Higher-order Motifs Research Articles

Finding synchronization state of higher-order motif networks by dynamic learning

Synchronization of action potentials is one of the important phenomena for neural networks to achieve biological functions. How to find the optimal parameter space of neural networks in a synchronized state is of great significance for studying their synchronization. This study explores the parameter space of triplet motif-based higher-order networks in a synchronized state through the dynamic learning of synchronization (DLS) technique, which dynamically modulates the connection weights between motifs to alter their firing patterns. Our study delves into regular, Erdós-Rënyi random graphs, small-world, and scale-free networks, emphasizing the high-order motif interactions that characterize these networks. Our key findings indicate that the DLS technique successfully promotes synchronization within high-order motif networks with various connection patterns, although the degree of synchronization in networks where motifs are interconnected by chemical synapses is slightly weaker than those interconnected by electrical synapses. Additionally, we demonstrate the pattern of weight changes during the regulation of network firing states by DLS, finding that the evolution of weight distributions correlates with the network's topological structure. This work might provide new insights into complex network synchronization and lays the foundation for further exploration of using DLS technology to synchronize higher-order networks through external factors. Published by the American Physical Society 2024

Evolutionary graph theory beyond pairwise interactions: Higher-order network motifs shape times to fixation in structured populations.

To design population topologies that can accelerate rates of solution discovery in directed evolution problems or for evolutionary optimization applications, we must first systematically understand how population structure shapes evolutionary outcome. Using the mathematical formalism of evolutionary graph theory, recent studies have shown how to topologically build networks of population interaction that increase probabilities of fixation of beneficial mutations, at the expense, however, of longer fixation times, which can slow down rates of evolution, under elevated mutation rate. Here we find that moving beyond dyadic interactions in population graphs is fundamental to explain the trade-offs between probabilities and times to fixation of new mutants in the population. We show that higher-order motifs, and in particular three-node structures, allow the tuning of times to fixation, without changes in probabilities of fixation. This gives a near-continuous control over achieving solutions that allow for a wide range of times to fixation. We apply our algorithms and analytic results to two evolutionary optimization problems and show that the rate of solution discovery can be tuned near continuously by adjusting the higher-order topology of the population. We show that the effects of population structure on the rate of evolution critically depend on the optimization landscape and find that decelerators, with longer times to fixation of new mutants, are able to reach the optimal solutions faster than accelerators in complex solution spaces. Our results highlight that no one population topology fits all optimization applications, and we provide analytic and computational tools that allow for the design of networks suitable for each specific task.

Strongly clustered random graphs via triadic closure: An exactly solvable model.

Triadic closure, the formation of a connection between two nodes in a network sharing a common neighbor, is considered a fundamental mechanism determining the clustered nature of many real-world topologies. In this work we define a static triadic closure (STC) model for clustered networks, whereby starting from an arbitrary fixed backbone network, each triad is closed independently with a given probability. Assuming a locally treelike backbone we derive exact expressions for the expected number of various small, loopy motifs (triangles, 4-loops, diamonds, and 4-cliques) as a function of moments of the backbone degree distribution. In this way we determine how transitivity and its suitably defined generalizations for higher-order motifs depend on the heterogeneity of the original network, revealing the existence of transitions due to the interplay between topologically inequivalent triads in the network. Furthermore, under reasonable assumptions for the moments of the backbone network, we establish approximate relationships between motif densities, which we test in a large dataset of real-world networks. We find a good agreement, indicating that STC is a realistic mechanism for the generation of clustered networks, while remaining simple enough to be amenable to analytical treatment.

Exact and sampling methods for mining higher-order motifs in large hypergraphs

Network motifs are recurrent, small-scale patterns of interactions observed frequently in a system. They shed light on the interplay between the topology and the dynamics of complex networks across various domains. In this work, we focus on the problem of counting occurrences of small sub-hypergraph patterns in very large hypergraphs, where higher-order interactions connect arbitrary numbers of system units. We show how directly exploiting higher-order structures speeds up the counting process compared to traditional data mining techniques for exact motif discovery. Moreover, with hyperedge sampling, performance is further improved at the cost of small errors in the estimation of motif frequency. We evaluate our method on several real-world datasets describing face-to-face interactions, co-authorship and human communication. We show that our approximated algorithm allows us to extract higher-order motifs faster and on a larger scale, beyond the computational limits of an exact approach.

A general higher-order supracentrality framework based on motifs of temporal networks and multiplex networks

As the dynamics and complexity of networks evolve, identifying the important nodes in temporal networks (i.e., the relationships change over time) and multiplex networks (i.e., the layers can encode different types of relationships) becomes a core topic in network analysis and data science. However, the existing researches merely focus on the first-order relationships (i.e., the interactions only lie in two nodes) regardless of the higher-order relationships (i.e., the interactions exist among multiple nodes) especially for intralayer node interactions, which has been proved to be significant to affect centralities of nodes in the temporal networks and multiplex networks. Motivated by this, by observing and analyzing the interaction patterns of nodes in the temporal networks and multiplex networks, and considering the importance of triadic closure, first we adopt triangular motifs, and reconstruct the real-world system as temporal networks or multiplex networks fused with higher-order motifs information to encode the interactions among multiple nodes. Then we design a higher-order supracentrality framework to mine critical nodes with higher-order features in temporal networks and multiplex networks. The framework we proposed can characterize how the importance of nodes changes over time or different relationships, and also illustrate a complicated dependency of the higher-order supracentrality on the network topology, tunable weights of interlayer coupling and layer teleportation probability. To demonstrate the effectiveness and superiority of higher-order supracentrality framework based on motifs, we provide theoretical analysis and sufficient experiments on two empirical datasets including temporal networks and multiplex networks, the experimental results show that the higher-order supracentrality framework can identify much more abundant important nodes accurately. More significantly, our framework can provide a unifying foundation for centrality analysis in the temporal networks and multiplex networks, and also serve as a guidance for practical applications.

CHIEF: Clustering With Higher-Order Motifs in Big Networks

Clustering network vertices is an enabler of various applications such as social computing and Internet of Things. However, challenges arise for clustering when networks increase in scale. This paper proposes CHIEF (Clustering with HIgher-ordEr motiFs), a solution which consists of two motif clustering techniques: standard acceleration CHIEF-ST and approximate acceleration CHIEF-AP. Both algorithms firstly find the maximal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-edge-connected subgraphs within the target networks to lower the network scale by optimizing the network structure with maximal <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-edge-connected subgraphs, and then use heterogeneous four-node motifs clustering in higher-order dense networks. For CHIEF-ST, we illustrate that all target motifs will be kept after this procedure when the minimum node degree of the target motif is equal or greater than <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>. For CHIEF-AP, we prove that the eigenvalues of the adjacency matrix and the Laplacian matrix are relatively stable after this step. CHIEF offers an improved efficiency of motif clustering for big networks, and it verifies higher-order motif significance. Experiments on real and synthetic networks demonstrate that the proposed solutions outperform baseline approaches in large network analysis, and higher-order motifs outperform traditional triangle motifs in clustering.

Higher-order motif analysis in hypergraphs

A deluge of new data on real-world networks suggests that interactions among system units are not limited to pairs, but often involve a higher number of nodes. To properly encode higher-order interactions, richer mathematical frameworks such as hypergraphs are needed, where hyperedges describe interactions among an arbitrary number of nodes. Here we systematically investigate higher-order motifs, defined as small connected subgraphs in which vertices may be linked by interactions of any order, and propose an efficient algorithm to extract complete higher-order motif profiles from empirical data. We identify different families of hypergraphs, characterized by distinct higher-order connectivity patterns at the local scale. We also propose a set of measures to study the nested structure of hyperedges and provide evidences of structural reinforcement, a mechanism that associates higher strengths of higher-order interactions for the nodes that interact more at the pairwise level. Our work highlights the informative power of higher-order motifs, providing a principled way to extract higher-order fingerprints in hypergraphs at the network microscale.

Tissue schematics map the specialization of immune tissue motifs and their appropriation by tumors.

A schematic of a biological system, i.e., a representation of its pieces, how they are combined, and what they do, would facilitate understanding its essential organization and alteration in pathogenesis or evolution. We present a computational approach for constructing tissue schematics (TSs) from high-parameter imaging data and a biological model for interpreting them. TSs map the spatial assembly of cellular neighborhoods into tissue motifs, whose modular composition, we propose, enables the generation of complex outputs. We developed our approach in human lymphoid tissue (HLT), identifying the follicular outer zone as a potential relay between neighboring zones and a core lymphoid assembly with modifications characteristic of each HLT type. Applying the TS approach to the tumor microenvironment in human colorectal cancer identified a higher-order motif, whose mutated assembly was negatively associated with patient survival. TSs may therefore elucidate how immune architectures can be specialized and become vulnerable to reprogramming by tumors.

The Mitotic Crosslinking Protein PRC1 Acts Like a Mechanical Dashpot to Resist Microtubule Sliding

Cell division in eukaryotes requires the regulated assembly of the spindle apparatus. The proper organization of microtubules within the spindle is driven by motor proteins that exert forces to slide filaments, while non-motor proteins crosslink filaments into higher order motifs such as overlapping bundles. It is not clear how active and passive forces are integrated to produce regulated mechanical outputs within spindles. Here we employ simultaneous optical trapping and TIRF microscopy to directly measure the frictional forces produced by the mitotic crosslinking protein PRC1 that resist microtubule sliding.

The Mitotic Crosslinking Protein PRC1 Acts Like a Mechanical Dashpot to Resist Microtubule Sliding.

Cell division in eukaryotes requires the regulated assembly of the spindle apparatus. The proper organization of microtubules within the spindle is driven by motor proteins that exert forces to slide filaments, whereas non-motor proteins crosslink filaments into higher-order motifs, such as overlapping bundles. It is not clear how active and passive forces are integrated to produce regulated mechanical outputs within spindles. Here, we employ simultaneous optical trapping and total internal reflection fluorescence (TIRF) microscopy to directly measure the frictional forces produced by the mitotic crosslinking protein PRC1 that resist microtubule sliding. These forces scale with microtubule sliding velocity and the number of PRC1 crosslinks but do not depend on overlap length or PRC1 density within overlaps. Our results suggest that PRC1 ensembles act similarly to a mechanical dashpot, producing significant resistance against fast motions but minimal resistance against slow motions, allowing for the integration of diverse motor activities into a single mechanical outcome.

The Mitotic Crosslinking Protein PRC1 Acts as a Mechanical Dashpot to Resist Microtubule Sliding

Cell division in eukaryotes requires the regulated assembly of the spindle apparatus. The proper organization of microtubules within the spindle is driven by motor proteins that exert forces to push and slide filaments, while non-motor proteins can crosslink filaments into higher order motifs such as overlapping bundles. It has not been clear how active and passive forces are integrated to produce regulated mechanical outputs within spindles. Here we employ a combined optical tweezers and TIRF microscopy instrument to directly measure the resistive forces produced by the mitotic crosslinking protein PRC1. We observe that PRC1 generates frictional forces that resist microtubule sliding. These forces scale with microtubule sliding velocity and the number of PRC1 crosslinks, but do not depend on overlap length or PRC1 density within overlaps. Our results suggest that PRC1 ensembles act like a mechanical dashpot, producing significant resistance against fast motions, but minimal resistance against slow motions, allowing for the integration of diverse motor activities into a single mechanical outcome.

Feedback through graph motifs relates structure and function in complex networks

In physics, biology and engineering, network systems abound. How does the connectivity of a network system combine with the behavior of its individual components to determine its collective function? We approach this question for networks with linear time-invariant dynamics by relating internal network feedbacks to the statistical prevalence of connectivity motifs, a set of surprisingly simple and local statistics of connectivity. This results in a reduced order model of the network input-output dynamics in terms of motifs structures. As an example, the new formulation dramatically simplifies the classic Erdos-Renyi graph, reducing the overall network behavior to one proportional feedback wrapped around the dynamics of a single node. For general networks, higher-order motifs systematically provide further layers and types of feedback to regulate the network response. Thus, the local connectivity shapes temporal and spectral processing by the network as a whole, and we show how this enables robust, yet tunable, functionality such as extending the time constant with which networks remember past signals. The theory also extends to networks composed from heterogeneous nodes with distinct dynamics and connectivity, and patterned input to (and readout from) subsets of nodes. These statistical descriptions provide a powerful theoretical framework to understand the functionality of real-world network systems, as we illustrate with examples including the mouse brain connectome.

Multilayer network decoding versatility and trust

In the recent years, the multilayer networks have increasingly been realized as a more realistic framework to understand emergent physical phenomena in complex real-world systems. We analyze massive time-varying social data drawn from the largest film industry of the world under a multilayer network framework. The framework enables us to evaluate the versatility of actors, which turns out to be an intrinsic property of lead actors. Versatility in dimers suggests that working with different types of nodes are more beneficial than with similar ones. However, the triangles yield a different relation between type of co-actor and the success of lead nodes indicating the importance of higher-order motifs in understanding the properties of the underlying system. Furthermore, despite the degree-degree correlations of entire networks being neutral, multilayering picks up different values of correlation indicating positive connotations like trust, in the recent years. The analysis of weak ties of the industry uncovers nodes from a lower-degree regime being important in linking Bollywood clusters. The framework and the tools used herein may be used for unraveling the complexity of other real-world systems.

Nucleic Acids Exert a Sequence-independent Cooperative Effect on Sequence-dependent Activation of Toll-like Receptor 9

Toll-like receptor 9 (TLR9) activates the innate immune system in response to microbial DNA or mimicking oligodeoxynucleotides. Although cell stimulation experiments demonstrate the preferential activation of TLR9 by CpG-containing nucleic acids, direct binding investigations have reached contradictory conclusions with respect to the ability of this receptor to bind nucleic acids in a sequence-specific manner. To address this apparent discrepancy, we report the purification of the soluble ectodomain of human TLR9 with characterization of its ligand binding properties. We observe that TLR9 has a high degree of specificity in its ability to bind nucleic acids that contain CpG dinucleotides as well as higher order motifs that mediate species-specific activation. However, TLR9 is also functionally influenced by nucleic acids in a sequence-independent fashion as both stimulatory and nonstimulatory nucleic acids sensitize TLR9 for in vitro ligand binding as well as in vivo activation. We propose a model in which receptor activation is achieved in a sequence-dependent manner, and sensitivity is modulated by the absolute concentration of nucleic acids in a sequence-independent fashion. This model bears resemblance to that recently proposed for Toll in that activation is a two-step process in which formation of a ligand-bound monomer precedes formation of the activated dimer. In each model receptor sensitivity is determined within the second step with the crucial distinction that Toll undergoes negative cooperativity, whereas TLR9 is sensitized through a positive cooperative effect.