A Bounded-Confidence Model of Opinion Dynamics on Hypergraphs
Abigail Hickok, Yacoub Kureh, Heather Z. Brooks, Michelle Feng, Mason A. Porter
AA BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ONHYPERGRAPHS ∗ ABIGAIL HICKOK † , YACOUB KUREH † , HEATHER Z. BROOKS ‡ , MICHELLE FENG § , AND
MASON A. PORTER † Abstract.
People’s opinions evolve over time as they interact with their friends, family, col-leagues, and others. In the study of opinion dynamics on networks, one often encodes interactionsbetween people in the form of dyadic relationships, but many social interactions in real life arepolyadic (i.e., they involve three or more people). In this paper, we extend an asynchronous bounded-confidence model (BCM) on graphs, in which nodes are connected pairwise by edges, to hypergraphs.We show that our hypergraph BCM converges to consensus under a wide range of initial conditionsfor the opinions of the nodes. We show that, under suitable conditions, echo chambers can formon hypergraphs with community structure. We also observe that the opinions of individuals cansometimes jump from one opinion cluster to another in a single time step, a phenomenon (which wecall “opinion jumping”) that is not possible in standard dyadic BCMs. We also show that there isa phase transition in the convergence time on the complete hypergraph when the variance σ of theinitial opinion distribution equals the confidence bound c . Therefore, to determine the convergenceproperties of our hypergraph BCM when the variance and the number of hyperedges are both large,it is necessary to use analytical methods instead of relying only on Monte Carlo simulations. Key words.
Hypergraphs, networks, continuous-valued opinion dynamics, bounded-confidencemodels, consensus, polarization
AMS subject classifications.
1. Introduction.
Social interactions with friends and acquaintances can leadpeople to change their opinions about public figures [27], social issues [41], economicpolicy [23], and more. In opinion dynamics, researchers study how people’s opinionsabout one or more topics evolve over time as they interact and influence each other[17]. Traditionally, one models entities as nodes in a graph and models the entities’social interactions as edges that encode pairwise interactions between them [4,36]. Theopinions of these entities can change as a result of such interactions. In the presentpaper, we build on these ideas by studying the effects of group interactions in opinionformation by modeling group social interactions as hyperedges in a hypergraph. Wefind that such polyadic interactions play a key role in whether consensus is reachedand how long it takes to reach it.We focus on continuous-opinion dynamics , in which nodes have continuous-valuedopinions. In our model, nodes hold opinions in R , and we denote the opinion state ofthe entire system by x ∈ R N , where N is the number of nodes. This is an appropriatemodel for opinions, such as the strength of support for a political candidate [28],that lie on a spectrum. By contrast, opinions such as whether one supports the LosAngeles Dodgers or San Francisco Giants may leave little or no room for any middleground.Bounded-confidence models (BCM) are continuous-opinion-state models in which ∗ Funding:
AH, MAP, MF, and YK acknowledge support from the National Science Foundation(Grant No. 1922952) through the Algorithms for Threat Detection (ATD) program. † Department of Mathematics, University of California, Los Angeles, CA, USA([email protected], [email protected], [email protected]). ‡ Department of Mathematics, Harvey Mudd College, Claremont, CA, USA([email protected]). § Department of Computing + Mathematical Sciences, California Institute of Technology,Pasadena, CA, USA ([email protected]). 1 a r X i v : . [ c s . S I] F e b A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER individuals are influenced only by neighbors who hold opinions that are within some confidence bound c of their own opinion [31, 38]. Individuals who disagree with eachother by too large an amount do not influence each other [43]. This models the conceptof selective exposure from social psychology; according to this principle, individualstend to ignore information that is contrary to their current viewpoint [29, 42]. Intraditional BCMs, each individual is a node in a graph and its neighbors are itsadjacent nodes. A BCM is asynchronous if only one pair of neighbors can interact at atime and is synchronous if all pairs of neighboring nodes interact during each time step.The two most commonly studied BCMs are the (asynchronous) Deffuant–Weisbuch(DW) model [7, 45] and the (synchronous) Hegselmann–Krause (HK) model [21]. SeeRef. [31] for a review and a comparison of these two models, and see the introductionof Ref. [34] for a recent summary of research on BCMs.An important limitation of graphs is that they force one to consider only pairwise(i.e., “dyadic”) interactions between nodes (as well as self-interactions, if one allowsself-edges), whereas many social interactions involve many nodes at once [1, 39]. Oneexample of such a polyadic (i.e., “higher-order”) social interaction is group messaging,such as group texting or e-mails with more than one recipient. We seek to examinethe effects of polyadic interactions on opinion dynamics, so we develop and analyzean extension of BCMs to hypergraphs. In a hypergraph, a hyperedge can connectan arbitrary number of nodes to each other, rather than just two. In the context ofopinion dynamics, one way to interpret such interactions is as a form of “peer pressure”[24, 33], but other interpretations are also possible. Importantly, it is not possible toreduce the higher-order interactions in our hypergraph BCM to an aggregation ofpairwise interactions.In our hypergraph BCM, we find that “large” hyperedges (i.e., hyperedges thatare incident to many nodes) are crucial for reaching consensus and that hypergraphsthat have large hyperedges behave rather differently than hypergraphs with only smallhyperedges. For the most part, other hypergraph extensions of opinion models haveconsidered interactions with three or fewer nodes (i.e., hypergraphs in which thehyperedge “sizes” are no larger than 3) [35,37]. To the best of our knowledge, opinionmodels that consider large hyperedges have been developed only for models (such asvoter models) with discrete-valued opinions (see, e.g., [5, 20, 22]).A key question in opinion dynamics is the examination of how model parameters,hypergraph structure (or the structure of other types of networks), and initial opinionstates influence the opinion state to which a model converges. By applying the resultsof [30], we show that the opinion state always converges to some limit state . In thestandard dyadic DW and HK models, the number of opinion clusters in the limit statedepends on the confidence bound c . We say that the opinion state is at consensus if there is a single opinion cluster; that is, every node has the same opinion. Mostwork on dyadic BCMs has drawn initial opinions uniformly at random from [0 , c is above a certainthreshold value [7, 14, 16, 18, 32, 45]. By contrast, we show in section 3 that there isno such confidence-bound threshold for our hypergraph BCM. In subsection 3.1, weprove this result for the complete hypergraph. In fact, we prove the following strongerstatement: if the initial opinion distribution is bounded, then the opinion state ona complete hypergraph converges to consensus almost surely for sufficiently manynodes. When one draws the initial opinions of the nodes in the complete hypergraphfrom a distribution with variance σ < c , we prove that the probability of consensusapproaches 1 as the number of nodes approaches infinity. We also present numericalevidence that even when c ≤ σ , opinions that are initially distributed normally BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS polarization for this phenomenon, and we use theterm echo chambers [6, 13, 44] for these different opinion clusters. In subsection 3.2,we study our BCM on hypergraphs with community structure. We prove that po-larization can occur if there is an upper bound on the sizes of the hyperedges thatconnect different communities. This provides a possible mechanism for the formationof echo chambers in hypergraphs. However, if there is no upper bound on the size ofinter-community hyperedges and each community forms a complete hypergraph (i.e.,a hyperclique ) and has sufficiently many nodes, we prove that our hypergraph BCMconverges to consensus.Using numerical simulations, we show that our results about reaching consensusare robust. The theorems in subsection 3.1 require that the hypergraph is complete,and some of the results in subsection 3.2 require that the communities form hyper-cliques. However, in practice, we can relax these conditions and the nodes’ opinionsstill eventually reach consensus on the hypergraph in the former case and on indi-vidual communities in the latter case. In subsection 3.3, we study the behavior ofour BCM on sparse Erd˝os–R´enyi-like hypergraphs by using Monte Carlo simulations.In subsection 3.4, we study the behavior of our model on the Enron e-mail hyper-graph [3], in which the nodes are Enron employees and hyperedges encode e-mailsbetween these employees. Hypergraphs that one constructs from empirical data areinteresting examples both because they are sparse and because their hyperedges areusually small in comparison to the number of nodes.The convergence time of our a BCM is a significant factor to consider when we arerunning numerical simulations of it. In section 4, we prove a partial characterizationof conditions under which our hypergraph BCM converges in finite time. In particular,we prove that it almost surely converges in finite time on the complete hypergraph.By comparison, the dyadic DW model for ordinary graphs usually does not convergein finite time, although the HK model always does [10]. We also show that there isa phase transition in the convergence time of our BCM on the complete hypergraphwhen the variance σ of the initial opinion distribution equals the confidence bound c . Meng et al. [34] demonstrated numerically that the standard dyadic DW modelalso has a phase transition in convergence time. It is important to understand thisphase transition because when one is running Monte Carlo simulations of a BCM,one chooses a finite cutoff time to stop the simulations. Without analysis of theconvergence time, one may accidentally cut off the numerical simulations too earlyand mistakenly conclude that there is a phase transition in the limit state when whathas actually occurred is a phase transition in convergence time.When studying opinion dynamics, one also wants to understand the evolutionof the opinion state before one reaches a limit state. In section 5, we investigate aphenomenon, which we call opinion jumping , in which the opinion of a node changesby more than c in a single time step. Opinion jumping allows nodes with extremeopinions to jump closer to the mean of the opinion distribution in a single time step.This behavior cannot occur in the classical dyadic DW or HK models because nodesin those BCMs interact only with neighbors whose opinions are sufficiently similar totheir own. A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER
Our paper proceeds as follows. In section 2, we give a formal definition of ourhypergraph BCM. In section 3, we present our results about its limit state. These arethe main results of our paper. In section 4, we discuss convergence time. In section 5,we examine opinion jumping and quantify how often it occurs. We conclude anddiscuss future work in section 6. Our code is available at https://bitbucket.org/ahickok/hypergraph-bcm.
2. A Bounded-Confidence Model on Hypergraphs.
In this section, wedevelop an extension of the Deffuant–Weisbuch (DW) model to hypergraphs. Westart by presenting the standard DW model on graphs.In the standard dyadic DW model, opinion dynamics occur on an unweighted andundirected graph whose edges encode social ties. At each discrete time t , one choosesan edge e = { i, j } uniformly at random. If the difference | x i ( t ) − x j ( t ) | of opinionsbetween nodes i and j is below some confidence bound c i,j , then nodes i and j adjusttheir opinions as follows: x i ( t + 1) = x i ( t ) + m i,j ( x j ( t ) − x i ( t )) ,x j ( t + 1) = x j ( t ) + m i,j ( x i ( t ) − x j ( t )) , (2.1)where m i,j is an element of the matrix of convergence parameters. Otherwise, theopinions of nodes i and j are too far apart at time t , so x i ( t +1) = x i ( t ) and x j ( t +1) = x j ( t ). The opinions of all other nodes do not change when we do this update. Withthis type of update rule, the mean opinion of the nodes in a network is a conservedquantity. The confidence bounds c i,j ∈ [0 , ∞ ) model how open-minded or close-minded individuals are to the opinions of others [11]. The convergence parameters m i,j ∈ [0 , .
5] (which resemble the trust parameters in DeGroot models [38]) controlthe rate at which individuals adjust their opinions [7, 45]. Using a single value c and m for all pairs leads to what is sometimes called the “homogeneous” DW model.We now define our BCM on hypergraphs as an extension of the homogeneous DWmodel. A hypergraph is a generalization of a graph that allows polyadic interactionsbetween arbitrarily many nodes. The space of possible hyperedges is the power set P ( V ) of the set V of nodes. Let N := | V | denote the number of nodes. In the hyper-graphs ( V, E ) that we consider, we restrict the hyperedge set E ⊂ { e ∈ P ( V ) | | e | ≥ } so that each hyperedge is incident to at least two nodes. Prohibiting hyperedgesthat are attached to only a single node (these are called “self-hyperedges”) affectsonly convergence time; it does not affect the limit state. All of our hypergraphs areunweighted and undirected. In our BCM, there is a time-dependent opinion state x ( t ) ∈ O N , where we take the opinion space O to be the real line R . We use x i ( t ) todenote the opinion of node i at time t .To generalize the notion of a confidence bound to hyperedges, we define a dis-cordance function d : E × O N → R ≥ that maps a hyperedge and opinion state to areal number. We use this function, which quantifies the level of disagreement amongthe nodes that are incident to a hyperedge, to determine whether or not these nodesupdate their opinions. Our discordance functions(2.2) d α ( e, x ) = (cid:18) | e | − (cid:19) α (cid:88) i ∈ e ( x i − ¯ x e ) are parameterized by α , where ¯ x e = ( (cid:80) i ∈ e x i ) / | e | . If the discordance d α ( e, x ( t )) isless than the confidence bound c , then we say that the hyperedge e is concordant attime t . Otherwise, we say that e is discordant . There are two special cases: α = 0 BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS α = 1. With α = 0, the function d ( e, x ) penalizes large hyperedges, in the sensethat(2.3) d ( e, x ) ≤ d ( e (cid:48) , x ) if e ⊂ e (cid:48) , with equality holding if and only if x i = ¯ x e for all i ∈ e (cid:48) \ e . We use the term hyperedgemonotonic for a discordance function that satisfies (2.3). This models a situation inwhich large groups tend to be less effective than small groups at changing opinions.With α = 1, the function, d ( e, x ) is equal to the unbiased sample variance of theopinions of the nodes that are incident to e .The scaling by 1 / ( | e | −
1) in (2.2) prevents hyperedges with many nodes frombeing disadvantaged relative to hyperedges with few nodes. Specifically, if the opinionsare independent and identically distributed, then the expected d -discordance of anysubset of nodes is(2.4) E [ d ( e, x )] = E [ d ( e (cid:48) , x )] for all e, e (cid:48) ∈ E .
Unlike d -discordance, d -discordance is not hyperedge monotonic. For example, let x = (0 , , .
5) and consider the hyperedges e = { , } and e (cid:48) = { , , } . We see that d ( e, x ) = 0 . > .
25 = d ( e (cid:48) , x ), even though e ⊂ e (cid:48) . One can interpret node 3’srole in the interaction as that of a mediator who reduces the amount of discordance,thereby potentially yielding an update that otherwise could not occur. We set thediscordance function to d = d for the remainder of this paper.We employ DW-type (i.e., asynchronous) updates. At each discrete time, werandomly select a hyperedge e from E according to some probability distribution. Formathematical convenience, we use the uniform distribution over E . If the discordance d ( e, x ) is less than the confidence bound c , the nodes i ∈ e update their opinions x i to the mean opinion ¯ x e ; otherwise, their opinions do not change. That is, if we selecthyperedge e at time t , then the update rule for each node i is(2.5) x i ( t + 1) = (cid:40) ¯ x e ( t ) , if i ∈ e and d ( e, x ) < cx i ( t ) , otherwise.The sequence x (0) , x (1) , x (2) , . . . of opinion states is a discrete-time Markov chainwith a continuous state space.If the hypergraph is a graph (i.e., if | e | = 2 for all e ∈ E ), our generalized BCMreduces to a standard DW model with a rescaled confidence bound c . This rescalingarises from the difference in discordance functions: the standard DW model uses theabsolute value | x i − x j | of the difference of opinions, whereas our model uses ( x i − x j ) from (2.2).
3. The Limit State of Our Hypergraph BCM.
We say that the opinionstate converges to x ∗ if lim t →∞ x ( t ) = x ∗ . We refer to x ∗ as the limit state . An opinion cluster in the limit state is a collection of nodes that all have the sameopinion in the limit state. The opinion value of an opinion cluster is the opinion γ ∈ R such that x ∗ i = γ for all nodes i in that cluster.The opinion state converges to consensus if there is a γ ∈ R such that x ∗ i = γ for all i . Equivalently, the opinion state converges to consensus if there is exactlyone opinion cluster in the limit state. If the opinion state converges to consensus, itis necessarily true that γ = N (cid:80) Ni =1 x i (0) because the mean opinion of the nodes isconstant with respect to time. A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER
We define an opinion state x to be an absorbing state if for all e ∈ E , either d ( e, x ) ≥ c or d ( e, x ) = 0 (i.e., x i = x j for all i, j ∈ e ). If x ( T ) is an absorbing state,then x ( t ) = x ( T ) for all t ≥ T . We will prove in Lemma 3.2 that the limit state isalmost surely an absorbing state.We begin by showing that starting from any initial opinion state x (0), the opinionstate of our hypergraph BCM converges in the limit t → ∞ . Theorem
Let x (0) be the initial opinion state, and update the opinion state x ( t ) according to (2.5) . It follows that the limit state x ∗ := lim t →∞ x ( t ) exists.Proof. Let A ( x ( t ) , t ) be the N × N matrix such that x ( t + 1) = A ( x ( t ) , t ) x ( t ),and let e t denote the hyperedge that we choose at discrete time t . If e t is discordant,then A ( x ( t ) , t ) = I N . If e t is concordant, then A ( x ( t ) , t ) is the matrix with entries A ( x ( t ) , t ) ij = (cid:40) / | e t | , i, j ∈ e t δ ij , otherwise . The matrix A ( x ( t ) , t ) satisfies the following conditions [30]:(1) Every agent has a bit of self-confidence : The diagonal entries of A ( x ( t ) , t ) arepositive.(2) Confidence is mutual : That is, for all i, j , we have that A ( x ( t ) , t ) ij > A ( x ( t ) , t ) ji > Positive weights do not converge to zero : There is a δ > A ( x ( t ) , t ) is at least δ . In our model, every positive entry isat least 1 /N .For any two times t and t with t < t , Lorenz [30] defined the accumulationmatrix A ( t , t ) := A ( x ( t − , t − A ( x ( t − , t − ×· · ·× A ( x ( t +1) , t +1) A ( x ( t ) , t ) . Using this notation, x ( t ) = A (0 , t ) x (0). He showed that if conditions (1)–(3) aresatisfied, then there is a time t and an ordering of the nodes such that(3.1) lim t →∞ A (0 , t ) = K
0. . .0 K p A (0 , t ) , where each K i is a row-stochastic matrix with equal rows. The DeGroot model [8],the dyadic DW model [7], and the HK model [21] all satisfy conditions (1)–(3).Let I i be the set of nodes in the block K i . Equation (3.1) implies that x ( t )converges to some opinion state x ∗ such that x ∗ j = x ∗ k for all j, k ∈ I i .We will use the following lemma repeatedly in the subsections that follow. Lemma
Let x (0) be the initial opinion state, and let x ( t ) be the opinion statedetermined by (2.5) . It follows that the limit state x ∗ := lim t →∞ x ( t ) is almost surelyan absorbing state.Proof. By Theorem 3.1, we know that x ∗ exists. If x ∗ is not an absorbing state,then there is a hyperedge e ∈ E such that d ( e, x ∗ ) < c and x ∗ i (cid:54) = x ∗ j for some i, j ∈ e .Let ¯ x ∗ e = | e | (cid:80) k ∈ e x ∗ k , and note that x ∗ i (cid:54) = ¯ x ∗ e . For all (cid:15) >
0, there is a time T such BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS | x i ( t + 1) − x i ( t ) | < (cid:15) , (3.2) | x i ( t ) − x ∗ i | < (cid:15) , (3.3) | d ( e, x ( t )) − d ( e, x ∗ ) | < (cid:15) , (3.4) | ¯ x e ( t ) − ¯ x ∗ e | < (cid:15) (3.5)for all t ≥ T . Choose (cid:15) < min { c − d ( e, x ∗ ) , | ¯ x ∗ e − x ∗ i | / } , and let T be a time thatsatisfies (3.2)–(3.5). With probability 1, we choose every hyperedge in E infinitelyoften (by the Borel–Cantelli lemma). Therefore, we choose e at some time t ≥ T almost surely. If this happens, then d ( e, x ( t )) < c by (3.4) and we update the nodesof e to obtain | x i ( t + 1) − x i ( t ) | = | ¯ x e ( t ) − x i ( t ) | > | ¯ x ∗ e − x ∗ i | − (cid:15) > (cid:15) by (3.3) and (3.5), contradicting (3.2).Lemma 3.2 implies that there are almost surely no hyperedges that are possibleto update in the limit state. In this sub-section, we study the limit state of our hypergraph BCM on the complete hypergraph.On the complete hypergraph, every possible subset of nodes can interact with one an-other. Some of our results apply more generally to any hypergraph that includes thehyperedge e = V . We begin by presenting several lemmas that we then use to proveTheorem 3.5. Lemma
If the opinion distribution at time t has finite variance σ , then (3.6) lim n →∞ P [ d ( e, x ( t )) < c | | e | = n ] = , c > σ , c = σ , c < σ . Proof.
The discordance of a hyperedge e at time t is the sample variance of theopinions { x j ( t ) | j ∈ e } . Let s n denote the sample variance of n opinions. Bydefinition of the discordance function d , it follows that P [ d ( e, x ( t )) < c | | e | = n ] = P [ s n < c ]. Because E [ s n ] = σ and lim n →∞ Var[ s n ] = 0, Chebyshev’s inequalityimplies that lim n →∞ P [ s n < c ] ≥ lim n →∞ − Var[ s n ] c − σ = 1 , c > σ , lim n →∞ P [ s n < c ] ≤ lim n →∞ Var[ s n ] σ − c = 0 , c < σ . Note that s n converges asymptotically to the normal distribution N ( σ , σ ( κ − /n ),where κ is the kurtosis of the initial opinion distribution. Because a normal distribu-tion is symmetric, lim n →∞ P [ s n < c ] = lim n →∞ P [ s n < E [ s n ]] = if c = σ .The following lemma says that if a hyperedge e has a nontrivial update at time t , then the discordance of each hyperedge e (cid:48) ⊃ e decreases [i.e., d ( e (cid:48) , x ( t + 1)) Lemma Let A = { x , x , . . . , x n } be a collection of n real numbers, and let A (cid:48) = { x i , x i , . . . , x i (cid:96) } be some subcollection of A . Construct a new collection B bytaking the union of A \ A (cid:48) and (cid:96) copies of the mean ¯ x A (cid:48) = (cid:96) ( (cid:80) (cid:96)j =1 x i j ) of A (cid:48) . Thesample variances satisfy s ( B ) ≤ s ( A ) , where equality holds if and only if A = B .Proof. The collections A and B have the same mean ¯ x A . We have(3.7) ( n − s ( A ) − s ( B )) = (cid:96) (cid:88) j =1 ( x i j − ¯ x ) − (cid:96) (¯ x A (cid:48) − ¯ x ) . Expanding the second term of the right-hand side of (3.7) yields (cid:96) (cid:88) j =1 ( x i j − ¯ x ) − (cid:96) (cid:96) (cid:88) j =1 ( x i j − ¯ x ) . We define y j := x i j − ¯ x to simplify the notation and write (cid:96) (cid:88) j =1 y j − (cid:96) (cid:96) (cid:88) j =1 y j . Expanding the second term of the right-hand side of (3.7) further and simplifyingyields 1 (cid:96) ( (cid:96) − (cid:96) (cid:88) j =1 y j − (cid:96) (cid:88) j =1 (cid:96) (cid:88) k = j +1 y j y k = 1 (cid:96) (cid:96) (cid:88) j =1 (cid:96) (cid:88) k = j +1 ( y j − y k ) ≥ . Equality occurs if and only if y = . . . = y (cid:96) , which proves the lemma. Theorem Suppose that H is an N -node hypergraph that includes the hyper-edge e N = V . Let x (0) be the initial opinion state, with opinions drawn independentlyfrom a distribution with a variance of σ < c , and let x ( t ) be the opinion state thatis determined by (2.5) . It then follows that the probability of reaching consensus ap-proaches as N → ∞ .Proof. By Lemma 3.4, the discordance function d ( e N , x ( t )) is nonincreasing intime. Therefore, if e N is concordant at time 0, it is concordant for all t . Additionally,if e N is concordant, H converges to consensus the first time that one selects thehyperedge e N . With probability 1, this selection occurs at some finite time. Thisshows that P [consensus] ≥ P [consensus | d ( e N , x (0)) < c ] P [ d ( e N , x (0)) < c ]= P [ d ( e N , x (0)) < c ] . By Lemma 3.3, lim N →∞ P [consensus] = 1. Remark In this subsubsection, we assume that wedraw the initial opinions of the nodes from some interval [ a, b ]. We present resultsabout the limit state for this important case, which includes drawing the initial opin-ions uniformly at random from [0 , 1] (a focal example of much prior work on thestandard dyadic DW model [7, 31, 45]) as a special case. BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS U (0 , c ∗ ≈ suchthat (1) the system converges to consensus with high probability for c > c ∗ and (2)the system converges to approximately (cid:98) c (cid:99) opinion clusters for c < c ∗ [7, 18, 45]. Aconsensus threshold also exists for the standard dyadic HK model, but it occurs ata smaller confidence bound (of about c ∗ = 0 . 19) [16, 32]. In Theorem 3.10, we provethat no such threshold exists for our hypergraph BCM and that the opinion stateconverges to consensus almost surely for large enough N whenever the initial opiniondistribution is bounded. Lemma Suppose that H is the complete hypergraph with N nodes and that c (cid:54) = 0 . Let x (0) be the initial opinion state, with x i (0) ∈ [ a, b ] for all i , and let x ( t ) be the opinion state that is determined by (2.5) . It is then the case that the numberof opinion clusters in the limit state is almost surely less than or equal to b − a √ c + 1 .Proof. Let x ∗ = lim t →∞ x ( t ) be the limit state, and let γ , . . . , γ m ∈ R be theopinion values of the m opinion clusters. Let ψ : V → { , . . . , m } map nodes to theopinion cluster to which they belong, such that x ∗ i = γ ψ ( i ) . By Lemma 3.2, it sufficesto show that m > b − a √ c implies that x ∗ is not an absorbing state.It must be the case that γ i ∈ [ a, b ] for all i because x j ( t ) ∈ [ a, b ] for all t and all j .If m > b − a √ c , then there is a pair γ i , γ j such that γ i (cid:54) = γ j and | γ i − γ j | < √ c . Let k i be a node in the i th opinion cluster ψ − ( i ), let k j be a node in the j th opinion cluster ψ − ( j ), and let e = { k i , k j } . The limit state x ∗ is not an absorbing state because0 < d ( e, x ∗ ) = ( γ i − γ j ) < c . The following lemma says that as N → ∞ , almost surely at least one of the followingthree outcomes occurs: (1) the opinion state converges to consensus, (2) the numberof opinion clusters approaches infinity, or (3) the difference between the opinion valuesof different opinion clusters approaches infinity. Lemma Suppose that H is the complete hypergraph with N nodes and that c (cid:54) = 0 . Let x (0) be the initial opinion state, and let x ( t ) be the opinion state that isdetermined by (2.5) . Let x ∗ be the limit state, which exists by Theorem , and let { γ , . . . , γ m } be the opinion values of the opinion clusters. It then follows that thenumber m of opinion clusters almost surely is either m = 1 (consensus) or satisfies m × (cid:16) max k,j ( γ k − γ j ) c − (cid:17) ≥ N . Proof. By Lemma 3.2, it suffices to show that if N > m × (cid:16) max k,j ( γ k − γ j ) c − (cid:17) and m > 1, then x ∗ is not an absorbing state. Let ψ : V → { , . . . , m } map nodes tothe opinion cluster to which they belong, such that x ∗ i = γ ψ ( i ) , and let N i = | ψ − ( i ) | be the size of the i th opinion cluster. If N > m × (cid:16) max k,j ( γ k − γ j ) c − (cid:17) , then thereexists an opinion cluster i such that N i > max k,j ( γ k − γ j ) c − 1. If m > 1, there is a γ j such that γ j (cid:54) = γ i . Let e be a hyperedge of size N i + 1 that is incident to the N i nodesin ψ − ( i ) and 1 node in ψ − ( j ). The limit-state sample mean of the nodes that are0 A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER incident to e is ¯ x ∗ e ( t ) = N i N i + 1 γ i + 1 N i + 1 γ j . The limit state x ∗ is not an absorbing state because0 < d ( e, x ∗ ) = N i ( γ i − ¯ x ∗ e ( t )) + ( γ j − ¯ x ∗ e ( t )) N i + 1 = ( γ i − γ j ) N i + 1 < c . Remark N . Theorem Suppose that H is the complete hypergraph with N nodes andthat c (cid:54) = 0 . Let x (0) be the initial opinion state, with x i (0) ∈ [ a, b ] for all i , and let x ( t ) be the opinion state that is determined by (2.5) . If N > ( b − a √ c + 1)( ( b − a ) c − ,then the opinion state converges to consensus almost surely.Proof. Let x ∗ be the limit state, which exists by Theorem 3.1, and let { γ , . . . , γ m } be the opinion values of the opinion clusters. By Lemma 3.7, m ≤ b − a √ c + 1 almostsurely. It is necessarily true that γ i ∈ [ a, b ] for all i , so max k,j ( γ k − γ j ) < ( b − a ) .By Lemma 3.8, m = 1 almost surely if N > ( b − a √ c + 1)( ( b − a ) c − Assume that the initial opin-ions are normally distributed with mean µ and variance σ . When σ < c , The-orem 3.5 implies that the probability of consensus for the complete hypergraph ap-proaches 1 as N → ∞ . Based on numerical evidence, we conjecture that the probabil-ity of reaching consensus for the complete hypergraph approaches 1 as N → ∞ evenwhen σ ≥ c , unless c = 0. Because the hypergraph is complete, note that Lemma 3.8also applies.In Figure 1, we show a typical simulation when σ > c . We have reduced thenumber of time steps by requiring that the hyperedge that we select at time 0 isconcordant. This requirement has no effect on the subsequent behavior or on thesystem’s limit state, but we will see why it is necessary in our proof of Theorem 4.7.Observe that the opinion state converges to consensus. In 1000 trials of our BCM fora confidence bound of c = 1, a complete hypergraph with N = 200 nodes, and aninitial opinion distribution with standard deviation σ = 1 . 2, we find that the opinionstate converges to consensus in every trial.The results of our Monte Carlo simulations lead us to conjecture that when c (cid:54) = 0and the initial opinion distribution is normal, the probability of consensus approaches1 even when σ > c . We now provide a heuristic explanation of this conjecture,although we do not have a mathematically rigorous proof of it.We fix the variance to be σ > c . At each discrete time, we select a hyperedgeuniformly at random. Let ˆ e be the first concordant hyperedge that we choose. Insection 4, we will show that if e is an arbitrary hyperedge, then P [ d ( e, x (0)) < c || e | = n ] ≤ r n − , where r = e (1 − c/σ ) (cid:112) ( c/σ ) < 1. If it is true (and we suspect thatit is) that P [ d ( e, x (0)) < c | | e | = n ] = ar n for some constants a and r < 1, then one BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS N = 500 nodes. Each curve traces the evolution of one node’s opinion. We draw theinitial opinions from N (0 , σ ) with σ = 1 . 2. We set the confidence bound to c = 1.We have reduced the number of time steps by requiring that the hyperedge that wechoose at time 0 is concordant. The opinion state converges to consensus.can calculate that E [ | ˆ e | ] = ( N (1 + r ) N − − N − (cid:16) r (1 + r ) N − N − r (cid:17) ∼ rr + 1 N as N → ∞ . That is, if we assume that P [ d ( e, x (0)) < c | | e | = n ] = ar n , then the expected sizeof ˆ e grows linearly with N . In Figure 2, we show the results of Monte Carlo simulationsto estimate E [ | ˆ e | ] as a function of N without using the assumption P [ d ( e, x (0)) < c || e | = n ] = ar n . As hypothesized, we observe a linear relationship.Because ˆ e is the first concordant hyperedge that we select, we update the nodesof ˆ e to the opinion ¯ x ˆ e (0), which is the mean of the initial opinions of the nodes thatare incident to e . Let µ be the mean of the opinion distribution. If e is an arbitraryhyperedge, then ¯ x e (0) → µ in distribution as | e | → ∞ . It is necessarily also true that¯ x ˆ e (0) → µ in distribution as | ˆ e | → ∞ because the sample mean and sample varianceof a normal distribution are independent of each other (by Basu’s theorem). Because E [ | ˆ e | ] → ∞ as N → ∞ , it follows that ¯ x ˆ e (0) → µ in distribution as N → ∞ . In otherwords, we are updating the opinions of the nodes in ˆ e to approximately µ .The observations above imply that (1) the first concordant hyperedge that weupdate includes a fraction of the nodes that is approximately constant as N → ∞ (even for large σ ) and (2) when we update all of those nodes, we are updating themto approximately µ , which is the mean of the distribution. This decreases the totalsample variance of all N nodes’ opinions and increases the clustering of opinions near µ , making it even more likely that the next hyperedge that we update will also be largeand have a mean opinion that is centered near µ . Eventually, the opinions convergeto a consensus value near µ . In this sub-section, we examine our BCM on hypergraphs with planted community structure.Suppose that we partition the set of nodes in a hypergraph into communities and2 A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER Fig. 2: An estimate of E [ | ˆ e | ] as a function of the number N of nodes, where ˆ e is the firstconcordant hyperedge that we select and the initial opinions are normally distributedwith a standard deviation of σ = 1 . 2. The confidence bound is c = 1. For each possiblehyperedge size n ∈ { , . . . , N } , we run 10 , 000 trials. For each trial, we randomlydraw n opinions from N (0 , σ ) and calculate the sample variance, which equals thediscordance of those n opinions. We approximate a n := P [ d ( e, x (0)) < c | | e | = n ] byletting (cid:99) a n be the fraction of trials that result in a concordant set of n opinions. Foreach N ∈ { , . . . , } , we have that E [ | ˆ e | ] ≈ (cid:80) Nn =2 n (cid:99) a n ( Nn ) (cid:80) Nn =2 (cid:99) a n ( Nn ) .that each community has its own independent distribution of initial opinions. Specif-ically, we study our BCM on hypergraphs that we generate using the hypergraphstochastic block model (HSBM) of [19]. An HSBM is a generative model for pro-ducing hypergraphs with community structure. Like a traditional SBM for ordinarygraphs [15, 40], the probability that a hyperedge exists depends on the communitymemberships of its nodes. In this HSBM, the probability that a hyperedge exists alsodepends on the size of the hyperedge and on the number of nodes in the hypergraph.More precisely, let ψ : V → { , . . . , k } be a partition of the set of nodes into k commu-nities, where we assume without loss of generality that every community is non-empty.We denote the i th community by C i := ψ − ( i ). For N = | V | and n ∈ { , . . . , N } , let α n,N ∈ [0 , n ∈ { , . . . , N } , let B n be a symmetric k -dimensional tensorof order n whose entries take values in [0 , e = { i , . . . , i n } ∈ P ( V ) of nodes, we include e in the hypergraphwith a probability of α n,N B nψ ( i ) ,...,ψ ( i n ) .In the simplest version of this HSBM, any intra-community hyperedge exists withan independent and uniform probability p and any inter-community hyperedge existswith an independent and uniform probability q . Definition Consider the HSBM of [19] with parameters α n,N = 1 for all n, N and B nψ ( i ) ,...,ψ ( i n ) = (cid:40) p , ψ ( i ) = · · · = ψ ( i n ) q , otherwisefor some p, q ∈ [0 , . We will refer to this HSBM as a ( p, q )-HSBM . If q = 0, the communities in an ( p, q )-HSBM are disjoint and the opinions in onecommunity cannot influence the opinions in other communities. BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS Definition Let ψ : V → { , . . . , k } be a partition of the set of nodes into k non-empty communities. The opinion state x ∗ is polarized if there are γ , . . . , γ k ∈ R , not all equal, such that x ∗ i = γ ψ ( i ) for all i . In other words, x ∗ is polarized if each community is at consensus but the communitiesare not at consensus with each other. For example, if q = 0 and it is not the case thatevery community has the same initial mean opinion, then the opinion state convergesto a limit state that is either polarized or such that there is a community whose nodesare not at consensus within the community.The following theorem says that if q (cid:54) = 0, then the probability that the limit stateof our BCM on a ( p, q )-HSBM hypergraph is polarized approaches 0 as N → ∞ . Theorem Suppose that H is generated from a ( p, q ) -HSBM with partition ψ : V → { , . . . , k } and that q (cid:54) = 0 . Additionally, suppose that c (cid:54) = 0 . Let x (0) be theinitial opinion state, with x i (0) ∈ [ a, b ] for all i , and let x ( t ) be the opinion state thatis determined by (2.5) . Let x ∗ be the limit state, which exists by Theorem . It thenfollows that the probability that x ∗ is polarized approaches as N → ∞ .Proof. Because x ∗ is almost surely an absorbing state by Lemma 3.2, it sufficesto show that P [ x ∗ polarized and absorbing] → N → ∞ . Suppose that x ∗ is apolarized, absorbing state. Because x ∗ is polarized, there are γ , . . . , γ k that are notall equal and that satisfy x ∗ i = γ ψ ( i ) . Without loss of generality, let C be the largestcommunity. It is necessarily true that | C | ≥ (cid:100) N/k (cid:101) . Because { γ i } are not all equal,there is a community j such that γ (cid:54) = γ j .To find a contradiction, suppose that there is a hyperedge e ∈ E of size n > | γ − γ j | /c such that e is incident to n − C and one node in C j . It followsthat 0 < d ( e, x ∗ ) = ( γ − γ j ) n < ( b − a ) n < c , which contradicts the assumption that x ∗ is an absorbing state. As N → ∞ , theprobability that there is no such e ∈ E islim N →∞ P [ (cid:64) e ∈ E ] = lim N →∞ (1 − q ) (cid:80) | C |− n i =max { , − n } ( | C | n i − ) ×| C j | ≤ lim N →∞ (1 − q ) (cid:100) N/k (cid:101) = 0 , where n = (cid:106) | γ − γ j | c (cid:107) .The following theorem says that if we also impose the condition p = 1 (so thateach community forms a hyperclique), then the probability of reaching consensusapproaches 1 as N → ∞ if all communities are sufficiently large. Theorem Suppose that we generate a hypergraph H from a ( p, q ) -HSBMwith partition ψ : V → { , . . . , k } and that p = 1 and q (cid:54) = 0 . Let x (0) be the initialopinion state, with x i (0) ∈ [ a, b ] for all i , and let x ( t ) be the opinion state that isdetermined by (2.5) . Additionally, suppose that | C i | > ( b − a √ c + 1)( ( b − a ) c − for all i ∈ { , . . . , k } and that c (cid:54) = 0 . It then follows that P [ consensus ] → as N → ∞ .Proof. By Theorem 3.1, the limit state x ∗ := lim t →∞ x ( t ) exists. By the sameargument as in the proof of Theorem 3.10, the nodes in each community converge toconsensus almost surely because | C i | > ( b − a √ c + 1)( ( b − a ) c − 1) for all i ∈ { , . . . , k } .That is, there exist γ , . . . , γ k ∈ [ a, b ] such that x ∗ i = γ j for all i ∈ C j . If γ i = γ j forall communities i and j , then the opinion state converges to consensus. Otherwise,4 A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER we assume without loss of generality that C is the largest community and we let C j be a community such that γ j (cid:54) = γ . Suppose that there is a hyperedge e ∈ E of size n > ( b − a ) c such that e is incident to n − C and one node in C j . By thesame argument as in the proof of Theorem 3.13, the probability that such a hyperedgeexists approaches 1 as N → ∞ ; additionally, if e does exist, we also have that x ∗ isnot an absorbing state. By Lemma 3.2, x ∗ is almost surely an absorbing state.In another version of the HSBM in [19], one requires that every inter-communityhyperedge is small. Definition Consider the HSBM of [19] with parameters α n,N = 1 for all n, N and B nψ ( i ) ,...,ψ ( i n ) = p , ψ ( i ) = · · · = ψ ( i n ) q , there exist j and k such that ψ ( i j ) (cid:54) = ψ ( i k ) and n ≤ M , otherwisefor some M ≥ and p, q ∈ [0 , . We will refer to this HSBM as a ( p, q, M )-HSBM . For fixed p, q, M and N → ∞ , the communities are “almost” disjoint; that is, theproportion of inter-community hyperedges (relative to all hyperedges) approaches 0.In contrast to Theorem 3.13, the following theorem gives conditions under whicha polarized opinion state is an absorbing state. Theorem 3.16 implies that echochambers can form when all of the inter-community edges are sufficiently small insize. Theorem Suppose that we generate a hypergraph H from a ( p, q, M ) -HSBMwith the partition ψ : V → { , . . . , k } . Let γ , . . . , γ k ∈ R , and let x ∗ be the polarizedopinion state with x ∗ i = γ ψ ( i ) . If min i,j { ( γ i − γ j ) | γ i (cid:54) = γ j } /M > c , then x ∗ is anabsorbing state.Proof. Without loss of generality, assume that γ i (cid:54) = γ j if i (cid:54) = j . (If not, wecombine any communities that hold the same opinion.) Let e ∈ E be a hyperedge.Let n = | e | , and let n i = |{ j ∈ e | ψ ( j ) = i }| be the number of nodes that are incidentto e and belong to community i . We have that d ( e, x ∗ ) = 1 n − (cid:32) (cid:88) i n i (cid:16) γ i − n (cid:88) j n j γ j (cid:17) (cid:33) = 1 n − (cid:32) (cid:88) i n i γ i − n (cid:16) (cid:88) i n i γ i (cid:17) (cid:33) = 1 n ( n − (cid:16) (cid:88) i n i γ i ( n − n i ) − (cid:88) i (cid:88) j
2. This polarized opinionstate is an absorbing state. n (cid:96) (cid:54) = n for all (cid:96) . Fixing n and enforcing the constraint that n (cid:96) (cid:54) = n for all (cid:96) , it followsthat (3.8) is minimized either when n (cid:96) = , (cid:96) (cid:54) = i ∗ , j ∗ n − , (cid:96) = i ∗ , (cid:96) = j ∗ or when n (cid:96) = , (cid:96) (cid:54) = i ∗ , j ∗ , (cid:96) = i ∗ n − , (cid:96) = j ∗ . Therefore, d ( e, x ∗ ) ≥ min { ( γ i − γ j ) | γ i (cid:54) = γ j } M > c if n (cid:96) (cid:54) = n for all n . We have thus shownthat for all e ∈ E , it must be the case that either d ( e, x ∗ ) = 0 or d ( e, x ∗ ) > c .In Figure 3, we show a typical simulation when the conditions of Theorem 3.16 aresatisfied. In the limit state, the communities are polarized. All nodes in community C i converge to opinion γ i = (cid:80) j ∈ C i x j (0), which is the mean of the initial opinionsin C i . By Theorem 3.16, we know that this polarized opinion state is an absorbingstate. Remark p, q, M )-HSBM if the initial opinions of the6 A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER(a) x i (0) ∼ U ( − , 2) (b) x i (0) ∼ N (0 , . Fig. 4: Typical simulations on sparse G ( N, m ) hypergraphs with a confidence boundof c = 1 and N = 1000 nodes that have (a) uniformly distributed initial opinionsand (b) normally distributed initial opinions. Both of these simulations converge toa consensus.different communities are sufficiently close to each other. If we draw the initial opin-ions from a bounded interval [ a, b ] with ( b − a ) /M < c and if | C i | > ( b − a √ c )( ( b − a ) c − i , then the probability of reaching consensus approaches 1 as N → ∞ . We now study our hypergraphBCM on sparse G ( N, m ) hypergraphs. The G ( N, m ) model is a generative hyper-graph model that is defined analogously to the Erd˝os–R´enyi G ( N, m ) generative graphmodel. Each hypergraph that one constructs from the G ( N, m ) model has N nodes;for each possible hyperedge size i ∈ { , . . . , N } , we choose m i hyperedges of that sizeuniformly at random to include in the hypergraph.In our Monte Carlo simulations, we set N = 1000 and m i = max { , (cid:0) Ni (cid:1) } forall i . We run 1000 simulations with a confidence bound of c = 1 and initial opinionsthat we draw uniformly at random from [ − , c = 1 and initial opinions that we draw from the normal distribution with mean µ = 0and standard deviation σ = 1 . 2. All of these trials also converge to a consensus. InFigure 4, we show a typical simulation for each of the two initial opinion distributions.As we increase N and increase the variance of the initial opinions, we observethat the time it takes to converge increases but that the opinion state still convergesto a consensus. In subsection 3.3, westudied our BCM on sparse hypergraphs. However, it is typically also the case that thehypergraphs that one constructs from empirical data are not merely sparse; they alsohave the property that their hyperedges are small in size in comparison to the numberof nodes. As one example, we use a hypergraph that Benson et al. [3] constructedfrom the well-known (and infamous) Enron e-mail data set [25]. In this Enron e-mail hypergraph, nodes represent Enron employees and hyperedges represent e-mailsbetween them. Each hyperedge is incident to the sender and recipients of one e-mailmessage. There are N = 143 nodes, but the maximum hyperedge size is only 18. BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS (a) x i (0) ∼ U (0 , 1) (b) x i (0) ∼ N (0 , Fig. 5: A typical simulation on the Enron e-mail hypergraph with a confidence boundof c = 1 and nodes with (a) uniformly distributed initial opinions and (b) normallydistributed initial opinions. Both of these simulations converge to a consensus.To examine our hypergraph BCM on the Enron e-mail hypergraph, we run 1000simulations with initial opinions that are uniformly distributed in [0 , µ = 0 and a varianceof σ = 1. The confidence bound is c = 1. In all trials, the opinion state converges toa consensus. In Figure 5b, we show the results of a typical simulation. 4. Convergence Time. In this section, we analyze the convergence time of ourhypergraph BCM. We say that the opinionstate converges in finite time if there is a time T such that x ( T ) is an absorbing state.At time T , no further opinion updates can occur. We use the following lemma toprove Theorem 4.2. Lemma Let ˜ e ∈ P ( V ) be a subset of nodes. There is a finite sequence { ˜ e i } mi =1 in P ( V ) such that (1) | ˜ e i | is prime for all i ≥ and (2) consecutively updating thenodes that are incident to ˜ e , . . . , ˜ e m to their respective mean opinions would resultin the same opinion state as updating the nodes that are incident to ˜ e to their meanopinion.Proof. Let n = | ˜ e | . If n is prime, we are done. This also proves the base case n = 2. If n is not prime, we can write n = pm , where p is prime and m < n . Withoutloss of generality, suppose that ˜ e = { , . . . , n } . Updating the nodes of ˜ e to the meanopinion in ˜ e at time t results in the opinion state(4.1) x i = (cid:40) n (cid:80) i ∈ ˜ e x i ( t ) = p (cid:80) p − k =0 (cid:16) m (cid:80) mj =1 x km + j ( t ) (cid:17) , i ≤ nx i ( t ) , i > n . Updating the nodes of { km + 1 , . . . , km + m } ∈ P ( V ) for each k ∈ { , . . . , p − } to8 A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER their respective mean opinions at time t results in the opinion state(4.2) x i = (cid:40) m (cid:80) m(cid:96) =1 x km + (cid:96) ( t ) , i = km + j , ≤ j ≤ m , ≤ k ≤ p − x i ( t ) , i > n . By induction (because m < n ), there is a sequence of elements of prime size in P ( V ) such that updating this sequence results in the same opinion state as updating { km + 1 , . . . , km + m } . Concatenating these p sequences (one for each k ) yields asequence ˜ e , . . . , ˜ e (cid:96) of elements of prime size in P ( V ) such that updating the nodes of˜ e , . . . , ˜ e (cid:96) to their respective mean opinions at time t results in the opinion state (4.2).Updating the sequence { , m + 1 , . . . , ( p − m + 1 } , { , m + 2 , . . . , ( p − m + 2 } ,. . . , { m, m, . . . , pm } of elements in P ( V ) then results in the opinion state (4.1).Therefore, ˜ e , . . . , ˜ e (cid:96) , { , . . . , ( p − m + 1 } , { , . . . , ( p − m + 2 } , . . . , { m, . . . , pm } isthe desired sequence of prime-sized elements in P ( V ). Theorem Let H be a hypergraph such that { e ∈ P ( V ) | | e | prime } ⊆ E . Let x (0) be any initial opinion state, and let x ( t ) be the opinion state that is determinedby (2.5) . It follows that x ( t ) converges in finite time almost surely.Proof. Let the time t , the matrices K , . . . , K p , and the node sets I , . . . , I p bedefined as in equation (3.1). This equation implies for all j, k ∈ I i that(4.3) lim t →∞ x j ( t ) = lim t →∞ x k ( t ) . Updating the nodes of I ∈ P ( V ) to their mean opinion results in consensus amongthe nodes of I . By Lemma 4.1, there is a sequence { e i } nj =1 of prime-sized elementsin P ( V ) such that consecutively updating { e i } results in the same opinion state asupdating I . By hypothesis, the hypergraph H includes e i for all i because e i is prime-sized. There is a time t ≥ t such that d ( e i , x ( t )) < c for all i for all times t ≥ t .The probability of consecutively choosing the hyperedges of { e i } ni =1 starting at a giventime is 1 / | E | n > 0, where E is the hyperedge set of H . Because this probability ispositive, the event of consecutively choosing these hyperedges occurs infinitely oftenalmost surely. Therefore, there is almost surely some time larger than t such that weconsecutively choose the hyperedges { e i } for updating. Let t > t be the time thatwe choose the last hyperedge e n of the sequence. At this time, x j ( t ) = x k ( t ) for all j, k ∈ I . Similarly, we can find times t ≤ · · · ≤ t p +1 for the nodes of I , . . . , I p . Byequation (3.1), any hyperedge e t that we choose at t ≥ t is discordant if e t is notcontained in some I i . Therefore, for t ≥ t p +1 , it follows that d ( e, x ( t )) = 0 if there isan i such that e ⊆ I i ; otherwise, d ( e, x ( t )) > c . Therefore, x ( t p +1 ) is an absorbingstate. Remark Theorem Let H be a hypergraph with N nodes and hyperedge set E . Supposethat there is a subset e = { i , . . . , i p } (cid:54)∈ E of nodes such that e has prime size p , andsuppose that the subhypergraph that is induced by { i , . . . , i p } is connected. (Theconnectivity requirement implies that p ≥ .) It then follows that there is an initialopinion state x (0) such that the opinion state x ( t ) , which is determined by (2.5) , doesnot converge in finite time.Proof. Without loss of generality, e = { , . . . , p } . By the continuity of thediscordance function, there is an (cid:15) > x (0) , . . . , x p (0) ∈ [0 , (cid:15) ], then BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS d ( e (cid:48) , x (0)) < c for all e (cid:48) ⊆ e . We choose r such that 1 / r < (cid:15) , and we let x (0) = 0and x i (0) = 1 / r for all i ∈ { , . . . , p } . Additionally, there is an M > (cid:15) such that if x p +1 (0) , . . . , x N (0) > M , then d ( e (cid:48) , x (0)) > c for all e (cid:48) ∈ E that satisfy e (cid:48) (cid:54)⊆ { , . . . , p } and e (cid:48) (cid:54)⊆ { p + 1 , . . . , n } . Let x p +1 (0) = · · · = x N (0) = M. Because of these initialconditions, the nodes in { , . . . p } never interact with the nodes in { p + 1 , . . . , n } .Therefore, x ( t ) , . . . , x p ( t ) ∈ [0 , (cid:15) ] for all t and every e (cid:48) ⊂ e is concordant for all t .Because the subhypergraph that is induced by the nodes { , . . . , p } is connected,lim t →∞ x i ( t ) = 1 p p (cid:88) j =1 x j (0) = p − r p for all i ∈ { , . . . , p } . Let e t = { i , . . . , i k } ∈ E be the hyperedge that we choose at time t . If e t (cid:54)⊆ e , then x i ( t + 1) = x i ( t ) for all i ∈ e . Otherwise, e t ⊂ e and for all i ∈ e t , we have x i ( t + 1) = 1 k k (cid:88) j =1 x j ( t ) , where 2 ≤ k ≤ p − | e | = p and e t is a strict subset of e . By induction on t ,there exist a ij ( t ) ∈ N ∪ { } and n k ( t ) ∈ N ∪ { } such that x i ( t + 1) = 1 (cid:81) p − k =2 k n k ( t ) p (cid:88) j =1 a ij ( t ) x j (0) = 12 r (cid:81) p − k =2 k n k ( t ) p (cid:88) j =2 a ij ( t ) for all i ∈ e . For all t , we have(4.4) ( p − r p − (cid:89) k =2 k n k ( t ) (cid:54) = 2 r p p (cid:88) j =2 a ij ( t )because the left-hand side of (4.4) is a non-zero integer without p in its prime fac-torization and the right-hand side of (4.4) is either 0 (if (cid:80) a i j ( t ) = 0) or a non-zerointeger with p in its prime factorization. Consequently, x i ( t ) (cid:54) = lim t →∞ x i ( t ) for all t and all i ∈ e . Therefore, x ( t ) does not converge in finite time.Together, Theorem 4.2 and Theorem 4.4 partially characterize the conditions forfinite-time convergence of our BCM. If H is a hypergraph whose hyperedge set includesall prime-sized subsets of nodes, then the opinion state almost surely converges in finitetime. However, if the hyperedge set is missing some prime-sized subset of nodes andthe subhypergraph that is induced by those nodes is connected, then the opinion statedoes not almost surely converge in finite time. We do not have a characterization ofthe convergence time in the missing case, in which this connectivity condition is notsatisfied for any of the prime-sized elements of P ( V ) that are not in the hyperedge set.However, we expect that “most” hypergraphs do not fall into this missing case. Theconnectivity condition is not hard to satisfy. For example, in the G ( N, m ) model, weexpect the subhypergraph that is induced by any subset of nodes { i , . . . , i p } to beconnected whenever p is sufficiently larger than the index of the first non-zero entryof m . The vast majority of hypergraphs that are produced by the G ( N, m ) modelsatisfy the conditions of either Theorem 4.2 or Theorem 4.4. σ = c . We now study the rate of convergenceof the opinion state in our hypergraph BCM. We focus on the complete hypergraph,0 A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER t * Fig. 6: Empirical convergence time for our BCM on a complete hypergraph with50 , 000 nodes and opinions that we seed independently using x i (0) ∼ N (0 , σ ). Weconsider all uniformly-spaced values σ ∈ [0 . , . 1] with a step size of ∆ σ = 0 . t ∗ is the first time that the discordance function of theopinion state satisfies d ( V, x ) < − . If the system does not reach such an opinionstate by t = 10 , we record t ∗ as 10 . We simulate 20 trials for each value of σ . Theblack curve gives the mean of t ∗ over the trials, and the blue area depicts one standarddeviation from the mean. We include a dashed red line at σ = 1 = c for reference.for which we observe that there is a phase transition in convergence time when theconfidence bound is c = σ .In Figure 6, we simulate our BCM with c = 1 on a complete hypergraph with50 , 000 nodes and initial opinions that we seed independently by setting x i (0) ∼N (0 , σ ) for σ ∈ [0 . , . t ∗ , which we set tobe the earliest time that the discordance function satisfies d ( e = V, x ) < − . Ourresults are consistent with the existence of a phase transition in convergence time.In Theorem 4.7, we prove that the convergence time grows at least exponentiallyfast as a function of N if c < σ . The proof relies on Lemma 3.3, where we calculatedthe value of lim n →∞ P [ d ( e, x (0)) < c | | e | = n ] for any initial opinion distributionwith finite variance σ and showed that there is a transition at σ = c . We also needa bound on the convergence rate in this limit. The inequalities that we derived inLemma 3.3 combined with the fact that Var[ s n ] = O ( n ) imply that the convergencerate is O ( n ) whenever σ (cid:54) = c . When the initial opinions are normally distributed,we can derive a much tighter bound on the convergence rate. Lemma Suppose that we draw the initial opinions from a normal distributionwith variance σ (cid:54) = c , and let λ = cσ . It follows that P [ d ( e, x (0)) < c | | e | = n ] ≥ − (cid:16) e (1 − λ ) √ λ (cid:17) n − , λ > , P [ d ( e, x (0)) < c | | e | = n ] ≤ (cid:16) e (1 − λ ) √ λ (cid:17) n − , λ < . Therefore, P [ d ( e, x (0)) < c | | e | = n ] converges exponentially fast as n → ∞ .Proof. The discordance of a hyperedge e at time 0 is the sample variance of theopinions { x j (0) | j ∈ e } . Let s n denote the sample variance of n opinions. We have BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS P [ d ( e, x (0)) < c | | e | = n ] = P [ s n < c ].Case 1. Suppose that λ > 1. By Chernoff’s bound [9],(4.5) P [ s n ≥ c ] = P (cid:104) n − σ s n ≥ λ ( n − (cid:105) ≤ E [ e t n − σ s n ] e tλ ( n − for all t > . By Cochran’s theorem, n − σ s n ∼ χ n − [26]. When t < , we have that E [ e t n − σ s n ] = 12 ( n − / Γ( n − ) (cid:90) ∞ e tx x n − − e − x/ dx = 1Γ( n − ) (cid:90) ∞ x n − − e − x (1 − t ) dx = 1Γ( n − ) (1 − t ) − n − (cid:90) ∞ x n − − e − x dx = (1 − t ) − n − . Therefore when 0 < t < , (4.5) becomes(4.6) P [ s n ≥ c ] ≤ (cid:16) e tλ √ − t (cid:17) n − . Setting t = (1 − λ ) in equation (4.6) yields P [ s n ≥ c ] ≤ (cid:16) e (1 − λ ) √ λ (cid:17) n − . Case 2. Suppose that λ < 1. By Chernoff’s bound,(4.7) P [ s n < c ] = P (cid:104) n − σ s n < λ ( n − (cid:105) ≤ E [ e − t n − σ s n ] e − tλ ( n − for all t > . Similarly to Case 1, we compute that E [ e − t n − σ s n ] = (cid:16) 11 + 2 t (cid:17) n − for t > 0. Therefore, when t > 0, (4.7) becomes(4.8) P [ s n < c ] ≤ (cid:32) e λt √ t (cid:33) n − . Setting t = ( λ − 1) in (4.8) yields P [ s n < c ] ≤ (cid:16) e (1 − λ ) √ λ (cid:17) n − . Remark σ = c and the initial opinions are normally distributed, wehave numerical evidence that P [ d ( e, x (0)) < c | | e | = n ] converges to exponentiallyfast as n → ∞ , but we do not have a mathematical proof of the convergence rate.2 A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER We now study the convergence time of our hypergraph BCM. We say that theopinion state converges with threshold (cid:15) at time T (cid:15) if for all e ∈ E , either d ( e, x ( T (cid:15) )) ≤ (cid:15) or d ( e, x ( T (cid:15) )) ≥ c . When (cid:15) = 0, the time T (cid:15) is exactly the convergence time. Theorem Let H be the complete hypergraph with N nodes. Let x (0) be theinitial opinion state and suppose that x i (0) ∼ N ( µ, σ ) , where σ > c and c (cid:54) = 0 .Finally, let (cid:15) < c and let T (cid:15),N be the convergence time with threshold (cid:15) . It thenfollows that E [ T (cid:15),N ] = Ω (cid:16) r (cid:104) r + 1 (cid:105) N (cid:17) , where r = e (1 − c/σ ) (cid:112) cσ < .Proof. Let A N be the Bernoulli random variable that equals 1 if there is a hyper-edge e ∈ E such that (cid:15) < d ( e, x (0)) < c and equals 0 if there is no such hyperedge.We have that E [ T (cid:15),N ] = E [ T (cid:15),N | A N = 1] P [ A N = 1] + E [ T (cid:15),N | A N = 0] P [ A N = 0] . If A N = 0, then T (cid:15),N = 0. As N → ∞ , we havelim N →∞ P [ A N = 0] = lim N →∞ N (cid:89) n =2 (1 − P [ (cid:15) < d ( e, x (0)) < c | | e | = n ])( Nn ) = 0 , where e denotes a hyperedge that we choose uniformly at random. Let s N be the firsttime that we select a concordant hyperedge. If A N = 1, then T (cid:15),N ≥ s N . Let X N bethe fraction of hyperedges in H that are concordant at time 0. We calculate E [ T (cid:15),N ] ≥ E [ T (cid:15),N | A N = 1] P [ A N = 1] ≥ E [ s N ] P [ A N = 1]= P [ A N = 1] E [ X N ] , E [ X N ] = (cid:80) Nn =2 (cid:0) Nn (cid:1) P [ d ( e, x (0)) < c | | e | = n ]2 N − N − ≤ (cid:80) Nn =2 (cid:0) Nn (cid:1) r n − N − N − r (( r + 1) N − N r − N − N − . Therefore, as N → ∞ , we obtain E [ T (cid:15),N ] ≥ r P [ A N = 1] 2 N − N − r + 1) N − N r − ∼ r (cid:16) r + 1 (cid:17) N . Remark T ,N ,which is almost surely finite by Theorem 4.2. 5. Opinion Jumping. We now study “opinion jumping”, a phenomenon thatoccurs in our hypergraph BCM that cannot occur in standard dyadic BCMs. An opinion jump occurs at time t if there is a node i such that | x i ( t + 1) − x i ( t ) | > c . The BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS t is the number of nodes i that satisfy | x i ( t + 1) − x i ( t ) | > c .An opinion jump can occur only if the size of the selected hyperedge is at least3. Therefore, this behavior requires polyadic interactions; it cannot occur on BCMson ordinary graphs. Moreover, we believe that it is one of the driving behaviors thatcauses our hypergraph BCM to converge to consensus so much more easily than isthe case for standard dyadic BCMs. For examples of opinion jumping, see Figure 1and Figure 5. In this section, we quantify how common it is for an opinion jump tooccur. Lemma Let J t be the number of opinion jumps that occur at time t . Supposethat the distribution of opinions at time t has a mean of µ and a variance of σ < ∞ .Let p n = P [ | x i − ¯ x e | > c | i ∈ e, | e | = n, d ( e, x ( t )) < c ] be the probability that anode’s opinion is farther than c from the mean opinion of the nodes in a concordant,size- n hyperedge that is incident to the node, and let p = P [ | x − µ | > c ] . Let a n = P [ d ( e, x ( t )) < c | | e | = n ] be the probability that a size- n hyperedge is concordant, andlet a = lim n →∞ a n be the limiting probability of concordance. Finally, let e t be thehyperedge that we select at time t . It then follows that E [ J t ] = (cid:32) pa E [ | e t | ] + p N (cid:88) n =2 ( a n − a ) P [ | e t | = n ] n + N (cid:88) n =2 ( p n − p ) a P [ | e t | = n ] n + N (cid:88) n =2 ( p n − p )( a n − a ) P [ | e t | = n ] n (cid:33) . Remark p n , p , a n , and a depend on the distribution of opin-ions at time t . The value of a is given by (3.6). Proof. For j ≥ 1, we have P [ J t = j ] = N (cid:88) n =2 P [ J t = j and | e t | = n ]= (cid:88) n P [ J t = j | | e t | = n ] P [ | e t | = n ]= (cid:88) n P [ J t = j and d ( e t , x ( t )) < c | | e t | = n ] P [ | e t | = n ] (because j ≥ (cid:88) n P [ J t = j | d ( e t , x ( t )) < c, | e t | = n ] a n P [ | e t | = n ]= (cid:88) n (cid:18) nj (cid:19) p jn (1 − p n ) j a n P [ | e t | = n ] . A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER Therefore, E [ J t ] = N (cid:88) j =0 P [ J t = j ] j = (cid:88) j (cid:88) n j (cid:18) nj (cid:19) p jn (1 − p n ) j a n P [ | e t | = n ]= (cid:88) n a n P [ | e t | = n ] (cid:88) j j (cid:18) nj (cid:19) p jn (1 − p n ) j = (cid:88) n a n P [ | e t | = n ] p n n = (cid:32) pa E [ | e t | ] + p N (cid:88) n =2 ( a n − a ) P [ | e t | = n ] n + N (cid:88) n =2 ( p n − p ) a P [ | e t | = n ] n + N (cid:88) n =2 ( p n − p )( a n − a ) P [ | e t | = n ] n (cid:33) . We use Lemma 5.1 to derive the asymptotic behavior of E [ J ]. The following proposi-tion says that, under certain conditions, E [ J ] grows linearly with the mean hyperedgesize of the hypergraph on which our BCM occurs. Proposition Let { H m } be a sequence of hypergraphs, and let { V m } and { E m } be the corresponding sequences of nodes and hyperedge sets, respectively. Let { g m } be the corresponding sequence of hyperedge size distributions, where g m ( n ) = |{ e ∈ E m | | e | = n }|| E m | . Suppose that we draw the initial opinions from the same distribution for all H m , andlet p, p n , a, a n be defined as in Lemma . Finally, let J m be the number of opinionjumps that occur at time for H m . If ( a − a n ) n → as n → ∞ , g m ( n ) → for all n as m → ∞ and a ( p n − p ) n → as n → ∞ , then E [ J m ] ∼ pa E [ | e | ] as m → ∞ .Remark H m is the complete hypergraph with m nodes,we have that g m ( n ) → n as m → ∞ . The values of a , a n , p , and p n dependonly on the opinion distribution at time t . The value of a is given by (3.6). If theinitial opinions are normally distributed, then Lemma 4.5 implies that ( a n − a ) n → σ (cid:54) = c . The exact value of p depends on the initial distribution, but it tendsto increase with σ . Our numerical computations suggest that a ( p n − p ) n → σ (cid:54) = c . Proof. Lemma 5.1 implies that E [ J m ] = pa E [ | e | ] + a (cid:88) n ( p n − p ) ng m ( n ) + p (cid:88) n ( a n − a ) ng m ( n )+ (cid:88) n ( p − p n )( a n − a ) ng m ( n ) . Let x n be any sequence such that x n → 0. For any m , the quantity (cid:80) n x n g m ( n ) is aweighted average of { x n } . As m → ∞ , the weights concentrate at larger values of n .Therefore, because x n → 0, it follows that (cid:80) n x n g m ( n ) → m → ∞ . BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS x n = a ( p n − p ) n , x n = ( a n − a ) n , and x n =( p − p n )( a n − a ) n to prove the proposition.In Figure 7, we present numerical results that support the claim that E [ J ] ≈ pa E [ | e | ] when the edge-size distribution concentrates at large edge sizes. We generatehypergraphs from the G ( N, m ) hypergraph model for different values of m . For eachhypergraph, we run trials of our hypergraph BCM and record the mean value of J for the hypergraph. We plot the mean value of J versus the mean hyperedge size E [ | e | ] in the hypergraph. We show results for initial opinions that are normallydistributed with standard deviations of σ = 0 . σ = 0 . σ = 1, and σ = 1 . 2. Weuse a confidence bound of c = 1 in all trials. The claim E [ J ] ≈ pa E [ | e | ] implies thatfor each σ , there should be a linear relationship with a slope of pa , where p and a depend on σ . Whenever σ < c , the limiting probability of concordance is a = 1. Theslope when σ = . σ = . p becomes largerfor progressively larger σ . When σ = 1, the value of p is larger than for the previousvalues of σ , but the limiting probability of concordance is only a = , and the slope pa is slightly less steep than when σ = . 8. We observe that the linear relationshipbetween E [ J ] and E [ | e | ] is not as strong when σ = c = 1 as when σ (cid:54) = c . Basedon numerical evidence, we suspect that this is because ( p n − p ) n (cid:54)→ σ = c .Finally, when σ = 1 . 2, we have that E [ J ] ≈ a = 0 whenever σ > c . 6. Conclusions and Discussion. We formulated a bounded-confidence model(BCM) for hypergraphs and explored its properties through both mathematical anal-ysis and Monte Carlo simulations. We showed that polyadic (i.e., “higher-order”)interactions play an important role in opinion dynamics and that one cannot reducesuch interactions to pairwise interactions on a graph. In our hypergraph BCM, we alsodemonstrated a novel phenomenon, which we called “opinion jumping”, that requirespolyadic interactions for it to manifest. Therefore, opinion jumping cannot occur instandard dyadic BCMs.We proved that our hypergraph BCM converges to consensus on the completehypergraph for a wide variety of initial conditions. This is very different from whatoccurs in standard dyadic BCMs, which usually converge to multiple opinion clusters.We also studied the effects of a variety of initial opinion distributions on the dynamicsof our BCM. In particular, we examined the convergence properties of our BCM whenthe initial opinion distribution is bounded, normally-distributed, or has a variance σ that is less than the confidence bound c . Based on our results, we expect that thelimit states of dyadic BCMs also depend on the initial opinion distribution (althoughthis is not something that has been studied in detail in prior research) and that thenumber of opinion clusters depends not only on the confidence bound c but also onthe relative sizes of c and the variance σ of the initial opinion distribution.We also explored the dependence of the limit state of our hypergraph BCM oncommunity structure. We proved that the opinion state can become polarized if theintra-community hyperedges are sufficiently small in size. This leads to the formationof echo chambers. However, we showed that if the intra-community hyperedges areunbounded in size and if the communities are large enough and form hypercliques,then the opinion state converges to consensus.We also showed that there is a phase transition in convergence time on the com-plete hypergraph when the confidence bound c equals the variance σ of the initialopinion distribution. When σ > c , the convergence time of our BCM on the completehypergraph depends exponentially on the number N of nodes. This has implications6 A. HICKOK, Y. KUREH, H. Z. BROOKS, M. FENG, M. A. PORTER Fig. 7: Empirical evidence for the linear relationship between E [ J ] and E [ | e | ], where J is the number of opinion jumps that occur at time 0 and the hyperedge e isthe hyperedge that we choose uniformly at random at time 0. In our numericalexperiment, initial opinions are normally distributed with a mean of 0 and standarddeviation of σ , which takes values of 0 . 6, 0 . 8, 1, and 1 . 2. For each σ , we generate 200hypergraphs with N = 1000 nodes. To construct the n th hypergraph, we choose x n ∈ [0 , 1] uniformly at random and set m ( n ) i = (cid:0) Ni (cid:1) x in . We generate the n th hypergraphfrom the G ( N, m ( n ) ) model, which has N nodes and m ( n ) i hyperedges of size i that wechoose uniformly at random. For each hypergraph, we run one step of our hypergraphBCM with confidence bound c = 1 and record the number J of opinion jumps. Werun 500 trials on each hypergraph; each time, we preserve the hypergraph structure,reset the initial opinions, and run one step of our hypergraph BCM. We record themean value of J over these 500 trials and we plot it versus the mean hyperedge size E [ | e | ] in the hypergraph. Each data point corresponds to the trial results for a singlehypergraph for a given value of σ . For each σ , we plot the line of best fit.for the feasibility of using Monte Carlo simulations for simulating our BCM on thecomplete hypergraph when σ > c and N is large (and, more generally, on any hyper-graph where large hyperedges make up a significant proportion of the hyperedge set,because large hyperedges are likely to be discordant and it thus takes many time stepsto choose a concordant hyperedge). It is fascinating that there is a phase transition inconvergence time, but not in the limit state. By contrast, in the standard dyadic DWmodel, there is a phase transition in convergence time at the same confidence boundthreshold c ∗ at which there is a phase transition in the limit state [34]. We also provedthat our hypergraph BCM converges in finite time on the complete hypergraph. Thisis similar to what occurs in the standard dyadic HK model, which converges in finitetime on the complete graph; however, it differs from the DW model, which tends not BOUNDED-CONFIDENCE MODEL OF OPINION DYNAMICS ON HYPERGRAPHS σ < c , we showedthat the number of opinion jumps in the first time step grows roughly linearly withthe mean hyperedge size in a hypergraph and that it becomes larger for progressivelylarger values of σ up to the value c . Interesting future work includes determining theprecise necessary conditions for opinion jumping in hypergraph BCMs.In our work, we made several modeling choices, and there are numerous alter-natives that are also worth studying. For example, one can formulate a hypergraphBCM that uses synchronous updates of opinions instead of asynchronous updates.For example, at each discrete time, suppose that each node updates its opinion to themean of the adjacent hyperedges’ mean opinions. We believe that such a synchronousmodel will have similar limit states as our asynchronous model, but we expect suchmodels to converge much more quickly to a limit state. One can also develop syn-chronous models in which each node updates its opinion to a weighted mean of theadjacent hyperedges’ mean opinions. Such heterogeneity models a situation in whichsome friendship groups have more influence on a person than others. This extends thenotion of trust from the dyadic DeGroot model [8]. Another of our model choices wasour specific choice of discordance function. Rather than setting d = d for the discor-dance function, it is worthwhile to study the entire family of discordance functions d α for α ∈ [0 , 1] that we defined in (2.2). The case α = 0 is particularly interestingbecause it models a scenario in which it is more difficult for large groups of peopleto agree than it is for small groups. Another variation of our model involves incor-porating heterogeneous confidence bounds, which models a situation in which someindividuals are more easily persuaded than others.There are a variety of other avenues to explore. For example, it is worth con-ducting a deeper investigation of the role of hypergraph topology on the limit statesof hypergraph BCMs, and one can also study BCMs on simplicial complexes (whichhave various constraints on the permissible polyadic interactions). We believe thatthe presence of large hyperedges that connect some subset of a hypergraph’s nodeswill facilitate the convergence of those nodes to a consensus. One can also developadaptive (i.e., coevolving) hypergraph BCMs, such as by modifying the hypergraphstructure at each discrete time in response to the current opinion state. For example,one can allow agents to strategically rewire in a way that maximizes their influenceor perhaps to simply leave a hyperedge when the other nodes that are incident toit become too annoying (as occurs sometimes in discussion groups on social media).From a control-theoretic perspective, one can ask how much control the agents (oran outside controller) can have in steering an opinion state towards a particular limitstate by choosing which edges to update or rewire. Appendix A. Continuum Formalism. Instead of running Monte Carlo sim-ulations, which are costly, one can study the “continuum” formalism of Ref. [2] usingnumerical integration. Consider a hypergraph in which every hyperedge is of size (cid:96) ∈ L ⊂ { , . . . , n } , and let P ( x, t ) dx be the probability density function that in-dicates how many nodes have opinions in the interval [ x, x + dx ] at time t . 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