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 recognize 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.

# Author Archives: Daniele Tantari

# P. Mazzarisi, P. Barucca, F. Lillo, D. Tantari (2020),

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A. Barra, G. Genovese, P. Sollich, D. Tantari, * Phase diagram of restricted Boltzmann machines and generalized Hopfield networks with arbitrary priors *, Physical Review E 97 (2), 022310, 2018

Restricted Boltzmann machines are described by the Gibbs measure of a bipartite spin glass, which in turn can be seen as a generalized Hopfield network. This equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities, both at low and high load, of pure states. We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern (i.e., weight) distribution and spin (i.e., unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region becomes larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.

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P.Mazzarisi, P.Barucca, F.Lillo, D.Tantari, *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.

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A.Barra, G.Genovese, P.Sollich, D.Tantari (2017), *Phase transitions in Restricted Boltzmann Machines with generic priors*

We study generalized restricted Boltzmann machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as generalized Hopfield models. We underline the role of the retrieval phase for both inference and learning processes and we show that retrieval is robust for a large class of weight and unit priors, beyond the standard Hopfield scenario. Furthermore, we show how the paramagnetic phase boundary is directly related to the optimal size of the training set necessary for good generalization in a teacher-student scenario of unsupervised learning.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.96.042156

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P.Barucca, P.Mazzarisi, F.Lillo, D.Tantari (2017), * Disentangling group and link persistence in dynamic stochastic block models*

We study the inference of a model of dynamic networks in which both communities and

links keep memory of previous network states. By considering maximum likelihood inference from

single snapshot observations of the network, we show that link persistence makes the inference of

communities harder, decreasing the detectability threshold, while community persistence tends to make

it easier. We analytically show that communities inferred from single network snapshot can share a

maximum overlap with the underlying communities of a specific previous instant in time. This leads

to time-lagged inference: the identification of past communities rather than present ones. Finally

we compute the time lag and propose a corrected algorithm, the Lagged Snapshot Dynamic (LSD)

algorithm, for community detection in dynamic networks. We analytically and numerically characterize

the detectability transitions of such algorithm as a function of the memory parameters of the model.

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P.Barucca, D.Tantari, F.Lillo (2016), *Centrality metrics and localization in core-periphery networks*

Two concepts of centrality have been defined in complex networks. The first considers the centrality of a node and many different metrics for it have been defined (e.g. eigenvector centrality, PageRank, non-backtracking centrality, etc). The second is related to large scale organization of the network, the core-periphery structure, composed by a dense core plus an outlying and loosely-connected periphery. In this paper we investigate the relation between these two concepts. We consider networks generated via the stochastic block model, or its degree corrected version, with a core-periphery structure and we investigate the centrality properties of the core nodes and the ability of several centrality metrics to identify them. We find that the three measures with the best performance are marginals obtained with belief propagation, PageRank, and degree centrality, while non-backtracking and eigenvector centrality (or MINRES [10], showed to be equivalent to the latter in the large network limit) perform worse in the investigated networks.

http://iopscience.iop.org/article/10.1088/1742-5468/2016/02/023401/meta