5,875 research outputs found
Dynamics of directed graphs: the world-wide Web
We introduce and simulate a growth model of the world-wide Web based on the
dynamics of outgoing links that is motivated by the conduct of the agents in
the real Web to update outgoing links (re)directing them towards constantly
changing selected nodes. Emergent statistical correlation between the
distributions of outgoing and incoming links is a key feature of the dynamics
of the Web. The growth phase is characterized by temporal fractal structures
which are manifested in the hierarchical organization of links. We obtain
quantitative agreement with the recent empirical data in the real Web for the
distributions of in- and out-links and for the size of connected component. In
a fully grown network of nodes we study the structure of connected clusters
of nodes that are accessible along outgoing links from a randomly selected
node. The distributions of size and depth of the connected clusters with a
giant component exhibit supercritical behavior. By decreasing the control
parameter---average fraction of updated and added links per time
step---towards the Web can resume a critical structure with
no giant component in it. We find a different universality class when the
updates of links are not allowed, i.e., for , corresponding to
the network of science citations.Comment: Revtex, 4 PostScript figures, small changes in the tex
The origin of bursts and heavy tails in human dynamics
The dynamics of many social, technological and economic phenomena are driven
by individual human actions, turning the quantitative understanding of human
behavior into a central question of modern science. Current models of human
dynamics, used from risk assessment to communications, assume that human
actions are randomly distributed in time and thus well approximated by Poisson
processes. In contrast, there is increasing evidence that the timing of many
human activities, ranging from communication to entertainment and work
patterns, follow non-Poisson statistics, characterized by bursts of rapidly
occurring events separated by long periods of inactivity. Here we show that the
bursty nature of human behavior is a consequence of a decision based queuing
process: when individuals execute tasks based on some perceived priority, the
timing of the tasks will be heavy tailed, most tasks being rapidly executed,
while a few experience very long waiting times. In contrast, priority blind
execution is well approximated by uniform interevent statistics. These findings
have important implications from resource management to service allocation in
both communications and retail.Comment: Supplementary Material available at http://www.nd.edu/~network
Deterministic Scale-Free Networks
Scale-free networks are abundant in nature and society, describing such
diverse systems as the world wide web, the web of human sexual contacts, or the
chemical network of a cell. All models used to generate a scale-free topology
are stochastic, that is they create networks in which the nodes appear to be
randomly connected to each other. Here we propose a simple model that generates
scale-free networks in a deterministic fashion. We solve exactly the model,
showing that the tail of the degree distribution follows a power law
Higher order clustering coefficients in Barabasi-Albert networks
Higher order clustering coefficients are introduced for random
networks. The coefficients express probabilities that the shortest distance
between any two nearest neighbours of a certain vertex equals , when one
neglects all paths crossing the node . Using we found that in the
Barab\'{a}si-Albert (BA) model the average shortest path length in a node's
neighbourhood is smaller than the equivalent quantity of the whole network and
the remainder depends only on the network parameter . Our results show that
small values of the standard clustering coefficient in large BA networks are
due to random character of the nearest neighbourhood of vertices in such
networks.Comment: 10 pages, 4 figure
The k-core and branching processes
The k-core of a graph G is the maximal subgraph of G having minimum degree at
least k. In 1996, Pittel, Spencer and Wormald found the threshold
for the emergence of a non-trivial k-core in the random graph ,
and the asymptotic size of the k-core above the threshold. We give a new proof
of this result using a local coupling of the graph to a suitable branching
process. This proof extends to a general model of inhomogeneous random graphs
with independence between the edges. As an example, we study the k-core in a
certain power-law or `scale-free' graph with a parameter c controlling the
overall density of edges. For each k at least 3, we find the threshold value of
c at which the k-core emerges, and the fraction of vertices in the k-core when
c is \epsilon above the threshold. In contrast to , this
fraction tends to 0 as \epsilon tends to 0.Comment: 30 pages, 1 figure. Minor revisions. To appear in Combinatorics,
Probability and Computin
- …