# Diffusion Dynamics with Changing Network Composition

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models of Information Diffusion

## 3. Data: Diffusion Events, Communication Networks, and Information Cascades

Protests 2011 | Protests 2012 | Intervening Period | |
---|---|---|---|

Date range | 25 April to 25 May | 30 April to 30 May | 1 June to 31 March |

Total number of messages | 581,749 | 1,026,292 | 555,521 |

Total number of unique users | 85,933 | 127,930 | 115,992 |

2011 | 2011–2012 | 2012 | ||||
---|---|---|---|---|---|---|

following/er (topological) | @s (dynamic) | following/er (topological) | @s (dynamic) | following/er (topological) | @s (dynamic) | |

N (# nodes) | 85,712 | 50,369 | 113,677 | 35,815 | 127,400 | 127,068 |

M (# arcs) | 6,030,459 | 135,637 | 10,191,085 | 98,709 | 7,459,518 | 522,430 |

<k> (avg degree) | 7.36 | 2.69 | 89.65 | 2.76 | 58.55 | 4.11 |

max(k_{in}) (max indegree) | 5,773 | 10,781 | 8,262 | 3,118 | 12,552 | 12,269 |

max(k_{out}) (max outdegree) | 31,798 | 245 | 37,810 | 651 | 34,892 | 658 |

C (clustering) | 0.022 | 0.002 | 0.028 | 0.015 | 0.026 | 0.013 |

l (path length) | 2.45 | 3.97 | 2.52 | 4.18 | 2.71 | 4.00 |

D (diameter) | 6 | 15 | 7 | 16 | 8 | 15 |

r (assortativity) | −0.13 | −0.07 | −0.11 | −0.09 | −0.13 | −0.08 |

# strong components | 3,392 | 23,445 | 10,871 | 20,309 | 12,151 | 59,792 |

N giant component | 82,253 | 26,881 | 102,750 | 15,572 | 115,105 | 67,331 |

N 2nd component | 4 | 2 | 3 | 2 | 4 | 2 |

_{in}+ k

_{out}, and max(k

_{in}); max(k

_{out}) for the maximum in-degree and out-degree respectively. The clustering coefficient (C) measures the degree to which nodes in the graph tend to cluster together; the average path length (l) measures the average distance in terms of links between two different nodes; the diameter (D) measures the longest shortest path, and the assortativity (r) gives the Pearson coefficient of degree between pairs of linked nodes. The number of strongly connected components, the size of the giant strongly connected component, and the size of the second largest connected component are also shown.

**Figure 1.**(

**a**) Diffusion curve of protest activity for both events showing the normalized cumulative fraction of users sending at least one message at a given day. (

**b**) Complementary cumulated (CCP) degree distribution for the dynamic network constructed using mentions and re-tweets between users. (

**c**) Complementary cumulated k-core distribution for the dynamic network.

**Figure 2.**(

**a**) Complementary cumulative probability distribution of cascade sizes for spreaders. (

**b**) Complementary cumulative probability distribution of cascade sizes for listeners. (

**c**) Correlation between degree centrality of the initial seed triggering the cascade and its final size comprising all nodes reached, rescaled by the network size (topological network). (

**d**) Correlation between k-core of the seed node and the final cascade size rescaled by the network size (topological network).

## 4. Changes in Network Composition and Visibility in Information Flow

**Figure 3.**(

**a**) Changes in the composition of the communication network, showing the percentages of users in different categories according to their presence in the different periods under consideration. (

**b**) Correlation of centrality measures (degree) of users present in 2011 and 2012 for the topological network (

**top**) and the dynamic network (

**bottom**). Hexagons bin data points, with darker colour indicating more users in that area of the scatterplot.

**Figure 4.**

**Upper panels**: Distribution of users in the three observation periods according to the relation between their inverse audience size, measured by the ratio of users being followed over the number of followers, and their protest visibility, defined as the ratio of mentions or RTs received over the number of mentions or RTs sent.

**Bottom panels**: allocation of visibility across the four categories in the different periods. Nodes sizes are proportional to the number of users in the corresponding group, and the width of the links between them is proportional to the number of mentions and RTs across groups.

**Figure 5.**(

**a**) Migration of users across categories from 2011 to 2012, indicated by a directed link between groups. (

**b**) Standardized residuals resulting from the comparison of observed frequencies and expected frequencies, which can be interpreted as z-scores measuring the distance from no difference; the values correspond to movement from rows to columns (that is, movements from the classification in 2011 to the classification in 2012).

**Figure 6.**Concentration of mentions and RTs received and sent by category of user. The inequality in the distribution of visibility (i.e., in how it is received and allocated) is measured by the Gini coefficient; the diagonal line is shown as a benchmark of perfect equality.

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix

Rank | Hashtag | Rank | Hashtag |
---|---|---|---|

1 | # acampadasol | 36 | # globalrevolution |

2 | # spanishrevolution | 37 | # acampadazaragoza |

3 | # nolesvotes | 38 | # acampadaparis |

4 | # 15 m | 39 | # takethesquare |

5 | # nonosvamos | 40 | # periodismoeticoya |

6 | # democraciarealya | 41 | # hastalasgenerales |

7 | # notenemosmiedo | 42 | # irishrevolution |

8 | # yeswecamp | 43 | # democraziarealeora |

9 | # 15mani | 44 | # democraciaparticipativa |

10 | # acampadasevilla | 45 | # 15mpamplona |

11 | # globalcamp | 46 | # barcelonarealya |

12 | # acampadavalencia | 47 | # dry_jaen |

13 | # acampadagranada | 48 | # usarevolution |

14 | # acampadamalaga | 49 | # dry_caceres |

15 | # acampadazgz | 50 | # dryasturies |

16 | # consensodeminimos | 51 | # democraziareale |

17 | # italianrevolution | 52 | # democratiereelle |

18 | # estonosepara | 53 | # dry_cadiz |

19 | # acampadaalicante | 54 | # dry_toledo |

20 | # tomalacalle | 55 | # acampadasvlla |

21 | # europeanrevolution | 56 | # drybizkaia |

22 | # acampadapamplona | 57 | # dry_santander |

23 | # worldrevolution | 58 | # 15mayovalencia |

24 | # acampadapalma | 59 | # dry_pisa |

25 | # tomalaplaza | 60 | # dryginebra |

26 | # acampadas | 61 | # DRY_Algeciras |

27 | # 15mpasalo | 62 | # demorealyaib |

28 | # cabemostodas | 63 | # DRYGipuzkoa |

29 | # nonosmovemos | 64 | # DryValladolid |

30 | # 3puntosbasicos | 65 | # ItalRevolution |

31 | # frenchrevolution | 66 | # BolognaDRY |

32 | # estonoseacaba | 67 | # DRY_Pavia |

33 | # acampadatoledo | 68 | # DRY_Almeria |

34 | # nonosrepresentan | 69 | # 15mayoCordoba |

35 | # acampadalondres | 70 | # ciudades-dry |

## References

- Easley, D.; Kleinberg, J. Networks, Crowds, and Markets: Reasoning about a Highly Connected World; Cambridge University Press: New York, NY, USA, 2010. [Google Scholar]
- Newman, M.E.J. Networks. An Introduction; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Wasserman, S.; Faust, K. Social Network Analysis: Methods and Applications; Cambridge University Press: Cambridge, UK, 1994. [Google Scholar]
- Watts, D.J. Six Degrees. The Science of a Connected Age; William Heinemann: London, UK, 2003. [Google Scholar]
- Borge-Holthoefer, J.; Rivero, A.; García, I.; Cauhé, E.; Ferrer, A.; Ferrer, D.; Francos, D.; Iñiguez, D.; Pérez, M.P.; Ruiz, G.; et al. Structural and dynamical patterns on online social networks: The spanish may 15th movement as a case study. PLoS One
**2011**, 6, e23883. [Google Scholar] [CrossRef] [PubMed] - Conover, M.D.; Davis, C.; Ferrara, E.; McKelvey, K.; Menczer, F.; Flammini, A. The geospatial characteristics of a social movement communication network. PLoS One
**2013**, 8, e55957. [Google Scholar] [CrossRef] [PubMed] - Conover, M.D.; Ferrara, E.; Menczer, F.; Flammini, A. The digital evolution of occupy wall street. PLoS One
**2013**, 8, e64679. [Google Scholar] [CrossRef] [PubMed] - González-Bailón, S.; Borge-Holthoefer, J.; Moreno, Y. Broadcasters and hidden influentials in online protest diffusion. Am. Behav. Sci.
**2013**, 57, 943–965. [Google Scholar] [CrossRef] - González-Bailón, S.; Borge-Holthoefer, J.; Rivero, A.; Moreno, Y. The dynamics of protest recruitment through an online network. Sci. Rep.
**2011**, 1. [Google Scholar] [CrossRef] [PubMed] - Centola, D.; Macy, M.W. Complex contagions and the weakness of long ties. Am. J. Sociol.
**2007**, 113, 702–734. [Google Scholar] [CrossRef] - Granovetter, M. Threshold models of collective behavior. Am. J. Sociol.
**1978**, 83, 1420–1443. [Google Scholar] [CrossRef] - Watts, D.J. A simple model of global cascades on random networks. Proc. Natl. Acad. Sci. USA
**2002**, 99, 5766–5771. [Google Scholar] [CrossRef] [PubMed] - Gleeson, J.P.; Cahalane, D.J. Seed size strongly affects cascades on random networks. Phys. Rev. E
**2007**, 75. [Google Scholar] [CrossRef] - Pastor-Satorras, R.; Vespignani, A. Epidemic spreading in scale-free networks. Phys. Rev. Lett.
**2001**, 86, 3200–3203. [Google Scholar] [CrossRef] [PubMed] - Moreno, Y.; Nekovee, M.; Pacheco, A.F. Dynamics of rumor spreading in complex networks. Phys. Rev. E
**2004**, 69. [Google Scholar] [CrossRef] - Borge-Holthoefer, J.; Baños, R.A.; González-Bailón, S.; Moreno, Y. Cascading behaviour in complex socio-technical networks. J. Complex Netw.
**2013**, 1, 3–24. [Google Scholar] [CrossRef] - Newman, M.; Barabási, A.-L.; Watts, D.J. The Structure and Dynamics of Networks; Princeton University Press: Princeton, NJ, USA, 2006. [Google Scholar]
- Albert, R.; Jeong, H.; Barabási, A.L. Error and attack tolerance of complex networks. Nature
**2000**, 406, 378–382. [Google Scholar] [CrossRef] [PubMed] - Borge-Holthoefer, J.; Moreno, Y. Absence of influential spreaders in rumor dynamics. Phys. Rev. E
**2012**, 85, 026116. [Google Scholar] [CrossRef] - Centola, D.; Eguiluz, V.M.; Macy, M.W. Cascade dynamics of complex propagation. Phys. A
**2007**, 374, 449–456. [Google Scholar] [CrossRef] - Goyal, A.; Bonchi, F.; Lakshmanan, L.V.S. Learning Influence Probabilities in Social Networks. In Proceedings of the Third ACM International Conference on Web Search and Data Mining; ACM: New York, NY, USA, 2010; pp. 241–250. [Google Scholar]
- Cha, M.; Haddadi, H.; Benevenuto, F.; Gummadi, K.P. Measuring User Influence in Twitter: The Million Follower Fallacy. In Proceedings of International AAAI Conference on Weblogs and Social Media (ICSWM), Washington, DC, USA, 23–26 May; AAAI: Washington, DC, USA, 2010. [Google Scholar]
- Romero, D.M.; Meeder, B.; Kleinberg, J. Differences in the Mechanics of Information Diffusion Across Topics: Idioms, Political Hashtags, and Complex Contagion on Twitter. In Proceedings of International World Wide Web Conference, Hyderabad, India, 28 March–1 April; ACM: Hyderabad, India, 2011. [Google Scholar]
- Granell, C.; Gomez, S.; Arenas, A. Dynamical interplay between awareness and epidemic spreading in multiplex networks. Phys. Rev. Lett. 2013. [CrossRef]
- Vanilla-Rodirguez, N.; Scellato, S.; Haddadi, H.; Forsell, C.; Crowcroft, J.; Mascolo, C. Los Twindignados: The Rise of the Indignants Movement. In Proceedings of ASE/IEEE International Conference on Social Computing (SocialCom), Amsterdam, The Netherlands, 3–5 September; ASE/IEEE: Amsterdam, The Netherlands, 2012. [Google Scholar]
- Baños, R.A.; Borge-Holthoefer, J.; Moreno, Y. The role of hidden influentials in the diffusion of online information cascades. EPJ Data Sci.
**2013**, 2, 6. [Google Scholar] [CrossRef] - González-Bailón, S.; Wang, N. The bridges and brokers of global campaigns in the context of social media. SSRN Work. Pap.
**2013**. Available online: http://ssrn.com/abstract=2268165 (accessed on 10 October 2013). [Google Scholar] [CrossRef] - Castells, M. Networks of Outrage and Hope. Social Movements in the Internet Age; Polity: Cambridge, UK, 2012. [Google Scholar]
- Gerbaudo, P. Tweets and the Streets. Social Media and Contemporary Activism; Pluto Books: London, UK, 2012. [Google Scholar]
- Morstatter, F.; Pfeffer, J.; Liu, H.; Carley, K.M. Is the Sample Good Enough? Comparing Data from Twitter’s Streaming Api with Twitter’s Firehose. In Proceedings of International AAAI Conference on Weblogs and Social Media (ICSWM), Boston, MA, 8–10 July; AAAI: Boston, MA, USA, 2013. [Google Scholar]
- Bonacich, P. Power and centrality: A family of measures. Am. J. Sociol.
**1987**, 92, 1170–1182. [Google Scholar] [CrossRef] - Seidman, S.B. Network structure and minimum degree. Soc. Netw.
**1983**, 5, 269–287. [Google Scholar] [CrossRef] - Borge-Holthoefer, J.; Rivero, A.; Moreno, Y. Locating privileged spreaders on an online social network. Phys. Rev. E
**2012**, 85, 066123. [Google Scholar] [CrossRef] - Cha, M.; Benevenuto, F.; Haddadi, H.; Gummadi, K.P. The world of connections and information flow in twitter. IEEE Trans. Syst. Man Cybern.
**2012**, 42, 991–998. [Google Scholar] - Hope, A.C.A. A simplified monte carlo significance test procedure. J. R. Stat. Soc. B
**1968**, 30, 582–598. [Google Scholar] - Barabási, A.L.; Albert, R. Emergence of scaling in random networks. Science
**1999**, 286, 509–512. [Google Scholar] [PubMed] - Rogers, E.M. Diffusion of Innovations, 5th ed.; Free Press: New York, NY, USA, 2003. [Google Scholar]
- Valente, T.W. Network Models of the Diffusion of Innovations; Hampton Press: Cresskill, NJ, USA, 1995. [Google Scholar]

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Baños, R.A.; Borge-Holthoefer, J.; Wang, N.; Moreno, Y.; González-Bailón, S. Diffusion Dynamics with Changing Network Composition. *Entropy* **2013**, *15*, 4553-4568.
https://doi.org/10.3390/e15114553

**AMA Style**

Baños RA, Borge-Holthoefer J, Wang N, Moreno Y, González-Bailón S. Diffusion Dynamics with Changing Network Composition. *Entropy*. 2013; 15(11):4553-4568.
https://doi.org/10.3390/e15114553

**Chicago/Turabian Style**

Baños, Raquel A., Javier Borge-Holthoefer, Ning Wang, Yamir Moreno, and Sandra González-Bailón. 2013. "Diffusion Dynamics with Changing Network Composition" *Entropy* 15, no. 11: 4553-4568.
https://doi.org/10.3390/e15114553