Biological networks share some similar globabl features with other complex networks, such as internet and social networks. These shared features include small-world property, broad-tailed distribution of the connectivities, and hiarchichal modularity.
These are some controvesies in the study on biological networks.
1) Does the broad-tail distribution of the connectivities follows power-law model?
2) Are hubs in protein interaction network tend to be essential proteins?
These controversies are rooted on different interpretations of statistical analyses. The matter is further complicated when people use different datasets and criteria in their analyses.
Despite the hype around biological networks, network researchs have not yet advanced biology in meaningful ways, partially because network analyses tend to focus on "global and systems level" pictures, whereas biology is best understood from the details (in my opinion, of course). In fact, "systems" biology is not well-received by biologists. I think that network analysis should be taken from the perspective of specific biological questions.
Small-world property
As is the case in other complex networks, biological networks are also "small-world", which means that on average, there is a small number of links between a random pair of nodes in biological networks. Because of this small-world property, clustering methods based on traversal paths do not give informative clustering results. This problem is not recognized by many people in the field, as evidenced by some publications in high-profiled journals.