Opinion Dynamics Incorporating Higher-Order Interactions
aa r X i v : . [ c s . S I] F e b IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 1
Opinion Dynamics Incorporating Higher-OrderInteractions
Zuobai Zhang, Wanyue Xu, Zhongzhi Zhang,
Member, IEEE, and Guanrong Chen,
Life Fellow, IEEE
Abstract —The issue of opinion sharing and formation has received considerable attention in the academic literature, and a few modelshave been proposed to study this problem. However, existing models are limited to the interactions among nearest neighbors, ignoringthose 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 randomwalks. We prove that the model converges to a fixed opinion vector, which may differ greatly from those models without higher-orderinteractions. Since direct computation of the equilibrium opinion is computationally expensive, which involves the operations ofhuge-scale matrix multiplication and inversion, we design a theoretically convergence-guaranteed estimation algorithm thatapproximates the equilibrium opinion vector nearly linearly in both space and time with respect to the number of edges in the graph. Weconduct extensive experiments on various social networks, demonstrating that the new algorithm is both highly efficient and effective.
Index Terms —Opinion dynamics; social network; computational social science; random walk; spectral graph theory ✦ NTRODUCTION R ECENT years have witnessed an explosive growth insocial media and online social networks, which haveincreasingly become an important part of our lives [1]. Forexample, online social networks can increase the diversity ofopinions, ideas, and information available to individuals [2],[3]. At the same time, people may use online social networksto broadcast information on their lives and their opinionsabout some topics or issues to a large audience. It hasbeen reported that social networks and social media haveresulted in a fundamental change of ways that people shareand shape opinions [4], [5], [6]. Recently, there have beena concerted effort to model opinion dynamics in socialnetworks, in order to understand the effects of variousfactors on the formation dynamics of opinions [7], [8], [9].One of the popular opinion dynamics models is theFriedkin-Johnsen (FJ) model [10]. Although simple andsuccinct, the FJ model can capture complex behavior ofreal social groups by incorporating French’s “theory ofsocial power” [11], and thus has been extensively studied.A sufficient condition for the stability of this standardmodel was obtained in [12], the average innate opinionwas estimated in [13], and the unique equilibrium expressedopinion vector was derived in [13], [14]. Some explanationsof this natural model were consequently explored fromdifferent perspectives [14], [15]. In addition, based on the • Zuobai Zhang, Wanyue Xu and Zhongzhi Zhang are with theShanghai Key Laboratory of Intelligent Information Processing,School of Computer Science, Fudan University, Shanghai 200433,China. Zhongzhi Zhang is also with the Shanghai EngineeringResearch Institute of Blockchains, Fudan University, Shanghai 200433,China. (e-mail: [email protected]; [email protected];[email protected]). • Guanrong Chen is with the Department of Electrical Engineer-ing, City University of Hong Kong, Hong Kong SAR, China (e-mail:[email protected]).(Corresponding author: Zhongzhi Zhang)Manuscript received November 27, 2020; revised November 27, 2020.
FJ opinion dynamics model, some social phenomena havebeen quantified and studied, including polarization [16],[17], disagreement [17], conflict [18], and controversy [18].Moreover, some optimization problems [19] for the FJ modelwere also investigated, such as opinion maximization [20].Other than studying the properties, interpretations andrelated quantities of the FJ model, many extensions orvariants of this popular model have been developed [7].In [19], the impact of susceptibility to persuasion on opin-ion dynamics were analyzed by introducing a resistanceparameter to modify the FJ model. In [21], a varying peer-pressure coefficient was introduced to the FJ model, aimingto explore the role of increasing peer pressure on opinionformation. In [22], the FJ model was augmented to includealgorithmic filtering, to analyze the effect of filter bubbleson polarization. Some multidimensional extensions weredeveloped for the FJ model [23], [24], [25], [26], extendingthe scalar opinion to vector-valued opinions correspondingto several settings, either independent [23] or interdepen-dent [24], [25], [26].The above related works for opinion dynamic modelsprovide deep insights into the understanding of opinionformulation, since they grasped various important aspectsaffecting opinion shaping, including individual’s attributes,interactions among individuals, and opinion update mech-anisms. However, existing models consider only the in-teractions among the nearest neighbors, neglecting thoseinteractions among second-order, third-order, and higher-order nearest neighbors, in spite of the fact that this situationis commonly encountered in real natural [27] and social [28],[29] networks. For example, in social networks an individualcan make use of the local, partial, or global knowledge corre-sponding to his direct, second-order, and even higher-orderneighbors to search for opinions about a concerned issue orto diffuse information and opinions in an efficient way. Todate, there still lack a comprehensive higher-order opiniondynamics model on social networks, although it has been
EEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 2 observed that long-range non-nearest-neighbor interactionscould play a fundamental role in opinion dynamics.In this paper, we make a natural extension of theclassical FJ opinion dynamics model to incorporate thehigher-order interactions between individuals and theirnon-nearest neighbors by leveraging higher-order randomwalks. We prove that the higher-order model converges toa unique equilibrium opinion vector, provided that each in-dividual has a non-zero resistance parameter measuring hissusceptibility to persuasion. We show that the equilibriumopinions of the higher-order FJ model differ greatly fromthose of the classical FJ model, demonstrating that higher-order interactions have a significant impact on opiniondynamics.Basically, the equilibrium opinions of the higher-order FJmodel on a graph are the same as those of the standard FJmodel on a corresponding dense graph with a loop at eachnode. That is, at each time step, every individual updateshis opinion according to his innate opinion, as well as thecurrently expressed opinions of his nearest neighbors onthe dense graph. Since the transition matrix of the densegraph is a combination of the powers of that on the originalgraph, direct construction of the transition matrix for thedense graph is computationally expensive. To reduce thecomputation cost, we construct a sparse matrix, which isspectrally close to the dense matrix, nearly linearly in bothspace and time with respect to the number of edges onthe original graph. This sparsified matrix maintains theinformation of the dense graph, such that the differencebetween the equilibrium opinions on the dense graph andthe sparsified graph is negligible.Based on the obtained sparsifed matrix, we furtherintroduce an iteration algorithm, which has a theoreticalconvergence and can approximate the equilibrium opinionsof the higher-order FJ model quickly. Finally, we performextensive experiments on different networks of variousscales, and show that the new algorithm achieves highefficiency and effectiveness. Particularly, this algorithm isscalable, which can approximate the equilibrium opinionsof the second-order FJ on large graphs with millions ofnodes. It is expected that the new model sheds light onfurther understanding of opinion formation, and that thenew algorithm can be helpful for various applications, suchas the computations of polarization and disagreement inopinion dynamics.A preliminary version of our work has been publishedin [30]. In this paper, we extend our preliminary results inseveral directions. First, we present proof details previouslyomitted in [30] for several important theorems, includingthe convergence analysis and approximation error boundof the proposed algorithm. Second, we add an illustrativeexample in Section HOFJ, in order to better understandand demonstrate the difference between the traditional FJmodel and the higher-order model. Finally, we provideadditional experimental results for different innate opiniondistributions and provide a thorough parameter analysis inSection 6.
RELIMINARIES
In this section, some basic concepts in graph and matrix the-ories, as well as the Friedkin-Johnsen (FJ) opinion dynamics model are briefly reviewed.
Consider a simple, connected, undirected social network(graph) G = ( V , E ) , where V = { , , ..., n } is the setof n agents and E = { ( i, j ) | i, j ∈ V} is the set of m edges describing relations among nearest neighbors. Thetopological and weighted properties of G are encoded inits adjacency matrix A = ( a ij ) n × n , where a ij = a ji = w e if i and j are linked by an edge e = ( i, j ) ∈ E with weight w e ,and a ij = 0 otherwise. Let N i = { j | ( i, j ) ∈ E} denote theset of neighbors of node i and d i = P j ∈N i w ij denote thedegree of i . The diagonal degree matrix of graph G is definedto be D = diag( d , d , ..., d n ) , and the Laplacian matrix of G is L = D − A . Let e i denote the i -th standard basisvector of appropriate dimension. Let ( ) be the vector withall entries being ones (zeros). Then, it can be verified that L = . The random walk transition matrix for G is definedas P = D − A , which is row-stochastic (i.e., each row-sumequals ). For a non-negative vector x , x max and x min denote themaximum and minimum entry, respectively. For an n × n matrix A , σ i ( A ) , i = 1 , , ..., n denote its singular values.Given a vector x , its -norm is defined as k x k = pP i | x i | and the ∞ -norm is defined as k x k ∞ = max i | x i | . It iseasy to verify that k x k ∞ ≤ k x k ≤ √ n k x k ∞ for any n -dimensional vector x . For a matrix A , its -norm isdefined to be k A k = max x k Ax k / k x k . By definition, k Ax k ≤ k A k k x k . It is known that the -norm of thematrix A is equal to its maximum singular value σ max ,satisfying k x ⊤ Ay k ≤ σ max k x k k y k for any vectors x and y [31]. The Friedkin-Johnsen (FJ) model is a classic opinion dynam-ics model [10]. For a specific topic, the FJ model assumesthat each agent i ∈ V is associated with an innate opinion s i ∈ [0 , , where higher values signify more favorable opin-ions, and a resistance parameter α i ∈ (0 , quantifying theagent’s stubbornness , with a higher value corresponding toa lower tendency to conform with his neighbors’ opinions.Let x ( t ) denote the opinion vector of all agents at time t ,with element x ( t ) i representing the opinion of agent i at thattime. At every timestep, each agent updates his opinion bytaking a convex combination of his innate opinion and theaverage of the expressed opinion of his neighbors in theprevious timestep. Mathematically, the opinion of agent i evolves according to the following rule: x ( t +1) i = α i s i + (1 − α i ) P j ∈N i w ij · x ( t ) j d i . (1)The evolution rule can be rewritten in matrix form as x ( t +1) = Λ s + ( I − Λ ) P x ( t ) , (2)where Λ denotes the diagonal matrix diag( α , α , ..., α n ) ,and I is the identity matrix. EEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 3
It has been proved [13] that the above opinion formationprocess converges to a unique equilibrium z when α i > for all i ∈ V . The equilibrium vector z can be obtained asthe unique fixed point of equation (2), i.e., z = ( I − ( I − Λ ) P ) − Λ s . (3)The i th entry z i of z is the expressed opinion of agent i .A straightforward way to calculate the equilibrium vec-tor z requires inverting a matrix, which is expensive andintractable for large networks. In [32], the iteration processof the opinion dynamics model is used to obtain an ap-proximation of vector z , which has a theoretical guaranteeof convergence. The method is very efficient, scalable tonetworks with millions of nodes. IGHER - ORDER O PINION D YNAMICS M ODEL
The classical FJ model has many advantages; for example, itcaptures some complex human behavior in social networks.However, this model considers only the interactions amongnearest neighbors, neglecting the higher-order interactionsexisting in social networks and social media. To fix thisdeficiency, in this section, we generalize the FJ model toa higher-order setting by using the random walk matrixpolynomials describing higher-order random walks.
For a network G , its random walk matrix polynomial isdefined as follows [33]: Definition 1.
Let A and D be, respectively, the adjacencymatrix and diagonal degree matrix of a graph G . Fora non-negative vector β = ( β , β , ..., β T ) satisfying P Tr =1 β r = 1 , the matrix L β ( G ) = D − T X r =1 β r D (cid:16) D − A (cid:17) r (4)is a T -degree random walk matrix polynomial of G .The Laplacian matrix L is a particular case of L β ( G ) ,which can be obtained from L β ( G ) by setting T = 1 and β = 1 . In fact, it can be proved that, for any β , therealways exists a graph G ′ with loops, whose Laplacian matrixis L β ( G ) , as characterized by the following theorem. Theorem 1 (Proposition 25 in [33]).
The random walk matrixpolynomial L β ( G ) is a Laplacian matrix.Define matrix L G r = D − D (cid:16) D − A (cid:17) r , which is aparticular case of matrix L β ( G ) corresponding to T = r and β r = 1 . In fact, L G r is the Laplacian matrix of graph G r , constructed from graph G by performing r -step randomwalks on graph G . The ij -th element of the adjacency matrix A G r for graph G r is equal to the product of the degree d i for node i in G and the probability that a walker startsfrom node i and ends at node j after performing r -steprandom walks in G . Thus, the matrix polynomial L β ( G ) is acombination of matrices L G r for r = 1 , , ..., T .Based on the random walk matrix polynomials, one candefine a generalized transition matrix P ∗ = P ∗ β for graph G as follows. Definition 2.
Given an undirected weighted graph G and acoefficient vector β = ( β , β , ..., β T ) with P Tr =1 β r = 1 ,the matrix P ∗ β = T X r =1 β r P r = I − D − L β ( G ) (5)is a T -order transition matrix of G with respect to vector β .Note that the generalized transition matrix P ∗ for graph G is actually the transition matrix for another graph G ′ . To introduce the higher-order FJ model, first modify theupdate rule in equation (2) by replacing P with P ∗ . In otherwords, the opinion vector evolves as follows: x ( t +1) = Λ s + ( I − Λ ) P ∗ x ( t ) (6) = Λ s + ( I − Λ ) h β P + β P + ... + β T P T i x ( t ) . In this way, individuals update their opinions by incor-pating those of their higher-order neighborhoods at eachtimestep. Moreover, by adjusting the coefficient vector β ,one can choose different weights for neighbors of differentorders.Note that for the case of P ∗ = P , the higher-order FJmodel is reduced to the classic FJ model. While for the caseof P ∗ = P , the higher-order FJ model can lead to verydifferent results, in comparison with the standard FJ model,as shown in the following example. Fig. 1. A tree with ten nodes.
Example.
Consider the tree shown in Figure 1. The nodesfrom the center to the periphery are colored in red, yellowand blue, respectively. Suppose that for the red node its ( s i , α i ) are given by (1 , , impling that the center node hasa favorable opinion and is insusceptible to be persuaded byothers. And, suppose that the yellow and blue nodes havevalues (0 , . and (0 , . , respectively. Then, calculate theequilibrium opinion vector for the following three cases:I. β = 1 , β = 0 . This case corresponds to the classicFJ model. At every timestep, the opinion of each nodeis influenced by the opinions of its nearest neighbors.At equilibrium, the expressed opinions of red, yellowand blue nodes are , . and . , respectively.This is consistent with the intuition, since the red andyellow nodes are stubborn and thus prone to theirinnate opinions, while the blue nodes are susceptibleto their neighboring nodes, the yellow ones.II. β = 0 , β = 1 . This case is associated with the second-order FJ model. In this case, only the influences of thesecond-order neighbors are considered. The equilib-rium expressed opinions of the red, yellow and bluenodes are , and . , respectively. This can be EEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 4 explained as follows. Since any yellow node is thesecond-order neighbor of the other two yellow nodes,they are influenced by each other, so they all stick totheir innate opinions. In contrast, the blue nodes arehighly affected by the center node.III. β = β = 0 . . This is in fact a hybrid case of theabove two cases, with interactions between a nodeand both of its first- and second-order neighbors withequal weights. For this case, the equilibrium opinionsfor red, yellow and blue nodes are , . and . ,respectively. The opinion of each node lies between theopinions of the above two cases.In addition to the expressed opinion of an individual, forthe above-considered three cases, the sum of their expressedopinions are also significantly different, which are equal to . , . and . , respectively.This example demonstrates that the interactions betweennodes and their higher-order neighbors can have substantialimpact on network opinions. Moreover, as will be seen inSection 6.2, the higher-order interactions also strongly affectthe opinion dynamics on real-world social networks. ONVERGENCE A NALYSIS
In this section, the convergence of the higher-order FJ modelis analyzed. It will be shown that if all α i are positive, themodel has a unique equilibrium and will converge to thatequilibrium after sufficiently many iterations.First, recall the Gershgorin Circle Theorem. Lemma 2 (Gershgorin Circle Theorem [34]).
Given a squarematrix A ∈ R n × n , let R i = P j = i | a ij | be the sum of theabsolute values of the non-diagonal entries in the i -throw and D ( a ii , R i ) ∈ C be a closed disc centered at a ii with radius R i . Then, every eigenvalue of A lies in atleast one of the discs D ( a ii , R i ) .Now, the following main result is established. Theorem 3.
The higher-order FJ model defined in (6) has aunique equilibrium if α i > for all i ∈ V . Proof.
Any equilibrium z ∗ ∈ R n of (6) must be a solutionof the following linear system: ( I − ( I − Λ ) P ∗ ) z ∗ = Λ s . (7)Let M = I − ( I − Λ ) P ∗ . It suffices to show that M isnon-singular. First, it is obvious that every diagonal entryof M is non-negative and every non-diagonal entry is non-positive, since every entry of P ∗ lies in the interval [0 , .Thus, for any row i of M , the sum of absolute values of itsnon-diagonal elements is R i = X j = i | M ij | = − e ⊤ i M + M ii = M ii − α i > , where M ij denotes the ( i, j ) th element of M . Then, accord-ing to Lemma 2, every eigenvalue λ of M lies within thediscs { z : | z − M ii | ≤ M ii − α i } . Since α i > , the set ofall eigenvalues for M excludes . Therefore, matrix M isinvertible, and thus the equilibrium is unique. (cid:3) Hence, z ∗ = ( I − ( I − Λ ) P ∗ ) − Λ s is the unique equi-librium of the opinion dynamics model defined by (6). Next,it will be proved that after sufficiently many iterations, the new higher-order FJ model will converge to this equilib-rium. Theorem 4. If α i > for all i ∈ V , then the higher-order FJ model converges to its unique equilibrium z ∗ = ( I − ( I − Λ ) P ∗ ) − Λ s . Proof.
Define the error vector ǫ ( t ) at the t -th iteration as ǫ ( t ) = x ( t ) − z ∗ . It can be proved that as t approachesinfinity, the error term ǫ ( t ) tends to zero. Substituting (6)into the formula of ǫ ( t ) , one obtains ǫ ( t +1) = ( I − Λ ) P ∗ ǫ ( t ) .In addition, because the sum of all the entries in each rowof P ∗ is one and (1 − α i ) ∈ [0 , , the elements of ǫ ( t ) become smaller as t grows. This can be explained as follows.Let ǫ ( t +1)max be the element of vector ǫ ( t ) that has the largestabsolute value. Then, (cid:12)(cid:12)(cid:12) ǫ ( t +1)max (cid:12)(cid:12)(cid:12) = max i =1 , ,...,n (1 − α i ) n X j =1 P ∗ ij ǫ ( t ) j ≤ max i =1 , ,...,n (1 − α i ) n X j =1 P ∗ ij (cid:12)(cid:12)(cid:12) ǫ ( t )max (cid:12)(cid:12)(cid:12) ≤ max i =1 , ,...,n n (1 − α i ) (cid:12)(cid:12)(cid:12) ǫ ( t )max (cid:12)(cid:12)(cid:12)o < (cid:12)(cid:12)(cid:12) ǫ ( t )max (cid:12)(cid:12)(cid:12) , which completes the proof of the theorem. (cid:3) AST E STIMATION OF EQUILIBRIUM OPINIONVECTOR
To compute the equilibrium expressed opinion vector z ∗ requires calculating matrix P ∗ and inverting a matrix, bothof which are time consuming. In general, for a network G ,sparse or dense, its r -step random walk graph G r could bevery dense. Particularly, for a small-world network witha moderately large r , its r -step random walk graph G r is a weighted and almost complete graph. This makes itinfeasible to compute the generalized transition matrix P ∗ for huge networks.In this section, the spectral graph sparsification tech-nique is utilized to obtain an approximation of matrix P ∗ .Then, a fast convergent algorithm is developed to approxi-mate the expressed opinion vector z ∗ , which avoids matrixinverse operation. The pseudocode of this new algorithm isshown in Algorithm 1. First, introduce the concept of spectral similarity and thetechnique of random-walk matrix polynomial sparsification.
Definition 3 (Spectral Similarity of Graphs [35]).
Considertwo weighted undirected networks G = ( V , E ) and ˜ G =( V , ˜ E ) . Let L and ˜L denote, respectively, their Laplacianmatrices. Graphs G and ˜ G are (1 + ǫ ) -spectrally similar if (1 − ǫ ) · x ⊤ ˜Lx ≤ x ⊤ Lx ≤ (1 + ǫ ) · x ⊤ ˜Lx , ∀ x ∈ R n . (8)Next, recall the sparsification algorithm [33]. For a givengraph G = ( V , E ) , start from an empty graph ˜ G with thesame node set V and an empty edge set. Then add M edgesinto the sparsifier ˜ G iteratively by a sampling technique. Ateach iteration, randomly pick an edge e = ( u, v ) from E as EEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 5 an intermediate edge and an integer r from { , , ..., T } asthe length of the random-walk path. To this end, run theP ATH S AMPLING ( e, r ) algorithm [33] to sample an edge byperforming r -step random walks, and add the sample edge,together with its corresponding weight, into the sparsifier ˜ G . Note that multiple edges will be merged into one singleedges by summing up their weights together. Finally, thealgorithm generates a sparsifier ˜ G for the original graph G with no more than M edges.In [33], an algorithm is designed to obtain a sparsifier ˜ G with O ( nǫ − log n ) edges for L β ( G ) , which consists of twosteps: The first step uses random walk path sampling to getan initial sparsifier with O ( T mǫ − log n ) edges. The secondstep utilizes the standard spectral sparsification algorithmproposed in [35] to further reduce the edge number to O ( nǫ − log n ) . Since a sparsifier with O ( T mǫ − log n ) edgesis sparse enough for the present purposes, only the firststep will be taken, while skipping the second step, to avoidunnecessary computations. Algorithm 1:
HOD
YNAMIC ( G , M, s , β , t ) Input : G : a connected undirected graph; M : the number of edges in sparsifier; s : the innate opinion vector; β : the coefficient vector of random walkmatrix polynomial; t : the number of iterations; Output : ˜x ( t ) : the approximate equilibrium vector; ˜ G = ( V , ∅ ) for i = 1 to M do Randomly pick an edge e = ( u, v ) ∈ E Select an integer r from { , , ..., T } at uniformas the length of the random-walk path Randomly pick an integer k ∈ { , , ..., r } Perform ( k − -step random walk from u to u Perform ( r − k ) -step random walk from v to u r Calculate Z ( p ) along the length- r path p between node u and node u r according to (9) Add an edge ( u , u r ) of weight rmβ r MZ to ˜ G ˜P = I − D − ˜L ( ˜ G ) ˜x (0) = s for i = 1 to t do ˜x ( i ) = Λ s + ( I − Λ ) ˜P˜x ( i − return ˜x ( t ) To sample an edge by performing r -step random walks,the procedure of P ATH S AMPLING algorithm [33] is charac-terized in Lines 5-9 of Algorithm 1. To sample an edge, firstdraw a random integer k from { , , ..., r } and then perform,respectively, ( k − -step and ( r − k ) -step walks startingfrom two end nodes of the edge e = ( u, v ) . This processsamples a length- r path p = ( u , u , ..., u r ) . At the sametime, compute Z ( p ) = r X i =1 a u i − ,u i . (9)The algorithm returns the two endpoints of path p asthe sample edge ( u , u r ) and the quantity Z ( p ) for thecalculation of weight. Theorem 5 (Spectral Sparsifiers of Random-Walk MatrixPolynomials [33]).
For a graph G with random-walkmatrix polynomial L β ( G ) = D − T X r =1 β r D (cid:16) D − A (cid:17) r , (10)where P Tr =1 β r = 1 and β r are non-negative, one canconstruct, in time O ( T mǫ − log n ) , a (1 + ǫ ) -spectralsparsifier, ˜L , with O ( nǫ − log n ) non-zeros.Now one can approximate the generalized transitionmatrix using the Laplacian ˜L ( ˜ G ) of the sparse graph ˜ G : P ∗ = I − D − L β ( G ) ≈ I − D − ˜L ( ˜ G ) = ˜P ∗ . (11)5.1.0.1 Complexity Analysis: Regarding the timeand space complexity of sparsification process of Algo-rithm 1, the main time cost of sparsification (Lines 2-9) is the M calls of the P ATH S AMPLING routine. In P
ATH S AMPLING ,it requires O (log n ) time to sample a neighbor from theweighted network, and thus takes O ( r log n ) time to samplea length- r path. Totally, the time complexity of Algorithm 1is O ( M T log n ) . As for space complexity, it takes O ( n + m ) space to store the original graph G and additional O ( M ) space to store the sparisifier ˜ G . Thus, for appropriate size M , the sparsifier is computable. With the spectral graph sparsification technique, it is possi-ble to approximate P ∗ with a sparse matrix. Nevertheless,directly computing the equilibrium still involves a matrixinverse operation, which is computationally expensive forlarge networks, such as those with millions of nodes. Toapproximate the equilibrium vector z ∗ using the recurrencedefined in (6) and multiple iterations, in this section, wedevelop a convergent approximation algorithm. For thispurpose, an important lemma is first introduced. Lemma 6 (Lemma 4 in [36]).
Let L = D − / LD − / and ˜ L = D − / ˜LD − / . Then all the singular values of ˜ L − L satisfy that for all i ∈ { , , ..., n } , σ i ( ˜ L − L ) < ǫ .Since D − L and D − ˜L are similar to L and ˜ L , respec-tively, the sets of eigenvalues for D − L and L are identical.The same is true for D − ˜L and ˜ L . Thus, the following resultis obvious. Corollary 1.
All the singular values of D − ˜L − D − L are smaller than ǫ , i.e., ∀ i ∈ { , , ..., n } , σ i ( D − ˜L − D − L ) < ǫ .Now, we are in position to introduce a new iterationmethod for approximating the equilibrium vector z ∗ . First,set ˜x (0) = x (0) = s . Then, in every timestep, update theopinion vector with the approximate transition matrix ˜P ∗ ,i.e., ˜x ( t +1) = Λ s + ( I − Λ ) ˜P ∗ ˜x ( t ) . Let α min be the smallestvalue among all i ∈ V . Lemma 7.
For every t ≥ , (cid:13)(cid:13)(cid:13) ˜x ( t ) − x ( t ) (cid:13)(cid:13)(cid:13) ∞ ≤ ǫ √ n · (1 − α min ) h − (1 − α min ) t i α min . (12) EEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 6
Proof.
Inequality (12) is proved by induction. The case of j = 0 is trivial, since ˜x (0) = x (0) = s . Assume that (12)holds for some integer t . Then, it needs to show that (12)also holds for t + 1 . To this end, split k ˜x ( t +1) − x ( t +1) k ∞ into two terms by using triangle inequality: (cid:13)(cid:13)(cid:13) ˜x ( t +1) − x ( t +1) (cid:13)(cid:13)(cid:13) ∞ (13) ≤ (cid:13)(cid:13)(cid:13) ( I − Λ ) D − (cid:16) L β − ˜L β (cid:17) x ( t ) (cid:13)(cid:13)(cid:13) ∞ + (cid:13)(cid:13)(cid:13) ( I − Λ ) (cid:16) I − D − ˜L β (cid:17) (cid:16) ˜x ( t ) − x ( t ) (cid:17)(cid:13)(cid:13)(cid:13) ∞ . (14)For every coordinate of the first term in (14), an upperbound can be derived as follows: (cid:12)(cid:12)(cid:12) e ⊤ i ( I − Λ ) D − (cid:16) L β − ˜L β (cid:17) x ( t ) (cid:12)(cid:12)(cid:12) ≤ (1 − α min ) · (cid:12)(cid:12)(cid:12) e ⊤ i D − (cid:16) L β − ˜L β (cid:17) x ( t ) (cid:12)(cid:12)(cid:12) ≤ (1 − α min ) · σ max (cid:16) D − (cid:16) L β − ˜L β (cid:17)(cid:17) k e i k (cid:13)(cid:13)(cid:13) x ( t ) (cid:13)(cid:13)(cid:13) ≤ ǫ (1 − α min ) √ n, (15)where the second inequality is obtained by using the in-equality that x ⊤ Ay ≤ σ max ( A ) k x k k y k for any matrix A , and the last inequality follows from Lemma 1 and thefact that x ( t ) i ≤ for all i ∈ { , , ..., n } .Next, consider the second term in (14). One has (cid:13)(cid:13)(cid:13) ( I − Λ ) (cid:16) I − D − ˜L β (cid:17) (cid:16) ˜x ( t ) − x ( t ) (cid:17)(cid:13)(cid:13)(cid:13) ∞ ≤ k I − Λ k ∞ (cid:13)(cid:13)(cid:13) I − D − ˜L β (cid:13)(cid:13)(cid:13) ∞ (cid:13)(cid:13)(cid:13) ˜x ( t ) − x ( t ) (cid:13)(cid:13)(cid:13) ∞ = (1 − α min ) · (cid:13)(cid:13)(cid:13) ˜x ( t ) − x ( t ) (cid:13)(cid:13)(cid:13) ∞ , (16)where the equality is due to the fact that k I − D − ˜L β k ∞ =1 , which can be understood as follows. Since every entry of I − D − ˜L β is non-negative and ˜L β = , one has (cid:13)(cid:13)(cid:13) I − D − ˜L β (cid:13)(cid:13)(cid:13) ∞ = (cid:13)(cid:13)(cid:13)(cid:16) I − D − ˜L β (cid:17) (cid:13)(cid:13)(cid:13) ∞ = 1 . Substituting (15) and (16) into (14), one obtains (cid:13)(cid:13)(cid:13) ˜x ( t +1) − x ( t +1) (cid:13)(cid:13)(cid:13) ∞ ≤ ǫ (1 − α min ) √ n + (1 − α min ) · (cid:13)(cid:13)(cid:13) ˜x ( t ) − x ( t ) (cid:13)(cid:13)(cid:13) ∞ = 4 ǫ √ n · (1 − α min ) h − (1 − α min ) t +1 i α min , as required. (cid:3) In order to show the convergence of this method, itneeds to prove that, after sufficiently many iterations, theerror between x ( t ) and z ∗ will be sufficiently small, ascharacterized by the following lemma. Lemma 8.
For every t ≥ , (cid:13)(cid:13)(cid:13) x ( t ) − z ∗ (cid:13)(cid:13)(cid:13) ∞ ≤ (1 − α min ) t α min . (17) Proof.
Expanding with the Neumann series ( I − ( I − Λ ) P ∗ ) − = P ∞ j =0 (( I − Λ ) P ∗ ) j leads to z ∗ − x ( t ) = ∞ X j = t (( I − Λ ) P ∗ ) j Λ s − (( I − Λ ) P ∗ ) t s . (18) Below, by induction, it will be shown that for any x ∈ [0 , n , the relation k (( I − Λ ) P ∗ ) j x k ∞ ≤ (1 − α min ) j holds for all j ≥ . Since every coordinate of x lies inthe interval [0 , , it is obvious that the above relation istrue for the case of t = 0 . Suppose that, for some j > ,every coordinate of y = (( I − Λ ) P ∗ ) j x has magnitude atmost (1 − α min ) j . Since P ∗ is row-stochastic, it follows that k P ∗ y k ∞ ≤ (1 − α min ) j . In addition, because α i ≥ α min for all i ∈ V , one has k ( I − Λ ) P ∗ y k ∞ ≤ (1 − α min ) j +1 ,completing the induction proof.Finally, since both P ∞ j = t (( I − Λ ) P ∗ ) j Λ s and (( I − Λ ) P ∗ ) t s have non-negative coordinates, onehas (cid:13)(cid:13)(cid:13) x ( t ) − z ∗ (cid:13)(cid:13)(cid:13) ∞ ≤ max ( (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X j = t (( I − Λ ) P ∗ ) j Λ s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ , (cid:13)(cid:13)(cid:13) (( I − Λ ) P ∗ ) t s (cid:13)(cid:13)(cid:13) ∞ ) ≤ ∞ X j = t (1 − α min ) j = (1 − α min ) t α min , as claimed by the lemma. (cid:3) Combining Lemmas 7 and 8, a convergent approximateiteration method can be summarized as stated in the follow-ing theorem.
Theorem 9 (Approximation Error).
For every t ≥ , (cid:13)(cid:13)(cid:13) ˜x ( t ) − z ∗ (cid:13)(cid:13)(cid:13) ∞ ≤ ǫ √ n · (1 − α min ) h − (1 − α min ) t i + (1 − α min ) t α min . In the sequel, this approximate iteration algorithm isreferred to as A
PPROX . It should be mentioned that Theo-rem 9 provides only a rough upper bound. The experimentsin Section 6.3 show that A
PPROX works well in practice,leading to very accurate results for real networks.
XPERIMENTS ON R EAL N ETWORKS
In this section, we conduct extensive experiments on real-world social networks to evaluate the performance of thealgorithm A
PPROX . Machine Configuration and Reproducibility.
Our ex-tensive experiments run on a Linux box with 16-core 3.00GHz Intel Xeon E5-2690 CPU and 64GB ofmain memory. All algorithms are programmed in
Ju-lia v1.3.1 . The source code is publicly available athttps://github.com/HODynamic/HODynamic.
Datasets.
We test the algorithm on a large set of realis-tic networks, all of which are collected from the KoblenzNetwork Collection [37] and Network Repository [38]. Forthose networks that are disconnected originally, we performexperiments on their largest connected components. Thestatistics of these networks are summarized in the first threecolumns of Table 1, where we use n ′ and m ′ to denote, EEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 7
TABLE 1Statistics of real networks used in experiments and comparison of running time (seconds, s ) between E XACT and A
PPROX for three innate opiniondistributions (uniform distribution, exponential distribution, and power-law distribution).
Running time ( s ) for E XACT and A
PPROX algorithmsNetwork n ′ m ′ Uniform distribution Exponential distribution Power-law distributionE
XACT A PPROX E XACT A PPROX E XACT A PPROX
HamstersterFriends 1788 12476 0.174 0.974 0.158 0.876 0.176 0.866HamstersterFull 2000 16098 0.303 1.540 0.316 1.568 0.317 1.547PagesTVshow 3892 17239 1.204 1.530 1.126 1.367 1.083 1.367Facebook (NIPS) 4039 88234 1.492 6.274 1.473 6.331 1.556 6.243PagesGovernment 7057 89429 5.857 7.679 5.682 7.316 5.682 7.353Anybeat 12645 49132 31.448 4.730 31.843 5.462 31.575 4.739PagesCompany 14113 52126 39.348 4.269 37.690 3.905 37.477 3.877Gplus 23613 39182 163.525 4.329 171.166 4.295 165.470 4.307GemsecRO 41773 125826 885.069 15.758 888.009 15.519 873.696 16.079GemsecHU 47538 222887 946.399 28.592 937.540 30.562 919.438 29.174PagesArtist 50515 819090 1160.469 139.565 1183.787 138.507 1167.293 148.941Brightkite 56739 212945 1913.246 27.351 1901.307 29.203 1895.432 28.361Livemocha* 104103 2193083 — 538.730 — 542.467 — 542.708Douban* 154908 327162 — 44.166 — 43.292 — 43.350Gowalla* 196591 950327 — 138.222 — 142.690 — 143.669TwitterFollows* 404719 713319 — 96.850 — 97.319 — 96.748Delicious* 536108 1365961 — 209.371 — 206.960 — 206.739YoutubeSnap* 1134890 2987624 — 663.090 — 667.921 — 667.770Hyves* 1402673 2777419 — 648.906 — 633.172 — 636.219 respectively, the numbers of nodes and edges in their largestconnected components. The smallest network consists of , nodes, while the largest network has more than onemillion nodes. In Table 1, the networks are listed in anincreasing order of the number of nodes in their largestconnected components. Input Generation.
For each dataset, we use the networkstructure to generate the input parameters in the followingway. The innate opinions are generated according to threedifferent distributions, that is, uniform distribution, expo-nential distribution, and power-law distribution, where thelatter two are generated by the randht.py file in [39]. Forthe uniform distribution, we generated the opinion s i ofnode i at random in the range of [0 , . For the exponentialdistribution, we use the probability density e x min e − x togenerate n ′ positive real numbers x with minimum value x min > . Then, we normalize these n ′ numbers to bewithin the range [0 , by dividing each x with the maximumobserved value. Similarly, for the power-law distribution,we choose the probability density ( α − x α − x − α with α = 2 . to generate n ′ positive real numbers, and thennormalize them to be within the interval [0 , as the innateopinions. We note that there is always a node with innateopinion due to the normalization operation for the lattertwo distributions. We generate the resistance parametersuniformly to be within the interval (0 , . To show the impact of higher-order interactions on the opin-ion dynamics, we compare the equilibrium expressed opin-ions between the second-order FJ model and the standardFJ model on four real networks: PagesTVshow, PagesCom-pany, Gplus, and GemsecRO. For both models, we generateinnate opinions and resistance parameters for each nodeaccording to the uniform distribution. We set β = 1 , β = 0 for the standard FJ model, and β = 0 , β = 1 for thesecond-order FJ model. > > > > > > @ ' L I I H U H Q F H R I ( T X L O L E U L X P ( [ S U H V V H G 2 S L Q L R Q V 3 H U F H Q W D J H R I 1 R G H V 3 D J H V 7 9 V K R Z 3 D J H V &