Four Degrees of Separation, Really

We recently measured the average distance of users in the Facebook graph, spurring comments in the scientific community as well as in the general press ("Four Degrees of Separation"). A number of interesting criticisms have been made about the meaningfulness, methods and consequences of the experiment we performed. In this paper we want to discuss some methodological aspects that we deem important to underline in the form of answers to the questions we have read in newspapers, magazines, blogs, or heard from colleagues. We indulge in some reflections on the actual meaning of "average distance" and make a number of side observations showing that, yes, 3.74 "degrees of separation" are really few

Four Degrees of Separation

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

Study of lee-side flows over conically cambered delta wings at supersonic speeds, part 1

An experimental investigation was performed in which surface pressure data, flow visualization data, and force and moment data were obtained on four conical delta wing models which differed in leading-edge camber only. Wing leading-edge camber was achieved through a deflection of the outboard 30% of the local wind semispan of a reference 75 degrees swept flat delta wing. The four wing models have leading-edge deflection angles delta sub F of 0, 5, 10, and 15 degrees measured streamwise. Data for the wings with delta sub F = 10 and 15 degrees showed that hinge-line separation dominated the lee-side wing loading and prohibited the develpment of leading-edge separation on the deflected portion of wing leading edge. However, data for the wing with delta sub F = 5 degrees, a vortex was positioned on the deflected leading edge with reattachment at the hinge line. Flow visualization results were presented which detail the influence of Mach number, angle of attack, and camber on the lee-side flow characteristics of conically cambered delta wings. Analysis of photgraphic data identified the existence of 12 distinctive lee-side flow types. In general, the aerodynamic force and moment data correlated well with the pressure and flow visualization data

Evaluation of panel code predictions with experimental results of inlet performance for a 17-inch ducted prop/fab simulator operating at Mach 0.2

An axisymmetric panel code was used to evaluate a series of ducted propeller inlets. The inlets were tested in the Lewis 9 by 15 Foot Low Speed Wind Tunnel. Three basic inlets having ratios of shroud length to propeller diameter of 0.2, 0.4, and 0.5 were tested with the Pratt and Whitney ducted prop/fan simulator. A fourth hybrid inlet consisting of the shroud from the shortest basic inlet coupled with the spinner from the largest basic inlet was also tested. This later configuration represented the shortest overall inlet. The simulator duct diameter at the propeller face was 17.25 inches. The short and long spinners provided hub-to-tip ratios of 0.44 at the propeller face. The four inlets were tested at a nominal free stream Mach number of 0.2 and at angles of attack from 0 degrees to 35 degrees. The panel code method incorporated a simple two-part separation model which yielded conservative estimates of inlet separation

What are the Best Hierarchical Descriptors for Complex Networks?

This work reviews several hierarchical measurements of the topology of complex networks and then applies feature selection concepts and methods in order to quantify the relative importance of each measurement with respect to the discrimination between four representative theoretical network models, namely Erd\"{o}s-R\'enyi, Barab\'asi-Albert, Watts-Strogatz as well as a geographical type of network. The obtained results confirmed that the four models can be well-separated by using a combination of measurements. In addition, the relative contribution of each considered feature for the overall discrimination of the models was quantified in terms of the respective weights in the canonical projection into two dimensions, with the traditional clustering coefficient, hierarchical clustering coefficient and neighborhood clustering coefficient resulting particularly effective. Interestingly, the average shortest path length and hierarchical node degrees contributed little for the separation of the four network models.Comment: 9 pages, 4 figure

Tachoastrometry: astrometry with radial velocities

Spectra of composite systems (e.g., spectroscopic binaries) contain spatial information that can be retrieved by measuring the radial velocities (i.e., Doppler shifts) of the components in four observations with the slit rotated by 90 degrees in the sky. By using basic concepts of slit spectroscopy we show that the geometry of composite systems can be reliably retrieved by measuring only radial velocity differences taken with different slit angles. The spatial resolution is determined by the precision with which differential radial velocities can be measured. We use the UVES spectrograph at the VLT to observe the known spectroscopic binary star HD 188088 (HIP 97944), which has a maximum expected separation of 23 milli-arcseconds. We measure an astrometric signal in radial velocity of 276 \ms, which corresponds to a separation between the two components at the time of the observations of 18 $\pm2$ milli-arcseconds. The stars were aligned east-west. We describe a simple optical device to simultaneously record pairs of spectra rotated by 180 degrees, thus reducing systematic effects. We compute and provide the function expressing the shift of the centroid of a seeing-limited image in the presence of a narrow slit.The proposed technique is simple to use and our test shows that it is amenable for deriving astrometry with milli-arcsecond accuracy or better, beyond the diffraction limit of the telescope. The technique can be further improved by using simple devices to simultaneously record the spectra with 180 degrees angles.With tachoastrometry, radial velocities and astrometric positions can be measured simultaneously for many double line system binaries in an easy way. The method is not limited to binary stars, but can be applied to any astrophysical configuration in which spectral lines are generated by separate (non-rotational symmetric) regions.Comment: Accepted for publication in A&

The Effect of Depression on Pattern Separation Task Performance

The aim of this study was to investigate a possible relationship between depression and memory performance, hypothesizing a decrease in performance among participants with depression tendencies. The cognitive memory task of pattern separation was tested with four different levels of difficulty by use of a computerized memory test and the self-rating MADRS-S depression scale was used to assess the degrees of depression among the non-clinical sample of participants. Pattern separation, the cognitive ability of processing new input and distinguishing it from already stored information was tested on four levels on all participants. Participants were 40 students from Lund University, 26 women and 14 men, with a mean age of 24 (SD 4.19). The levels of depression were then analyzed and compared to the pattern separation task performance. The results proved the experiments ability to successfully test four levels of pattern separation demand, as every level showed an increase in difficulty. Although no significance was found between pattern separation and degree of depression, the results showed a tendency toward the hypothesized direction