The study of complex networks has a huge and growing literature, and has even been called the field of network science . Much of our research is motivated by the need to formulate mathematical principles that are common to various types of complex networks. For example, cell biology, communication webs, and search engines all need to process noisy, uncertain, and incomplete data that is potentially stochastic. We may describe the cell cycle with empirical distributions driven only by partially-known environmental cues and intracellular checkpoints. In turn, search engines employ empirical distributions based on the very limited sampling of features that are often not well understood. This motivates problems that include the characterization of robustness, reliability, and possible uncertainty principles. Our goal is to investigate how curvature and other intrinsic geometric/topological properties affect these features. In short, we want to develop the necessary theory and tools that will permit the understanding and management of network dynamics at various scales.
Figure 1. Network curvature, robustness, and entropy are all positively correlated. Image credit: Liangjia Zhu.
Network geometry, and curvature in particular, is intimately related to network entropy (see Figure 1). In fact, one way to generalize curvature to rather broad metric measure spaces is to exploit entropy’s convexity properties along geodesic paths defined at the level of the associated space of probability measures with an induced Riemannian structure.
We are very interested in applications of this geometric network framework, particularly to cancer. The connection to cancer biology arises from one’s ability to model many cellular gene and protein networks as weighted graphs, whose edges reflect interaction strengths/rates between the corresponding nodes (genes or proteins). As a concrete example, let us consider a genetic regulatory network. The expression of a gene, i.e., the production of a protein that the given gene encodes, is regulated by other proteins. Thus, one may model the genomic machine as a graph (network), with vertices representing the genes and edges depicting the correlation (dependence) of a given protein’s production by the corresponding gene on additional proteins produced by other genes in the genome.
Accordingly, much of our research focuses on employing network geometrical concepts to quantify (and therefore predict) pathway-related robustness/fragility in a given cancer system. This helps uncover hypothesized sets of targets that can properly disrupt alternative signaling cascades contributing to drug resistance. Our program’s mathematical component builds on and relies upon several observations with far-reaching physical (statistical mechanics) and information theoretic significance. More specifically, one can begin by placing a probability structure on a graph (e.g., representing expression levels of genes). This space of probabilities on graphs has several properties that enrich the structure of the underlying discrete space, based on the fact that a Riemannian structure may be endowed on the associated probability measures. Geodesic paths ensue, and convexity properties of the entropy along such paths reflect the space’s geometric features.
Entropy’s close relation to network topology and robustness has been noted in evolutionary biology literature . For example, Lloyd Demetrius uses Darwinian principles to argue that entropy is a selective criterion that may account for the robustness and heterogeneity of both man-made and biological networks . In our research program, we observe that curvature from network geometry is also strongly related to biological functional robustness. Biological networks seem to display a greater degree of robustness—as exhibited by higher curvature—than random networks.
Based on this observation, we are developing analytical methods for quantitatively describing the functional robustness of cancer networks to identify targets (genes/proteins) of opportunity. We hope that the analytical methods will empower treatments involving targeted drug agents and the combination of immunotherapy with more traditional chemical agents. This will help optimize the efficacy of certain immunotherapy methodologies for the alteration or upregulation of tumor cell antigens. Such an approach involves the use of graph theoretic techniques to identify key cancer hubs by partitioning the network into dense, highly-connected subgraphs. In order to account for both the activation and inhibition properties of the various complex interactions, one must extend existing theory to the case of directed graphs .
We are also studying possible mechanisms of resistance. For instance, it seems that the inhibition of certain key pathways (i.e., by making them more fragile) can increase robustness in neighboring pathways and thus contribute to an escape route from a given therapy. Network robustness may also indicate resistance to treatment, while fragility reflects sensitivity. The notion of graph curvature can be quite valuable in quantifying such phenomena. Cancer cells exhibit fate plasticity and are able to shift along a spectrum of differentiation in response to changes in gene expression caused by various genetic assaults (radiotherapy/chemotherapy/immunotherapy) or environmental stresses (hypoxia, reactive oxygen species). The methods we propose can also characterize both the processes that lead to differentiation and targeted anticancer therapies that must account for not only the differentiation state of the tumor as a whole, but also the likelihood that drug-resistant subclones will emerge.
In summary, our work uses ideas from geometric network mathematics in the battle against cancer. Our research is part of the emerging field of mathematical oncology, and hopefully will help in the development of new treatments for this deadly disease.
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 Barabasi, A.-L. (2012). The network takeover. Nat. Phys., 8, 14-16.
 Demetrius, L.A. (2013). Boltzmann, Darwin and directionality theory. Phys. Rep., 530, 1-85.