53 research outputs found
The birth of the strong components
Random directed graphs undergo a phase transition around the point
, and the width of the transition window has been known since the
works of Luczak and Seierstad. They have established that as
when , the asymptotic probability that the strongly
connected components of a random directed graph are only cycles and single
vertices decreases from 1 to 0 as goes from to .
By using techniques from analytic combinatorics, we establish the exact
limiting value of this probability as a function of and provide more
properties of the structure of a random digraph around, below and above its
transition point. We obtain the limiting probability that a random digraph is
acyclic and the probability that it has one strongly connected complex
component with a given difference between the number of edges and vertices
(called excess). Our result can be extended to the case of several complex
components with given excesses as well in the whole range of sparse digraphs.
Our study is based on a general symbolic method which can deal with a great
variety of possible digraph families, and a version of the saddle-point method
which can be systematically applied to the complex contour integrals appearing
from the symbolic method. While the technically easiest model is the model of
random multidigraphs, in which multiple edges are allowed, and where edge
multiplicities are sampled independently according to a Poisson distribution
with a fixed parameter , we also show how to systematically approach the
family of simple digraphs, where multiple edges are forbidden, and where
2-cycles are either allowed or not.
Our theoretical predictions are supported by numerical simulations, and we
provide tables of numerical values for the integrals of Airy functions that
appear in this study.Comment: 62 pages, 12 figures, 6 tables. Supplementary computer algebra
computations available at https://gitlab.com/vit.north/strong-components-au
Measuring social dynamics in a massive multiplayer online game
Quantification of human group-behavior has so far defied an empirical,
falsifiable approach. This is due to tremendous difficulties in data
acquisition of social systems. Massive multiplayer online games (MMOG) provide
a fascinating new way of observing hundreds of thousands of simultaneously
socially interacting individuals engaged in virtual economic activities. We
have compiled a data set consisting of practically all actions of all players
over a period of three years from a MMOG played by 300,000 people. This
large-scale data set of a socio-economic unit contains all social and economic
data from a single and coherent source. Players have to generate a virtual
income through economic activities to `survive' and are typically engaged in a
multitude of social activities offered within the game. Our analysis of
high-frequency log files focuses on three types of social networks, and tests a
series of social-dynamics hypotheses. In particular we study the structure and
dynamics of friend-, enemy- and communication networks. We find striking
differences in topological structure between positive (friend) and negative
(enemy) tie networks. All networks confirm the recently observed phenomenon of
network densification. We propose two approximate social laws in communication
networks, the first expressing betweenness centrality as the inverse square of
the overlap, the second relating communication strength to the cube of the
overlap. These empirical laws provide strong quantitative evidence for the Weak
ties hypothesis of Granovetter. Further, the analysis of triad significance
profiles validates well-established assertions from social balance theory. We
find overrepresentation (underrepresentation) of complete (incomplete) triads
in networks of positive ties, and vice versa for networks of negative ties...Comment: 23 pages 19 figure
Threshold phenomena involving the connected components of random graphs and digraphs
We consider some models of random graphs and directed graphs and investigate their behavior near thresholds for the appearance of certain types of connected components. Firstly, we look at the critical window for the appearance of a giant strongly connected component in binomial random digraphs. We provide bounds on the probability that the largest strongly connected component is very large or very small. Next, we study the configuration model for graphs and show new upper bounds on the size of the largest connected component in the subcritical and barely subcritical regimes. We also show that these bounds are tight in some instances. Finally we look at the configuration model for random digraphs. We investigate the barely sub-critical region and show that this model behaves similarly to the binomial random digraph whose barely sub- and super-critical behaviour was studied by Luczak and Seierstad. Moreover, we show the existence of a threshold for the existence of a giant weak component, as predicted by Kryven.En aquesta tesi considerem diversos models de grafs i graf dirigits aleatoris, i investiguem el seu comportament a prop dels llindars per l'aparició de certs tipus de components connexes. En primer lloc, estudiem la finestra crÃtica per a l'aparició d'una component fortament connexa en dÃgrafs aleatoris binomials (o d'Erdos-Rényi). En particular, provem diversos resultats sobre la probabilitat lÃmit que la component fortament connexa sigui sigui molt gran o molt petita. A continuació, estudiem el model de configuració per a grafs no dirigits i mostrem noves cotes superiors per la mida de la component connexa més gran en els règims sub-crÃtics i quasi-subcrÃtics. També demostrem que, en general, aquestes cotes no poden ser millorades. Finalment, estudiem el model de configuració per a dÃgrafs aleatoris. Ens centrem en la regió quasi-subcrÃtica i demostrem que aquest model es comporta de manera similar al model binomial, el comportament del qual va ser estudiat per Luczak i Seierstad en les regions quasi-subcrÃtica i quasi-supercrÃtica. A més a més, demostrem l'existència d'una funció llindar per a l'existència d'una component feble gegant, tal com va predir Kryven.Postprint (published version
A classification of isomorphism-invariant random digraphs
We classify isomorphism-invariant random digraphs \linebreak (IIRDs) according to where randomness lies, namely, on arcs, vertices, vertices and arcs together as arc random digraphs (ARD), vertex random digraphs (VRD), vertex-arc random digraphs (VARD) as an extension of the classification of isomorphism-invariant random graphs (IIRGs) \cite{beer:2011}, and introduce randomness in direction (together with arcs, vertices, etc.) also which in turn yield direction random digraphs (DRDs) and its variants, respectively. We demonstrate that for the number of vertices , ARDs and VRDs are mutually exclusive and are both proper subsets of VARDs, and also demonstrate the existence of VARDs which are neither ARDs nor VRDs, and the existence of IIRDs that are not VARDs (e.g., random nearest neighbor digraphs(RNNDs)). We demonstrate that to obtain a DRD as an IIRD, one has to start with an IIRG and insert directions randomly. Depending on the type of IIRG, we obtain direction-edge random digraphs (DERDs), direction-vertex random digraphs (DVRDs), and direction-vertex-edge random digraphs (DVERDs), and demonstrate that DERDs and DVRDs have an overlap but are mutually exclusive for , and both are proper subsets of DVERDs which is a proper subset of DRDs and also the complement of DRDs in IIRDs is nonempty (e.g., RNNDs). We also study the relation of DRDs with VARDs, VRDs, and ARDs and show that for , the intersection of DERDs and VARDs is ARDs; we provide some results and open problems and conjectures. For example, the relation of DVRDs and DVERDs with the VARDs (hence with ARDs and VRDs) are still open problems for . We also show positive dependence between the arcs of a VARD whose tails are same which implies the asymptotic distribution of the arc density of VRDs and ARDs has nonnegative variance
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