arXiv subjects

Yuhao Yi

Publications and source records attributed to Yuhao Yi.

29 records · Page 2Linked to original sources

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↗

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

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

Current Flow Group Closeness Centrality for Complex Networks

Current flow closeness centrality (CFCC) has a better discriminating ability than the ordinary closeness centrality based on shortest paths. In this paper, we extend this notion to a group of vertices in a weighted graph, and then study the problem of finding a subset $S$ of $k$ vertices to maximize its CFCC $C(S)$, both theoretically and experimentally. We show that the problem is NP-hard, but propose two greedy algorithms for minimizing the reciprocal of $C(S)$ with provable guarantees using the monotoncity and supermodularity. The first is a deterministic algorithm with an approximation factor $(1-\frac{k}{k-1}\cdot\frac{1}{e})$ and cubic running time; while the second is a randomized algorithm with a $(1-\frac{k}{k-1}\cdot\frac{1}{e}-ε)$-approximation and nearly-linear running time for any $ε> 0$. Extensive experiments on model and real networks demonstrate that our algorithms are effective and efficient, with the second algorithm being scalable to massive networks with more than a million vertices.

cs.DS↗

arXiv2018

Scale-free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks

The vast majority of real-world networks are scale-free, loopy, and sparse, with a power-law degree distribution and a constant average degree. In this paper, we study first-order consensus dynamics in binary scale-free networks, where vertices are subject to white noise. We focus on the coherence of networks characterized in terms of the $H_2$-norm, which quantifies how closely agents track the consensus value. We first provide a lower bound of coherence of a network in terms of its average degree, which is independent of the network order. We then study the coherence of some sparse, scale-free real-world networks, which approaches a constant. We also study numerically the coherence of Barabási-Albert networks and high-dimensional random Apollonian networks, which also converges to a constant when the networks grow. Finally, based on the connection of coherence and the Kirchhoff index, we study analytically the coherence of two deterministically-growing sparse networks and obtain the exact expressions, which tend to small constants. Our results indicate that the effect of noise on the consensus dynamics in power-law networks is negligible. We argue that scale-free topology, together with loopy structure, is responsible for the strong robustness with respect to noisy consensus dynamics in power-law networks.

eess.SY↗

arXiv2017

Consensus in Self-similar Hierarchical Graphs and Sierpiński Graphs: Convergence Speed, Delay Robustness, and Coherence

The hierarchical graphs and Sierpiński graphs are constructed iteratively, which have the same number of vertices and edges at any iteration, but exhibit quite different structural properties: the hierarchical graphs are non-fractal and small-world, while the Sierpiński graphs are fractal and "large-world". Both graphs have found broad applications. In this paper, we study consensus problems in hierarchical graphs and Sierpiński graphs, focusing on three important quantities of consensus problems, that is, convergence speed, delay robustness, and coherence for first-order (and second-order) dynamics, which are, respectively, determined by algebraic connectivity, maximum eigenvalue, and sum of reciprocal (and square of reciprocal) of each nonzero eigenvalue of Laplacian matrix. For both graphs, based on the explicit recursive relation of eigenvalues at two successive iterations, we evaluate the second smallest eigenvalue, as well as the largest eigenvalue, and obtain the closed-form solutions to the sum of reciprocals (and square of reciprocals) of all nonzero eigenvalues. We also compare our obtained results for consensus problems on both graphs and show that they differ in all quantities concerned, which is due to the marked difference of their topological structures.

eess.SY↗

arXiv2017

Biharmonic Distance and the Performance of Second-Order Consensus Networks with Stochastic Disturbances

We study second order consensus dynamics with random additive disturbances. We investigate three different performance measures: the steady-state variance of pairwise differences between vertex states, the steady-state variance of the deviation of each vertex state from the average, and the total steady-state variance of the system. We show that these performance measures are closely related to the biharmonic distance; the square of the biharmonic distance plays similar role in the system performance as resistance distances plays in the performance of first-order noisy consensus dynamics. We further define the new concepts of biharmonic Kirchhoff index and vertex centrality based on the biharmonic distance. Finally, we derive analytical results for the performance measures and concepts for complete graphs, star graphs, cycles, and paths, and we use this analysis to compare the asymptotic behavior of the steady-variance in first- and second-order systems.

math.OC↗

arXiv2017

A Resistance Distance-Based Approach for Optimal Leader Selection in Noisy Consensus Networks

We study the performance of leader-follower noisy consensus networks, and in particular, the relationship between this performance and the locations of the leader nodes. Two types of dynamics are considered (1) noise-free leaders, in which leaders dictate the trajectory exactly and followers are subject to external disturbances, and (2) noise-corrupted leaders, in which both leaders and followers are subject to external perturbations. We measure the performance of a network by its coherence, an $H_2$ norm that quantifies how closely the followers track the leaders' trajectory. For both dynamics, we show a relationship between the coherence and resistance distances in an a electrical network. Using this relationship, we derive closed-form expressions for coherence as a function of the locations of the leaders. Further, we give analytical solutions to the optimal leader selection problem for several special classes of graphs.

math.OC↗

arXiv2016

Corona graphs as a model of small-world networks

We introduce recursive corona graphs as a model of small-world networks. We investigate analytically the critical characteristics of the model, including order and size, degree distribution, average path length, clustering coefficient, and the number of spanning trees, as well as Kirchhoff index. Furthermore, we study the spectra for the adjacency matrix and the Laplacian matrix for the model. We obtain explicit results for all the quantities of the recursive corona graphs, which are similar to those observed in real-life networks.

cs.DM↗

arXiv2015

Anomalous behavior of trapping in extended dendrimers with a perfect trap

Compact and extended dendrimers are two important classes of dendritic polymers. The impact of the underlying structure of compact dendrimers on dynamical processes has been much studied, yet the relation between the dynamical and structural properties of extended dendrimers remains not well understood. In this paper, we study the trapping problem in extended dendrimers with generation-dependent segment lengths, which is different from that of compact dendrimers where the length of the linear segments is fixed. We first consider a particular case that the deep trap is located at the central node, and derive an exact formula for the average trapping time (ATT) defined as the average of the source-to-trap mean first passage time over all starting points. Then, using the obtained result we deduce a closed-form expression for the ATT to an arbitrary trap node, based on which we further obtain an explicit solution to the ATT corresponding to the trapping issue with the trap uniformly distributed in the polymer systems. We show that the trap location has a substantial influence on the trapping efficiency measured by the ATT, which increases with the shortest distance from the trap to the central node, a phenomenon similar to that for compact dendrimers. In contrast to this resemblance, the leading terms of ATTs for the three trapping problems differ drastically between extended and compact dendrimers, with the trapping processes in the extended dendrimers being less efficient than in compact dendrimers.

cond-mat.soft↗