Simple model for directed networks
Luis G. Morelli*
Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100Trieste, Italy
Received 24 January 2003; published 17 June 2003
We study a model for directed networks based on the Watts-Stogatz model for small-world phenomena. We
focus on some topological aspects of directed networks inspired in food web theory, namely, the fraction of
basal and top nodes in the network and node level distributions. We argue that in directed networks basal nodes
play an important role, collecting information or resources from the environment. We give analytical expres-
sions for the fraction of basal and top nodes for the model, and study the node level distributions with
numerical simulations.
DOI: 10.1103/PhysRevE.67.066107 PACS numbers: 02.50.-r, 89.75.Hc, 05.10.-a
I. INTRODUCTION
The subject of complex networks has become relevant in
several branches of science. Sometimes, complex systems
can be described as a large set of interacting units. Regard-
less of the nature of these units and the way they interact
with each other, it is possible to define a network describing
the interactions present in the system, thinking of the units as
the nodes of the network and placing a link between two
units whenever an interaction is present. The interest in com-
plex networks has been triggered both by novel theoretical
approaches 1–3 and by the availability of large datasets for
networks of different origins. In the recent literature, a sys-
tematic study of the topological properties of neural 1,4,
ecological 5–7, metabolic 8, transport 9, technological
1,10,11 and social 1,9,12,22,23 networks has been carried
out, revealing a rich subject with a wide scope of applica-
tions.
Most of the models proposed to account for the observed
features in real networks consist of undirected networks 13,
i.e., networks in which the links connecting two nodes have
no definite direction. However, in many cases of interest the
interactions are not symmetric. A natural way to describe
these networks is to consider directed links. One example of
directed networks comes from ecology, where ecosystems
can be represented by food webs 14. In predator-prey food
webs, each species is represented as a node of the network,
and a link is placed between two species whenever one of
them feeds on the other. Food webs describe in this way the
who-eats-who interactions in the ecosystem. The natural way
to describe the flow of resources from prey to predators is to
consider a directed network.
1
Neural networks provide an-
other example of directed networks. In fact, neurons connect
to other cells with their axon, and receive connections from
other axon cells through their dendrites. This asymmetry of
the cell can be accounted for with a directed link. In ener-
getic networks as the power grid, the energy flows in some
definite direction, from providers to consumers. Also in
economy, directed networks play an important role. The net-
work of goods necessary to produce other goods is an ex-
ample. Hierarchical social networks, in which the relation-
ships between people are not symmetric, give yet another
example of directed networks.
The relevance of directed networks has already been rec-
ognized and emphasized 13,17. A model for the World
Wide Web with directed links has been proposed 16 and a
spreading process occurring on a directed network has been
studied 18. The problem of percolation in a directed scale-
free network 19 and the dynamics of a spin model in a
directed small world network 20 have also been consid-
ered.
In this paper, we propose a model for directed networks
based on the Watts-Stogatz model for small-world phenom-
ena. It is a simple extension of the model, modified to ac-
count for the direction of the links. We focus our study on
some static topological properties inspired in food web
theory 15. In the following section, we introduce some
definitions concerning the topology of directed networks. In
the following, we present the model and give analytical ex-
pressions for the fraction of basal and top nodes. Then we
show numerical results for the fraction of basal nodes and the
node level distribution. We finish the paper with a discussion
of the results and some remarks.
II. SOME ASPECTS OF FOOD WEB TOPOLOGY
We therefore begin with a brief discussion of some as-
pects of network topology inspired in food web theory. A
very coarse grained classification of the species in a food
web is based on the fraction of basal species, the fraction of
top species, and the fraction of intermediate species in the
web. Basal species feed only on the environment, and have
no prey. Top species have no predators. Intermediate species
have both preys and predators. Here we define basal, top, and
intermediate nodes in an analogous way. Basal nodes have
only outgoing links, intermediate nodes have both outgoing
and incoming links, and top nodes have only incoming links.
The fractions of basal, intermediate, and top nodes in the
network are noted as B, I, and T. Note that by definition, B
+T +I =1.
In food web models, basal species feed only on the envi-
ronment. The ecosystem is an open system, and resources
*Electronic address: morelli@ictp.trieste.it
1
In some contexts, though, food webs can be represented by un-
directed networks because a fluctuation in the population of a spe-
cies has consequences both on the populations of its preys and its
predators.
PHYSICAL REVIEW E 67, 066107 2003
1063-651X/2003/676/0661077/$20.00 ©2003 The American Physical Society 67 066107-1