Complex Networks Theory

At the outset of my PhD studies, there was a rapid development of networks theory that continues onto this day. A considerable amount of the work was heavily devoted to the structural properties of networks, and questions such as why are there so many networks with fat-tailed distributions of connectivity instead of these being rare. Could this be the consequence of some underlying optimization process that led to more efficient networks? But efficient for what purpose?

In the meantime, my own research was already developing in the direction of flow. But the flows I was interested in were in disordered media embedded in low dimension, such as in oil extraction in an underground oil field which can effectively be represented in most cases by a 2-dimensional situation. I suppose such interests may have had to do with my being from Venezuela, where oil is by far the most critical economic driver.

But this newly developing field intrigued me. I was curious to know if the high dimensional flows of networks could somehow teach us something we did not know about low dimension. To my great surprise, when I started to learn about the work being done in networks, I found that very little of it concerned flows. So I rolled up my sleeves and started to work.

My first article on networks was concerned with characterizing steady state flows on networks of two kinds, trying to determine what role the so-called heavy tails could play. What I found was that the transport properties of random networks are heavily mean-field: in most non-pathological cases (i.e. no unexplained presence of bottlenecks for instance) , transport ability of a network is dominated by the vicinities of the source and destination nodes. This work has become of the standard references of other work regarding flow on networks. Article here.

But structural properties of networks do matter. And the most essential of those is connectivity. If an intended flow in a network is meant to occur between two disconnected nodes, then this flow will not occur. This is what happens during flight cancelations, where it becomes impossible to reach a destination. The overarching theory that deals with such problems is called Percolation Theory, and it asks questions such as the amount of typical disruption of paths necessary to disconnect a network. However, percolation theory is in some sense an upper bound: it accepts that any path between two nodes is sufficient for there to be communication. In reality, a commuter cannot make it to work if she/he would need to travel all day to reach work (much less than that even). Therefore a more general notion of reachability needs to be applied. This is what I address in my model of limited path percolation where I find that reachability is a much more realistic and fragile notion that leads to the disruption of a system a lot earlier than predicted by connectivity. This finding is bad for communications and commuting, and good for disrupting a disease. Article here.

Other kinds of flows that I have been interested in include data transfer in networks, and nutrient flows in fungal networks. As an example of the former, in the article Structural Bottlenecks for Communication in Networks, we find limits on how much routing protocols can be improve to increase flow and avoid data packet delivery collapse in a network. In fungal networks, one can apply methods such as those used in my first articles to relate fungal growth to the delivery of nutrients to parts of the network. In the article Growth-Induced Mass Flow in Fungal Networks, we find that fungi literally grow to feed themselves, an evolutionary strategy that serves the fungus well, as if it does not grow, it cannot find new sources of food.

Finally, I have also dedicated some attention to the origin of weights on networks that may be relevant for social contexts. In this case, I have explored the possible impact that affiliation effects may play on the weights observed in otherwise pairwise network data for ties. In Weighted Projected Networks: Mapping Hypergraphs to Networks, I use ensemble theory of hypergraphs (adapted from both statistics and statistical mechanics) to determine some statistical properties of weights of networks that may emerge as a consequence of different interaction mechanisms of social groups dominated by triadic (or higher) relations. Such underlying multiway relations should produce detectable traces in the weights of a network underpinned by them, and my work finds that as such networks grow or decay, the traces should be observed.