# Complex Contagion Features without Social Reinforcement in a Model of Social Information Flow

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Measuring Information Flow

**cross-entropy**${h}_{\times}$ between two texts A and B [11,18]:

#### 2.2. Quoter Model

#### 2.3. Other Models of Information Flow

## 3. Materials and Methods

#### 3.1. The Quoter Model

#### 3.2. Measuring Information Flow over the Network

#### 3.3. Simulating Contagion Models

#### 3.4. Assessing the Impact of Structure on Dynamics

#### 3.5. Network Datasets

## 4. Results

#### 4.1. Information Flow and Models of Contagion

#### 4.2. Interplay of Clustering and Information Flow

#### 4.3. Community Structure and the Weakness of Long Ties

#### 4.4. The Role of Dynamic Heterogeneity

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ASPL | Average Shortest Path Length |

BA | Barabási-Albert |

ER | Erdos-Rényi |

SBM | Stochastic Block Model |

SI | Susceptible-Infected |

SIR | Susceptible-Infected-Recovered |

WS | Watts–Strogatz |

## Appendix A. Further Exploring Quoter Model Parameters

**Figure A1.**Trends in information flow in ER, BA, and small-world networks for $q\in \{0.1,0.5,0.9\}$. Except for very low quote probabilities, we see qualitatively similar trends. (

**A**) ER & BA networks of size $N=100$ with varying average degree. Each point constitutes 200 simulations. (

**B**) Small-world networks of size $N=200$ with $k=6$ with varying rewiring probability. Each point constitutes 500 simulations.

**Figure A2.**Effects of quoter model parameter choices on observed trends. Information flow is lower for denser ER and BA networks across a range of q and $\lambda $ with the effect being more pronounced at higher values of q and $\lambda $. Likewise, for small-world networks, more clustering (lower p) exhibits higher ${h}_{\times}$ than less clustering (higher p), with the effect being most pronounced at $q>0.5$ regardless of $\lambda $. Here, ER & BA networks had $N=100$ and small-world networks had $N=200$ and $k=6$. Each cell constitutes 100 simulations.

## Appendix B. Summarizing ${h}_{\times}$

**Figure A3.**The distributions of ${h}_{\times}$ for quoter model simulations on various networks. Examining the distributions supports using $\u2329{h}_{\times}\u232a$ and $\mathrm{Var}\left({h}_{\times}\right)$ as summary statistics, although some real networks show a small bimodality (an excess of ${h}_{\times}<3$ bits). We also remark that the mean and median are approximately equal (solid line shows $\langle {h}_{\times}\rangle $, dashed line shows median ${h}_{\times}$) for all networks. ER & BA networks have $N=1000$ nodes with $\langle k\rangle =12$, and 200 simulations as in Figure 1. Small-world networks have $N=200$ nodes with $k=6$ and $p={10}^{-4}$, and 500 simulations as in Figure 4A. Real-world networks are from 300 simulations as in Figure 2 and Figure 4B,C. Quoter model parameters are given in Section 3.1.

## Appendix C. Network Corpus

- Les Miserables co-appearances [44] [Undirected, Weighted].
- Hollywood film music [45] [Undirected, Weighted]. This is a bipartite network; we converted it to a one-mode projection (nodes are composers and two composers are linked if they worked with the same producer).
- Freeman’s EIES dataset [46] [Directed, Weighted]. We used the “personal relationships (time 1)” network.
- Sampson’s monastery [47] [Directed, Weighted]. We used the Pajek dataset. The weight of a directed link represents how an individual rates the other. The rating can be positive (1,2,3 = top 3 ranked) or negative (-1,-2,-3 = worst 3 ranked). We chose to only keep links which were positive.
- Golden Age of Hollywood [48] [Directed, Weighted]. We used the aggregated network over 1909-2009.
- 9-11 terrorist network [49] [Undirected, Unweighted].
- CKM physicians social network [50] (1966) [Directed, Unweighted]. We used “CKM physicians Freeman” networks hosted by Linton Freeman, and chose the “friend” network (i.e., the third adjacency matrix). We took only the giant component.
- Kapferer tailor shop [51] (1972) [Undirected, Unweighted]. We used the “Kapferer tailor shop 1” Pajek dataset (kapfts1.dat).
- Dolphin social network [52] (1994-2001) [Undirected, Unweighted].
- Email network (Uni. R-V, Spain, 2003) [53] [Directed, Unweighted]. We used the “email-uni-rv-spain-arenas” network.

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**Figure 1.**Denser networks are associated with higher information flow for simple contagion but lower information flow for both complex contagion and the quoter model. Here density is measured by average degree $\u2329k\u232a$ for Erdős-Rényi (ER) & Barabási-Albert (BA) model networks. (

**A**) Simple contagion. (

**B**) Complex contagion (

**C**) Quoter model. (Panel C, inset) Average cross-entropy on links; higher cross-entropies correspond to lower predictabilities and lower information flow, unlike for contagions where higher average peak sizes correspond to higher information flow. Networks consisted of $N=1000$ nodes and each point constitutes 200 simulations; parameters for simulating information flow in these models are described in Section 3.

**Figure 2.**Information flow on real-world networks. (

**A**) Simple contagion. (

**B**) Complex contagion. (

**C**) Quoter model. Here information flow measures (average peak size, average text predictability) are compared to network density $M/\left(\genfrac{}{}{0pt}{}{N}{2}\right)$. The association between information flow and density, either positive (simple contagion) or negative (complex contagion, quoter model), is significant (Wald test on non-zero regression slope, $p<0.05$). Each point constitutes 300 simulations.

**Figure 3.**Exploring the variance of information flow. (

**A**) Variance of cross-entropy is higher at low densities for BA than ER networks despite the average ${h}_{\times}$ being similar (Figure 1C). (

**B–D**) Information flow on dichotomous networks (random networks where all nodes have degree ${k}_{1}$ or degree ${k}_{2}$, allowing tunable degree heterogeneity) of size $N\in \{500,1000\}$ with $\langle k\rangle \in \{16,32\}$. Each point constitutes 500 trials. (

**B**) Average cross-entropy versus ${k}_{1}/{k}_{2}$. Degree heterogeneity does not affect average cross-entropy, supporting Figure 1C. Network size has a smaller effect on ${h}_{\times}$ compared to the average degree. (

**C**) Variance of cross-entropy versus ${k}_{1}/{k}_{2}$. Higher degree heterogeneity (lower ${k}_{1}/{k}_{2}$) leads to higher variation in ${h}_{\times}$ over links, indicating the existence of highly predictive nodes and nodes that contribute little predictive information within heterogeneous networks. (

**D**) Dichotomous networks of size $N=1000$ and $\langle k\rangle =16$. Average cross-entropy over links conditioned on degrees of endpoints (predicting ego from alter). Only the degree of the ego matters, approximately, not the degree of the alter.

**Figure 4.**Mixed effects of clustering on information flow. (

**A**) Information flow on small-world networks of size $N\in \{200,400\}$ and average degree $k\in \{6,12\}$. As network rewiring increases (and clustering decreases) ${h}_{\times}$ increases. This suggests that clustered networks promote information flow. Rewiring a small-world network changes the diameter (L) as well the clustering (panel A, bottom); however, ${h}_{\times}$ begins to increase primarily when the clustering begins to drop, not when diameter begins to drop. Each point constitutes 300 trials. (

**B**) Average cross-entropy versus transitivity for real-world networks. By randomizing networks using the standard “x-swap” method (Section 3.4), we can lower the transitivity and investigate how ${h}_{\times}$ changes. Some networks show little change in ${h}_{\times}$ on randomized networks compared with the original networks, while others show a slight decrease in ${h}_{\times}$. This is especially visible in the inset comparing ${h}_{\times}$ directly. Each point constitutes 300 simulations. (

**C**) Several network properties before and after the x-swap method. While the x-swap method lowers transitivity, it also alters other important network properties, making it challenging to isolate the role of clustering from other properties.

**Figure 5.**Information flow within the stochastic block model (SBM) of $N=100$ (two blocks of size $N=50$). Each point constitutes 10k trials. (

**A**) Average cross-entropy on within-block edges and between-block edges as a function of the within-block connection probability ${p}_{0}$ for different between-block connection probabilities ${p}_{1}$. (

**B**,

**C**) Examining the cross-entropy difference $\Delta {h}_{\times}\equiv \langle {h}_{\times}\left(\mathrm{between}\right)\rangle -\langle {h}_{\times}\left(\mathrm{within}\right)\rangle $ across (

**B**) connection probabilities and (

**C**) modularity Q. Examining $\Delta {h}_{\times}$ as a function of modularity Q shows a clear collapse across values of SBM probabilities. Interestingly, anti-community structure ($Q<0$) still leads to positive $\Delta {h}_{\times}$, indicating that information flow is still more prevalent within blocks.

**Figure 6.**Effects of dynamic heterogeneity on information flow in the stochastic block model. Nodes in group A have Zipfian vocabulary distribution with exponent ${\alpha}_{A}$ while nodes in B have exponent ${\alpha}_{B}$. The between-block connection probability is fixed (${p}_{1}=0.15$) and the within-block connection probability ${p}_{0}$ is varied to generate a range of modularities. Since the structure is symmetric (subgraphs A and B have the same size and expected density), we only show the result of fixing ${\alpha}_{A}=2$ and varying ${\alpha}_{B}$. Each point constitutes 150 trials. (

**A**) The vocabulary distribution of group A has a lower Shannon entropy than of B, and this difference is visible from examining links $A\to A$ and $B\to B$. When examining links $A\to B$ and $B\to A$, the cross-entropy is mainly dependent on the vocabulary distribution of the alter. As modularity increases, differences between the predictabilities of various nodes are exaggerated. (

**B**) In homogeneous communities, the cross-entropy does not vary with modularity at such a scale. (

**C**) The vocabulary distribution of group A has a higher Shannon entropy than of B. Similar mirror results are seen as in panel A.

**Table 1.**Descriptive statistics for real-world networks used in this study. ASPL: Average Shortest Path Length. Modularity computed using the Louvain method [40].

Network | $\left|\mathit{V}\right|$ | $\left|\mathit{E}\right|$ | $\langle \mathit{k}\rangle $ | Density | Transitivity | ASPL | Modularity | Assortativity |
---|---|---|---|---|---|---|---|---|

Sampson’s monastery | 18 | 71 | 7.9 | 0.464 | 0.53 | 1.54 | 0.29 | −0.07 |

Freeman’s EIES | 34 | 415 | 24.4 | 0.740 | 0.82 | 1.26 | 0.07 | −0.15 |

Kapferer tailor | 39 | 158 | 8.1 | 0.213 | 0.39 | 2.04 | 0.32 | −0.18 |

Hollywood music | 39 | 219 | 11.2 | 0.296 | 0.56 | 1.86 | 0.20 | −0.08 |

Golden Age | 55 | 564 | 20.5 | 0.380 | 0.53 | 1.64 | 0.45 | −0.13 |

Dolphins | 62 | 159 | 5.1 | 0.084 | 0.31 | 3.36 | 0.52 | −0.04 |

Terrorist | 62 | 152 | 4.9 | 0.080 | 0.36 | 2.95 | 0.52 | −0.08 |

Les Miserables | 77 | 254 | 6.6 | 0.087 | 0.50 | 2.64 | 0.56 | −0.17 |

CKM physicians | 110 | 193 | 3.5 | 0.032 | 0.16 | 4.24 | 0.61 | −0.11 |

Email Spain | 1133 | 5452 | 9.6 | 0.009 | 0.17 | 3.61 | 0.57 | −0.08 |

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**MDPI and ACS Style**

Pond, T.; Magsarjav, S.; South, T.; Mitchell, L.; Bagrow, J.P.
Complex Contagion Features without Social Reinforcement in a Model of Social Information Flow. *Entropy* **2020**, *22*, 265.
https://doi.org/10.3390/e22030265

**AMA Style**

Pond T, Magsarjav S, South T, Mitchell L, Bagrow JP.
Complex Contagion Features without Social Reinforcement in a Model of Social Information Flow. *Entropy*. 2020; 22(3):265.
https://doi.org/10.3390/e22030265

**Chicago/Turabian Style**

Pond, Tyson, Saranzaya Magsarjav, Tobin South, Lewis Mitchell, and James P. Bagrow.
2020. "Complex Contagion Features without Social Reinforcement in a Model of Social Information Flow" *Entropy* 22, no. 3: 265.
https://doi.org/10.3390/e22030265