Network constraints on learnability of probabilistic motor sequences

Human learners are adept at grasping the complex relationships underlying incoming sequential input. In the present work, we formalize complex relationships as graph structures derived from temporal associations in motor sequences. Next, we explore the extent to which learners are sensitive to key variations in the topological properties inherent to those graph structures. Participants performed a probabilistic motor sequence task in which the order of button presses was determined by the traversal of graphs with modular, lattice-like, or random organization. Graph nodes each represented a unique button press and edges represented a transition between button presses. Results indicate that learning, indexed here by participants' response times, was strongly mediated by the graph's meso-scale organization, with modular graphs being associated with shorter response times than random and lattice graphs. Moreover, variations in a node's number of connections (degree) and a node's role in mediating long-distance communication (betweenness centrality) impacted graph learning, even after accounting for level of practice on that node. These results demonstrate that the graph architecture underlying temporal sequences of stimuli fundamentally constrains learning, and moreover that tools from network science provide a valuable framework for assessing how learners encode complex, temporally structured information.Comment: 29 pages, 4 figure

A note on dominating cycles in 2-connected graphs

Let G be a 2-connected graph on n vertices such that d(x) + d(y) + d(z) n for all triples of independent vertices x, y, z. We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs

Long cycles in graphs containing a 2-factor with many odd components

We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+d(c)+d(w)n+2 for every tiple u, v, w of independent vertices. As a corollary we obtain the follwing improvement of a conjectre of Häggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume G has a 2-factor with at least k+1 odd components. Then G is hamiltonian

Long cycles, degree sums and neighborhood unions

AbstractFor a graph G, define the parameters α(G)=max{|S| |S is an independent set of vertices of G}, σk(G)=min{∑ki=1d(vi)|{v1,…,vk} is an independent set} and NCk(G)= min{|∪ki=1 N(vi)∥{v1,…,vk} is an independent set} (k⩾2). It is shown that every 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,n+NCr+5+∈(n+r)(G)-α(G)}, where ε(i)=3(⌈13i⌉−13i). This result extends previous results in Bauer et al. (1989/90), Faßbender (1992) and Flandrin et al. (1991). It is also shown that a 1-tough graph G of order n⩾3 with σ3(G)⩾n+r⩾n has a cycle of length at least min{n,2NC⌊18(n+6r+17)⌋(G)}. Analogous results are established for 2-connected graphs

Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction

We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping