Analytical Results

A REPULSIVE BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS IN POLARIZED COMMUNITIES

3.4 Analytical Results

−2.5 0.0 2.5 5.0

Time

P osition

Figure 3.3: This image shows a simulation where final opinion width was wider than initial opinion width. The edges are created as described in Equation 3.6 where 𝑝₁=40%, 𝑝₂=80% and a confidence bound of 1.6.

convergence𝑇,

|𝑥₀(𝑇) −𝑥₁(𝑇) | ≤max{𝑐,|𝑥₀(0) −𝑥₁(0) |}.

Proof. Suppose that there are no edges, or 𝐸 = ∅. Then the model converges at time𝑇 =1, and

|𝑥₀(𝑇) −𝑥₁(𝑇) | =|𝑥₀(0) −𝑥₁(0) |.

Now suppose that 𝐸 = {(𝑥₀, 𝑥₁)}, and |𝑥₀(0) −𝑥₁(0) | ≥ 𝑐. Then the update rule will result in no changes, the model converges at𝑇 =1

|𝑥₀(𝑇) −𝑥₁(𝑇) | =|𝑥₀(0) −𝑥₁(0) |.

Now suppose that|𝑥₀(0) −𝑥₁(0) | < 𝑐. If𝐴₀₁=−1, the two nodes repel each other.

Then

𝑥₁(1) =𝑥₁(0) + (𝑥₁(0) −𝑥₁(0)) +𝑐− (𝑥₁(0) −𝑥₀(1)) 2

𝑥₀(1) =𝑥₀(0) + (𝑥₀(0) −𝑥₀(0)) − (𝑐− (𝑥₁(0) −𝑥₀(0)) 2

𝑥₁(1) −𝑥₀(1) =𝑥₁(0) −𝑥₀(0) + 2(𝑐− (𝑥₁(0) −𝑥₀(0))) 2

=𝑐

and𝑥₁(1) −𝑥₀(1) >=𝑐, so that after this time, these two nodes will no longer affect each other, and cannot push each other further, so the model has converged, and max𝑖, 𝑗|𝑥₀(𝑇) −𝑥₁(𝑇) | ≤𝑐.

If 𝐴₀₁ =1, the two nodes attract each other, and the model is equivalent to standard Hegselmann–Krause, so that we have convergence to a single point and

|𝑥₀(𝑇) −𝑥₁(𝑇) | =0.

This covers all possible cases, and the proposition is proven. □ The main point to note from this two-node proof is that the repulsive forces between any two nodes will contribute to pushing them apart to a distance of precisely 𝑐. Also note that any node cannot move more than𝑐 in any direction over the course of one timestep, because|𝑀_{𝑖 𝑗}(𝑡) | ≤ 𝑐. In order to prove Theorem 1 we need several lemmas and corollaries. The proofs of these can be found in Appendix C.1. We state them here so that we can use them for the final proof.

Lemma 1. Suppose𝑖 ∈𝑉 a node. Define the following sets:

𝑉⁺

𝑖 (𝑡)=

𝑗 ∈𝑉 : 𝐴_{𝑖 𝑗} =1and|𝑥_𝑗(𝑡) −𝑥_𝑖(𝑡) | < 𝑐 𝑈_𝑖(𝑡)=

𝑗 ∈𝑉 : 𝐴_{𝑖 𝑗} =−1and

(0< 𝑥_𝑗(𝑡) −𝑥_𝑖(𝑡) < 𝑐)or 𝑥_𝑗(𝑡) =𝑥_𝑖(𝑡) and 𝑗 > 𝑖 𝐿_𝑖(𝑡)=

𝑗 ∈𝑉 : 𝐴_{𝑖 𝑗} =−1and

(0< 𝑥_𝑖(𝑡) −𝑥_𝑗(𝑡) < 𝑐)or 𝑥_𝑗(𝑡) =𝑥_𝑖(𝑡) and𝑖 > 𝑗 .

Then

𝑥_𝑖(𝑡+1) = Í

𝑗∈𝑉⁺

𝑖 (𝑡)𝑥_𝑗(𝑡) +Í

𝑗∈𝑈𝑖(𝑡)(𝑥_𝑗(𝑡) −𝑐) +Í

𝑗∈𝐿𝑖(𝑥_𝑗(𝑡) +𝑐)

|𝑉⁺

𝑖 (𝑡) | + |𝑈_𝑖(𝑡) + |𝐿_𝑖(𝑡) | . (3.5)

Intuitively, this lemma tells us that the update rule moves𝑥_𝑖(𝑡)to𝑥_𝑖(𝑡+1)by taking an average of several opinions. The set𝑉⁺

𝑖 (𝑡)contains nodes𝑖is attracted to at time 𝑡. The set𝑈_𝑖(𝑡) contains nodes which repulse𝑖 at time 𝑡, and which will push𝑖’s opinion lower. The set 𝐿_𝑖(𝑡) contains nodes which repulse 𝑖 at time𝑡, and which will push 𝑖’s opinion higher. Equation 3.5 tells us that we can take the average of 𝑥_𝑗(𝑡) for 𝑗 ∈𝑉⁺

𝑖 (𝑡),𝑥_𝑗(𝑡) −𝑐for 𝑗 ∈𝑈_𝑖(𝑡), and𝑥_𝑗(𝑡) +𝑐for 𝑗 ∈𝐿_𝑖(𝑡) to determine 𝑥_𝑖(𝑡+1).

Lemma 2. Let𝑖 ∈𝑉 at time𝑡, and let𝑊(𝑡) ⊂𝑉 be a set of nodes such that𝑊(𝑡)is completely contained in𝑉⁺

𝑖 (𝑡) ∪𝑈_𝑖(𝑡) ∪𝐿_𝑖(𝑡). Define 𝑊(𝑡)=

Í

𝑗∈𝑊(𝑡)𝑥_𝑗(𝑡)

|𝑊(𝑡) |

to be the average of𝑥_𝑗(𝑡) for all 𝑗 ∈𝑊(𝑡). Then we can rewrite Equation3.5as

𝑥_𝑖(𝑡+1) = Í

𝑗∈(^𝑉𝑖⁺(𝑡)∪𝑈𝑖(𝑡)∪𝐿𝑖(𝑡))^\𝑊(𝑡)𝑥_𝑗(𝑡) +Í

𝑗∈𝑊(𝑡)𝑊(𝑡) + (|𝐿_𝑖(𝑡) | − |𝑈_𝑖(𝑡) |)𝑐

|𝑉⁺

𝑖 (𝑡) | + |𝑈_𝑖(𝑡) + |𝐿_𝑖(𝑡) | .

This lemma allows us to replace a group of opinion values of individual nodes with the average of opinion values across the group, in certain situations.

Lemma 3. Let𝐺 = (𝑉 , 𝐸)be a network with𝑛nodes and𝑚edges with confidence bound𝑐. Suppose that every edge in𝐺 is repulsive. At time𝑡, suppose𝑥_𝑖(𝑡) > 𝑥_𝑗(𝑡) for all other nodes 𝑗, so that𝑖 is the node with the highest opinion value at time𝑡. Then𝑥_𝑖(𝑡+1) > 𝑥_𝑗(𝑡+1)for all 𝑗.

Corollary 1. Let𝐺 =(𝑉 , 𝐸)be a network with𝑛nodes and𝑚edges with confidence bound𝑐. Suppose that every edge in𝐺 is repulsive. At time𝑡, let 𝑀 = {𝑖 : 𝑥_𝑖(𝑡) ≥ 𝑥_𝑗(𝑡)∀𝑗 ∈𝑉}. Then𝑥_max

𝑀𝑖(𝑡+1) > 𝑥_𝑗(𝑡+1)∀𝑗 ∈𝑉.

Corollary 2. Let𝐺 = (𝑉 , 𝐸)be the complete network with𝑛nodes with confidence bound𝑐. Suppose that every edge in𝐺 is repulsive. At time𝑡, let 𝑀 = {𝑖 : 𝑥_𝑖(𝑡) ≤ 𝑥_𝑗(𝑡)∀𝑗 ∈𝑉}. Then𝑥_min

𝑀𝑖(𝑡+1) < 𝑥_𝑗(𝑡+1)∀𝑗 ∈𝑉.

Lemma 4. Let𝐺 = (𝑉 , 𝐸)be a network with𝑛nodes and𝑚edges with confidence bound𝑐. Suppose that every edge in𝐺 is repulsive. At time𝑡, suppose𝑥_𝑖(𝑡) > 𝑥_𝑗(𝑡) for all other nodes 𝑗 ∈ 𝑉, so that𝑖 is the node with the highest-valued opinion at time𝑡. Suppose that there is some node 𝑗 such that𝑥_𝑖(𝑡) −𝑥_𝑗(𝑡) < 𝑐, and that 𝑗 has the highest-valued opinion of all such nodes. Then

2𝑐 2+ |𝐿_{𝑖 𝑗}(𝑡) | + |𝐿^′

𝑗 𝑖(𝑡) | ≤ 𝑥_𝑖(𝑡+1) −𝑥_𝑗(𝑡+1) ≤

(|𝐿^′

𝑗 𝑖(𝑡) | +2)𝑐 2+ |𝐿_{𝑖 𝑗}(𝑡) | + |𝐿^′

𝑗 𝑖(𝑡) |.

Corollary 3. Let𝐺 =(𝑉 , 𝐸)be a network with𝑛nodes and𝑚edges with confidence bound𝑐. Suppose that every edge in𝐺 is repulsive. At time𝑡, suppose𝑥_𝑖(𝑡) < 𝑥_𝑗(𝑡) for all other nodes 𝑗 ∈𝑉, so that𝑖is the node with the lowest-valued opinion at time 𝑡. Suppose that there is some node 𝑗 such that𝑥_𝑗(𝑡) −𝑥_𝑖(𝑡) < 𝑐, and that 𝑗 has the lowest-valued opinion of all such nodes. Then

2𝑐 2+ |𝑈_{𝑖 𝑗}(𝑡) | + |𝑈^′

𝑖 𝑗(𝑡) | ≤ 𝑥_𝑖(𝑡+1) −𝑥_𝑗(𝑡+1) ≤

(|𝑈^′

𝑖 𝑗(𝑡) | +2)𝑐 2+ |𝑈_{𝑖 𝑗}(𝑡) | + |𝑈^′

𝑖 𝑗(𝑡) |.

Lemma 4 and Corollary 3 give us precise conditions under which the nodes with the most extreme opinions will no longer be within confidence bound of any other nodes.

Specifically, in order for the node with the highest-value opinion to lose connection with all other nodes, it must be true that the only node it is still influenced by is the node with the second-highest-value opinion, and that neither of the two nodes is influenced by any other nodes. Otherwise, they will remain within confidence of each other, even as the node with highest-value opinion remains the most extreme node and continues to have its opinion pushed upward.

We conclude with one more lemma about the bound on the width of the gap between consecutive nodes.

Lemma 5. Let𝐺 = (𝑉 , 𝐸) be the complete network with 𝑛nodes and confidence bound𝑐. Suppose that every edge in𝐺 is repulsive. At time𝑡, suppose that𝑖and 𝑗 are nodes such that(𝑖, 𝑗) ∈ 𝐸,𝑥_𝑖(𝑡) > 𝑥_𝑗(𝑡), and𝑥_𝑖(𝑡) −𝑥_𝑗(𝑡) < 𝑐, and there exist no nodes𝑘 connected to𝑖or 𝑗 such that𝑥_𝑖(𝑡) > 𝑥_𝑘(𝑡) > 𝑥_𝑗(𝑡). Then

|𝑥_𝑖(𝑡+1) −𝑥_𝑗(𝑡+1) | ≤𝑐 .

At this point we have the tools to return us to Theorem 1. As a reminder, the statement of the theorem was:

Theorem 1. Let𝐺 = (𝑉 , 𝐸) be a network with 𝑛 nodes and confidence bound𝑐. Suppose that 𝐺 is the complete graph, and that every edge (𝑖, 𝑗) ∈ 𝐸 is repulsive (that is 𝐴_{𝑖 𝑗} = −1). Suppose also that we have initial opinions 𝑥_𝑖(0) such that

|𝑥_𝑖(0)−𝑥_𝑗(0) | < 𝑐. Then the model converges, andmax𝑖, 𝑗

𝑥_𝑖(𝑇) −𝑥_𝑗(𝑇)

=(𝑛−1)𝑐. Proof. The intuition for this theorem is as follows: for any repulsive edge (𝑖, 𝑗), nodes𝑖and 𝑗 will repel each other until

|𝑥_𝑖(𝑡) −𝑥_𝑗(𝑡) | >=𝑐

at some future time𝑡. If every edge is repulsive, we must have a distance at least 𝑐 between every pair of nodes connected by an edge in order for the model to converge. Intuitively, the nodes will always continue to push each other outward until they reach a distance of𝑐, and no further, so that the final convergent state of the model will occur when there are gaps of at least𝑐between all of the𝑚edges in the original graph. However, from Lemma 4, the gaps will have precisely width𝑐, so that the bound holds.

From Corollary 1 and Corollary 2, at time 1, there must be a highest and lowest-value opinion node. By Lemma 3, for𝑡 > 1, these nodes will always be the highest and lowest-value opinion nodes. Call these nodes𝑖_{𝑚 𝑎𝑥}, 𝑖_𝑚𝑖𝑛.

Because𝐺 is the complete graph, and all edges are repulsive, we can observe that 𝑖_{𝑚 𝑎𝑥} and 𝑖_𝑚𝑖𝑛 will have their opinions pushed outward, since initially every node is within confidence of every node. Additionally, from Lemma 2, we can observe that𝑖_{𝑚 𝑎𝑥} will be pushed in the direction of {𝑗 ≠𝑖}(0) +𝑐, so that the nodes with opinions much lower valued than the average will start to drop out of confidence of 𝑖_{𝑚 𝑎𝑥}. Further, from Lemma 4, 𝑖_{𝑚 𝑎𝑥} will remain within confidence of at least one

node as long as it is within confidence of at least 2 nodes in the previous timestep.

Combining these lemmas, we can see that eventually at time𝑡^′,𝑖_{𝑚 𝑎𝑥} will be within confidence of exactly one other node.

Let𝑖^′

𝑚 𝑎𝑥be the singular node for which𝑥_𝑖

𝑚𝑎 𝑥(𝑡^′) −𝑥_𝑖′

𝑚𝑎 𝑥(𝑡^′) < 𝑐. Then we can follow the same proof procedure as in Lemma 3 to prove that 𝑥_𝑖′

𝑚𝑎 𝑥(𝑡^′+1) > 𝑥_𝑗(𝑡^′+1) for all 𝑗 ∈ 𝑉 other than 𝑗 = 𝑖_{𝑚 𝑎𝑥}, and that none of the remaining nodes can be pushed into confidence of𝑖_𝑚𝑎𝑥. We do not include the procedure here because of its similarity to Lemma 3, but the key observation that drives the proof is that there is only a single node𝑖_{𝑚 𝑎𝑥} exerting downward pressure on𝑖^′

𝑚 𝑎𝑥 (if a very high number of nodes were exerting downward pressure on𝑖^′

𝑚 𝑎𝑥, it would be possible for𝑖^′

𝑚 𝑎𝑥 to

lose its position as the node with second-highest-value opinion). This allows us to rewrite the𝑥_𝑖′

𝑚𝑎 𝑥 as an average of values which preserve the order of𝑖_{𝑚 𝑎𝑥}, 𝑖^′

𝑚 𝑎𝑥,and

the remaining nodes. Similarly, we can show that there is some time after which the node with the second-lowest-value opinion will always remain the node with the second-lowest-value opinion.

We continue in this manner, proceeding from the nodes with the highest and lowest- value opinions inwards until we show that after some time, the nodes’ opinions must remain in a fixed order.

From this point on, we observe that from Lemma 5, the gap between any two consecutive nodes is bounded by𝑐. Because of our initial conditions on𝑥_𝑖(𝑡), it is impossible for any gap between consecutive nodes to be larger at any point. If any two nodes have a gap smaller than𝑐, we will not have converged, as the repulsion between the two nodes will push them apart in the next time step. All nodes will push each other apart until the gap between any two consecutive nodes is precisely 𝑐, at which point the model has converged. Because there are𝑛nodes, this tells us

max{|𝑥_𝑖(𝑇) −𝑥_𝑗(𝑇) |}= (𝑛−1)𝑐 .

□ The proof for Theorem 1 relies on all edges being repulsive, thereby preserving the ordering of the nodes. This property does not necessarily hold when there are both attractive and repulsive edges. However, we suspect based on numerics that the following theorem is also true:

Theorem 2. Suppose𝐺 = (𝑉 , 𝐸)is a network with𝑛nodes,𝑚edges, and confidence bound𝑐. Let𝑚_𝑟 be the number of repulsive edges in the network. Then the model

converges and

max𝑖, 𝑗∈𝑉{𝑥_𝑖(𝑇) −𝑥_𝑗(𝑇)} ≤ max{max

𝑖, 𝑗∈𝑉{𝑥_𝑖(0) −𝑥_𝑗(0)}, 𝑚 𝑐}.

Intuition. The worst case for this model assumes that all repulsive nodes end up at least𝑐 apart from each other, so if all nodes start out within confidence of each other, the worst case is one in which all nodes with repulsive edges are chained together in consecutive order along a line of𝑚 edges, in which case the width of their opinions cannot exceed 𝑚 𝑐, since the bounds in Lemma 5 should apply and prevent any individual gap from growing wider. The only way a gap could grow wider is if there are attractive nodes pulling the repulsed nodes further apart, in which case those attractive nodes either have repulsive forces between them, and have already been considered in the line, or must have started farther apart to begin with, in which case we look at max𝑖, 𝑗∈𝑉{𝑥_𝑖(0) −𝑥_𝑗(0)}.

Because we cannot rely on nodes remaining in fixed order in this case, we cannot use the same technique as in Theorem 1 to prove convergence and a bound. However, in practice, we observe that the range of final opinions increases with the number of repulsive edges, and that in practice the bound of𝑚 𝑐is not very tight (this is to be expected, as, for example, in the case of the complete graph in Theorem 1, the bound is considerably smaller). To see numerics showing that the range of final opinions scales with number of repulsive edges and𝑐, see Figure 3.5 and associated

discussion. □