This morning I had the pleasure of reading Barabási and Bonabeau’s article Scale Free Networks, published in Scientific American’s May 2003 issue. Yes, in the scientific world it may be ancient history, but I found it a very appealing article due to its readability and discussion of the real world implications of scale free networks. As I have recently been trying to better understand the topology, particularly the degree distribution, of the Preliminaries network–and I am way overdue for some blogging–I thought I would take this opportunity to further reflect on the true nature of Preliminaries database and it implications for early modern cultural systems.
Before I begin this discussion, I need to be very clear on one point: I have not proved, mathematically, the scale free nature of the Preliminaries network. A power-law degree distribution (typical of scale free networks), p(k) ∝ k-α, when presented in a log-log plot, is a straight line for all values:
This image is simply the Prelims network degree distribution fed into the above equation using ~ -1.94 as the exponent parameter. However, most empirical data does not truly follow this distribution, and power-law tail can be tricky to prove (Clauset et al., 2009), especially for the mathematically uninitiated. For example, if we look at a degree distribution probability plotted in log-log, we do not see a straight line, instead we see variation, especially at very low and very high values of x:
This can be explained quite simply. First of all due to the Preliminaries database methodology and data set, we see no nodes with a degree of 0 and relatively few with a degree of 1. A node is not created unless it is found in the preliminaries of an edition, and therefore all node in the database have a degree of at least 1. Also, due to the nature of the data, we find that most names, places, etc. are recurring; most works have multiple editions,etc; so generally most nodes have at least two or three connections.
For the high values of x, due to the method of calculating probability of degree occurrence, we find a minimum y value of 0.00061881188, or 1/1616. The preliminaries graph has 1616 nodes, and for the unique nodes with very high degree, we can only say that there is a 1 in 1616 chance that a node of that degree exists. Eventually I would like to employ a more sophisticated analysis, such as proposed in Clauset et al., to determine if these high x values truly fit a power law distribution.
Currently, just to investigate a bit, I have experimented a bit with making a rough estimate for the exponential parameter. First I calculated the -α in p(k) ∝ k-α for each degree value in the prelims graph. This produces a range of values between -2.356155 and -1.241804 with a rough fit at -1.945687. I also have calculated, following Ghoshal and Barabasi 2011, the relative degree distance between the top two nodes (Madrid and Lope de Vega), Δk = 1.3375. While not as great a distance observed in the much larger scale free networks, it is considerably larger than those observed in large exponential networks. Finally, I plotted the power-law distribution and the real world data together:
Due to the general good fit found with these simple calculations, the relatively large distance between the top two nodes, and the general look of the network when represented visually (think hubs), I would like to simply state here that the data may suggest that the Prelims graph is scale free. (code for visualization)
So what are the implications of this? As far as I can tell, after a very brief and perhaps elementary review of the literature on scale free graphs, there are two primary factors: growth and robustness. Here I hope to briefly discuss each of these factors and its relevance to cultural networks.
A scale free network has two principle driving growth factors: continuous expansion and preferential attachment (Barabasi, Albert 1999). Nodes are continuously added to the network in an incremental fashion, but they do not connect to other nodes randomly. Instead, they connect preferentially to nodes that are well connected. Let’s think about this in terms of a network of cultural objects, people, institutions, and places such as the Preliminaries network. A new writer comes on the scene. First of all, if he wants to actually publish, he will somehow connect himself, perhaps based on his own geographic limitations, to a major publishing center, let’s say Madrid. Now, publishing a book in 1607 in Madrid was no simple process. First you need to pass through a process of censorship: ecclesiastic approval, licensing, revisions, pricing. In an ideal world, this process should be completed based on the merit of the work, however, in the bureaucratic and favor currying world of early modern Spain, it seems that being well connected could greatly expedite this process. Considering that much licencing was done directly from the Consejo de Castilla, being well connected at court could never hurt. The practice of authors soliciting the patronage of powerful nobles at this time is well documented, and it appears that their patronage not only protected and funded them, but also opened doors into the world of publication.
For example, for a number of years Lope de Vega was the personal secretary of Pedro Fernandez de Castro y Andrade, VII Count of Lemos, who was also the recipient of regular dedications in from important writers by Cervantes, Gongora, Quevedo, and Lope himself. The prestige and influence of these authors cannot be questioned, and their relationship with the Count of Lemos is hardly coincidental. This intuitive association is well grounded, and has been the subject of serious study. However, in terms of network growth, this kind of preferential attachment, through jockeying for position within a network of cultural influence, can help to explain the heterogeneous nature of literary influence and publication we see in this time period. Referring to their model for scale free growth, Barabási and Albert anticipate this phenomenon in their 1999 article:
“Similar mechanisms could explain the origin of the social and economic disparities governing competitive systems, because the scale-free inhomogeneities are the inevitable consequence of self-organization due to the local decisions made by the individual vertices, based on information that is biased toward the more visible(richer) vertices, irrespective of the nature and origin of this visibility.” (512)
Now in terms of the durability and influence of a certain novel, it is perhaps hasty to say that 17th century network structure has resulted in the enduring popularity and fame of a work such as Don Quixote. Even in Cervantes day, the fact that Don Quixote was so widely published and appreciated obviously is the result of varied factors. But I believe that it is very important to recognize that this imbalance, or heterogeneity, is at least partially a natural consequence of self organizing scale free networks.
Scale free networks are generally robust, unless they are subject to a targeted attack (Barabási, Bonabeau 2003). What does this mean exactly? Well first of all, most nodes are not hubs, therefore removing a single node from the network generally does not really affect the structure of the network. Indeed even removing a hub will generally not cause the network to fracture, because of the existence of other hubs. However, a coordinated attack on a scale free network that removes a significant number of hubs can threaten the integrity of the entire system. In terms of cultural networks, I believe that this robustness is crucial. Culture as a system is very resilient, and does not rely too heavily on any one person, object or location. After removing Madrid from the Prelims network, the graph is still connected (there is a path between each set of nodes in the network). However, thinking about a cultural network, removing a physical hub like Madrid, or an author like Lope, does not reflect the reality of the situation. If we remove Madrid, wouldn’t we have to remove all of the nodes associated with Madrid? All of the editions published there, all of the editors who lived there, are somehow inextricably linked to this hub. But if Madrid had never existed, would these same editions have been published in different places? Would the people associated with Madrid exist? Would they have existed in another space and attached themselves differently to the network? These questions, although perhaps impossible to answer, must be posed when considering the robustness of a cultural network.
The other side of the robustness issue, the vulnerability to concentrated attacks, has interesting implications when considering the idea of cultural conquest. If, during the expansion of the Spanish Empire, the hubs of the indigenous American culture were attacked, destroyed, or taken over, this would certainly be the most effective way of causing failure in the previous cultural network. For example, when Cortés took Tenochtitlan, he was taking control of one of the most important cultural hubs in the Aztec empire. However, he did not remove this node, causing cultural collapse. Instead, this became the beginning of a process of new information being passed through the network using an old hub as a key location. Today, Mexico City continues to be a center for Mexican culture. It is interesting to observe, however effective the Spaniards cultural campaign was, that the resulting Mexican culture was somehow a mix of Prehispanic and European cultural practices. This seems to suggest that even under directed cultural attack, cultural information present in a network has lasting effects, something that speaks to the ability of information to continue to circulate through a network, even in the presence of new, sometimes conflicting information, especially, as many authors claim, when there is some degree of compatibility between the two types of information.
These ideas are just the beginning of a reflection on the implications of scale free structures in cultural systems. To fully develop this idea, I must continue to research, as I have only just begun to understand the science and history that I am writing about. Until next time…