Journal of Physics Complexity最新文献

Sparsification aims at extracting a reduced core of associations that best preserves both the dynamics and topology of networks while reducing the computational cost of simulations. We show that the semi-metric topology of complex networks yields a natural and algebraically-principled sparsification that outperforms existing methods on those goals. Weighted graphs whose edges represent distances between nodes are semi-metric when at least one edge breaks the triangle inequality (transitivity). We first confirm with new experiments that the metric backbone-a unique subgraph of all edges that obey the triangle inequality and thus preserve all shortest paths-recovers susceptible-infected dynamics over the original non-sparsified graph. This recovery is improved when we remove only those edges that break the triangle inequality significantly, i.e. edges with large semi-metric distortion. Based on these results, we propose the new semi-metric distortion sparsification method to progressively sparsify networks in decreasing order of semi-metric distortion. Our method recovers the macro- and micro-level dynamics of epidemic outbreaks better than other methods while also yielding sparser yet connected subgraphs that preserve all shortest paths. Overall, we show that semi-metric distortion overcomes the limitations of edge betweenness in ranking the dynamical relevance of edges not participating in any shortest path, as it quantifies the existence and strength of alternative transmission pathways.

Khovanov homology has been the subject of much study in knot theory and low dimensional topology since 2000. This work introduces a Khovanov Laplacian and a Khovanov Dirac to study knot and link diagrams. The harmonic spectrum of the Khovanov Laplacian or the Khovanov Dirac retains the topological invariants of Khovanov homology, while their non-harmonic spectra reveal additional information that is distinct from Khovanov homology.

Recent work has emphasized the ubiquity of higher-order interactions in brain function. These interactions can be characterized as being either redundancy or synergy-dominated by applying tools from multivariate information theory. Though recent work has shown the importance of both synergistic and redundant interactions to brain function, their dynamic structure is still unknown. Here we analyze the moment-to-moment synergy and redundancy dominance of the fMRI BOLD signal during rest for 95 unrelated subjects to show that redundant and synergistic interactions have highly structured dynamics across many interaction sizes. The whole brain is strongly redundancy-dominated, with some subjects never experiencing a whole-brain synergistic moment. In small sets of brain regions, our analyses reveal that subsets which are redundancy dominated on average exhibit the most complex dynamic behavior as well as the most synergistic and most redundant time points. In accord with previous work, these regions frequently belong to a single coherent functional system, and our analysis reveals that they become synergistic when that functional system becomes momentarily disintegrated. Although larger subsets cannot be contained in a single functional network, similar patterns of instantaneous disintegration mark when they become synergistic. At all sizes of interaction, we find notable temporal structure of both synergy and redundancy-dominated interactions. We show that the interacting nodes change smoothly in time and have significant recurrence. Both of these properties make time-localized measures of synergy and redundancy highly relevant to future studies of behavior or cognition as time-resolved phenomena.

Topological data analysis (TDA) has made significant progress in developing a new class of fundamental operators known as the Dirac operator, particularly in topological signals and molecular representations. However, the current approaches being used are based on the classical case of chain complexes. The present study establishes Mayer Dirac operators based on N-chain complexes. These operators interconnect an alternating sequence of Mayer Laplacian operators, providing a generalization of the classical result $D^2=L$ . Furthermore, the research presents an explicit formulation of the Laplacian for N-chain complexes induced by vertex sequences on a finite set. Weighted versions of Mayer Laplacian and Dirac operators are introduced to expand the scope and improve applicability, showcasing their effectiveness in capturing physical attributes in various practical scenarios. The study presents a generalized version for factorizing Laplacian operators as an operator's product and its 'adjoint'. Additionally, the proposed persistent Mayer Dirac operators and extensions are applied to biological and chemical domains, particularly in the analysis of molecular structures. The study also highlights the potential applications of persistent Mayer Dirac operators in data science.

Minimum spanning trees and forests are powerful sparsification techniques that remove cycles from weighted graphs to minimize total edge weight while preserving node reachability, with applications in computer science, network science, and graph theory. Despite their utility and ubiquity, they have several limitations, including that they are only defined for undirected networks, they significantly alter dynamics on networks, and they do not generally preserve important network features such as shortest distances, shortest path distribution, and community structure. In contrast, distance backbones, which are subgraphs formed by all edges that obey a generalized triangle inequality, are well defined in directed and undirected graphs and preserve those and other important network features. The backbone of a graph is defined with respect to a specified path-length operator that aggregates weights along a path to define its length, thereby associating a cost to indirect connections. The backbone is the union of all shortest paths between each pair of nodes according to the specified operator. One such operator, the max function, computes the length of a path as the largest weight of the edges that compose it (a weakest link criterion). It is the only operator that yields an algebraic structure for computing shortest paths that is consistent with De Morgan's laws. Applying this operator yields the ultrametric backbone of a graph in that (semi-triangular) edges whose weights are larger than the length of an indirect path connecting the same nodes (i.e. those that break the generalized triangle inequality based on max as a path-length operator) are removed. We show that the ultrametric backbone is the union of minimum spanning forests in undirected graphs and provides a new generalization of minimum spanning trees to directed graphs that, unlike minimum equivalent graphs and minimum spanning arborescences, preserves all $max-min$ shortest paths and De Morgan's law consistency.