Define Future challenges for dynamical network?
Graph theory is an old field of mathematics where biological applications are driving new advances. Mathematicians are supplementing famous classes such as random or Markovian networks with new classes of networks such as "small world" networks (where most nodes are locally connected but a few have long- range links) (Watts and Strogatz 1998) and "scale-free" networks (where node degree follows a power-law distribution: most nodes are connected to only a few neighbors but a few nodes are connected to many neighbors: Albert and Barabasi 2002). In addition to knowing the mathematical properties of such networks, biologists will need to develop new statistical methods to supplement existing methods from sociology for reconstructing the structure of a network from observations on individuals (Morris 1993). Perhaps the biggest open mathematical challenges, though, are in understanding the dynamic properties of networks that cannot be derived from static measures of their structure. Networks may have evolved dynamically by the non-random addition and subtraction of links to their current state, as in the case of food webs (Dunne et al. 2002); they may continue to change over the period of observation, as in the case of contact networks for sexually transmitted disease (Altmann 1995); or they may be relatively static, but form the substrate for dynamic changes in the states of their nodes, as in the case of measles passing among school children or foot-and-mouth disease virus epidemics passing among farms (Keeling 1999, Keeling et al. 2001). All of these cases require mathematical, statistical, and computational methods well further than our current knowledge base before we can fully understand the emergent properties of the biological networks concerned.