More than numbers

Numbers are one way we measure the world. From the negative temperatures on a thermometer, to our bank account balances precise to the nearest cent, or to the number of followers we have on Twitter, we use numbers routinely in our everyday lives. Our early education reflects the importance of numbers, and we learn arithmetic before we learn other, more sophisticated kinds of mathematics.

Mathematics, however, is more than about numbers. A fundamental aspect of nature is the relationship between objects. For example, consider cities on a map and roads connecting them. Or imagine the vast social network of Facebook representing our on-line friendships. Within the cells of our body, there are thousands of proteins interacting with each other biochemically. Networks pervade our world even though they may be hidden from sight.

The role of networks

Networks matter. They allow us to quantify and visualize relationships. A network consists of set of objects or nodes, which may or may not be connected be lines or edges. There are different kinds of networks, just like there are different kinds of numbers. Edges in a network may be directed, where information or traffic flows one way (as in one way streets).

Mathematicians studied networks for centuries, but the subject flourished in the 21^st century. In the era of big data, we can now map out networks with trillions of nodes, just as Google does when it crawls web sites gathering information for its search engine ranking. Social networks such as Facebook and Instagram, protein interaction networks in living cells, and stocks in a financial market and their correlations, are all examples of complex networks. Mathematicians work alongside computer scientists, physicists, and biologists to develop models to better understand complex networks.

We’ve made amazing strides in the modelling and mining of complex networks; however, the subject is in its infancy. For example, we don’t have rigorous mathematical definition of what forms a community in a network. Further, we don’t understand precisely the mechanisms for how complex networks evolve over time.  Even the mining complex networks is a challenge, and is especially so in the biological sciences, where protein data sets are notoriously noisy.

Hidden reality of networks

My research on complex networks aims at uncovering what we call the hidden reality of networks. A powerful tool for this direction is to visualize the network as geometric, where nodes live in some higher dimensional space, and nodes are close in the space if they share similar features. The notion of Blau space in sociology takes that view, where nodes (who are people, in this case) are more likely connected if they share similar socio-demographic features.

We are just beginning to learn how to quantify Blau space; for example, how do we measure distance in the space, and what is the dimension of the space? My research attempts to answer these questions, using a synthesis of mathematical modelling and machine learning. The spread of memes in a network, mapping out character networks in fictional works such as novels and films, and modelling negatively correlated stocks are a few of the interdisciplinary projects on complex networks I’ve worked on with my students and collaborators.

Networks are all around us. The next time you are in a crowd, say at the mall or downtown Toronto, try to imagine the invisible lines connecting the people there. Within the variously sized social clusters, there may be leaders or hubs who glue the groups together. The clusters can form disparate islands, or connect to the larger crowd.  When you critically view your social interaction in this way, it will help remind you that our world is networked—just as frigid, winter mornings remind us of the prevalence of negative numbers. Networks form the backbone of our connected world, and play a central role in the mathematics of the 21^st century.

Anthony Bonato