes N u ∩ v , N u \ v , and N v \ u of the common and the exclusiveneighborhoods, respectively. As each of the n − R u ∩ v withprobability p , we have R u ∩ v ∼ Bin( n − , p ). For N u \ v and N v \ u , we already know that v is aneighbor of u and vice versa. Thus, ( N u \ v − ∼ Bin( n − , q ) and ( N v \ u − ∼ Bin( n − , q ).Moreover, the three random variables are not independent, as each vertex lies in only exactlyone of the four regions, i.e., N u ∩ v , ( N u \ v − N v \ u − n − , p ) with p = ( p, q, q, − p − q ).The following lemma shows that these dependencies are small if p and q are sufficientlysmall. This lets us assume that N u ∩ v , ( N u \ v − N v \ u −
1) are independent randomvariables following binomial distributions if the expected average degree is not too large. ▶ Lemma 12.
Let X = ( X , X , X , X ) ∼ Multi ( n, p ) with p = ( p, q, q, − p − q ) . Thenthere exist independent random variables Y ∼ Bin ( n, p ) , Y ∼ Bin ( n, q ) and Y ∼ Bin ( n, q ) such that Pr [( X , X , X ) = ( Y , Y , Y )] ≥ − n · max( p, q ) . Proof.
Let Y ∼ Bin ( n, p ), and Y , Y ∼ Bin ( n, q ) be independent random variables. Wedefine the event B to hold, if each of the n individual trials increments at most one ofthe random variables Y , Y , or Y . More formally, for i ∈ [3] and j ∈ [ n ], let Y i,j be theindividual Bernoulli trials of Y i , i.e., Y i = P j ∈ [ n ] Y i,j . For j ∈ [ n ], we define the event B j tobe Y ,j + Y ,j + Y ,j ≤
1, and the event B = T j ∈ [ n ] B j .Based on this, we now define the random variables X , X , X , and X as follows.If B holds, we set X i = Y i for i ∈ [3] and X = n − X − X − X . Otherwise, if B ,we draw X = ( X , X , X , X ) ∼ Multi ( n, p ) independently from Y , Y , and Y with p = ( p, q, q, − p − q ). Note that X clearly follows Multi ( n, p ) if B . Moreover, conditionedon B , each individual trial increments exactly one of the variables X , X , X , or X withprobabilities p , q , q , and 1 − p − q , respectively, i.e., X ∼ Multi ( n, p ).Thus, we end up with X ∼ Multi ( n, p ). Additionally, we have three independent randomvariables Y ∼ Bin ( n, p ), and Y , Y ∼ Bin ( n, q ) with ( X , X , X ) = ( Y , Y , Y ) if B holds. . Bläsius, T. Friedrich, M. S. Krejca and L. Molitor 11 Thus, to prove the lemma, it remains to show that Pr [ B ] ≥ − n max( p, q ) . For j ∈ [ n ],the probability that the j th trial goes wrong isPr (cid:2) B j (cid:3) = 1 − (cid:0) (1 − p )(1 − q ) (cid:1) − (cid:0) p (1 − q ) (cid:1) − q (1 − p )(1 − q ))= 2 pq − pq + q ≤ pq + q ≤ · max( p, q ) . Using the union bound it follows that Pr (cid:2) B (cid:3) ≤ P j ∈ [ n ] Pr (cid:2) B j (cid:3) ≤ n · max( p, q ) . ◀ As mentioned before, we are interested in the event N u ∩ v ≥ N u \ v (and likewise N u ∩ v ≥ N v \ u ),in order to apply Theorem 11. Moreover, due to Lemma 12, we know that N u ∩ v and ( N u \ v − n − , p ) and Bin( n − , q ),respectively. The following lemma helps to bound the probability for N u ∩ v ≥ N u \ v . Notethat it gives a bound for the probability of achieving strict inequality (instead of just ≥ ),which accounts for the fact that ( N u \ v −
1) and not N u \ v itself follows a binomial distribution. ▶ Lemma 13.
Let n ∈ N with n ≥ , and let p, q ∈ (0 , ] with p ≥ q . Further, let X ∼ Bin( n, p ) and Y ∼ Bin( n, q ) be independent, let d = ⌊ p ( n + 1) ⌋ , and let d = o ( √ n ) ,then Pr [
X > Y ] ≥ (cid:0) − / √ π d (cid:1) (1 − o (1)) . Proof.
By Lemma 1, we get Pr [ X ≥ Y ] ≥ , and we boundPr [ X > Y ] = Pr [ X ≥ Y ] − Pr [ X = Y ] ≥ − Pr [ X = Y ] , leaving us to bound Pr [ X = Y ] from above. By independence of X and Y , we getPr [ X = Y ] = X i ∈ [ n ] Pr [ X = i ] · Pr [ Y = i ] . (6)Note that, by Lemma 2, for all i ∈ [0 ..n ], it holds that Pr [ X = i ] ≤ Pr [ X = d ]. Assume thatwe have a bound B such that Pr [ X = d ] ≤ B . Substituting this into Equation (6) yieldsPr [ X = Y ] ≤ B X i ∈ [ n ] Pr [ Y = i ] = B, resulting in Pr [ X > Y ] ≥ − B . Thus, we now derive such a bound for B and apply theinequality that for all x ∈ R , it holds that 1 + x ≤ e x , as well as Equation (1). We get (cid:18) nd (cid:19) p d (1 − p ) n − d ≤ n d d ! (cid:18) dn (cid:19) d (cid:18) − dn (cid:19) n (cid:18) − dn (cid:19) − d ≤ d d d ! e − d (cid:18) − dn (cid:19) − d ≤ d d √ π d d +1 / e − d e − d (cid:18) − dn (cid:19) − d = 1 √ π d − d/n ) d . (7)By Bernoulli’s inequality, we bound (1 − d/n ) d ≥ − d /n = 1 − o (1) by the assumption d = o ( √ n ). Substituting this back into Equation (7) concludes the proof. ◀ Finally, in order to apply Theorem 11, we have to make sure not to end up in the specialcase where a = b ≤
1, i.e., we have to make sure that the common neighborhood includes atleast two vertices. The probability for this to happen is given by the following lemma. R u ∩ v R u \ v R v \ u u v (a) The geometric re-gions corresponding tothe common and exclu-sive neighborhoods, re-spectively, with yellowillustrating R u ∩ v andblue illustrating R u \ v and R v \ u . xαji (b) Let α be the centralangle determined bythe intersection points i and j , and let x bethe corresponding circu-lar sector (illustrated inyellow). y (cid:96) (c) Let y be a trianglein the intersection (il-lustrated in green) de-termined by the radicalaxis ℓ and the centralangle α , cf. Fig 2b. h (cid:96)r (d) The height h di-vides the area µ ( y )(illustrated in green)of the triangle y , cf.Fig 2c, into two subar-eas of equal size, sinceadjacent and oppositelegs have the samelength r . Figure 2
The neighborhood of two adjacent vertices u and v in a random geometric graph. ▶ Lemma 14.
Let X ∼ Bin( n, p ) and let c = pn ∈ o ( n ) . Then it holds that Pr [
X > ≥ (1 − e − c (1 + c )) (1 − o (1)) . Proof. As X > X ̸ = 0 and X ̸ = 1, we getPr [ X >
1] = 1 − Pr [ X = 0] − Pr [ X = 1] = 1 − (1 − p ) n − n · p · (1 − p ) n − . Using that for all x ∈ R it holds that 1 − x ≤ e − x , we getPr [ X > ≥ − e − pn − n · p · e − p ( n − = 1 − e − c − c · e c/n · e − c = 1 − e − c (cid:16) c · e c/n (cid:17) . As e c/n goes to 1 for n → ∞ , we get the claimed bound. ◀ In Section 3.3, we derived a lower bound on the probability that N u ∩ v ≥ N u \ v provided thatthe probability for a vertex to end up in the shared region R u ∩ v is sufficiently large comparedto R u \ v . In the following, we estimate the measures of these regions depending on the distancebetween u and v . Then, we give a lower bound on the probability that µ ( R u ∩ v ) ≥ µ ( R u \ v ). ▶ Lemma 15.
Let G ∼ G ( n, r ) be a random geometric graph with expected average degree δ ,let { u, v } ∈ E be an edge, and let τ := dist( u,v ) r . Then, µ ( R u ∩ v ) = δ ( n − π (cid:16) (cid:16) τ (cid:17) − sin (cid:16) (cid:16) τ (cid:17)(cid:17)(cid:17) and (8) µ ( R u \ v ) = µ ( R v \ u ) = δn − − µ ( R u ∩ v ) . (9) Proof.
We start with proving Equation (8). Let i and j be the two intersection points of thedisks of u and v , let α be the central angle enclosed by i and j , and let x be the corresponding . Bläsius, T. Friedrich, M. S. Krejca and L. Molitor 13 circular sector, cf. Fig 2b. Moreover, let the triangle y be a subarea of x determined by α and the radical axis ℓ , cf. Fig 2c. Let h denote the height of the triangle y , cf. Fig 2d.For our calculations, we restrict the length of ℓ by the intersection points i and j . Sincewe consider the intersection between disks and thus ℓ divides the area µ ( R u ∩ v ) into twosubareas of equal sizes, it holds that µ ( R u ∩ v ) = 2 ( µ ( x ) − µ ( y )). Considering the two areas µ ( x ) and µ ( y ), it holds that µ ( x ) = α r and µ ( y ) = h · ℓ (cid:16) α (cid:17) r · sin (cid:16) α (cid:17) r = sin( α )2 r . (10)For the central angle α we know cos ( α/
2) = h/r = τ / α = 2 arccos (cid:0) τ (cid:1) .Together with Equation (10), we obtain µ ( R u ∩ v ) = 2 ( µ ( x ) − µ ( y )) = 2 (cid:0) τ (cid:1) r − sin (cid:0) (cid:0) τ (cid:1)(cid:1) r ! . (11)The area of a general circle is equal to π r , the area of one disk in the random geometricgraph equals δn − , i.e., r = δ ( n − π . Together with Equation (11), we obtain Equation (8).Equation (9): We get the claimed equality by noting that µ ( R u ∩ v ) + µ ( R u \ v ) = π r . ◀▶ Lemma 16.
Let G ∼ G ( n, r ) be a random geometric graph, and let { u, v } ∈ E be an edge.Then Pr (cid:2) µ ( R u ∩ v ) ≥ µ ( R u \ v ) (cid:3) ≥ (cid:0) (cid:1) . Proof.
Let τ = dist( u,v ) r . By Lemma 15 with µ ( R u ∩ v ) ≥ µ ( R v \ u ), we get (cid:16) (cid:16) τ (cid:17) − sin (cid:16) (cid:16) τ (cid:17)(cid:17)(cid:17) ≥ π , which is true for τ ≥ . The area of a disk of radius r is (cid:0) π ( r ) (cid:1) / (cid:0) π r (cid:1) = (cid:0) (cid:1) timesthe area of a disk of radius r . Hence, the fraction of edges with distance at most r is atleast (cid:0) (cid:1) , concluding the proof. ◀ By Theorem 7, the probability that a random edge { u, v } is monochrome is at least +Pr [ D ] /
2, where D is the event that the common neighborhood of u and v is more decisivethan each exclusive neighborhood. It remains to bound Pr [ D ]. Existence of an edge yields a large shared region.
Let R be the event that µ ( R u ∩ v ) ≥ µ ( R u \ v ). Note that this also implies µ ( R u ∩ v ) ≥ µ ( R v \ u ) as µ ( R u \ v ) = µ ( R v \ u ). Due to thelaw of total probability, we havePr [ D ] ≥ Pr [ R ] · Pr [ D | R ] . Due to Lemma 16, we have Pr [ R ] ≥ (cid:0) (cid:1) . By conditioning on R in the following, we canassume that µ ( R u ∩ v ) ≥ δ n ≥ µ ( R u \ v ) = µ ( R v \ u ), where δ is the expected average degree. Neighborhood sizes are roughly binomially distributed.
The next step is to go from thesize of the regions to the number of vertices in these regions. Each of the remaining n ′ = n − R u ∩ v , R u \ v , R v \ u , or R u ∪ v .Denote the resulting numbers of vertices with X , X , X , and X , respectively. Then ( X , X , X , X ) follows a multinomial distribution with parameter p = ( p, q, q, − p − q )for p = µ ( R u ∩ v ) and q = µ ( R u \ v ) = µ ( R v \ u ). Note that N u ∩ v = X , N u \ v = X + 1, and N v \ u = X + 1 holds for the sizes of the common and exclusive neighborhoods, where the +1comes from the fact that v is always a neighbor of u and vice versa.We apply Lemma 12 to obtain independent binomially distributed random variables Y , Y , and Y that are likely to coincide with X = N u ∩ v , X = N u \ v −
1, and X = N v \ u − B denote the event that ( N u ∩ v , N u \ v − , N v \ u −
1) = ( Y , Y , Y ). Again,using the law of total probabilities and due to the fact that R and B are independent, we getPr [ D | R ] ≥ Pr [ B | R ] · Pr [ D | R ∩ B ] = Pr [ B ] · Pr [ D | R ∩ B ] . Note that p, q ≤ δn for the expected average degree δ . Thus, Lemma 12 implies thatPr [ B ] ≥ (cid:16) − δ /n (cid:17) . Conditioning on B makes it correct to assume that N u ∩ v ∼ Bin( n ′ , p ),( N u \ v − ∼ Bin( n ′ , q ), ( N v \ u − ∼ Bin( n ′ , q ) are independently distributed. Additionallyconditioning on R gives us p ≥ δ n ≥ q . A large shared region yields a large shared neighborhood.
In the next step, we consideran event that makes sure that the number N u ∩ v of vertices in the shared neighborhood issufficiently large. Let N , N , and N be the events that N u ∩ v ≥ N u \ v , N u ∩ v ≥ N v \ u , and N u ∩ v >
1, respectively. Let N be the intersection of N , N , and N . We obtainPr [ D | R ∩ B ] ≥ Pr [ N | R ∩ B ] · Pr [ D | R ∩ B ∩ N ] ≥ Pr [ N | R ∩ B ] · Pr [ N | R ∩ B ] · Pr [ N | R ∩ B ] · Pr [ D | R ∩ B ∩ N ] , where the last step follows from Lemma 5 as the inequalities in N , N , and N all go inthe same direction. Note that N u ∩ v ≥ N u \ v is equivalent to N u ∩ v > N u \ v −
1. Due to thecondition on B , N u ∩ v and N u \ v − n ′ , p )and Bin( n ′ , q ), respectively, with p ≥ q due to the condition on R . Thus, we can applyLemma 13, to obtainPr [ N | R ∩ B ] = Pr [ N | R ∩ B ] ≥ − q π ⌊ δ/ ⌋ (1 − o (1)) , and Lemma 14 gives the boundPr [ N | R ∩ B ] ≥ − e − δ/ (cid:18) δ · (1 + o (1)) (cid:19) . Note that both of these probabilities are bounded away from 0 for δ ≥
2. Conditioning on N lets us assume that the shared neighborhood of u and v contains at least two vertices andthat it is at least as big as each of the exclusive neighborhoods. A large shared neighborhood yields high decisiveness.
The last step is to actually boundthe remaining probability Pr [ D | R ∩ B ∩ N ]. Note that, once we know the number of verticesin the shared and exclusive neighborhoods, the decisiveness no longer depends on R or B , i.e.,we can bound Pr [ D | N ] instead. For this, let D and D be the events that D u ∩ v > D u \ v and D u ∩ v > D v \ u , respectively. Note that D is their intersection. Moreover, due to Lemma 5,we have Pr [ D | N ] ≥ Pr [ D | N ] · Pr [ D | N ]. To bound Pr [ D | N ] = Pr [ D | N ], we useTheorem 11. Note that the b and a in Theorem 11 correspond to N u ∩ v and N u \ v + 1 (the +1coming from the fact that N u \ v does not count the vertex v ). Moreover conditioning on N implies that a ≤ b and b >
1. Thus, Theorem 11 implies Pr [ D | N ] ≥ . . Bläsius, T. Friedrich, M. S. Krejca and L. Molitor 15 Conclusion.
The above arguments gives us that the fraction of monochrome edges is12 + Pr [ D ]2 ≥
12 + 12 · Pr [ R ] | {z } ≥ ( ) · Pr [ B ] | {z } − o(1) · (cid:0) Pr [ N | R ∩ B ] | {z } ≥ − √ π ⌊ δ/ ⌋ (cid:1) · Pr [ N | R ∩ B ] | {z } ≥ − e − δ/ (cid:0) δ (cid:1) · (cid:0) Pr [ D | N ] | {z } ≥ (cid:1) , where we omitted the o (1) terms for Pr [ N | R ∩ B ] and Pr [ N | R ∩ B ], as they are alreadycovered by the 1 + o (1) coming from Pr [ B ]. This yields the bound stated in Theorem 6:12 + 9800 · − q π ⌊ δ/ ⌋ · (cid:18) − e − δ/ (cid:18) δ (cid:19)(cid:19) · (1 − o (1)) . In the following, we are interested in the probability that an edge { u, v } is monochromeafter the FSP on Erdős–Rényi graphs. In contrast to geometric random graphs, we provean upper bound. To this end, we show that it is likely that the common neighborhood isempty and therefore u and v choose their types to be the predominant type in their exclusiveneighborhood, which is t + and t − with probability , each. ▶ Theorem 17.
Let G ∼ G ( n, p ) be an Erdős–Rényi graph with expected average degree δ = o ( √ n ) . The expected fraction of monochrome edges after the FSP is at most + o (1) . Proof.
Given an edge { u, v } , let M be the event that { u, v } is monochrome. We first split M into disjoint sets with respect to the size of the common neighborhood and apply the law oftotal probability and getPr [ M ] = Pr [ M | N u ∩ v = 0 ] · Pr [ N u ∩ v = 0] + Pr [ M | N u ∩ v > · Pr [ N u ∩ v > ≤ Pr [ M | N u ∩ v = 0 ] · · Pr [ N u ∩ v > . We bound each of the summands separately. For estimating Pr [ M | N u ∩ v = 0 ], we notethat the types of u and v are determined by the predominant type in disjoint vertex sets. Bydefinition of the FSP this implies that the probability of an monochrome edge is equal to .We are left with bounding Pr [ N u ∩ v > N u ∩ v ∼ Bin (cid:0) n, p (cid:1) . Thus, byBernoulli’s inequality we get Pr [ N u ∩ v >
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