arXiv subjects

Zhongzhi Zhang

Publications and source records attributed to Zhongzhi Zhang.

At least 55 records · Page 3Linked to original sources

arXiv2022

Resistance Distances in Simplicial Networks

It is well known that in many real networks, such as brain networks and scientific collaboration networks, there exist higher-order nonpairwise relations among nodes, i.e., interactions between among than two nodes at a time. This simplicial structure can be described by simplicial complexes and has an important effect on topological and dynamical properties of networks involving such group interactions. In this paper, we study analytically resistance distances in iteratively growing networks with higher-order interactions characterized by the simplicial structure that is controlled by a parameter q. We derive exact formulas for interesting quantities about resistance distances, including Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index, as well as average resistance distance, which have found applications in various areas elsewhere. We show that the average resistance distance tends to a q-dependent constant, indicating the impact of simplicial organization on the structural robustness measured by average resistance distance.

cs.SI↗

arXiv2022

Effects of Stubbornness on Opinion Dynamics

As an important factor governing opinion dynamics, stubbornness strongly affects various aspects of opinion formation. However, a systematically theoretical study about the influences of heterogeneous stubbornness on opinion dynamics is still lacking. In this paper, we study a popular opinion model in the presence of inhomogeneous stubbornness. We show analytically that heterogeneous stubbornness has a great impact on convergence time, expressed opinion of every node, and the overall expressed opinion. We provide an explanation of the expressed opinion in terms of stubbornness-dependent spanning diverging forests. We propose quantitative indicators to quantify some social concepts, including conflict, disagreement, and polarization by incorporating heterogeneous stubbornness, and develop a nearly linear time algorithm to approximate these quantities, which has a proved theoretical guarantee for the error of each quantity. To demonstrate the performance of our algorithm, we perform extensive experiments on a large set of real networks, which indicate that our algorithm is both efficient and effective, scalable to large networks with millions of nodes.

cs.SI↗

arXiv2021

Fast Evaluation for Relevant Quantities of Opinion Dynamics

One of the main subjects in the field of social networks is to quantify conflict, disagreement, controversy, and polarization, and some quantitative indicators have been developed to quantify these concepts. However, direct computation of these indicators involves the operations of matrix inversion and multiplication, which make it computationally infeasible for large-scale graphs with millions of nodes. In this paper, by reducing the problem of computing relevant quantities to evaluating $\ell_2$ norms of some vectors, we present a nearly linear time algorithm to estimate all these quantities. Our algorithm is based on the Laplacian solvers, and has a proved theoretical guarantee of error for each quantity. We execute extensive numerical experiments on a variety of real networks, which demonstrate that our approximation algorithm is efficient and effective, scalable to large graphs having millions of nodes.

cs.SI↗

arXiv2021

Maximizing Influence of Leaders in Social Networks

The operation of adding edges has been frequently used to the study of opinion dynamics in social networks for various purposes. In this paper, we consider the edge addition problem for the DeGroot model of opinion dynamics in a social network with $n$ nodes and $m$ edges, in the presence of a small number $s \ll n$ of competing leaders with binary opposing opinions 0 or 1. Concretely, we pose and investigate the problem of maximizing the equilibrium overall opinion by creating $k$ new edges in a candidate edge set, where each edge is incident to a 1-valued leader and a follower node. We show that the objective function is monotone and submodular. We then propose a simple greedy algorithm with an approximation factor $(1-\frac{1}{e})$ that approximately solves the problem in $O(n^3)$ time. Moreover, we provide a fast algorithm with a $(1-\frac{1}{e}-ε)$ approximation ratio and $\tilde{O}(mkε^{-2})$ time complexity for any $ε>0$, where $\tilde{O}(\cdot)$ notation suppresses the ${\rm poly} (\log n)$ factors. Extensive experiments demonstrate that our second approximate algorithm is efficient and effective, which scales to large networks with more than a million nodes.

cs.SI↗

arXiv2021

Edge Domination Number and the Number of Minimum Edge Dominating Sets in Pseudofractal Scale-Free Web and Sierpiński Gasket

As a fundamental research object, the minimum edge dominating set (MEDS) problem is of both theoretical and practical interest. However, determining the size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and it thus makes sense to find special graphs for which the MEDS problem can be exactly solved. In this paper, we study analytically the MEDS problem in the pseudofractal scale-free web and the Sierpiński gasket with the same number of vertices and edges. For both graphs, we obtain exact expressions for the edge domination number, as well as recursive solutions to the number of distinct MEDSs. In the pseudofractal scale-free web, the edge domination number is one-ninth of the number of edges, which is three-fifths of the edge domination number of the Sierpiński gasket. Moreover, the number of all MEDSs in the pseudofractal scale-free web is also less than that corresponding to the Sierpiński gasket. We argue that the difference of the size and number of MEDSs between the two studied graphs lies in the scale-free topology.

math.CO↗

arXiv2021

Coherence Scaling of Noisy Second-Order Scale-Free Consensus Networks

A striking discovery in the field of network science is that the majority of real networked systems have some universal structural properties. In generally, they are simultaneously sparse, scale-free, small-world, and loopy. In this paper, we investigate the second-order consensus of dynamic networks with such universal structures subject to white noise at vertices. We focus on the network coherence $H_{\rm SO}$ characterized in terms of the $\mathcal{H}_2$-norm of the vertex systems, which measures the mean deviation of vertex states from their average value. We first study numerically the coherence of some representative real-world networks. We find that their coherence $H_{\rm SO}$ scales sublinearly with the vertex number $N$. We then study analytically $H_{\rm SO}$ for a class of iteratively growing networks -- pseudofractal scale-free webs (PSFWs), and obtain an exact solution to $H_{\rm SO}$, which also increases sublinearly in $N$, with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally study $H_{\rm SO}$ for Sierpinśki gaskets, for which $H_{\rm SO}$ grows superlinearly in $N$, with a power exponent much larger than 1. Sierpinśki gaskets have the same number of vertices and edges as the PSFWs, but do not display the scale-free and small-world properties. We thus conclude that the scale-free and small-world, and loopy topologies are jointly responsible for the observed sublinear scaling of $H_{\rm SO}$.

math.NA↗

arXiv2020

Edge corona product as an approach to modeling complex simplical networks

Many graph products have been applied to generate complex networks with striking properties observed in real-world systems. In this paper, we propose a simple generative model for simplicial networks by iteratively using edge corona product. We present a comprehensive analysis of the structural properties of the network model, including degree distribution, diameter, clustering coefficient, as well as distribution of clique sizes, obtaining explicit expressions for these relevant quantities, which agree with the behaviors found in diverse real networks. Moreover, we obtain exact expressions for all the eigenvalues and their associated multiplicities of the normalized Laplacian matrix, based on which we derive explicit formulas for mixing time, mean hitting time and the number of spanning trees. Thus, as previous models generated by other graph products, our model is also an exactly solvable one, whose structural properties can be analytically treated. More interestingly, the expressions for the spectra of our model are also exactly determined, which is sharp contrast to previous models whose spectra can only be given recursively at most. This advantage makes our model a good test-bed and an ideal substrate network for studying dynamical processes, especially those closely related to the spectra of normalized Laplacian matrix, in order to uncover the influences of simplicial structure on these processes.

cs.DM↗

arXiv2020

Diffusion and Consensus in a Weakly Coupled Network of Networks

We study diffusion and consensus dynamics in a Network of Networks model. In this model, there is a collection of sub-networks, connected to one another using a small number of links. We consider a setting where the links between networks have small weights, or are used less frequently than links within each sub-network. Using spectral perturbation theory, we analyze the diffusion rate and convergence rate of the investigated systems. Our analysis shows that the first order approximation of the diffusion and convergence rates is independent of the topologies of the individual graphs; the rates depend only on the number of nodes in each graph and the topology of the connecting edges. The second order analysis shows a relationship between the diffusion and convergence rates and the information centrality of the connecting nodes within each sub-network. We further highlight these theoretical results through numerical examples.

cs.SI↗

arXiv2019

The k-Power Domination Number in Some Self-Similar Graphs

The $k$-power domination problem is a problem in graph theory, which has applications in many areas. However, it is hard to calculate the exact $k$-power domination number since determining k-power domination number of a generic graph is a NP-complete problem. We determine the exact $k$-power domination number in two graphs which have the same number of vertices and edges: pseudofractal scale-free web and Sierpiński gasket. The $k$-power domination number becomes 1 for $k\ge2$ in the Sierpiński gasket, while the $k$-power domination number increases at an exponential rate with regard to the number of vertices in the pseudofractal scale-free web. The scale-free property may account for the difference in the behavior of two graphs.

math.CO↗

arXiv2019

Exact calculations of first-passage properties on the pseudofractal scale-free web

In this paper, we consider discrete time random walks on the pseudofractal scale-free web (PSFW) and we study analytically the related first passage properties. First, we classify the nodes of the PSFW into different levels and propose a method to derive the generation function of the first passage probability from an arbitrary starting node to the absorbing domain, which is located at one or more nodes of low-level (i.e., nodes with large degree). Then, we calculate exactly the first passage probability, the survival probability, the mean and the variance of first passage time by using the generating functions as a tool. Finally, for some illustrative examples corresponding to given choices of starting node and absorbing domain, we derive exact and explicit results for such first passage properties. The method we propose can as well address the cases where the absorbing domain is located at one or more nodes of high-level on the PSFW, and it can also be used to calculate the first passage properties on other networks with self-similar structure, such as $(u, v)$ flowers and recursive scale-free trees.

cond-mat.stat-mech↗

arXiv2018

The resistance distance and Kirchhoff index on quadrilateral graph and pentagonal graph

The quadrilateral graph Q(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 3, whereas the pentagonal graph W(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 4. In this paper, closed-form formulas of resistance distance and Kirchhoff index for quadrilateral graph and pentagonal graph are obtained whenever G is an arbitrary graph.

math.CO↗

arXiv2018

Hitting times and resistance distances of $q$-triangulation graphs: Accurate results and applications

Graph operations or products, such as triangulation and Kronecker product have been extensively applied to model complex networks with striking properties observed in real-world complex systems. In this paper, we study hitting times and resistance distances of $q$-triangulation graphs. For a simple connected graph $G$, its $q$-triangulation graph $R_q(G)$ is obtained from $G$ by performing the $q$-triangulation operation on $G$. That is, for every edge $uv$ in $G$, we add $q$ disjoint paths of length $2$, each having $u$ and $v$ as its ends. We first derive the eigenvalues and eigenvectors of normalized adjacency matrix of $R_q(G)$, expressing them in terms of those associated with $G$. Based on these results, we further obtain some interesting quantities about random walks and resistance distances for $R_q(G)$, including two-node hitting time, Kemeny's constant, two-node resistance distance, Kirchhoff index, additive degree-Kirchhoff index, and multiplicative degree-Kirchhoff index. Finally, we provide exact formulas for the aforementioned quantities of iterated $q$-triangulation graphs, using which we provide closed-form expressions for those quantities corresponding to a class of scale-free small-world graphs, which has been applied to mimic complex networks.

math.CO↗

arXiv2018

Spectra, hitting times, and resistance distances of $q$-subdivision graphs

Graph operations or products play an important role in complex networks. In this paper, we study the properties of $q$-subdivision graphs, which have been applied to model complex networks. For a simple connected graph $G$, its $q$-subdivision graph $S_q(G)$ is obtained from $G$ through replacing every edge $uv$ in $G$ by $q$ disjoint paths of length 2, with each path having $u$ and $v$ as its ends. We derive explicit formulas for many quantities of $S_q(G)$ in terms of those corresponding to $G$, including the eigenvalues and eigenvectors of normalized adjacency matrix, two-node hitting time, Kemeny constant, two-node resistance distance, Kirchhoff index, additive degree-Kirchhoff index, and multiplicative degree-Kirchhoff index. We also study the properties of the iterated $q$-subdivision graphs, based on which we obtain the closed-form expressions for a family of hierarchical lattices, which has been used to describe scale-free fractal networks.

math.CO↗

arXiv2018

Maximizing the Number of Spanning Trees in a Connected Graph

We study the problem of maximizing the number of spanning trees in a connected graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm that, using submodularity, obtains an approximation ratio of $(1 - 1/e - ε)$ in the exponent of the number of spanning trees for any $ε> 0$ in time $\tilde{O}(m ε^{-1} + (n + q) ε^{-3})$, where $m$ and $q$ is the number of edges in the original graph and the candidate edge set, respectively. Our running time is optimal with respect to the input size up to logarithmic factors, and substantially improves upon the $O(n^3)$ running time of the previous proposed greedy algorithm with approximation ratio $(1 - 1/e)$ in the exponent. Notably, the independence of our running time of $k$ is novel, comparing to conventional top-$k$ selections on graphs that usually run in $Ω(mk)$ time. A key ingredient of our greedy algorithm is a routine for maintaining effective resistances under edge additions in an online-offline hybrid setting. - We show the exponential inapproximability of this problem by proving that there exists a constant $c > 0$ such that it is NP-hard to approximate the optimum number of spanning trees in the exponent within $(1 - c)$. This inapproximability result follows from a reduction from the minimum path cover in undirected graphs, whose hardness again follows from the constant inapproximability of the Traveling Salesman Problem (TSP) with distances 1 and 2. Thus, the approximation ratio of our algorithm is also optimal up to a constant factor in the exponent. To our knowledge, this is the first hardness of approximation result for maximizing the number of spanning trees in a graph, or equivalently, by Kirchhoff's matrix-tree theorem, maximizing the determinant of an SDDM matrix.

cs.DS↗

arXiv2018

Normalized Laplacian spectrum of some generalized subdivision-corona of two regular graphs

In this paper, we determine the two normalized Laplacian spectrum of generalized subdivision- vertex corona, subdivision-edge corona for a connected regular graph with an arbitrary reg- ular graph in terms of their normalized Laplacian eigenvalues. Moreover, applying these results, we find some non-regular normaliaed Laplacian cospectral graphs. These results generalize the existing results in [11].

math.CO↗

arXiv2018

Improving information centrality of a node in complex networks by adding edges

The problem of increasing the centrality of a network node arises in many practical applications. In this paper, we study the optimization problem of maximizing the information centrality $I_v$ of a given node $v$ in a network with $n$ nodes and $m$ edges, by creating $k$ new edges incident to $v$. Since $I_v$ is the reciprocal of the sum of resistance distance $\mathcal{R}_v$ between $v$ and all nodes, we alternatively consider the problem of minimizing $\mathcal{R}_v$ by adding $k$ new edges linked to $v$. We show that the objective function is monotone and supermodular. We provide a simple greedy algorithm with an approximation factor $\left(1-\frac{1}{e}\right)$ and $O(n^3)$ running time. To speed up the computation, we also present an algorithm to compute $\left(1-\frac{1}{e}-ε\right)$-approximate resistance distance $\mathcal{R}_v$ after iteratively adding $k$ edges, the running time of which is $\widetilde{O} (mkε^{-2})$ for any $ε>0$, where the $\widetilde{O} (\cdot)$ notation suppresses the ${\rm poly} (\log n)$ factors. We experimentally demonstrate the effectiveness and efficiency of our proposed algorithms.

cs.SI↗

arXiv2018

Non-Backtracking Centrality Based Random Walk on Networks

Random walks are a fundamental tool for analyzing realistic complex networked systems and implementing randomized algorithms to solve diverse problems such as searching and sampling. For many real applications, their actual effect and convenience depend on the properties (e.g. stationary distribution and hitting time) of random walks, with biased random walks often outperforming traditional unbiased random walks (TURW). In this paper, we present a new class of biased random walks, non-backtracking centrality based random walks (NBCRW) on a network, where the walker prefers to jump to neighbors with high non-backtracking centrality that has some advantages over eigenvector centrality. We study some properties of the non-backtracking matrix of a network, on the basis of which we propose a theoretical framework for fast computation of the transition probabilities, stationary distribution, and hitting times for NBCRW on the network. Within the paradigm, we study NBCRW on some model and real networks and compare the results with those corresponding to TURW and maximal entropy random walks (MERW), with the latter being biased random walks based on eigenvector centrality. We show that the behaviors of stationary distribution and hitting times for NBCRW widely differ from those associated with TURW and MERW, especially for heterogeneous networks.

cs.SI↗

arXiv2018

Independence number and the number of maximum independent sets in pseudofractal scale-free web and Sierpiński gasket

As a fundamental subject of theoretical computer science, the maximum independent set (MIS) problem not only is of purely theoretical interest, but also has found wide applications in various fields. However, for a general graph determining the size of a MIS is NP-hard, and exact computation of the number of all MISs is even more difficult. It is thus of significant interest to seek special graphs for which the MIS problem can be exactly solved. In this paper, we address the MIS problem in the pseudofractal scale-free web and the Sierpiński gasket, which have the same number of vertices and edges. For both graphs, we determine exactly the independence number and the number of all possible MISs. The independence number of the pseudofractal scale-free web is as twice as the one of the Sierpiński gasket. Moreover, the pseudofractal scale-free web has a unique MIS, while the number of MISs in the Sierpiński gasket grows exponentially with the number of vertices.

cs.DS↗