Modelling Non-Linear Consensus Dynamics on Hypergraphs
MModelling Non-Linear Consensus Dynamics on Hypergraphs
Rohit Sahasrabuddhe
Indian Institute of Science Education and Research, Pune, India
Leonie Neuh¨auser
Hertie School Data Science Lab, Berlin, Germany
Renaud Lambiotte
Mathematical Institute, University of Oxford, United Kingdom (Dated: July 21, 2020)The basic interaction unit of many dynamical systems involves more than two nodes. In suchsituations where networks are not an appropriate modelling framework, it has recently becomeincreasingly popular to turn to higher-order models, including hypergraphs. In this paper, we explorethe non-linear dynamics of consensus on hypergraphs, allowing for interactions within hyperedgesof any cardinality. After discussing the different ways in which non-linearities can be incorporatedin the dynamical model, building on different sociological theories, we explore its mathematicalproperties and perform simulations to investigate them numerically. After focussing on synthetichypergraphs, namely on block hypergraphs, we investigate the dynamics on real-world structures,and explore in detail the role of involvement and stubbornness on polarisation.
Keywords: consensus, diffusion, higher-order, hypergraphs, non-linear, networks, group dynamics, multi-body interactions a r X i v : . [ phy s i c s . s o c - ph ] J u l I. INTRODUCTION
Networks provide a powerful language to model systems made of interacting elements, as observed in many scientificdomains. Within the social sciences, for instance, their significance has increased in recent years with the emergenceof online social networks. In particular, social network analysis has become a common tool to help model and extractinformation from the myriad of social interactions in a system. In addition to algorithms aiming at extracting centralnodes or clusters of similar nodes, much research has focussed on the impact of the network structure on the dynamicsof opinion formation [1], looking at a variety of models, from linear ones, like the Voter model, to non-linear ones, likethe de Groot model [2], bounded-confidence models [3] and threshold models [4].Nevertheless, networks can serve as a model of reality, and several studies have shown, more recently, that theymay often be inadequate to capture critical aspects of interacting systems [5, 6]. In particular, the basic interactingunits of a network are pairs of nodes, an assumption that is not verified in situations when multi-body or groupinteractions take place, such as in neural activity [7–9], robotics [10] or scientific collaborations [11]. The importanceof irreducible group interactions, that is interactions that can not be built from a combination of pairwise interactions,is especially relevant for opinion dynamics. Experiments in social psychology such as the conformity experiment [12]indicate that multiple exposures might be necessary for an agent to adopt a certain opinion state. This dependenceon multiple contacts can be seen as a first step towards group effect. Threshold models, in which the state of agentsswitches if a certain fraction of their neighbours agree, have been developed to describe this phenomenon and arealready inherently different from simple models of epidemic spread. However, these models are based on independent,pairwise interactions that are linearly accumulated and therefore do not account for truly higher-order effects[13].Popular choices for modelling genuine group interactions include hypergraphs [14] and simplicial complexes. Thelatter has opened the doors to the use of algebraic topology in the analysis and study of complex systems [15–22].Several works have attempted to extended social dynamics to either of those structures [20, 23, 24]. We considerhere the case of hypergraphs and generalise the so-called three-body consensus model on hypergraphs (3CM) [25]. Inthe case of a non-linear interaction function that captures reinforcing group dynamics, i.e. the reinforcing effect ofsimilar nodes on their connections through peer pressure, group interactions can lead to shifts in the average stateof the system that would not be captured by pairwise interactions alone. Generally, the study of 3CM revealed thatmulti-body dynamical effects that go beyond rescaled pairwise interactions can only appear if the interaction functionis non-linear, regardless of the underlying multi-body structure. If the interaction function is linear, the system canalways be written as a linear, pairwise interaction system on a rescaled network. Therefore, non-linearity is theessential ingredient to make genuine group dynamics appear [25].In this work, we explore the different effects of peer pressure and homophily, i.e. the tendency for similar nodesto interact more often, for opinion dynamics on hypergraphs. We first generalise 3CM by proposing a Multi-bodyConsensus Model (MCM) for opinion consensus on hypergraphs of arbitrary group size, and explore different versionsof interaction functions emphasising different sociological mechanisms. We then present the combined effects of thesephenomena in numerical simulations on empirical hypergraphs. The rest of this paper is structured as follows. InSection II, we introduce the process of modelling multi-body interactions, and derive some useful analytical quantities.In Section III, we present and discuss MCM. Section IV contains simulations on real-world hypergraphs and SectionV is devoted to discussing further possibilities and avenues of research.
II. MODELLING MULTI-BODY DYNAMICAL SYSTEMSA. Structure of a hypergraph
The hypergraph H is a set V ( H ) = { , , . . . , N } of N nodes, and a set E ( H ) = { E , E , . . . , E M } of M hyperedges. Each hyperedge E α is a set of nodes, i.e. E α ⊆ V ( H ) ∀ α = 1 , , . . . , M . We denote by E c ( H ) the setof all hyperedges of cardinality c (henceforth referred to as c -edges).We describe the structure of H using the adjacency tensors A c ∈ R N c , c = 2 , . . . , N , where A c represents theconnections made by c -edges. A cij... = (cid:40) { i, j . . . } ∈ E c ( H )0 otherwise (1)Thus A c is symmetric in all dimensions, and A ij... = 1 ⇒ i (cid:54) = j (cid:54) = . . . . Each node i ∈ V ( H ) has a dynamical variable x i ∈ R associated with it. x i is termed the ’state’ of i , and represents the notion of an opinion of an individual i in ahypergraph of social interactions. B. General diffusion-like processes on hypergraphs
We denote by ˙ x iE α the effect of E α on ˙ x i . We have that ˙ x iE α = 0 when i (cid:54)∈ E α , and for i ∈ E α , we write:˙ x iE α = (cid:88) j ∈ E α s ijE α ( x i , x j , . . . ) ( x j − x i ) i, j, . . . ∈ E α (2)This models the influence of j ∈ E α , j (cid:54) = i on i via the linear term ( x j − x i ), modulated by the function s ijE α . Themodulating function is symmetric in all k ∈ E α , k (cid:54) = i, j . The complete ’diffusion-like’ process is then obtained bylinearly combining the effect of each hyperedge [25, 26], and one thus obtains the system of equations˙ x i = (cid:88) α ˙ x iE α . (3)The special case where the modulating function for a node i is the same for all E α , and symmetric in all k ∈ E α , k (cid:54) = i allows us to derive some interesting analytical results. Under this assumption, we denote the modulating function as s i and we can write the effect of all c -edges on i as:˙ x ic = (cid:88) jk... A cijk... s i ( x i , x j , . . . ) ( x j − x i + . . . ) × c − (cid:88) jk... A cijk... s i ( x i , x j , . . . ) ( x j − x i ) × c − x i is given by: ˙ x i = N (cid:88) c =2 (cid:88) jk... A cijk... s i ( x i , x j , . . . ) ( x j − x i ) × c − C. Deriving the Laplacian
We now derive the Laplacian for this process. Along the lines of [25], we define weight matrices W c and the degreematrix D c as: W cij = (cid:88) kl... A cijk... s i ( x i , x j , . . . ) 1( c − D cii = (cid:88) jkl... A cijk... s i ( x i , x j , . . . ) 1( c − (cid:88) j W cij ( c −
1) (6)Here, D cij = 0 ∀ i (cid:54) = j . This allows us to write ˙ x ic = − (cid:88) j L cij x j (7)where L cij = ( c − D cij − W ij . Eq.(5) can now be written as:˙ x i = − N (cid:88) c =2 (cid:88) j L cij x j = − (cid:88) j L ij x j (8)where L ij = (cid:80) Nc =2 L cij .We see that when the modulation function s i is a constant, i.e. when the interactions are linear, the dynamicsreduce to those of a static weighted network. However, when the interactions are non-linear, the corresponding networkis time-dependent. Thus, we conclude as in [25] that irreducible multi-body effects are created only by non-linearinteractions. III. MULTI-BODY CONSENSUS MODELA. Definition of the model
Following the discussions of the previous section, we introduce a general form of Multi-body Consensus Model(MCM) as follows: ˙ x iE α = s Ii (cid:18) | (cid:80) j ∈ E α x j | E α | − x i | (cid:19) ) × (cid:88) j ∈ E α s IIi (cid:18) | (cid:80) k ∈ E α ,k (cid:54) = i x k | E α | − − x j | (cid:19) ( x j − x i ) (9)The effect of E α on i ∈ E α is modulated by two functions - s Ii and s IIi . They define the two ’facets’ of MCM, whichcan be studied separately in two simplified models that we call MCM I, with˙ x iE α = s Ii (cid:18) | (cid:80) j ∈ E α x j | E α | − x i | (cid:19) (cid:88) j ∈ E α ( x j − x i ) , (10)and MCM II, with ˙ x iE α = (cid:88) j ∈ E α s IIi (cid:18) | (cid:80) k ∈ E α ,k (cid:54) = i x k | E α | − − x j | (cid:19) ( x j − x i ) . (11)The difference in their sociological motivation and mathematical properties is captured by the arguments of theirmodulating functions. • s Ii is a function of the distance of x i to the mean state of the hyperedge, and determines the rate at which thathyperedge influences its state. • s IIi is a function of the distance of a participating node j from the mean state of the hyperedge excluding i .Thus, while MCM I modulates the competing effect of different hyperedges on the state an incident node, MCM IIdetermines which nodes inside a single hyperedge are the most influential. It is also important to emphasise thatthe effect of s IIi only appears in hyperedges of cardinality larger than 3, and this aspect of the model was thus notpresent in the original 3CM. In the next two sections, we discuss the mathematical and sociological differences ofthese functions further.
B. Mathematical Differences
1. MCM I: Analysis in the mean field
Both s Ii and s IIi are invariant under translations and rotation, implying that the dynamics are independent ofthe global reference frame [27]. Since the argument of s IIi is dependent on both i and j , it does not lend itself tomathematical analysis. In the case of s Ii , in contrast, as it is symmetric in all k ∈ E α , k (cid:54) = i , its Laplacian can bederived as in Section II C. Further, we can analytically derive some properties in the mean field.We assume that all nodes have the same modulating function, and denote it as s I . Under the homogeneous mixinghypothesis, consider a hypergraph H with m k hyperedges of cardinality k for k = 2 , , . . . , N . Our assumptions implythat each node participates in km k N hyperedges of cardinality k and that the mean of every hyperedge is the globalmean ¯ x . In order to examine the time evolution of ¯ x in the mean field, we proceed as follows. For i ∈ E α , we have˙ x iE α = s I ( | ¯ x − x i | ) × | E α | (¯ x − x i ) (12)Using this, we can write ˙ x i = N (cid:88) k =2 k m k N s I ( | ¯ x − x i | )(¯ x − x i ) (13) ̄x̄at̄t = 0 ̄ x ̄ a t ̄ c o n s e n s u s ̄x̄at̄t = 0λ = − 10λ = − 1λ = 1λ = 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ̄x̄at̄t = 0 ̄ x ̄ a t ̄ c o n s e n s u s δ = − 10δ = − 1δ = 1δ = 10 FIG. 1: Numerical simulations to compare the evolution of MCM I (left) and II (right) on a fully connected hypergraph.
Thus, the mean opinion evolves as ˙¯ x = 1 N (cid:32) N (cid:88) k =2 k m k (cid:33) (cid:32) N (cid:88) i =1 s I ( | ¯ x − x i | )(¯ x − x i ) (cid:33) (14)and we observe that in a homogeneously mixed system, the mean does not shift if the distribution of x i about themean is symmetric.We can also investigate the effect of asymmetry in the initial distribution of the states. Consider a situation wherethe initial states are binary (either 1 or 0). Suppose at t = 0, f fraction of the nodes have state 0, and the rest( f = 1 − f ) have state 1. From Eq.14, we can write˙¯ x = 1 N (cid:32) N (cid:88) k =2 k m k (cid:33) (cid:32) N (cid:88) i =1 s I ( | f − x i | )( f − x i ) (cid:33) = 1 N (cid:32) N (cid:88) k =2 k m k (cid:33) f f ( s I ( f ) − s I ( f ))If s I is monotonically increasing, f > f implies that ˙¯ x > f < f that ˙¯ x <
0, i.e. ¯ x shifts towards the majority.Similarly, ¯ x shifts towards the minority for monotonically decreasing s I .
2. Numerical Simulations
To demonstrate the fundamental differences between MCM I and II, we run numerical simulations of each model onidentical topologies, and with the same choice of modulating function. As a natural choice for monotonic modulation,we consider for this illustration exponential functions. s Ii ( x ) = e λx ∀ i ∈ V ( H ) s IIi ( x ) = e δx ∀ i ∈ V ( H ) (15)We define a hypergraph with N nodes as fully connected if all 2 N − N − ≥ N = 10 nodes by initialising node states as binary numbers(0 or 1), with n nodes of state 0. Since the modulating functions are always positive, a global consensus is reachedwithout oscillations as expected. In the simulations, we define the global consensus as the average state when thestandard deviation σ ( { x i } ) ≤ . λ > λ < Time N o d e S t a t e FIG. 2: MCM I with the step modulating function ( φ = 0 .
1) for a fully connected hypergraph with N = 10 shows an opposite behaviour, despite the same choice function for the modulating function. These drastic differencesunderline the distinct nature of the two modulating functions, – and of their associated models MCM I and MCM II–, as well as the huge effect of the argument in the modulation function on the modelling outcome. After this firstanalysis of the mathematical differences between the two variations of MCM, let us now turn to their sociologicalmotivation, in order to chose appropriate modulating functions. C. Sociological Differences
1. MCM I
Homophily is a central concept in sociology describing the tendency of like-minded individuals to interact [28]. Thetopology of social interactions is often influenced heavily by homophily. In sociological terms, the argument to s Ii quantifies the difference between the opinion of individual i and the average opinion of group E α that i belongs to.The influence of a group on a node is thus determined by the proximity of its average state to the state of the node.For instance, consider the step function: s Ii ( x ) = (cid:40) x ≤ φ i otherwise (16)Individual i is influenced by a group only if their opinions differ by a value less than the threshold φ i . This mechanismis reminiscent of threshold models [29] [30] and bounded confidence models like the Deffuant model [31], with theimportant difference that it is group-specific, i.e. each hyperedge corresponds to one group, while network-basedmodels usually consider the whole neighbourhood of a node as its single group. As an illustration, in Fig. 2, wepresent the evolution of MCM I on a fully connected hypergraph ( N = 10) initialised as two tight groups around 0 . .
95 with φ i = 0 . s Ii is a monotonically decreasing function, so that an individual i is less influenced bygroups with opinions very different from its own than by groups with similar opinions. This can be thought of asindividual i resisting change, or some form of ’stubbornness’. Consider the exponential function: s Ii ( x ) = e λ i x (17)For λ i < > λ i < > N = 10) with binary symmetric initialisation. The nodes initialised to 1 (0) have λ i = − ∆ (∆). Numerical results show that consensus shifts towards the opinion of stubborn individuals.An important effect of MCM I is that it makes it possible for an individual to heavily influence other members ofa group while being resistant to their influence. This allows certain individuals to be ’trendsetters’ and to pull entiregroups towards their opinion. Stubborn individuals in a group of people with whom they disagree can be trendsetters. Time N o d e S t a t e ̄x (a) Evolution with ∆ = 1 −10.0 −7.5 −5.0 −2.5 0.0 2.5 5.0 7.5 10.0 Δ C o n s e n s u s Δ S t a t e (b) Consensus vs ∆FIG. 3: Evolution of MCM I on a fully connected hypergraph of 10 nodes initialised with 5 nodes each of opinions 0 (with λ i = ∆) and 1 (with λ i = − ∆).
2. MCM II
The pressure to conform is an important sociological and psychological phenomenon [32], at the root of mostmodels of opinion dynamics. ’Conformity’ is used to describe the tendency of an individual to align its beliefs tothose of its peers, and is usually affected by the reinforcing nature of shared opinions (peer pressure). The sociologicalinterpretation of the argument of s IIi is the difference between the opinion of individual j to the average opinion of thegroup except individual i . Thus, in MCM II, the influence exerted by j inside a hyperedge depends on the proximityof its opinion to those of the rest of the group. For instance, consider the exponential function: s IIi ( x ) = e δ i x (18)When δ i <
0, individual i tends to be more influenced by individuals who agree with the rest of the group. In contrast,for δ i > δ i >
0. In Fig. 1 (b), we see that in a population ofconformers (contrarians), the mean shifts towards the initial majority (minority), as expected.
D. Defining s Ii and s IIi for MCM
The previous sections considered MCM I and MCM II separetedly. Let us now consider a general model integratingboth types of mechanisms. To do so, we build on Social Judgement Theory [33] where the responses of people todifferent opinions are categorised as: • The latitude of acceptance - where the other opinion is sufficiently close to their own belief and is accepted.Their beliefs shift towards the new opinion (assimilation). • The latitude of rejection - where the other opinion is too far from theirs and is therefore unacceptable. Thisleads to their opinion shifting away from the other (contrast). • The latitude of non-commitment - where the other opinion is neither sufficiently close to be accepted nor differentenough to be repulsive.According to Social Judgement Theory, the more the individual is personally involved in an issue, the smaller theirlatitudes of acceptance and non-commitment. Inspired by this categorisation, and drawing from the Jager-Amblardmodel [34], we define for each node i , two thresholds - that of acceptance ( φ Ai ) and that of rejection ( φ Ri ), and wepropose s Ii ( x ) = e λ i x x ≤ φ Ai φ Ai < x < φ Ri − e λ i x x ≥ φ Ri (19) N u m b e r o f H y p e r e dg e s p out FIG. 4: Cardinality distribution for a block model with N = 5 for various values of p out . φ A φ R Fraction of extreme nodes0.10 0.15 0 . ± . . ± . . ± . N = 5, p out = 0 . For MCM II, we keep an exponential modulating function. s IIi ( x ) = e δ i x (20) E. Evolution of MCM on a block hypergraph
In the previous sections, simulations were run on fully connected hypergraphs. Let us now turn our attention tomore nuanced hypergraph models in order to uncover the impact of structure on dynamics. As a first step, we considera block hypergraph model, inspired by the stochastic block network model for networks, and defined as a block modelfor a hypergraph with 2 blocks of N nodes each. Both the blocks are fully connected, which means that all possibleintra-block hyperedges are present. Inter-block hyperedges with n and n nodes from the two blocks respectively arecreated with probability p n n out . Here, p out is a parameter for the model. In Fig. 4, we plot the cardinality distributionof a block model of a hypergraph averaged over 1000 instances with N = 5.The system has several parameters, some associated with its structure, and other to the dynamical model. As afirst descriptive study, we study the effect of φ Ai and φ Ri , we fix δ i = − λ i = − N = 5 and p out = 0 .
4, with the nodes in the two blocks initialised with values from a uniformdistribution over [0 , .
5] and [0 . ,
1] respectively. Further we set φ Ai = φ A and φ Ri = φ R as the same for all nodes. Foreach choice of the parameters φ Ai and φ Ri , we simulate 500 realisations of the evolution of MCM. Unlike the numericalsimulations performed previously, MCM is not guaranteed to reach a global consensus. Here, we stop the simulationwhen the difference in standard deviation over 500 timesteps is less than 0 . φ R , which represents intolerance in the social context, createpolarisation. In contrast, a large value of φ A encourages consensus. In all the simulations, the range of possibleopinions is restricted to [0 , > . < . φ A and φ R , there is polarisation (consensus). For intermediate values, the two blocksreach an internal consensus first, before converging to the global consensus. Due to the small size of the blocks, aswell as the presence of several inter-block hyperedges, we do not see a co-existence of extreme and non-extreme nodes.However, for large group sizes and more sparsely connected blocks, this coexistence is indeed possible.In the next sections, we will explore the dynamics of MCM for these specific choices of interaction function, with a N o d e S t a t e (a) φ A = 0 . , φ R = 0 . N o d e S t a t e (b) φ A = 0 . , φ R = 0 . N o d e S t a t e (c) φ A = 0 . , φ R = 0 . N = 5, p out = 0 . specific focus on the emergence of polarisation and the importance of stubbornness. IV. NUMERICAL SIMULATIONS ON REAL-WORLD HYPERGRAPHSA. Datasets
We study the dynamics of MCM on empirical hypergraphs of social interaction. To create hypergraphs describingreal-world interactions, we use the publicly available SocioPatterns dataset [35]. To represent a diverse variety ofsocial situations, we use the following datasets: • Primary Schools dataset [36][37] - schoolchildren and teachers at a primary school in France • Conference dataset [38] - participants at the 2009 SFHH conference in Nice, France • Workplace dataset [38] - the staff at an office building in France (2015) • High School dataset [39] - students at a high school in Marseilles, France (2012)The SocioPatterns datasets record face-to-face interactions with a temporal resolution of 20 seconds. This allows usto check whether individuals are truly interacting as a group [40]. For every 20 second window, we create a networkof interactions and catalogue all the maximal cliques. If a clique interacts often, we assume that it represents a socialgroup where the interactions are multi-body, and are distinguishable from the pairwise interactions of its members.0
Hypergraph
N M m m m m m m m m Primary School 242 3463 3118 338 7 0 0 0 0 0Conference 392 1710 1441 224 29 8 3 2 2 1Workplace 212 1703 1606 97 0 0 0 0 0 0High School 177 866 795 69 2 0 0 0 0 0TABLE II: Details of hypergraphs created from real-world data - number of nodes N , number of hyperedges M , number of k -edges m k In practice, we set an arbitrary threshold and include a maximal clique as a hyperedge if it is observed ≥ A ij is given by thenumber of hyperedges containing both i and j .The resulting hypergraphs show very different types of structures, including different degree distributions. Thisdiversity is an asset in order to test the qualitative features of the dynamics of MCM. B. Role of involvement
We initialise the nodes with states drawn from a uniform distribution in [0 , λ i = − δ i = −
5, we considerthree scenarios for (19): • High involvement - φ Ai = 0 . φ Ri = 0 . • Medium involvement - φ Ai = 0 . φ Ri = 0 . • Low involvement - φ Ai = 0 . φ Ri = 0 . C. Role of stubbornness
As a second step, we investigate the impact of stubbornness. To facilitate the understanding of the results, werestrict the scope to situations where all nodes are subject solely to acceptance, and not rejection, by imposing that φ A , φ R >
1. We also fix the same value of δ i for all the nodes, with δ i = − λ i defines the stubbornness for each node i , such that λ i < λ i . Numericalsimulations reveal that the effect of changing λ i is sensitive to the topology, initialisation as well as the values of theother parameters. To fix the topology, we consider the Primary School hypergraph. Fig. 6 shows that the PrimarySchool hypergraph is reminiscent of a block model. Students interact much more with their classmates than withothers. In order to ensure that the effect of stubborness is present in every block, we fix the stubborn nodes - choosingall 10 teachers and picking 3 students randomly from each class (50 in total). We initialise all the nodes with binarystates - 1 for the stubborn nodes ( λ i = λ ∗ ), and 0 for the others ( λ i = − x ( t = 0) = 0 . λ ∗ shows that the effect of stubborn individuals increases with their stubbornness.1
10 20 30 40 50 60
Degree F r a c t i o n o f n o d e s (a) Primary School Degree F r a c t i o n o f n o d e s (b) Conference Degree F r a c t i o n o f n o d e s (c) Workplace Degree F r a c t i o n o f n o d e s (d) High SchoolFIG. 6: Degree distribution and adjacency matrices of the hypergraphs. In the adjacency matrices, A ij = 0 , , > Time N o d e S t a t e Time N o d e S t a t e Time N o d e S t a t e (a) Primary School Time N o d e S t a t e Time N o d e S t a t e Time N o d e S t a t e (b) Conference Time N o d e S t a t e Time N o d e S t a t e Time N o d e S t a t e (c) Workplace Time N o d e S t a t e Time N o d e S t a t e Time N o d e S t a t e (d) High School Time F r a c t i o n o f e x t r e m e n o d e s Primary SchoolConferenceWorkplaceHigh School 0.0 0.5 1.0 1.5 2.0
Time F r a c t i o n o f e x t r e m e n o d e s Primary SchoolConferenceWorkplaceHigh School 0.0 0.1 0.2 0.3 0.4
Time F r a c t i o n o f e x t r e m e n o d e s Primary SchoolConferenceWorkplaceHigh School (e) Fraction of extreme nodesFIG. 7: Polarisation in High (left), medium (center) and low (right) involvement scenarios. Time M e a n S t a t e λ * = − 1λ * = − 5λ * = − 10λ * = − 15 −14 −12 −10 −8 −6 −4 −2 λ * C o n s e n s u s S t a t e ̄x at t̄0 FIG. 8: Stubbornness in the Primary Schools hypergraph (a) evolution of the mean state with time and (b) consensus state vs λ ∗ V. DISCUSSION
Networks are a powerful language to model interacting systems. Yet one of its core assumptions, that connectivityemerges from the combination of pairwise interactions, is not necessarily verified in empirical systems. In this paper, wehave explored the problem of consensus in situations such that basic interaction units are composed of more than twonodes. Representing the underlying structure as a hypergraph, we propose a general non-linear model for consensusdynamics, called MCM, where different model ingredients are associated with different sociological mechanisms. Wehave studied certain aspects of the models mathematically and have explored its behaviour on artificial as well ason real-life hypergraphs, revealing a rich phenomenology and the strong interplay between the structure and thedynamics. Yet, this paper only scratches the surface of this dynamical system, and many research venues remainopen. First, we have explored only a limited part of the parameter space, and additional theoretical and numericalresults would be required to improve our understanding of the different regimes supported by the model and the natureof the transitions between them. Second, though we define multi-body interactions in a general way, we analyse onlya special class of them. Finding different functional forms of group interactions relevant to other complex processesis a possible direction of research. Another interesting line of inquiry lies in extending and generalising multi-bodyinteractions to adaptive hypergraphs. Most dynamical processes (including MCM) are sensitive to the topology, andthus making models to capture the dynamics of the underlying hypergraph promises to be an important area. Otherpossible extensions could be studying opinion dynamics on weighted or directed hypergraphs, or exploring the effectsof stochasticity on the consensus.
VI. ACKNOWLEDGEMENTS
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