The majority of current research efforts that explore network-based biological questions make a series of simplifying assumptions about the nature of the networks themselves. The primary purpose of such assumptions has been to enable the application of existing techniques from subjects such as graph theory and linear algebra rather than to enable either biological accuracy or rigorous examination of the mathematical impact of the simplifications. Within this class of assumptions, four assumptions have the greatest potential to compromise the generation of valid, novel biological understanding:
- The network has only a single type of interaction among entities
- The interactions among entities can be completely described as pairwise
- The network is static over time
- The impact of noise can be ignored
To describe and study many networks in almost all areas of biology --- from protein interaction networks to functional brain networks and animal behavior networks --- it is necessary to relax these assumptions. It is important to capture time-dependence among interaction strengths and the presence of interactions themselves, explicitly consider interactions among arbitrary numbers of entities in a mathematically natural way, and make explicit the impact of noise in both descriptive and predictive methods. Generalizing existing diagnostics and analytical techniques to make them suitable for these more general situations is also necessary.
For example, as with human social networks, animals communicate over multiple channels and in ways that are not aptly described as pairwise. Indeed, each of the assumptions (1)-(4) is violated. Social insects communicate with each other using a mixture of chemical signaling, direct physical contact, visual cues, and auditory cues. These modes of communication occur simultaneously, reach different numbers of insects over different ranges, and are time-dependent. Often, individual behaviors are a function of many different types of received communications (and how they change) over time. Traditional dyadic representations fail to provide natural mathematical tools by which to characterize and predict how information flows, individuals make decisions, and consensus building is achieved.
Recent efforts, often employing differential-equation models and/or methods from statistical mechanics, have begun to study network dynamics both in time-dependent network structures and in the biological processes that operate on static network structures. There have also been many recent attempts at generalizing "ordinary" network theory and diagnostics --- both independent of, and in application to, a diverse array of biological systems. For example, hypergraphs, which provide a means to go beyond pairwise interactions, have been used to study local and global alignment across multiple species in protein-protein interaction networks. Simplicial complexes have provided novel insights into electrophysiological networks. Tensors, which provide a means to consider multiple types of interactions simultaneously, have been used to examine the learning of simple motor skills in functional brain networks. Moreover, all of these mathematical structures should incorporate noise, because noise in real biological networks has fundamental effects. The development of these new mathematical structures with built-in stochasticity will enable the translation of mathematical theory to empirically falsifiable biological hypotheses.
Existing and anticipated efforts to relax the standard four limiting assumptions represent valuable attempts at bringing a deeper and more accurate mathematical framework for the investigation of networked biological systems. The richest insights from the interface between biological and mathematical questions often arise when mathematical tools are used to make comparisons across different biological systems. Thus, by developing more general mathematical descriptions, one can enable cross-comparison of networks and thereby use theoretical and computational tools to facilitate technology transfer among different areas of biology. This makes it possible to simultaneously capture fundamental properties of biological networks (and the systems that they represent) and opens myriad possibilities for generating new, deep problems in mathematics.