Detectability of Macroscopic Structures in Directed Asymmetric Stochastic Block Model

We study the problem of identifying macroscopic structures in networks, characterizing the impact of introducing link directions on the detectability phase transition. To this end, building on the stochastic block model, we construct a class of hardly detectable directed networks. We find closed form solutions by using belief propagation method showing how the transition line depends on the assortativity and the asymmetry of the network. Finally, we numerically identify the existence of a hard phase for detection close to the transition point.Comment: 9 pages, 7 figure

A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market

We propose a dynamic network model where two mechanisms control the probability of a link between two nodes: (i) the existence or absence of this link in the past, and (ii) node-specific latent variables (dynamic fitnesses) describing the propensity of each node to create links. Assuming a Markov dynamics for both mechanisms, we propose an Expectation-Maximization algorithm for model estimation and inference of the latent variables. The estimated parameters and fitnesses can be used to forecast the presence of a link in the future. We apply our methodology to the e-MID interbank network for which the two linkage mechanisms are associated with two different trading behaviors in the process of network formation, namely preferential trading and trading driven by node-specific characteristics. The empirical results allow to recognise preferential lending in the interbank market and indicate how a method that does not account for time-varying network topologies tends to overestimate preferential linkage.Comment: 19 pages, 6 figure

A dynamical systems approach to systemic risk

In the last years, many physical works have achieved important results in interdisciplinary fields, such as finance, applying the methodologies of dynamical systems theory, networks theory, and statistical mechanics. A topic of growing interest concerns the study of the dynamics of financial systems, and specifically the study of their stability. Typically, this problem is approached in the field of complex systems. In fact, financial markets exhibit several of the properties that characterize complex systems. They are open systems in which many subunits, financial institutions, interact nonlinearly in the presence of feedbacks. The aim of the thesis concerns the development of a theoretical model based on empirical evidences, in order to explain the intrinsic features of the dynamics of financial systems. Particularly, we focus on the problem of systemic risk and systemic financial stability, a problem of renewed interest after the financial crisis of 2007-2009. In financial markets, systemic risk is in effect an emergent phenomenon and can be seen as a phase transition from stability to instability. Specifically, systemic risk refers to a financial instability that affects initially a micro-region of the system, but it has bad consequences at the macro level, inducing potentially a catastrophic collapse of the entire financial market. Although systemic financial instability is usually triggered by a stochastic event, there are many empirical evidences, which lead to consider systemic risk as related to the network structure and dynamical properties of financial systems. A financial market can be seen as a bipartite network characterized by financial institutions (representing one type of nodes) owning a portfolio (i.e. creating links between nodes) of some risky investment assets (the other type of nodes), whose values evolve stochastically in time. Each investment asset is characterized by a risk, that is the diffusion rate of its price. In order to maximize their profit minimizing the risk, financial institutions in creating a portfolio, choose the optimal values of diversification (number of assets in the portfolio) and financial leverage (the ratio between invested capital and initial equity). The major financial institutions operating in real financial markets, adopt a strategy based on the periodical rebalancing of portfolio, in order to maintain fixed the chosen (target) leverage. The portfolio rebalancing induces feedback effects on asset prices, because in financial markets the demand (supply) for an asset tends to put as upward (downward) pressure on its price. The larger are leverage and diversification, the larger is the impact of rebalancing feedback effects. This mechanism is in effect a positive feedback. Since the positive feedback has the effect of amplifying asset price movements, it increases the risk of assets in financial markets and, as a consequence, has the potential of leading to the breaking of systemic financial stability. In this thesis, we propose a dynamical systems approach to systemic risk, in order to answer the following question: What is the relationship between the role of feedback effects, defining the dynamical behavior of financial institutions at the micro level, and the emerging macro consequences on systemic financial stability? Specifically, we focus on how financial institutions respond to an increase of risk due to the positive feedback. In solving the problem of portfolio optimization, financial institutions form expectations about future asset risks through estimates of observed past risks. We consider two types of expectations scheme: naive expectations, i.e. when financial institutions forecast future risk to be equal to the last observed one, and adaptive expectations, i.e. when financial institutions forecast future risk to be equal to a weighted average, with geometrically declining weights, of all past risks. According to the forecasting strategy, they define periodically the optimal portfolio configuration. In our dynamical approach, the key point is that the portfolio choices based on the expectations scheme adopted by financial institutions together with the impact on risk due to feedback effects arising from the target leverage strategy, drive the dynamics of financial market. Depending on diversification costs, under naive expectations, at a given threshold the positive feedback triggers the appearance of financial cycles characterized by a sequence of speculative periods and non-speculative ones. During financial cycles, we can notice how the financial leverage switches from aggressive configurations (speculative periods) to cautious ones (non- speculative periods). The financial leverage cycles reflect the occurrence of periods characterized by a macro-component of risk, due to an higher impact of feedback effects, followed by periods in which feedback effects do not affect importantly the asset risk. When financial cycles appear, the amplitude of cycles can be interpreted as a measure of systemic risk in financial market. Furthermore, under adaptive expectations, the financial system exhibits a dynamical transition from a periodic cyclical behavior to (deterministic) chaos. In the chaotic regime, in addition to the occurrence of highly risky periods identified by a very large macro-component of risk, the chaotic dynamical behavior of financial system is characterized by positive entropy, suggesting how much an improvement of expectations scheme by financial institutions may be hard due to missing information about financial market dynamics. Although feedback effects have been recognized as an important source of systemic risk in financial markets, this thesis represents an original dynamical systems approach to the considered problem. Particularly, our work focuses on the possible dynamical outcomes displayed by a financial system due to rebalancing feedback, and not only on the consequences on asset prices and risks due to the feedback effects. We believe that our original results, which especially suggest the possibility that chaos may occur in financial markets, indicate that our model deserves attention. Furthermore, a result of this type bypasses the specific dynamical model under consideration, since the occurrence of chaos in financial markets may be the consequence of universal aspects related to the nonlinearity of the feedback effects. Finally, from the point of view of financial market policy, we believe that our original results deserve attention because they highlight how the dynamical properties of financial markets may drastically change when market conditions change. Specifically, a decrease of diversification costs in the presence of strong feedback effects may lead to the breaking of systemic financial stability

When panic makes you blind: A chaotic route to systemic risk

We present an analytical model to study the role of expectation feedbacks and overlapping portfolios on systemic stability of financial systems. Building on Corsi et al. (2016), we model a set of financial institutions having Value-at-Risk capital requirements and investing in a portfolio of risky assets, whose prices evolve stochastically in time and are endogenously driven by the trading decisions of financial institutions. Assuming that they use adaptive expectations of risk, we show that the evolution of the system is described by a slow-fast random dynamical system, which can be studied analytically in some regimes. The model shows how the risk expectations play a central role in determining the systemic stability of the financial system and how wrong risk expectations may create panic-induced reduction or over-optimistic expansion of balance sheets. Specifically, when investors are myopic in estimating the risk, the fixed point equilibrium of the system breaks into leverage cycles and financial variables display a bifurcation cascade eventually leading to chaos. We discuss the role of financial policy and the effects of some market frictions, as the cost of diversification and financial transaction taxes, in determining the stability of the system in the presence of adaptive expectations of risk

Dynamic network models with applications to finance

This thesis provides new contributions to the field of network models, in two directions. On one hand, we study statistical models of static networks, in particular by contributing to the problem of community detection when link direction is taken into account, thus identifying what are the macroscopic structures of interest for the problem and the conditions for detectability [Wilinski et al., 2019]. Then, we introduce novel statistical models of dynamic networks which are able to capture simultaneously latent dynamics for node-specific characteristics together with link-specific persistence patterns. While the latent dynamics drives the evolution of the network topologies, such as the node degree, i.e. the number of incident links to the node, or the community structure, i.e. how nodes connect each other in forming groups, link persistence preserves the past structure of the network. Within this context, the contribution of the thesis is twofold, both theoretical and empirical [Mazzarisi et al., 2019a, Barucca et al., 2018]. We develop novel methodologies to disentangle the two linkage mechanisms in order to learn correctly both latent variables and static parameters of the models. And we consider also applications to financial data to reveal genuine patterns of persistence, which reflects the role both nodes and links have in the process of network formation and evolution. On the other hand, with a focus on the systemic risk of financial systems, we present a theoretical study of the expectation feedback mechanism which governs the dynamics of a financial network, thus determining its dynamical stability [Mazzarisi et al., 2019b]. Any financial system is an expectation feedback system: the current decisions of financial agents depend on what they expect will occur in the future. Agents\u2019 decisions affect the price dynamics in illiquid markets. Then, when expectations are formed by using models of past observations, the price dynamics itself feeds back on agents\u2019 expectations. This is in effect a feedback dynamics. Interestingly, the process of expectation formation by agents and the price dynamics act on different time scales. In our modeling, it is slow for the agents\u2019 expectations and fast for the price dynamics. Moreover, the agents\u2019 decisions, given the expectations formed on the basis of the random price dynamics, is to some extent deterministic, because they represent the optimal portfolio choice in a heavily regulated market. This separation of time scales is crucial and we are able to characterize analytically the feedback dynamics in the asymptotic limit of one time scale infinitely larger than the other one. Hence, we contribute to the research field of systemic risk with the first analytical proof (to the best of our knowledge) of how expectation feedbacks in relation to the estimation of investments\u2019 risk and dependencies determine the dynamical instability of a financial system. In line with the two research directions, the thesis is divided in two parts. [...

Trip Centrality: walking on a temporal multiplex with non-instantaneous link travel time

In complex networks, centrality metrics quantify the connectivity of nodes and identify the most important ones in the transmission of signals. In many real world networks, especially in transportation systems, links are dynamic, i.e. their presence depends on time, and travelling between two nodes requires a non-vanishing time. Additionally, many networks are structured on several layers, representing, e.g., different transportation modes or service providers. Temporal generalisations of centrality metrics based on walk-counting, like Katz centrality, exist, however they do not account for non-zero link travel times and for the multiplex structure. We propose a generalisation of Katz centrality, termed Trip Centrality, counting only the paths that can be travelled according to the network temporal structure, i.e. "trips", while also differentiating the contributions of inter- and intra-layer walks to centrality. We show an application to the US air transport system, specifically computing airports' centrality losses due to delays in the flight network

Betweenness centrality for temporal multiplexes

Betweenness centrality quantifies the importance of a vertex for the information flow in a network. We propose a flexible definition of betweenness for temporal multiplexes, where geodesics are determined accounting for the topological and temporal structure and the duration of paths. We propose an algorithm to compute the new metric via a mapping to a static graph. We show the importance of considering the temporal multiplex structure and an appropriate distance metric comparing the results with those obtained with static or single-layer metrics on a dataset of $\sim 20$k European flights

When panic makes you blind: a chaotic route to systemic risk

We present an analytical model to study the role of expectation feedbacks and overlapping portfolios on systemic stability of financial systems. Building on [Corsi et al., 2016], we model a set of financial institutions having Value at Risk capital requirements and investing in a portfolio of risky assets, whose prices evolve stochastically in time and are endogenously driven by the trading decisions of financial institutions. Assuming that they use adaptive expectations of risk, we show that the evolution of the system is described by a slow-fast random dynamical system, which can be studied analytically in some regimes. The model shows how the risk expectations play a central role in determining the systemic stability of the financial system and how wrong risk expectations may create panic-induced reduction or over-optimistic expansion of balance sheets. Specifically, when investors are myopic in estimating the risk, the fixed point equilibrium of the system breaks into leverage cycles and financial variables display a bifurcation cascade eventually leading to chaos. We discuss the role of financial policy and the effects of some market frictions, as the cost of diversification and financial transaction taxes, in determining the stability of the system in the presence of adaptive expectations of risk.Comment: 24 pages, 10 figure

Trip Centrality: walking on a temporal multiplex with non-instantaneous link travel time

In complex networks, centrality metrics quantify the connectivity of nodes and identify the most important ones in the transmission of signals. In many real world networks, especially in transportation systems, links are dynamic, i.e. their presence depends on time, and travelling between two nodes requires a non-vanishing time. Additionally, many networks are structured on several layers, representing, e.g., different transportation modes or service providers. Temporal generalisations of centrality metrics based on walk-counting, like Katz centrality, exist, however they do not account for non-zero link travel times and for the multiplex structure. We propose a generalisation of Katz centrality, termed trip Centrality, counting only the walks that can be travelled according to the network temporal structure, i.e. \u201ctrips\u201d, while also differentiating the contributions of inter- and intra- layer walks to centrality. We show an application to the US air transport system, specifically computing airports\u2019 centrality losses due to delays in the flight network

On the equivalence between the Kinetic Ising Model and discrete autoregressive processes

Binary random variables are the building blocks used to describe a large variety of systems, from magnetic spins to financial time series and neuron activity. In Statistical Physics the Kinetic Ising Model has been introduced to describe the dynamics of the magnetic moments of a spin lattice, while in time series analysis discrete autoregressive processes have been designed to capture the multivariate dependence structure across binary time series. In this article we provide a rigorous proof of the equivalence between the two models in the range of a unique and invertible map unambiguously linking one model parameters set to the other. Our result finds further justification acknowledging that both models provide maximum entropy distributions of binary time series with given means, auto-correlations, and lagged cross-correlations of order one. We further show that the equivalence between the two models permits to exploit the inference methods originally developed for one model in the inference of the other