Recently Watts and Strogatz have given an interesting model of small-world
networks. Here we concretise the concept of a ``far away'' connection in a
network by defining a {\it far edge}. Our definition is algorithmic and
independent of underlying topology of the network. We show that it is possible
to control spread of an epidemic by using the knowledge of far edges. We also
suggest a model for better advertisement using the far edges. Our findings
indicate that the number of far edges can be a good intrinsic parameter to
characterize small-world phenomena.Comment: 9 pages and 6 figure

D. J. Watts

J. Gaure

J. J. Collins

J. J. Hopfield

R. E. Amritkar

S. A. Pandit

S. Milgram

English

arXiv

Recently, Watts and Strogatz [Nature (London) 393, 440 (1998)] offered an interesting model of small-world networks. Here we concretize the concept of a "faraway" connection in a network by defining a far edge. Our definition is algorithmic and independent of any external parameters such as topology of the underlying space of the network. We show that it is possible to control the spread of an epidemic by using the knowledge of far edges. We also suggest a model for better product advertisement using the far edges. Our findings indicate that the number of far edges can be a good intrinsic parameter to characterize small-world phenomena

Pandit, S. A.

Amritkar, R. E.

Indian Academy of Sciences

Characterization and control of small-world networks

Crossref

Publications of the IAS Fellows

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Characterization and control of small-world networks
S. A. Pandit∗ and R. E. Amritkar†
Physical Research Laboratory, Navarangpura, Ahmedabad, 380 009, India
Abstract
Recently Watts and Strogatz have given an interesting model of small-world
networks. Here we concretise the concept of a “far away” connection in a
network by defining a far edge. Our definition is algorithmic and independent
of underlying topology of the network. We show that it is possible to control
spread of an epidemic by using the knowledge of far edges. We also suggest
a model for better advertisement using the far edges. Our findings indicate
that the number of far edges can be a good intrinsic parameter to characterize
small-world phenomena.
Typeset using REVTEX
∗e-mail: sagar@prl.ernet.in
†e-mail: amritkar@prl.ernet.in
1
The properties of very large networks are mainly determined by the way the connections
between the vertices are made. At one extreme are the regular networks where only the
“local” vertices are inter-connected and the “far away” vertices are not connected while at
the other extreme are the random networks where the vertices are connected at random.
The regular networks display a high degree of local clustering and the average distance be-
tween vertices is quite large. On the other hand, the random networks show negligible local
clustering and the average distance between vertices is quite small. The small-world net-
works [1,2] have intermediate connectivity properties but exhibit a high degree of clustering
as in regular networks and small average distance between vertices as in random networks.
A very interesting model for small-world networks was recently proposed by Watts and Stro-
gatz [3]. They found that a regular network acquires the properties of a small-world network
with only a very small fraction of connections or edges (about 1%) rewired to “far away”
vertices. They demonstrated that several diverse phenomena like neural networks [5], power
grids and collaboration graphs of film actors [4] can be modeled using small-world networks.
Also the spread of an epidemic is much faster in small-world networks than in the regular
networks and almost close to that of random networks.
In this paper we suggest a possible way of characterizing small-world networks. The
basic ingredients of small-world networks are the “far away” connections. We introduce a
notion of far edges in a network to identify these “far away” connections. Our definition
of a far edge is independent of any underlying topology for a network and depends only on
the way connections or edges are made. We claim that the rapid spread of an epidemic
in small-world network as found by Watts and Strogatz [3] is due to these far edges. This
allows us to propose a mechanism to control the epidemic using the same far edges which
are responsible for the rapid spread. We further demonstrate the utility of our notion of far
edges by giving an better method of advertisement.
Consider a graph (network) with n vertices and E edges. Let N νij denote the number of
distinct paths of length ν between the vertices i and j. For a simple graph, N 1ij is one if
there is an edge between vertices i and j else it is zero. We now concretise the idea of “far
2
away” connections by defining a far edge. Let an edge eij between vertices i and j be a far
edge of order µ if it is an edge for which N µ+1ij = 0 and N
l
ij 6= 0 for all l ≤ µ.
Fig. 1 shows an example of a far edge of order one. We note that none of the edges in
a completely connected graph are far edges, while all edges in a tree are far edges of order
one. Hence forth we will assume that a far edge has order one unless stated otherwise.
To generate small-world networks and also other type of networks we follow the procedure
given in Ref. [3]. We start with a regular network consisting of a ring of n vertices with edges
connecting each vertex to its k nearest neighbours. Each edge is rewired with probability p
avoiding multiple edges. The p = 1 case corresponds to a random network. The networks
obtained with p ≈ 0.01 correspond to small-world networks [3].
We have generated several networks from regular (p = 0) to random (p = 1) case. For
each network we calculate the average path length L(p) and clustering coefficient C(p). The
quantity L(p) denotes the average length of the shortest path between two vertices, and
C(p) denotes the average of Cv over all the vertices v, where Cv is the number of edges
connecting the neighbours of v normalized with respect to the maximum number of possible
edges between these neighbours [3]. Next we determine the far edges in these networks. Let
F denote the ratio of number of far edges with the total number of edges. We find that
initially, to a good approximation, F is equal to p for p ≤ 0.1 and then it increases slowly
till it saturates to a value of about 0.2 for p = 1. It turns out that the number far edges of
order higher than one are negligible.
In Fig. 2 we plot C(F)/C(0) and L(F)/L(0) as functions of F . This figure is similar
in nature to the plot of C(p)/C(0) and L(p)/L(0) as functions of p (Fig. 2 of Ref. [3]).
The small-world networks can be identified as those with C(p)/C(0) ≈ 1 and L(p)/L(0) ≈
L(1)/L(0). From Fig. 2 we see that this corresponds to F ≈ 0.01. Thus F can be used as
a parameter to characterize networks which interpolate between regular and random cases.
We note that F is an intrinsic quantity and does not depend on the procedure of generating
networks and hence should prove to be a better parameter than p.
To further investigate the importance of far edges, we consider the problem of spread of
3
an epidemic. Consider an epidemic starting from a random vertex (seed). We assume that
at each time step all the neighbours of infected vertices are affected with probability one,
which is the most infectious case, and the vertices which are already affected die and play
no further role in the spread of the epidemic. Here, neighbours of a given vertex means all
the vertices which are joined to it by edges. As found by Watts and Strogatz [3], the spread
of an epidemic in small-world networks is almost as fast as that in the random case. We
propose that the mechanism for the rapid spread of epidemic in small-world networks is due
to the traversal of the disease along the far edges. Each such traversal opens a virgin area
for the spread of epidemic leading to a rapid growth.
Clearly if the far edges are responsible for the rapid growth of epidemic then we should
be able to effectively control the spread by preventing the traversal of epidemic along the far
edges. To test this hypothesis, we propose the following mechanism to control an epidemic.
We assume that we have sufficient knowledge of the network and we have identified all the
far edges. We note that identification of far edges requires only the knowledge of vertices
and edges and hence should be possible in many practical situations. Let τ denote the time
steps elapsed between the beginning of the epidemic and its detection. Let m denote the
number of vertices that can be immunized at each time step. To block a far edge we first
immunize one of the two vertices connected by this far edge. Immunization is carried out by
first blocking all the far edges and then immunizing at random. If the number of far edges
is greater than m then blocking all the far edges will take more than one time step.
In Fig. 3 we show the fraction of vertices affected as a function of time steps for a small-
world network. Curve (a) shows the uncontrolled spread of the epidemic. Curves (d) and
(g) show the spread of epidemic with the control method suggested above for τ = 7 and 2
respectively. For comparison we show, by curves (c) and (f), the epidemic with only random
immunization for τ = 7 and 2 respectively. It is obvious that the far edge control mechanism
proposed here is very effective. For larger τ some of the far edges are already traversed
by the epidemic, decreasing the efficiency of our control mechanism. Comparing the far
edge immunization and the random immunization, we find that the far edge immunization
4
decreases the rate of spread of epidemic more effectively but takes longer time for completely
stopping the spread (See Fig. 3, curves (d) and (g)). Further, to test the effectiveness of
our method we compare the results with another method of immunization. We order the
vertices by their degree. Immunization is carried out by starting with the vertex with the
largest degree and then going down the degree. The results for τ = 7 and 2 are shown as
curves (b) and (e) in Fig. 3 respectively. We note that results for immunization using degree
are similar to that of the random immunization.
Let d denote the asymptotic difference between the number of affected vertices in random
and far edge immunization. We plot d as a function of m for three different values of F (or
p) in Fig. 4. The plot shows that the far edge immunization is most effective when m is
about half the number of far edges. The reason for the decrease of d for large m is that the
probability that random immunization blocks a far edge, keeps on increasing as m increases,
thereby decreasing the difference between the two methods. The plot of d as a function of F
for different values of m is shown in Fig. 5. The figure shows that the far edge immunization
is more effective for small-world networks. Also from Figs. 4 and 5 it is clear that the far
edge immunization gives a substantial benefit in terms of number of unaffected vertices in
the small-world case and this number can be as large as 410 which is more than 40% of the
total number of vertices.
Now, we consider an interesting model of advertisement. Let r be the number of vertices
or centers from where a product is advertised. The information about the product spreads
by word of mouth to the neighbours with the probability qt where t is the time elapsed
from the initial advertisement. We compare the results of two different ways of choosing the
initial centers. In one way the centers are chosen at random and in the other they are chosen
as one of the vertex in a far edge. Fig. 6 shows the number of people informed about the
product as a function of t. It is clear that the choice of centers using far edges has definite
advantage over that of random choice.
To conclude we have introduced the concept of far edges in networks. Our definition
of a far edge is in accordance with the intuitive idea of a “far away” connection between
5
two vertices. The advantage of our definition of far edge is that it is independent of the
underlying topology of the network. Also the definition is algorithmic in nature, and allows
the determination of far edges only from the knowledge of vertices and edges. We have also
applied the idea of far edges to the networks which are not generated by the algorithm given
in Ref. [3] and arrived at similar conclusions [6].
We have demonstrated the use of far edges in the control of the spread of an epidemic
and the advertisement of products. Our simulations show that the far edges are indeed
important in the spread of epidemic, particularly in the small-world networks. We have
shown that the knowledge of far edges can be fruitfully utilized to control the spread of
epidemic and better advertisement. Our results strongly indicate that the far edges are the
key elements responsible for the special properties of small world phenomena.
6
REFERENCES
[1] S. Milgram, Psychol. Today 2, 60-67 (1967)
[2] J. Gaure in Six Degrees of separation: A Play (Vintage Books, New York, 1990)
[3] D. J. Watts, S. H. Strogatz, Nature 393 , 440 (1998)
[4] J. J. Collins, C. C. Chow, Nature 393, 409 (1998)
[5] J. J. Hopfield, A. V. M. Herz, Proc. Natl. Acad. Sci. USA 92, 6655 (1995)
[6] S. A. Pandit, R. E. Amritkar, Unpublished
7
Figure Captions:
Figure 1 An example of a network consisting of a far edge. The edge between vertices ‘a’
and ‘b’ is a far edge of order one.
Figure 2 The graph of C(F)/C(0) and L(F)/L(0) as a function of F , where C is the
clustering coefficient, L is the average path length and F is the ratio of the number of
far edges with the total number of edges. This figure is similar in nature to the plot
of C(p)/C(0) and L(p)/L(0) as functions of p. The small-world networks lie around
F = 0.01.
Figure 3 The graphs of fraction of vertices affected as a function of time steps. The curve
(a) is the epidemic spread without immunization, the curves (c) and (f) represent
the spread when the random immunization is applied (see text) for τ = 7 and 2
respectively, the curve (b) and (e) shows the spread if the immunization is carried out
for the vertices with highest degree first and then in descending degree for τ = 7 and 2
respectively and the curves (d) and (g) are the spread when the far edge immunization
is used τ = 7 and 2 respectively. The simulations are carried out on a small-world
network of 1000 vertices and 10000 edges. The plotted results are averaged quantities
over 500 seeds for epidemic.
Figure 4 The graph of the asymptotic difference between the number of affected vertices
in random and far edge immunization, d as function of number of vertices immunized
in one time step, m. The three curves (a), (b) and (c) are for F = 0.0022, 0.0084
and 0.0162 respectively. The curve (b) corresponds to small-world network. The other
parameters are as in Fig. 3.
Figure 5 The graph of d as function of F . The three curves (a), (b) and (c) are plotted
for m = 30, 10 and 80 respectively. The figure shows that the immunization method
suggested here is most effective in small-world networks.
8
Figure 6 The graph of number people informed as function of t. The curves (a) and (b)
show the result for far edge centers and random centers respectively. The simulation
is carried out on a small-world network with 1000 vertices and 10000 edge. The
initial advertisement is done from five centers. The probability function qt is chosen
as q1 = 0.8 and qi = 0.18, where i ≥ 2.
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Characterization and control of small-world networks

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