The phenomenon of opinion disagreement has been empirically observed and reported in the literature, which is affected by various factors, such as the structure of social networks. An important discovery in network science is that most real-life networks, including social networks, are scale-free and sparse. In this paper, we study noisy opinion dynamics in sparse scale-free social networks to uncover the influence of power-law topology on opinion disagreement. We adopt the popular discrete-time DeGroot model for opinion dynamics in a graph, where nodes' opinions are subject to white noise. We first study opinion disagreement in many realistic and model networks with a scale-free topology, which approaches a constant, indicating that a scale-free structure is resistant to noise in the opinion dynamics. Moreover, existing algorithms for estimating opinion disagreement are computationally impractical for large-scale networks due to their high computational complexity. To solve this challenge, we introduce a sublinear-time algorithm to approximate this quantity with a theoretically guaranteed error. This algorithm efficiently simulates truncated random walks starting from a subset of nodes while preserving accurate estimation. Extensive experiments demonstrate its efficiency, accuracy, and scalability.

For random walks on graph \(\mathcal{G}\) with \(n\) vertices and \(m\) edges, the mean hitting time \(H_{j}\) from a vertex chosen from the stationary distribution to vertex \(j\) measures the importance for \(j\) , while the Kemeny constant \(\mathcal{K}\) is the mean hitting time from one vertex to another selected randomly according to the stationary distribution. In this article, we first establish a connection between the two quantities, representing \(\mathcal{K}\) in terms of \(H_{j}\) for all vertices. We then develop an efficient algorithm estimating \(H_{j}\) for all vertices and \(\mathcal{K}\) in nearly linear time of \(m\) . Moreover, we extend the centrality \(H_{j}\) of a single vertex to \(H(S)\) of a vertex set \(S\) , and establish a link between \(H(S)\) and some other quantities. We further study the NP-hard problem of selecting a group \(S\) of \(k\ll n\) vertices with minimum \(H(S)\) , whose objective function is monotonic and supermodular. We finally propose two greedy algorithms approximately solving the problem. The former has an approximation factor \((1-\frac{k}{k-1}\frac{1}{e})\) and \(O(kn^{3})\) running time, while the latter returns a \((1-\frac{k}{k-1}\frac{1}{e}-\epsilon)\) -approximation solution in nearly-linear time of \(m\) , for any parameter \(0{\lt}\epsilon{\lt}1\) . Extensive experiment results validate the performance of our algorithms.

Abstract Graph products have been extensively applied to model complex networks with striking properties observed in real-world complex systems. In this paper, we study the hitting times for random walks on a class of graphs generated iteratively by edge corona product. We first derive recursive solutions to the eigenvalues and eigenvectors of the normalized adjacency matrix associated with the graphs. Based on these results, we further obtain interesting quantities about hitting times of random walks, providing iterative formulas for two-node hitting time, as well as closed-form expressions for the Kemeny’s constant defined as a weighted average of hitting times over all node pairs, as well as the arithmetic mean of hitting times of all pairs of nodes.

Opinion dynamics is a central subject of computational social science, and various models have been developed to understand the evolution and formulation of opinions. Existing models mainly focus on opinion dynamics on graphs that only capture pairwise interactions between agents. In this article, we extend the popular Friedkin–Johnsen model for opinion dynamics on graphs to hypergraphs, which describe higher order interactions occurring frequently on real networks, especially social networks. To achieve this, based on the fact that for linear dynamics, the multiway interactions can be reduced to effective pairwise node interactions, we propose a method to decode the group interactions encoded in hyperedges by undirected edges or directed edges in graphs. We then show that higher order interactions play an important role in the opinion dynamics since the overall steady-state expressed opinion and polarization differ greatly from those without group interactions. We also provide an interpretation of the equilibrium expressed opinion from the perspective of the spanning converging forest, based on which we design a fast sampling algorithm to approximately evaluate the overall opinion and opinion polarization on directed weighted graphs. Finally, we conduct experiments on real-world hypergraph datasets, demonstrating the performance of our algorithm.

A signed graph offers richer information than an unsigned graph, since it describes both collaborative and competitive relationships in social networks. In this paper, we study opinion dynamics on a signed graph, based on the Friedkin-Johnsen model. We first interpret the equilibrium opinion in terms of a defined random walk on an augmented signed graph, by representing the equilibrium opinion of every node as a combination of all nodes’ internal opinions, with the coefficient of the internal opinion for each node being the difference of two absorbing probabilities. We then quantify some relevant social phenomena and express them in terms of the$\ell _{2}$norms of vectors. We also design a nearly-linear time signed Laplacian solver for assessing these quantities, by establishing a connection between the absorbing probability of random walks on a signed graph and that on an associated unsigned graph. We further study the opinion optimization problem by changing the initial opinions of a fixed number of nodes, which can be optimally solved in cubic time. We provide a nearly-linear time algorithm with an error guarantee to approximately solve the problem. Finally, we execute extensive experiments on sixteen real-life signed networks, which show that both of our algorithms are effective and efficient, and are scalable to massive graphs with over 20 million nodes.

The operation of creating edges has been widely applied to optimize relevant quantities of opinion dynamics. In this paper, we consider a problem of polarization optimization for the leader-follower opinion dynamics in a noisy social network with n nodes and m edges, where a group Q of q nodes are leaders, and the remaining n-q nodes are followers. We adopt the popular leader-follower DeGroot model, where the opinion of every leader is identical and remains unchanged, while the opinion of every follower is subject to white noise. The polarization is defined as the steady-state variance of the deviation of each node's opinion from leaders' opinion, which equals one half of the effective resistance RQ between the node group Q and all other nodes. Concretely, we propose and study the problem of minimizing RQ by adding k new edges with each incident to a node in Q. We show that the objective function is monotone and supermodular. We then propose a simple greedy algorithm with an approximation factor 1-1/e that approximately solves the problem in O((n-q)3) time. To speed up the computation, we also provide a fast algorithm to compute (1-1/e-ε)-approximate effective resistance RQ, the running time of which is O~ (mkε-2) for any ε>0, where the O~(·) notation suppresses the poly(log n) factors. Extensive experiment results show that our second algorithm is both effective and efficient.

Abstract 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 more 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.

The cover time of random walks on a graph has found wide practical applications in different fields of computer science, such as crawling and searching on the World Wide Web and query processing in sensor networks, with the application effects dependent on the behavior of the cover time: the smaller the cover time, the better the application performance. It was proved that over all graphs with N nodes, complete graphs have the minimum cover time N log N . However, complete graphs cannot mimic real-world networks with small average degree and scale-free small-world properties, for which the cover time has not been examined carefully, and its behavior is still not well understood. In this article, we first experimentally evaluate the cover time for various real-world networks with scale-free small-world properties, which scales as N log N . To better understand the behavior of the cover time for real-world networks, we then study the cover time of three scale-free small-world model networks by using the connection between cover time and resistance diameter. For all the three networks, their cover time also behaves as N log N . This work indicates that sparse networks with scale-free and small-world topology are favorable architectures with optimal scaling of cover time. Our results deepen understanding the behavior of cover time in real-world networks with scale-free small-world structure, and have potential implications in the design of efficient algorithms related to cover time.

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.

Abstract Edge centrality has found wide applications in various aspects. Many edge centrality metrics have been proposed, but the crucial issue that how good the discriminating power of a metric is, with respect to other measures, is still open. In this paper, we address the question about the benchmark of the discriminating power of edge centrality metrics. We first use the automorphism concept to define equivalent edges, based on which we introduce a benchmark for the discriminating power of edge centrality measures and develop a fast approach to compare the discriminating power of different measures. According to the benchmark, for a desirable measure, equivalent edges have identical metric scores, while inequivalent edges possess different scores. However, we show that even in a toy graph, inequivalent edges cannot be discriminated by three existing edge centrality metrics. We then present a novel edge centrality metric called forest centrality (FC). Extensive experiments on real-world networks and model networks indicate that FC has better discriminating power than three existing edge centrality metrics.

Abstract The power-law behavior is ubiquitous in a majority of real-world networks, and it was shown to have a strong effect on various combinatorial, structural and dynamical properties of graphs. For example, it has been shown that in real-life power-law networks, both the matching number and the domination number are relatively smaller, compared with homogeneous graphs. In this paper, we study analytically several combinatorial problems for two power-law graphs with the same number of vertices, edges and the same power exponent. For both graphs, we determine exactly or recursively their matching number, independence number, domination number, the number of maximum matchings, the number of maximum independent sets and the number of minimum dominating sets. We show that power-law behavior itself cannot characterize the combinatorial properties of a heterogenous graph. Since the combinatorial properties studied here have found wide applications in different fields, such as structural controllability of complex networks, our work offers insight in the applications of these combinatorial problems in power-law graphs.

Abstract 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.

A striking discovery in the field of network science is that the majority of real networked systems have some universal structural properties. In general, they are simultaneously sparse, scale-free, small-world, and loopy. In this article, we investigate the second-order consensus of dynamic networks with such universal structures subject to white noise at vertices. We focus on the network coherenceHSOcharacterized 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 coherenceHSOscales sublinearly with the vertex number$N$. We then study analyticallyHSOfor a class of iteratively growing networks—pseudofractal scale-free webs (PSFWs), and obtain an exact solution toHSO, which also increases sublinearly in$N$, with an exponent much smaller than 1. To explain the reasons for this sublinear behavior, we finally studyHSOfor Sierpinśki gaskets, for whichHSOgrows 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, small-world, and loopy topologies are jointly responsible for the observed sublinear scaling ofHSO.

It has been recently established that for second-order consensus dynamics with additive noise, the performance measures, including the vertex coherence and network coherence defined, respectively, as the steady-state variance of the deviation of each vertex state from the average and the average steady-state variance of the system, are closely related to the biharmonic distances. However, direct computation of biharmonic distances is computationally infeasible for huge networks with millions of vertices. In this article, leveraging the implicit fact that both vertex and network coherence can be expressed in terms of the diagonal entries of pseudoinverse$\boldsymbol {{L}}^{2\dagger }$of the square of graph Laplacian, we develop a nearly linear-time algorithm to approximate all diagonal entries of$\boldsymbol {{L}}^{2\dagger }$, which has a theoretically guaranteed error for each diagonal entry. The key ingredient of our approximation algorithm is an integration of the Johnson–Lindenstrauss lemma and Laplacian solvers. Extensive numerical experiments on real-life and model networks are presented, which indicate that our approximation algorithm is both efficient and accurate and is scalable to large-scale networks with millions of vertices.

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 ℓ2 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.

Abstract The mixing time of random walks on a graph has found broad applications across both theoretical and practical aspects of computer science, with the application effects depending on the behavior of mixing time. It is extensively believed that real-world networks, especially social networks, are fast mixing with their mixing time at most $O(\log N)$ where $N$ is the number of vertices. However, the behavior of mixing time in the real-life networks has not been examined carefully, and exactly analytical research for mixing time in models mimicking real networks is still lacking. In this paper, we first experimentally evaluate the mixing time of various real-world networks with scale-free small-world properties and show that their mixing time is much higher than anticipated. To better understand the behavior of the mixing time for real-world networks, we then analytically study the mixing time of the Apollonian network, which is simultaneously scale-free and small-world. To this end, we derive the recursive relations for all eigenvalues, especially the second largest eigenvalue modulus of the transition matrix, based on which we deduce a lower bound for the mixing time of the Apollonian network, which approximately scales sublinearly with $N$. Our results indicate that real-world networks are not always fast mixing, which has potential implications in the design of algorithms related to mixing time.

The issue of opinion sharing and formation has received considerable attention in the academic literature, and a few models have been proposed to study this problem. However, existing models are limited to the interactions among nearest neighbors, ignoring those second, third, and higher-order neighbors, despite the fact that higher-order interactions occur frequently in real social networks. In this paper, we develop a new model for opinion dynamics by incorporating long-range interactions based on higher-order random walks. We prove that the model converges to a fixed opinion vector, which may differ greatly from those models without higher-order interactions. Since direct computation of the equilibrium opinions is computationally expensive, which involves the operations of huge-scale matrix multiplication and inversion, we design a theoretically convergence-guaranteed estimation algorithm that approximates the equilibrium opinion vector nearly linearly in both space and time with respect to the number of edges in the graph. We conduct extensive experiments on various social networks, demonstrating that the new algorithm is both highly efficient and effective.

For random walks on a graph, the mean hitting time $H_j$ from a vertex i chosen from the stationary distribution to the target vertex j can be used as a measure of importance for vertex j, while the Kemeny constant K is the mean hitting time from a vertex i to a vertex j selected randomly according to the stationary distribution. Both quantities have found a large variety of applications in different areas. However, their high computational complexity limits their applications, especially for large networks with millions of vertices. In this paper, we first establish a connection between the two quantities, representing K in terms of $H_j$ for all vertices. We then express both quantities in terms of quadratic forms of the pseudoinverse for graph Laplacian, based on which we develop an efficient algorithm that provides an approximation of $H_j$ for all vertices and K in nearly linear time with respect to the edge number, with high probability. Extensive experiment results on real-life and model networks validate both the efficiency and accuracy of the proposed algorithm.

The mean hitting time from a node i to a node j selected randomly according to the stationary distribution of random walks is called the Kemeny constant, which has found various applications. It was proved that over all graphs with N vertices, complete graphs have the exact minimum Kemeny constant, growing linearly with N. Here we study numerically or analytically the Kemeny constant on many sparse real-world and model networks with scale-free small-world topology, and show that their Kemeny constant also behaves linearly with N. Thus, sparse networks with scale-free and small-world topology are favorable architectures with optimal scaling of Kemeny constant. We then present a theoretically guaranteed estimation algorithm, which approximates the Kemeny constant for a graph in nearly linear time with respect to the number of edges. Extensive numerical experiments on model and real networks show that our approximation algorithm is both efficient and accurate.