A small world network is
characterized by a low average path length between any two nodes, and high
clustering. Most real networks, including biological networks like protein
interaction networks, are known to have
this property. Since random networks are also known to have the property of low
diameter, one might think that real world networks have even shorter diameters
than their random counterparts. The reason for this is that short diameters can
potentially increase the network efficiency of exchanging mass and/or
information, not only in biological networks (e.g. the metabolic network) but
also in transportation, communication and computer networks. However, studies
at the Department of Ecology and
Evolutionary Biology, University of Michigan have shown that the observed diameter in real world networks is greater than the random expectation. The following are some of the salient observations of this study.

The diameter of a real
network was compared to that of its randomly rewired network in which the
connections between nodes were randomized while the degree of every node
remained unchanged. A total of 13 real networks examined, including two
linguistic, three technological, four social, and four biological networks,
were found to have diameters greater than their random expectations (Table 1). An analysis of the frequency
distribution of diameters showed that the greater than expected diameters are
not caused by presence of some extremely long paths, but due to the presence of
many elongated minimal paths. Overall, the diameters of real networks were
found to be 2.3–128.3% greater than
their random expectations, with a median difference of 17.4%. Albert and Barabási had also noted earlier
that many real networks have longer diameters than those computed under the
power-law degree distribution.

Table 1

**Real networks have greater-than-expected clustering coefficients**

Small world networks have
high clustering coefficients. We all have seen this in the case of protein
interaction networks. In 12 of the 13 real networks examined (except the
dolphin association network), the clustering coefficient C was
found to be greater than the expected value determined from its randomly
rewired networks, and this difference is statistically significant in 7 cases (Table 2). However, there is no consistent
relationship between clustering coefficient and diameter among the randomly
rewired networks of each real network (Table 2), suggesting that the
greater-than-expected diameters of real networks cannot be explained by the
greater-than-expected clustering coefficients. Furthermore, across the 13 real
networks, the correlation between the expected

*C*and expected*D*from their randomly rewired networks appears negatively (Spearman's*R*= −0.57,*P*= 0.047), while the correlation between the*Z*-score for*C*and*Z*-score for*D*is not significant (Spearman's*R*= −0.13,*P*= 0.68).
Table 2

**Network modularization enlarges the diameter**

If shorter diameters are
beneficial to at least some networks, why do all networks have longer diameters
than expected by chance? It was hypothesized that this phenomenon might relate
to modularity (Newman and Girvan) in networks. Simulations were conducted to find the relation
between diameter and modularity for a network with a fixed amount of nodes and
edges. It was observed that the network
diameter increases as the modularity increases in these simulated networks (Fig 1a). But the relationship between
diameter and modularity is not linear; when the diameter is short, a small
percentage increase in diameter allows a substantial percentage increase in
modularity (Fig 1a). A similar concave curve is observed
when the increases in diameter and modularity are measured by

*Z*-scores, rather than the absolute values (Fig 1b). Using a similar simulation, the relationship between modularity and diameter using networks with a fixed number of modules but different mean degrees was also confirmed.
If modularization is
truly the cause of the higher-than-expected diameters of real networks, all
real networks should have modularities greater than expected from their
randomly rewired networks. This is indeed the case (Table 1). The percentage excess in modularity
(compared to the random expectation) ranges from 4.8 to 206.9% for the 13
networks, with a median of 40.8%. This percentage excess exceeds that for
diameter in 10 of the 13 networks, a nonrandom pattern that is consistent with
the simulation result in Fig 1 (

*P*= 0.046, one-tail binomial test).**Implications**

All networks studied,
including biological networks, have diameters greater than their random
expectations. This suggests that modularization may well be a universal
characteristic of real networks, due to the advantages it brings to network
multi-functionality, robustness and evolvability. As a consequence, the network
diameter has to be sacrificed to accommodate modular structures. Because
shorter diameters could provide higher functional efficiency, the result
suggests a tradeoff between network efficiency and multi-functionality,
robustness, and/or evolvability. Although there are many networks unstudied in
this work, this analysis covers major types of networks and the results are
likely to reflect a general pattern of real networks. It would be an
interesting study to look for those rare real world networks for whom the diameter
is actually less than their random expections and examine in what way are they different from general real world networks.

References:

[1] Zhang Z, Zhang J (2009) A Big World Inside Small-World Networks. PLoS ONE 4(5): e5686. doi:10.1371/journal.pone.0005686

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