Frigyes Karinthy, in his 1929 short story "L\'aancszemek" ("Chains") suggested that any two persons are distanced by at most six friendship links. (The exact wording of the story is slightly ambiguous: "He bet us that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual [...]". It is not completely clear whether the selected individual is part of the five, so this could actually allude to distance five or six in the language of graph theory, but the "six degrees of separation" phrase stuck after John Guare's 1990 eponymous play. Following Milgram's definition and Guare's interpretation, we will assume that "degrees of separation" is the same as "distance minus one", where "distance" is the usual path length-the number of arcs in the path.) Stanley Milgram in his famous experiment challenged people to route postcards to a fixed recipient by passing them only through direct acquaintances. The average number of intermediaries on the path of the postcards lay between 4.4 and 5.7, depending on the sample of people chosen. We report the results of the first world-scale social-network graph-distance computations, using the entire Facebook network of active users (\approx721 million users, \approx69 billion friendship links). The average distance we observe is 4.74, corresponding to 3.74 intermediaries or "degrees of separation", showing that the world is even smaller than we expected, and prompting the title of this paper. More generally, we study the distance distribution of Facebook and of some interesting geographic subgraphs, looking also at their evolution over time. The networks we are able to explore are almost two orders of magnitude larger than those analysed in the previous literature. We report detailed statistical metadata showing that our measurements (which rely on probabilistic algorithms) are very accurate.
At the 20th World-Wide Web Conference, in Hyderabad, India, one of the authors (Sebastiano) presented a new tool for studying the distance distribution of very large graphs: Hy-perANF [3]. Building on previous graph compression [4] work and on the idea of diffusive computation pioneered in [21], the new tool made it possible to accurately study the distance distribution of graphs orders of magnitude larger than it was previously possible.
One of the goals in studying the distance distribution is the identification of interesting statistical parameters that can be used to tell proper social networks from other complex networks, such as web graphs. More generally, the distance distribution is one interesting global feature that makes it possible to reject probabilistic models even when they match local features such as the in-degree distribution.
In particular, earlier work had shown that the spid2 , which measures the dispersion of the distance distribution, appeared to be smaller than 1 (underdispersion) for social networks, but larger than one (overdispersion) for web graphs [3]. Hence, during the talk, one of the main open questions was “What is the spid of Facebook?”.
Lars Backstrom happened to listen to the talk, and suggested a collaboration studying the Facebook graph. This was of course an extremely intriguing possibility: beside testing the “spid hypothesis”, computing the distance distribution of the Facebook graph would have been the largest Milgramlike [20] experiment ever performed, orders of magnitudes larger than previous attempts (during our experiments Facebook has ≈ 721 million active users and ≈ 69 billion friendship links).
This paper reports our findings in studying the distance distribution of the largest electronic social network ever created. That world is smaller than we thought: the average distance of the current Facebook graph is 4.74. Moreover, the spid of the graph is just 0.09, corroborating the conjecture [3] that proper social networks have a spid well below one. We also observe, contrary to previous literature analysing graphs orders of magnitude smaller, both a stabilisation of the average distance over time, and that the density of the Facebook graph over time does not neatly fit previous models.
Towards a deeper understanding of the structure of the Facebook graph, we also apply recent compression techniques that exploit the underlying cluster structure of the graph to increase locality. The results obtained suggests the existence of overlapping clusters similar to those observed in other social networks.
Replicability of scientific results is important. While for obvious nondisclosure reasons we cannot release to the public the actual 30 graphs that have been studied in this paper, we distribute freely the derived data upon which the tables and figures of this papers have been built, that is, the Web-Graph properties, which contain structural information about the graphs, and the probabilistic estimations of their neighbourhood functions (see below) that have been used to study their distance distributions. The software used in this paper is distributed under the (L)GPL General Public License. 3
The most obvious precursor of our work is Milgram’s celebrated “small world” experiment, described first in [20] and later with more details in [23]: Milgram’s works were actually following a stream of research started in sociology and psychology in the late 50s [12]. In his experiment, Milgram aimed at answering the following question (in his words): “given two individuals selected randomly from the population, what is the probability that the minimum number of intermediaries required to link them is 0, 1, 2, . . . , k?”.
The technique Milgram used (inspired by [22]) was the following: he selected 296 volunteers (the starting population) and asked them to dispatch a message to a specific individual (the target person), a stockholder living in Sharon, MA, a suburb of Boston, and working in Boston. The message could not be sent directly to the target person (unless the sender knew him personally), but could only be mailed to a personal acquaintance who is more likely than the sender to know the target person. The starting population was selected as follows: 100 of them were people living in Boston, 100 were Nebraska stockholders (i.e., people living far from the target but sharing with him their profession) and 96 were Nebraska inhabitants chosen at random.
In a nutshell, the results obtained from Milgram’s experiments were the following: only 64 chains (22%) were completed (i.e., they reached the target); the average number of intermediaries in these chains was 5.2, with a marked difference between the Boston group (4.4) and the rest of the starting population, whereas the difference between the two other subpopulations was not statistically significant; at the other end of the spectrum, the random (and essentially clueless) group from Nebraska needed 5.7 intermediaries
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