Optimal Hamilton covers and linear arboricity for random graphs

In his seminal 1976 paper, P\'osa showed that for all $p\geq C\log n/n$, the binomial random graph $G(n,p)$ is with high probability Hamiltonian. This leads to the following natural questions, which have been extensively studied: How well is it typically possible to cover all edges of $G(n,p)$ with Hamilton cycles? How many cycles are necessary? In this paper we show that for $p\geq C\log n/n$, we can cover $G\sim G(n,p)$ with precisely $\lceil\Delta(G)/2\rceil$ Hamilton cycles. Our result is clearly best possible both in terms of the number of required cycles, and the asymptotics of the edge probability $p$, since it starts working at the weak threshold needed for Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szab\'o, and improves upon previous work of Hefetz, K\"uhn, Lapinskas and Osthus, and of Ferber, Kronenberg and Long, essentially closing a long line of research on Hamiltonian packing and covering problems in random graphs.Comment: 13 page

Finding long cycles in graphs

We analyze the problem of discovering long cycles inside a graph. We propose and test two algorithms for this task. The first one is based on recent advances in statistical mechanics and relies on a message passing procedure. The second follows a more standard Monte Carlo Markov Chain strategy. Special attention is devoted to Hamiltonian cycles of (non-regular) random graphs of minimal connectivity equal to three

Phase-locked Loop Dynamics in the Presence of Noise by Fokker-planck Techniques

Phase error behavior of phase-locked loop tracking system in presence of gaussian noise determined by fokker-planck equatio

Cycle factors and renewal theory

For which values of $k$ does a uniformly chosen $3$-regular graph $G$ on $n$ vertices typically contain $n/k$ vertex-disjoint $k$-cycles (a $k$-cycle factor)? To date, this has been answered for $k=n$ and for $k \ll \log n$; the former, the Hamiltonicity problem, was finally answered in the affirmative by Robinson and Wormald in 1992, while the answer in the latter case is negative since with high probability most vertices do not lie on $k$-cycles. Here we settle the problem completely: the threshold for a $k$-cycle factor in $G$ as above is $\kappa_0 \log_2 n$ with $\kappa_0=[1-\frac12\log_2 3]^{-1}\approx 4.82$. Precisely, we prove a 2-point concentration result: if $k \geq \kappa_0 \log_2(2n/e)$ divides $n$ then $G$ contains a $k$-cycle factor w.h.p., whereas if $k<\kappa_0\log_2(2n/e)-\frac{\log^2 n}n$ then w.h.p. it does not. As a byproduct, we confirm the "Comb Conjecture," an old problem concerning the embedding of certain spanning trees in the random graph $G(n,p)$. The proof follows the small subgraph conditioning framework, but the associated second moment analysis here is far more delicate than in any earlier use of this method and involves several novel features, among them a sharp estimate for tail probabilities in renewal processes without replacement which may be of independent interest.Comment: 45 page

Income Distribution, Credit and Fiscal Policies in an Agent-Based Keynesian Model

This work studies the interactions between income distribution and monetary and fiscal policies in terms of ensuing dynamics of macro variables (GDP growth, unemployment, etc.) on the grounds of an agent-based Keynesian model. The direct ancestor of this work is the "Keynes meeting Schumpeter" formalism presented in Dosi et al. (2010). To that model, we add a banking sector and a monetary authority setting interest rates and credit lending conditions. The model combines Keynesian mechanisms of demand generation, a "Schumpeterian" innovation-fueled process of growth and Minskian credit dynamics. The robustness of the model is checked against its capability to jointly account for a large set of empirical regularities both at the micro level and at the macro one. The model is able to catch salient features underlying the current as well as previous recessions, the impact of financial factors and the role in them of income distribution. We find that different income distribution regimes heavily affect macroeconomic performance: more unequal economies are exposed to more severe business cycles fluctuations, higher unemployment rates, and higher probability of crises. On the policy side, fiscal policies do not only dampen business cycles, reduce unemployment and the likelihood of experiencing a huge crisis. In some circumstances they also affect positively long-term growth. Further, the more income distribution is skewed toward profits, the greater the effects of fiscal policies. About monetary policy, we find a strong non-linearity in the way interest rates affect macroeconomic dynamics: in one "regime" with low rates, changes in interest rates are ineffective up to a threshold beyond which increasing the interest rate implies smaller output growth rates and larger output volatility, unemployment and likelihood of crises.agent-based Keynesian models, multiple equilibria, fiscal and monetary policies, income distribution, transmission mechanisms, credit constraints

Income Distribution, Credit and Fiscal Policies in an Agent-Based Keynesian Model

This work studies the interactions between income distribution and monetary and fiscal policies in terms of ensuing dynamics of macro variables (GDP growth, unemployment, etc.) on the grounds of an agent-based Keynesian model. The direct ancestor of this work is the "Keynes meeting Schumpeter" formalism presented in Dosi et al. (2010). To that model, we add a banking sector and a monetary authority setting interest rates and credit lending conditions. The model combines Keynesian mechanisms of demand generation, a "Schumpeterian" innovation-fueled process of growth and Minskian credit dynamics. The robustness of the model is checked against its capability to jointly account for a large set of empirical regularities both at the micro level and at the macro one. The model is able to catch salient features underlying the current as well as previous recessions, the impact of financial factors and the role in them of income distribution. We find that different income distribution regimes heavily affect macroeconomic performance: more unequal economies are exposed to more severe business cycles fluctuations, higher unemployment rates, and higher probability of crises. On the policy side, fiscal policies do not only dampen business cycles, reduce unemployment and the likelihood of experiencing a huge crisis. In some circumstances they also affect positively long-term growth. Further, the more income distribution is skewed toward profits, the greater the effects of fiscal policies. About monetary policy, we find a strong non-linearity in the way interest rates affect macroeconomic dynamics: in one "regime" with low rates, changes in interest rates are ineffective up to a threshold beyond which increasing the interest rate implies smaller output growth rates and larger output volatility, unemployment and likelihood of crises.agent-based Keynesian models, multiple equilibria, fiscal and monetary policies, income distribution, transmission mechanisms, credit constraints

Income Distribution, Credit and Fiscal Policies in an Agent-Based Keynesian Model

This work studies the interactions between income distribution and monetary and fiscal policies in terms of ensuing dynamics of macro variables (GDP growth, unemployment, etc.) on the grounds of an agent-based Keynesian model. The direct ancestor of this work is the "Keynes meeting Schumpeter" formalism presented in Dosi et al. (2010). To that model, we add a banking sector and a monetary authority setting interest rates and credit lending conditions. The model combines Keynesian mechanisms of demand generation, a "Schumpeterian" innovation-fueled process of growth and Minskian credit dynamics. The robustness of the model is checked against its capability to jointly account for a large set of empirical regularities both at the micro level and at the macro one. The model is able to catch salient features underlying the current as well as previous recessions, the impact of financial factors and the role in them of income distribution. We find that different income distribution regimes heavily affect macroeconomic performance: more unequal economies are exposed to more severe business cycles fluctuations, higher unemployment rates, and higher probability of crises. On the policy side, fiscal policies do not only dampen business cycles, reduce unemployment and the likelihood of experiencing a huge crisis. In some circumstances they also affect positively long-term growth. Further, the more income distribution is skewed toward profits, the greater the effects of fiscal policies. About monetary policy, we find a strong non-linearity in the way interest rates affect macroeconomic dynamics: in one "regime" with low rates, changes in interest rates are ineffective up to a threshold beyond which increasing the interest rate implies smaller output growth rates and larger output volatility, unemployment and likelihood of crises.agent-based Keynesian models, multiple equilibria, fiscal and monetary policies, income distribution, transmission mechanisms, credit constraints