Numerical Results on Synthetic Networks - A REPULSIVE BOUNDED-CONFIDENCE MODEL OF OPINION DYNAM

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

3.5 Numerical Results on Synthetic Networks

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. □

nodes can be written as:

𝑃( (𝑖, 𝑗) ∈𝐸) =













0 (1− 𝑝₁) (1− 𝑝₂) +𝑝₁𝑝₂ 1 𝑝₁(1− 𝑝₂)

−1 (1− 𝑝₁)𝑝₂.

(3.6)

(a) G1: positive node network with 𝑝₁ =0.6

(b) G2: negative node network with𝑝₂ =0.2

Figure 3.4: An example of the generation of the ER network with attractive and repulsive edges

To create the simulation results, 100 trials were run with all combinations of the following parameters:

𝑝₁∈ (0.2,0.4,0.6,0.8,1) 𝑝₂∈ (0.0,0.2,0.4,0.6,0.8)

𝑐 ∈ (0.05,0.4,0.8,1.2,1.6).

For each trial, a random set of initial opinions is generated and the model is applied for 10000 iterations. In Figure 3.5, one trial is shown for each set of parameters when𝑝₁is set to 0.4. This trial was chosen randomly and all other trials qualitatively look the same.

The final range of opinions gets wider with both 𝑝₂ and𝑐 once repulsive edges are included. These results are in line with expectations. As 𝑝₂increases, so does the number of negative connections, resulting in more repulsive forces between nodes, pushing opinions apart. As𝑐increases, nodes have more neighbors. For𝑝₂<< 𝑝₁, the attractive forces overpower the repulsive ones, so that higher 𝑐 leads to more consensus, as in standard HK models. For𝑝₂ >> 𝑝₁, the opposite is true—repulsive forces overpower attractive ones, and nodes push each other further apart for higher 𝑐, resulting in a wider spread of opinions.

In particular, we observe that the proportion of ^𝑝_𝑝²

1 seems to be the driving factor in the range of final opinions. To draw clearer conclusions, we look atopinion spread, which we define as the following quantity:

0 0.2 0.4 0.6 0.8

0.050.40.81.21.6

−2.5 0.0 2.5 5.0

Time

Position

Figure 3.5: For all plots, 𝑝₁ = .4. The horizontal axis represents 𝑝₂, while the vertical axis represents𝑐. Note that as𝑝₂increases, the range of final opinions gets wider. For low values of 𝑝₂, as 𝑐 increases, the range of final opinions becomes narrower (closer to consensus). By contrast, for high values of 𝑝₂, as𝑐 increases, the range of final opinions becomes wider.

max𝑖, 𝑗|𝑥_𝑖(𝑇) −𝑥_𝑗(𝑇) |

max𝑖, 𝑗|𝑥_𝑖(0) −𝑥_𝑗(0) | (3.7)

0.05 0.4 0.8 1.2 1.6

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8 1

Negative Connection Probability

Positive Connection Probability

3 6 9 Spread

Figure 3.6: Heat map of opinion spread as a function of the probability of a connection in𝐺₁and𝐺₂iterated over confidence bound𝑐. Data drawn from the mean of 100 trials for each set of parameters with parameters 𝑝₁ ∈ {0.2,0.4,0.6,0.8,1}, 𝑝₂ ∈ {0,0.2,0.4,0.6,0.8,1}, and𝑐 ∈ {0.05,0.4,0.8,1.2,1.6}.

In Figure 3.6, we plot average opinion spread across trials as a function of the proportion ^𝑝_𝑝²

1 and confidence bound𝑐in a heat map. We observe similar trends as in Figure 3.5, with higher proportions ^𝑝_𝑝²

1 leading to higher values of opinion spread,

and the influence of𝑐on opinion spread depending on^𝑝_𝑝²

1. In the following examples, we will similarly see that opinion spread is largely controlled by the negative edges in the network, but that the addition of more structure to the network will influence opinion formation in interesting ways.

Stochastic Block Models

Next, we adapt a Stochastic Block Model (SBM) in order to incorporate both positive and negative edges. In these networks, each node is assigned to a group 𝑘 ∈ 𝐾. The probabilities of connections when𝑖_𝑘 = 𝑗_𝑘 is different than when𝑖_𝑘 ≠ 𝑗_𝑘. This enforces structure within the network. As in Section 3.5, we generate this network through two sub networks. In this case, the process begins with two SBM networks, 𝐺₁=𝐺(𝑛, 𝑝₁, 𝜌)and𝐺₂=𝐺(𝑛, 𝑝₂, 𝜌), with associated adjacency matrices𝐴₁and 𝐴₂. The variable 𝑝 is the probability of having a connection with another node in the same cluster while 𝑝₁₍₂₎𝜌is the probability of having a connection with a node in a different cluster. If G is network represented by the adjacency matrix given by 𝐴₁−𝐴₂we have the generated network edge probabilities:

𝑃( (𝑖, 𝑗) ∈ 𝐸)𝑖𝑘=𝑖𝑗 =













0 (1−𝑝₁) (1−𝑝₂) + 𝑝₁𝑝₂ 1 𝑝₁(1−𝑝₂)

−1 (1− 𝑝₁)𝑝₂

(3.8)

𝑃( (𝑖, 𝑗) ∈𝐸)𝑖_𝑘≠𝑖_𝑗 =













0 (1−𝑝₁𝜌) (1− 𝑝₂𝜌) +𝜌²𝑝₁𝑝₂ 1 𝑝₁𝜌(1− 𝑝₂𝜌)

−1 (1−𝑝₁𝜌)𝑝₂𝜌 .

(3.9)

A sample of the generative process can be seen in Figure 3.7, where the blue edges represent positive edges and the black negative. Each color of nodes represents a group𝑘 ∈𝐾.

Again, in order to create simulation results, 100 trials are run for all combinations of the parameters

𝜌 ∈ (0,0.2,0.4,0.6,0.8,1) 𝑝₁∈ (0.2,0.4,0.6,0.8,1) 𝑝₂∈ (0.0,0.2,0.4,0.6,0.8)

𝑐 ∈ (0.05,0.4,0.8,1.2,1.6).

(a) G1: positive node network with 𝑝₁ = 0.85 and 𝜌=0.05

(b) G2: negative node network with 𝑝₂ = 0.3 and 𝜌=0.05

Figure 3.7: An example of the generation of the SBM network with attractive and repulsive edges

For each trial, a random set of initial opinions is generated and the model is applied.

The full histories of an example run when 𝑝₁ = 0.8 and 𝑝₂ = 0.2 can be seen in Figure 3.8. It can be seen that each group fully converges on itself, while the confidence interval determines how dispersed the groups are from each other. As 𝜌increases, the full set begins to converge. As is the case with the ER models, for certain combinations the spread at steady-state is larger than that at the start. From these values we can see that when 𝑝₁and 𝑝₂are locked, higher confidence bounds result in a larger terminal spread, as does lower percentages of cross-group edges (𝜌).

As with the ER model, with the SBM we first look at steady-state opinion spread.

The results can be seen in Figure 3.9. Note, that when 𝜌 = 1 the SBM model is equivalent to the ER model for the same parameters. We therefore have that the final row is identical to Figure 3.6. We note that as the value of 𝜌shrinks, the final spread increases. Otherwise, the trends found for the ER model are consistent.

In the case of SBMs, since each vertex 𝑖 ∈ 𝑉 is assigned to a group in 𝑘, we are also interested in clustering in addition to opinion spread. We introduce a measures of how close vertices are to in-group vertices versus out-group vertices. In order to calculate this, which we call proportional spread, first we find the average in and out-group distances as:

I𝑇 = 1

|𝑘|

∑︁

𝑘_𝑖∈𝑘

|𝑘_𝑖|²

∑︁

𝑗∈𝑘_𝑖⊂𝑉

∑︁

ℓ∈𝑘_𝑖∈⊂𝑉

|𝑥_𝑗(𝑇) −𝑥_ℓ(𝑇) | (3.10)

O𝑇 = 1

|𝑘|

∑︁

𝑘_𝑖∈𝑘

|𝑘_𝑖|

∑︁

𝑗∈𝑘_𝑖⊂𝑉

|𝑘| − |𝑘_𝑖|

∑︁

ℓ∈⊂𝑉/𝑘𝑖

|𝑥_𝑗(𝑇) −𝑥_ℓ(𝑇) |. (3.11)

0 0.2 0.4 0.6 0.8 1

0.050.40.81.21.6

−1 0 1 2

Time

Position

Figure 3.8: Example of the paths taken when the graph is distributed according to the Stochastic Block Model scheme laid out in Equation 3.9. In this case 𝑝₁ =0.8 and𝑝₂=0.2 are both locked the confidence bound varies between the rows and the percent of cross group links varies over columns. Each color represents a different group. There are five groups in these examples.

Since the end-spread of the samples differ, in order to appropriately compare them we look at the ratio, that is

PS𝑇 = I𝑇

O𝑇

. (3.12)

Smaller values of proportional spread imply same-group nodes are significantly closer to each other than different-group nodes. Thus, small values imply increased separation by group. In Figure 3.10 the spread as well asO₀andI₀can be seen. It is clear that the values range most of the space, in additionO₀ ≈ I₀. Thus, uniformly, at the beginning of the trials we havePS₀=1.

The plots of proportional spread at steady-state over the trials can be seen in Fig- ure 3.11. There are many things to note in this figure. First, we observe that clustering is most clear with higher values of 𝑝₁and lower values of 𝜌. The former

0.05 0.4 0.8 1.2 1.6

00.20.40.81

0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.2

0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

Negative Connection Probability

Positive Connection Probability

3 6 9 Spread

Figure 3.9: Heat map of the opinion spread of Stochastic Block Model trials. The rows represent 𝜌, the percent of positive(negative) connection values that are out (in) group, while the columns are𝑐the confidence bound.

0.999000.999250.999500.999751.00000 Range of Original Values

0.330 0.333 0.336 Mean Out Group Distance

0.330 0.333 0.336 Mean In Group Distance

Figure 3.10: Histograms of metrics at the the start of each trial, useful for comparing the final results.

indicates higher level of in-group positive connections while, in terms of in group connections, the latter represents fewer out-group positive connections. This trend

0.2 0.4 0.6 0.8 1

00.20.40.60.8

0.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.00 0.00

0.25 0.50 0.75 1.00

0.00 0.25 0.50 0.75 1.00

Percent Cross Group Edges

Proportional Spread

CI 0.05 0.4 0.8 1.2 1.6

Figure 3.11: Proportional spread for Stochastic Block Model trials. The rows represent 𝑝₂, the base probability of an out-group negative connection, while the columns are 𝑝₁the base in-group positive probability. The x-axis is 𝜌 the percent of cross group edges and the color represents the confidence bound. A value of 1 means that nodes are equally as close to in and out-group nodes. Smaller values mean groups are increasingly clustered.

thus is consistent with expectations—higher group connectivity implies a larger probability that the entire group is connected and attracting each other. The low out-group positive connections means that there is a weaker countervailing force pulling nodes towards nodes of opposing groups.

The values of𝑝₂are measures of repulsive forces. To a lesser extent, clustering also increases with lower values of 𝑝₂. This implies that repulsive connections between groups increases polarization. This was the major result we were searching for by including the repulsive elements in the model to begin with. However, the relative importance of positive versus negative connections was not expected. While the

negative connection are necessary to create spread in this way, the density of positive connections seems to be more important in determining the level of polarization.

Dalam dokumen Computational Methods in the Study of Political Behavior (Halaman 79-87)