arXiv subjects

Dinghua Shi

Publications and source records attributed to Dinghua Shi.

15 recordsLinked to original sources

arXiv2025

Node-neighbor subnetworks and Hk-core decomposition

The network homology Hk-core decomposition proposed in this article is similar to the k-core decomposition based on node degrees of the network. The C. elegans neural network and the cat cortical network are used as examples to reveal the symmetry of the deep structures of such networks. First, based on the concept of neighborhood in mathematics, some new concepts are introduced, including such as node-neighbor subnetwork and Betti numbers of the neighbor subnetwork, among others. Then, the Betti numbers of the neighbor subnetwork of each node are computed, which are used to perform Hk-core decomposition of the network homology. The construction process is as follows: the initial network is referred to as the H0-core; the H1-core is obtained from the H0-core by deleting some nodes of certain properties; the H2-core is obtained from the H1-core by deleting some nodes or edges of certain properties; the H3-core is obtained from the H2-core by deleting some nodes of certain properties or by retaining the nodes of certain properties, and so on, which will be described in detail in the main text. Throughout the process, the index of node involved in deleting edge needs to be updated in every step. The Hk-core decomposition is easy to implement in parallel. It has a wide range of applications in many fields such as network science, data science, computational topology, and artificial intelligence. In this article, we also show how to use it to simplify homology calculation, e.g. for the C. elegans neural network, whereas the results of decomposition are the H1-core, the H2-core, and the H3-core. Thus, the simplexes consisting of four highest-order cavities in the H3-core subnetwork can also be directly obtained.

cs.CG↗

arXiv2023

Computing persistent homology by spanning trees and critical simplices

Topological data analysis can extract effective information from higher-dimensional data. Its mathematical basis is persistent homology. The persistent homology can calculate topological features at different spatiotemporal scales of the dataset; that is, establishing the integrated taxonomic relation among points, lines and simplices. Here, the simplicial network composed of all-order simplices in a simplicial complex is essential. Because the sequence of nested simplicial subnetworks can be regarded as a discrete Morse function from the simplicial network to real values, a method based on the concept of critical simplices can be developed by searching all-order spanning trees. Employing this new method, not only the Morse function values with the theoretical minimum number of critical simplices can be obtained, but also the Betti numbers and composition of all-order cavities in the simplicial network can be calculated quickly. Finally, this method is used to analyze some examples and compared with other methods, showing its effectiveness and feasibility.

math.AT↗

arXiv2021

Computing Cliques and Cavities in Networks

Complex networks contain complete subgraphs such as nodes, edges, triangles, etc., referred to as simplices and cliques of different orders. Notably, cavities consisting of higher-order cliques play an important role in brain functions. Since searching for maximum cliques is an NP-complete problem, we use k-core decomposition to determine the computability of a given network. For a computable network, we design a search method with an implementable algorithm for finding cliques of different orders, obtaining also the Euler characteristic number. Then, we compute the Betti numbers by using the ranks of boundary matrices of adjacent cliques. Furthermore, we design an optimized algorithm for finding cavities of different orders. Finally, we apply the algorithm to the neuronal network of C. elegans with data from one typical dataset, and find all of its cliques and some cavities of different orders, providing a basis for further mathematical analysis and computation of its structure and function.

cs.NE↗

arXiv2021

Optimizing higher-order network topology for synchronization of coupled phase oscillators

Networks in nature have complex interactions among agents. One significant phenomenon induced by interactions is synchronization of coupled agents, and the interactive network topology can be tuned to optimize synchronization. The previous studies showed that the optimized conventional network with pairwise interactions favors a homogeneous degree distribution of nodes when the interaction is undirected, and is always structurally asymmetric when the interaction is directed. However, the optimal control on synchronization for networks with prevailing higher-order interactions is less explored. Here, by considering the higher-order interactions in hypergraph and the Kuramoto model with 2-hyperlink interactions, we find that the network topology with optimized synchronizability may have distinct properties. For the undirected interaction, optimized networks with 2-hyperlink interactions by simulated annealing tend to become homogeneous in the nodes' generalized degree, consistent with 1-hyperlink (pairwise) interactions. We further define the directed hyperlink, and rigorously demonstrate that for the directed interaction, the structural symmetry can be preserved in the optimally synchronizable network with 2-hyperlink interactions, in contrast to the conclusion for 1-hyperlink interactions. The results suggest that controlling the network topology of higher-order interactions leads to synchronization phenomena beyond pairwise interactions.

physics.soc-ph↗

arXiv2021

Characterizing cycle structure in complex networks

Cycle is the simplest structure that brings redundant paths in network connectivity and feedback effects in network dynamics. Focusing on cycle structure, this paper defines a new matrix, named cycle number matrix, to represent cycle information of a network, and an index, named cycle ratio, to quantify the node importance. Experiments on real networks suggest that cycle ratio contains rich information in addition to well-known benchmark indices, for example, the node rankings by cycle ratio are largely different from rankings by degree, H-index, coreness, betweenness and articulation ranking, while the rankings by degree, H-index, coreness are very similar to each other. Extensive experiments on identifying vital nodes that maintain network connectivity, facilitate network synchronization and maximize the early reach of spreading show that cycle ratio is competitive to betweenness and overall better than other benchmarks. We believe the in-depth analyses on cycle structure may yield novel insights, metrics, models and algorithms for network science.

physics.soc-ph↗

arXiv2020

Totally Homogeneous Networks

In network science, the non-homogeneity of node degrees has been a concerned issue for study. Yet, with the modern web technologies today, the traditional social communication topologies have evolved from node-central structures to online cycle-based communities, urgently requiring new network theories and tools. Switching the focus from node degrees to network cycles, it could reveal many interesting properties from the perspective of totally homogeneous networks, or sub-networks in a complex network, especially basic simplexes (cliques) such as links and triangles. Clearly, comparing to node degrees it is much more challenging to deal with network cycles. For studying the latter, a new clique vector space framework is introduced in this paper, where the vector space with a basis consisting of links has the dimension equal to the number of links, that with a basis consisting of triangles has the dimension equal to the number of triangles, and so on. These two vector spaces are related through a boundary operator, e.g., mapping the boundary of a triangle in one space to the sun of three links in the other space. Under the new framework, some important concepts and methodologies from algebraic topology, such as characteristic number, homology group and Betti number, will have a play in network science leading to foreseeable new research directions. As immediate applications, the paper illustrates some important characteristics affecting the collective behaviors of complex networks, some new cycle-dependent importance indexes of nodes, and implications for network synchronization and brain network analysis.

cs.SI↗

arXiv2019

Towards the cycle structures in complex network: A new perspective

Stars and cycles are basic structures in network construction. The former has been well studied in network analysis, while the latter attracted rare attention. A node together with its neighbors constitute a neighborhood star-structure where the basic assumption is two nodes interact through their direct connection. A cycle is a closed loop with many nodes who can influence each other even without direct connection. Here we show their difference and relationship in understanding network structure and function. We define two cycle-based node characteristics, namely cycle number and cycle ratio, which can be used to measure a node's importance. Numerical analyses on six disparate real networks suggest that the nodes with higher cycle ratio are more important to network connectivity, while cycle number can better quantify a node influence of cycle-based spreading than the common star-based node centralities. We also find that an ordinary network can be converted into a hypernetwork by considering its basic cycles as hyperedges, meanwhile, a new matrix called the cycle number matrix is captured. We hope that this paper can open a new direction of understanding both local and global structures of network and its function.

physics.soc-ph↗

arXiv2009

Degree-distribution Stability of Evolving Networks

In this paper, we study a class of stochastic processes, called evolving network Markov chains, in evolving networks. Our approach is to transform the degree distribution problem of an evolving network to a corresponding problem of evolving network Markov chains. We investigate the evolving network Markov chains, thereby obtaining some exact formulas as well as a precise criterion for determining whether the steady degree distribution of the evolving network is a power-law or not. With this new method, we finally obtain a rigorous, exact and unified solution of the steady degree distribution of the evolving network.

math-ph↗

arXiv2009

Degree-distribution Stability of Growing Networks

In this paper, we abstract a kind of stochastic processes from evolving processes of growing networks, this process is called growing network Markov chains. Thus the existence and the formulas of degree distribution are transformed to the corresponding problems of growing network Markov chains. First we investigate the growing network Markov chains, and obtain the condition in which the steady degree distribution exists and get its exact formulas. Then we apply it to various growing networks. With this method, we get a rigorous, exact and unified solution of the steady degree distribution for growing networks.

math-ph↗

arXiv2008

Stability of the LCD Model

In this paper, first-passage probability of Markov chains is used to get a strict proof of the existence of degree distribution of the LCD model presented by Bollobas (Random Structures and Algorithms 18(2001)). Also, a precise expression of degree distribution is presented.

math.PR↗

arXiv2008

Markov chain-based stability analysis of growing networks

From the perspective of probability, the stability of growing network is studied in the present paper. Using the DMS model as an example, we establish a relation between the growing network and Markov process. Based on the concept and technique of first-passage probability in Markov theory, we provide a rigorous proof for existence of the steady-state degree distribution, mathematically re-deriving the exact formula of the distribution. The approach based on Markov chain theory is universal and performs well in a large class of growing networks.

math-ph↗

arXiv2008

Degree-distribution stability of scale-free networks

Based on the concept and techniques of first-passage probability in Markov chain theory, this letter provides a rigorous proof for the existence of the steady-state degree distribution of the scale-free network generated by the Barabasi-Albert (BA) model, and mathematically re-derives the exact analytic formulas of the distribution. The approach developed here is quite general, applicable to many other scale-free types of complex networks.

math.PR↗

arXiv2005

Evolving Networks and Birth-and-Death Processes

Using a simple model with link removals as well as link additions, we show that an evolving network is scale free with a degree exponent in the range of (2, 4]. We then establish a relation between the network evolution and a set of non-homogeneous birth-and-death processes, and, with which, we capture the process by which the network connectivity evolves. We develop an effective algorithm to compute the network degree distribution accurately. Comparing analytical and numerical results with simulation, we identify some interesting network properties and verify the effectiveness of our method.

math-ph↗

arXiv2004

A Markov Chain-Based Numerical Method for Calculating Network Degree Distributions

This paper establishes a relation between scale-free networks and Markov chains, and proposes a computation framework for degree distributions of scale-free networks. We first find that, under the BA model, the degree evolution of individual nodes in a scale-free network follows some non-homogeneous Markov chains. Exploring the special structure of these Markov chains, we are able to develop an efficient algorithm to compute the degree distribution numerically. The complexity of our algorithm is O(t^2), where $t$ is the number of time steps for adding new nodes. We use three examples to demonstrate the computation procedure and compare the results with those from the existing methods.

math-ph↗

arXiv2004

A Simple Model of Scale-free Networks Driven by both Randomness and Adaptability

In this paper, we present a simple model of scale-free networks that incorporates both preferential & random attachment and anti-preferential & random deletion at each time step. We derive the degree distribution analytically and show that it follows a power law with the degree exponent in the range of (2,infinity). We also find a way to derive an expression of the clustering coefficient for growing networks and compute the average path length through simulation.

physics.data-an↗