First principles and effective theory approaches to dynamics of complex networks

This dissertation concerns modeling two aspects of dynamics of complex networks: (1) response dynamics and (2) growth and formation. A particularly challenging class of networks are ones in which both nodes and links are evolving over time – the most prominent example is a financial network. In the first part of the dissertation we present a model for the response dynamics in networks near a metastable point. We start with a Landau-Ginzburg approach and show that the most general lowest order Lagrangians for dynamical weighted networks can be used to derive conditions for stability under external shocks. Using a closely related model, which is easier to solve numerically, we propose a powerful and intuitive set of equations for response dynamics of financial networks. We find the stability conditions of the model and find two phases: “calm” phase , in which changes are sub-exponential and where the system moves to a new, close-by equilibrium; “frantic” phase, where changes are exponential, with negative blows resulting in crashes and positive ones leading to formation of "bubbles". We empirically verify these claims by analyzing data from Eurozone crisis of 2009-2012 and stock markets. We show that the model correctly identifies the time-line of the Eurozone crisis, and in the stock market data it correctly reproduces the auto-correlations and phases observed in the data. The second half of the dissertation addresses the following question: Do networks that form due to local interactions (local in real space, or in an abstract parameter space) have characteristics different from networks formed of random or non-local interactions? Using interacting fields obeying Fokker-Planck equations we show that many network characteristics such as degree distribution, degree-degree correlation and clustering can either be derived analytically or there are analytical bounds on their behaviour. In particular, we derive recursive equations for all powers of the ensemble average of the adjacency matrix. We analyze a few real world networks and show that some networks that seem to form from local interactions indeed have characteristics almost identical to simulations based on our model, in contrast with many other networks

Crises and physical phases of a bipartite market model

We analyze the linear response of a market network to shocks based on the bipartite market model we introduced in an earlier paper, which we claimed to be able to identify the time-line of the 2009-2011 Eurozone crisis correctly. We show that this model has three distinct phases that can broadly be categorized as "stable" and "unstable". Based on the interpretation of our behavioral parameters, the stable phase describes periods where investors and traders have confidence in the market (e.g. predict that the market rebounds from a loss). We show that the unstable phase happens when there is a lack of confidence and seems to describe "boom-bust" periods in which changes in prices are exponential. We analytically derive these phases and where the phase transition happens using a mean field approximation of the model. We show that the condition for stability is αβ<1 with α being the inverse of the "price elasticity" and β the "income elasticity of demand", which measures how rash the investors make decisions. We also show that in the mean-field limit this model reduces to the Langevin model by Bouchaud et al. for price returns.First author draf

Crises and physical phases of a bipartite market model

We analyze the linear response of a market network to shocks based on the bipartite market model we introduced in an earlier paper, which we claimed to be able to identify the time-line of the 2009-2011 Eurozone crisis correctly. We show that this model has three distinct phases that can broadly be categorized as "stable" and "unstable". Based on the interpretation of our behavioral parameters, the stable phase describes periods where investors and traders have confidence in the market (e.g. predict that the market rebounds from a loss). We show that the unstable phase happens when there is a lack of confidence and seems to describe "boom-bust" periods in which changes in prices are exponential. We analytically derive these phases and where the phase transition happens using a mean field approximation of the model. We show that the condition for stability is αβ<1 with α being the inverse of the "price elasticity" and β the "income elasticity of demand", which measures how rash the investors make decisions. We also show that in the mean-field limit this model reduces to the Langevin model by Bouchaud et al. for price returns.First author draf

Generative Adversarial Symmetry Discovery

Despite the success of equivariant neural networks in scientific applications, they require knowing the symmetry group a priori. However, it may be difficult to know which symmetry to use as an inductive bias in practice. Enforcing the wrong symmetry could even hurt the performance. In this paper, we propose a framework, LieGAN, to automatically discover equivariances from a dataset using a paradigm akin to generative adversarial training. Specifically, a generator learns a group of transformations applied to the data, which preserve the original distribution and fool the discriminator. LieGAN represents symmetry as interpretable Lie algebra basis and can discover various symmetries such as the rotation group $\mathrm{SO}(n)$, restricted Lorentz group $\mathrm{SO}(1,3)^+$ in trajectory prediction and top-quark tagging tasks. The learned symmetry can also be readily used in several existing equivariant neural networks to improve accuracy and generalization in prediction

3D Topology Transformation with Generative Adversarial Networks

Generation and transformation of images and videos using artificial intelligence have flourished over the past few years. Yet, there are only a few works aiming to produce creative 3D shapes, such as sculptures. Here we show a novel 3D-to-3D topology transformation method using Generative Adversarial Networks (GAN). We use a modified pix2pix GAN, which we call Vox2Vox, to transform the volumetric style of a 3D object while retaining the original object shape. In particular, we show how to transform 3D models into two new volumetric topologies - the 3D Network and the Ghirigoro. We describe how to use our approach to construct customized 3D representations. We believe that the generated 3D shapes are novel and inspirational. Finally, we compare the results between our approach and a baseline algorithm that directly convert the 3D shapes, without using our GAN

Classical mechanics of economic networks

Financial networks are dynamic. To assess their systemic importance to the world-wide economic network and avert losses we need models that take the time variations of the links and nodes into account. Using the methodology of classical mechanics and Laplacian determinism we develop a model that can predict the response of the financial network to a shock. We also propose a way of measuring the systemic importance of the banks, which we call BankRank. Using European Bank Authority 2011 stress test exposure data, we apply our model to the bipartite network connecting the largest institutional debt holders of the troubled European countries (Greece, Italy, Portugal, Spain, and Ireland). From simulating our model we can determine whether a network is in a "stable" state in which shocks do not cause major losses, or a "unstable" state in which devastating damages occur. Fitting the parameters of the model, which play the role of physical coupling constants, to Eurozone crisis data shows that before the Eurozone crisis the system was mostly in a "stable" regime, and that during the crisis it transitioned into an "unstable" regime. The numerical solutions produced by our model match closely the actual time-line of events of the crisis. We also find that, while the largest holders are usually more important, in the unstable regime smaller holders also exhibit systemic importance. Our model also proves useful for determining the vulnerability of banks and assets to shocks. This suggests that our model may be a useful tool for simulating the response dynamics of shared portfolio networks

Systemic stress test model for shared portfolio networks

We propose a dynamic model for systemic risk using a bipartite network of banks and assets in which the weight of links and node attributes vary over time. Using market data and bank asset holdings, we are able to estimate a single parameter as an indicator of the stability of the financial system. We apply the model to the European sovereign debt crisis and observe that the results closely match real-world events (e.g., the high risk of Greek sovereign bonds and the distress of Greek banks). Our model could become complementary to existing stress tests, incorporating the contribution of interconnectivity of the banks to systemic risk in time-dependent networks. Additionally, we propose an institutional systemic importance ranking, BankRank, for the financial institutions analyzed in this study to assess the contribution of individual banks to the overall systemic risk.Published versio