Topology of networks

Topology of Networks: Beyond Dimension One.

Networks permeate our life. The most immediate examples are the World Wide Web, the electric grid, the nervous system, metabolic networks, proteins in our bodies, collaboration and citation relations in the sciences, the network of film actors, and so on. The traditional approach to modeling and studying networks is to consider all participants as vertices of a graph and relations between pairs of them as edges of this graph. This approach may be inadequate as it often ignores important features of the network. We suggest studying networks via mathematical models that have multidimensional structure. These objects are called simplicial complexes (also hypergraphs). Just as graphs they are built of vertices and edges but also of higher dimensional simplices, i.e., triangles, tetrahedra, etc.

Consider, for example, three molecules, A, B, and C, that participate, in separate pairs, in three different chemical reactions, P, Q, and R. Then, following the traditional approach one connects pairs of vertices by edges, AB, BC, and CA, representing these reactions. As a result we have a triangular frame with vertices A, B, C as a model of this elementary network. Now suppose alternatively that all of these molecules participate in a single reaction R. The graph representation of this network is still the same triangular frame with each of the three edges (inadequately) representing the same reaction R. On the other hand, the simplicial representation of this network is the whole triangle ABC spanned on this frame. This triangle, just as each edge in the previous example, uniquely corresponds to a chemical reaction. Therefore unlike the traditional model, this approach distinguishes these two distinctly different situations by preserving all information about the network.

In general, the model of a network will be a combination of simplices adjacent to each other. Meanwhile its graph representation is just the (one-dimensional) “frame” of this more general, and more complete, simplicial representation. The multidimensionality of this representation is a crucial advantage over the representation via graphs or even bipartite graphs as it provides a better setting for studying processes on networks, such as the flow of information, spread of a disease or a computer virus.

Mathematically, ours is a study of topology of random simplicial complexes. The tools will be the ones of algebraic topology which studies simplicial complexes by assigning to them certain algebraic structures (such as homology groups). These structures numerically represent the connectivity of the complex not only in terms of components as graphs theory does, but also for loops, holes, voids, etc. In general, instead of considering the place of each vertex in the network topology studies the “big picture”.

This is a truly multidisciplinary research project. Indeed, the expertise for this study is necessary in the following disciplines: algebraic topology, combinatorics, statistics, computer science, as well as the application areas. To study real life networks one needs to collect data from the Internet, the World Wide Web, as well as various databases. To process this information, effective algorithms need to be designed and implemented for computation of the relevant topological and other invariants. Another approach to real life networks will be computer modelling and statistical study of random simplicial complexes.