6,510 research outputs found
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
On Computing the Diameter of (Weighted) Link Streams
A weighted link stream is a pair (V,?) comprising V, the set of nodes, and ?, the list of temporal edges (u,v,t,?), where u,v are two nodes in V, t is the starting time of the temporal edge, and ? is its travel time. By making use of this model, different notions of diameter can be defined, which refer to the following distances: earliest arrival time, latest departure time, fastest time, and shortest time. After proving that any of these diameters cannot be computed in time sub-quadratic with respect to the number of temporal edges, we propose different algorithms (inspired by the approach used for computing the diameter of graphs) which allow us to compute, in practice very efficiently, the diameter of quite large real-world weighted link stream for several definitions of the diameter. Indeed, all the proposed algorithms require very often a very low number of single source (or target) best path computations. We verify the effectiveness of our approach by means of an extensive set of experiments on real-world link streams. We also experimentally prove that the temporal version of the well-known 2-sweep technique, for computing a lower bound on the diameter of a graph, is quite effective in the case of weighted link stream, by returning very often tight bounds
The phase transition in inhomogeneous random graphs
We introduce a very general model of an inhomogenous random graph with
independence between the edges, which scales so that the number of edges is
linear in the number of vertices. This scaling corresponds to the p=c/n scaling
for G(n,p) used to study the phase transition; also, it seems to be a property
of many large real-world graphs. Our model includes as special cases many
models previously studied.
We show that under one very weak assumption (that the expected number of
edges is `what it should be'), many properties of the model can be determined,
in particular the critical point of the phase transition, and the size of the
giant component above the transition. We do this by relating our random graphs
to branching processes, which are much easier to analyze.
We also consider other properties of the model, showing, for example, that
when there is a giant component, it is `stable': for a typical random graph, no
matter how we add or delete o(n) edges, the size of the giant component does
not change by more than o(n).Comment: 135 pages; revised and expanded slightly. To appear in Random
Structures and Algorithm
Synchronization in complex networks
Synchronization processes in populations of locally interacting elements are
in the focus of intense research in physical, biological, chemical,
technological and social systems. The many efforts devoted to understand
synchronization phenomena in natural systems take now advantage of the recent
theory of complex networks. In this review, we report the advances in the
comprehension of synchronization phenomena when oscillating elements are
constrained to interact in a complex network topology. We also overview the new
emergent features coming out from the interplay between the structure and the
function of the underlying pattern of connections. Extensive numerical work as
well as analytical approaches to the problem are presented. Finally, we review
several applications of synchronization in complex networks to different
disciplines: biological systems and neuroscience, engineering and computer
science, and economy and social sciences.Comment: Final version published in Physics Reports. More information
available at http://synchronets.googlepages.com
Parallel Graph Algorithms in Constant Adaptive Rounds: Theory meets Practice
We study fundamental graph problems such as graph connectivity, minimum
spanning forest (MSF), and approximate maximum (weight) matching in a
distributed setting. In particular, we focus on the Adaptive Massively Parallel
Computation (AMPC) model, which is a theoretical model that captures
MapReduce-like computation augmented with a distributed hash table.
We show the first AMPC algorithms for all of the studied problems that run in
a constant number of rounds and use only space per machine,
where . Our results improve both upon the previous results in
the AMPC model, as well as the best-known results in the MPC model, which is
the theoretical model underpinning many popular distributed computation
frameworks, such as MapReduce, Hadoop, Beam, Pregel and Giraph.
Finally, we provide an empirical comparison of the algorithms in the MPC and
AMPC models in a fault-tolerant distriubted computation environment. We
empirically evaluate our algorithms on a set of large real-world graphs and
show that our AMPC algorithms can achieve improvements in both running time and
round-complexity over optimized MPC baselines
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