Heterogeneous Retirement Savings Strategy Selection with Reinforcement Learning.

Saving and investment behaviour is crucial for all individuals to guarantee their welfare during work-life and retirement. We introduce a deep reinforcement learning model in which agents learn optimal portfolio allocation and saving strategies suitable for their heterogeneous profiles. The environment is calibrated with occupation- and age-dependent income dynamics. The research focuses on heterogeneous income trajectories dependent on agents' profiles and incorporates the parameterisation of agents' behaviours. The model provides a new flexible methodology to estimate lifetime consumption and investment choices for individuals with heterogeneous profiles.

Simplicial Persistence of Financial Markets: Filtering, Generative Processes and Structural Risk.

We introduce simplicial persistence, a measure of time evolution of motifs in networks obtained from correlation filtering. We observe long memory in the evolution of structures, with a two power law decay regimes in the number of persistent simplicial complexes. Null models of the underlying time series are tested to investigate properties of the generative process and its evolutional constraints. Networks are generated with both a topological embedding network filtering technique called TMFG and by thresholding, showing that the TMFG method identifies high order structures throughout the market sample, where thresholding methods fail. The decay exponents of these long memory processes are used to characterise financial markets based on their efficiency and liquidity. We find that more liquid markets tend to have a slower persistence decay. This appears to be in contrast with the common understanding that efficient markets are more random. We argue that they are indeed less predictable for what concerns the dynamics of each single variable but they are more predictable for what concerns the collective evolution of the variables. This could imply higher fragility to systemic shocks.

Cross-correlations of American baby names.

The quantitative description of cultural evolution is a challenging task. The most difficult part of the problem is probably to find the appropriate measurable quantities that can make more quantitative such evasive concepts as, for example, dynamics of cultural movements, behavioral patterns, and traditions of the people. A strategy to tackle this issue is to observe particular features of human activities, i.e., cultural traits, such as names given to newborns. We study the names of babies born in the United States from 1910 to 2012. Our analysis shows that groups of different correlated states naturally emerge in different epochs, and we are able to follow and decrypt their evolution. Although these groups of states are stable across many decades, a sudden reorganization occurs in the last part of the 20th century. We unambiguously demonstrate that cultural evolution of society can be observed and quantified by looking at cultural traits. We think that this kind of quantitative analysis can be possibly extended to other cultural traits: Although databases covering more than one century (such as the one we used) are rare, the cultural evolution on shorter timescales can be studied due to the fact that many human activities are usually recorded in the present digital era.

Resolution of ranking hierarchies in directed networks.

Identifying hierarchies and rankings of nodes in directed graphs is fundamental in many applications such as social network analysis, biology, economics, and finance. A recently proposed method identifies the hierarchy by finding the ordered partition of nodes which minimises a score function, termed agony. This function penalises the links violating the hierarchy in a way depending on the strength of the violation. To investigate the resolution of ranking hierarchies we introduce an ensemble of random graphs, the Ranked Stochastic Block Model. We find that agony may fail to identify hierarchies when the structure is not strong enough and the size of the classes is small with respect to the whole network. We analytically characterise the resolution threshold and we show that an iterated version of agony can partly overcome this resolution limit.

Spectral partitioning in equitable graphs.

Graph partitioning problems emerge in a wide variety of complex systems, ranging from biology to finance, but can be rigorously analyzed and solved only for a few graph ensembles. Here, an ensemble of equitable graphs, i.e., random graphs with a block-regular structure, is studied, for which analytical results can be obtained. In particular, the spectral density of this ensemble is computed exactly for a modular and bipartite structure. Kesten-McKay's law for random regular graphs is found analytically to apply also for modular and bipartite structures when blocks are homogeneous. An exact solution to graph partitioning for two equal-sized communities is proposed and verified numerically, and a conjecture on the absence of an efficient recovery detectability transition in equitable graphs is suggested. A final discussion summarizes results and outlines their relevance for the solution of graph partitioning problems in other graph ensembles, in particular for the study of detectability thresholds and resolution limits in stochastic block models.

Localization in covariance matrices of coupled heterogenous Ornstein-Uhlenbeck processes.

We define a random-matrix ensemble given by the infinite-time covariance matrices of Ornstein-Uhlenbeck processes at different temperatures coupled by a Gaussian symmetric matrix. The spectral properties of this ensemble are shown to be in qualitative agreement with some stylized facts of financial markets. Through the presented model formulas are given for the analysis of heterogeneous time series. Furthermore evidence for a localization transition in eigenvectors related to small and large eigenvalues in cross-correlations analysis of this model is found, and a simple explanation of localization phenomena in financial time series is provided. Finally we identify both in our model and in real financial data an inverted-bell effect in correlation between localized components and their local temperature: high- and low-temperature components are the most localized ones.

Temperature chaos and quenched heterogeneities.

We present a treatable generalization of the Sherrington-Kirkpatrick (SK) model which introduces correlations in the elements of the coupling matrix through multiplicative disorder on the single variables and investigate the consequences on the phase diagram. We define a generalized qEA parameter and test the structural stability of the SK results in this correlated case evaluating the de Almeida-Thouless line of the model. As a main result we demonstrate the increase of temperature chaos effects due to heterogeneities.