# The Spread of Ideas in a Network—The Garbage-Can Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. The Deterministic Picture

#### 3.2. The Stochastic Picture

^{F}. Consider the case K = 2, where all strings within each group are unified during the first t* time steps. The distance between the strings in different groups is equal to F with probability P, which decreases with F and increases with q.

^{F}, then Smax = F ln(q). For the results shown below in Table 1 we assigned an average value of s = S/Smax instead of S, to extract the dependence on F and q which comes from different values of Ω. This average is calculated over 50 single trajectories of the system. In particular, for a state where all strings happen to be identical in both groups, its contribution to the average entropy is zero, despite the fact that such a state can be different for different simulations.

^{3}(not shown here) allow us to draw the following conclusions:

- -
- A change in q mostly changes the normalization constant Smax; after excluding this dependence, <s> remains almost the same for different q. In particular, the <s> values in the first two rows in the table are similar to those in the next two rows. The same can be observed in the four last rows;
- -
- On the contrary, the consequences of change in F cannot be reduced to a variation of the normalization constant. When F increases from 3 to 4, <s> is reduced more than twice;
- -
- The results clearly depend on the threshold dc. For dc = F, entropy is about five times less than for dc = F − 1. More generally, with smaller dc the unification is more limited, and the diversity of the states is greater.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Two bands evolve without contact. These bands remain concentrated around different initial values. Their widths decrease as exp(−t)/2. (

**b**) At time t*, mutual contact between the communities is enabled if their distance is smaller than H. For a higher value of H, the entire range of opinions varies in width as exp[−2(t − t*)].

**Figure 2.**In cases where all the symbols in both strings are different, they remain unchanged. However, for dc = F (here F = 5) and if in both strings the symbols are the same in at least one cell, the strings unify, i.e., all the symbols in the corresponding cells are set equally. Such a case is presented here, as the symbols in the last cells of both strings on the left are α. On the right the strings are shown after their unification.

**Figure 3.**An example of a numerical solution obtained using the computational method for two groups, each of N/2 = 50 strings. The mean distance between strings within groups (g1 and g2, lower panels), between groups (g12, upper right panel) and taken as a whole (upper left panel) versus time. The parameters are as follows: F = 4, q = 5, dc = 4, t* = 10

^{6}; the latter parameter is marked as a vertical black line.

**Figure 4.**An example of a numerical solution obtained using the computational method for two groups, each of N/2 = 50 strings. The mean distance between strings within groups (g1 and g2, lower panels), between groups (g12, upper right panel) and as a whole (upper left panel) versus time. The parameters are: F = 4, q = 5, dc = 4, t* = 10

^{5}; the latter parameter is marked as a vertical black line.

**Table 1.**The average value of normalized entropy <s>, calculated over single trajectories, taking into account parameters F, q and d

_{c}. The states are stored for time t* = 10

^{6}time steps. For completeness, the last column is added to show normalization constant Smax.

F | q | dc | <s> | F lnq |
---|---|---|---|---|

3 | 4 | 2 | 0.460 | 4.159 |

3 | 4 | 3 | 0.077 | 4.159 |

3 | 5 | 2 | 0.474 | 4.828 |

3 | 5 | 3 | 0.099 | 4.828 |

4 | 4 | 3 | 0.174 | 5.545 |

4 | 4 | 4 | 0.032 | 5.545 |

4 | 5 | 3 | 0.208 | 6.438 |

4 | 5 | 4 | 0.035 | 6.438 |

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Żuchowska-Skiba, D.; Stojkow, M.; Krawczyk, M.J.; Kułakowski, K.
The Spread of Ideas in a Network—The Garbage-Can Model. *Entropy* **2021**, *23*, 1345.
https://doi.org/10.3390/e23101345

**AMA Style**

Żuchowska-Skiba D, Stojkow M, Krawczyk MJ, Kułakowski K.
The Spread of Ideas in a Network—The Garbage-Can Model. *Entropy*. 2021; 23(10):1345.
https://doi.org/10.3390/e23101345

**Chicago/Turabian Style**

Żuchowska-Skiba, Dorota, Maria Stojkow, Malgorzata J. Krawczyk, and Krzysztof Kułakowski.
2021. "The Spread of Ideas in a Network—The Garbage-Can Model" *Entropy* 23, no. 10: 1345.
https://doi.org/10.3390/e23101345