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Balanced and Fragmented Phases in Societies with Homophily and Social Balance

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Balanced and Fragmented Phases in Societies with Homophily and Social Balance Balanced and fragmented phases in societies with homophily and social balance Tuan Minh Pham,1, 2 Andrew C. Alexander,3 Jan Korbel,1, 2 Rudolf Hanel,1, 2 and Stefan Thurner1, 2, 4 1Section for the Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090, Vienna, Austria 2Complexity Science Hub, Vienna Josefst¨adterstrasse 39, A-1090 Vienna, Austria 3Department of Mathematics, Princeton University, NJ 08544, Princeton, USA 4Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA (Dated: December 22, 2020) Recent attempts to understand the origin of social fragmentation are based on spin models which include terms accounting for two social phenomena: homophily|the tendency for people with similar opinions to establish positive relations|and social balance|the tendency for people to establish balanced triadic relations. Spins represent attribute vectors that encode multiple (binary) opinions of individuals and social interactions between individuals can be positive or negative. Recent work suggests that large systems of N 1 individuals never reach a balanced state (where unbalanced triads with one or three hostile links remain), provided the number of attributes for each agent is less than O(N 2) [Phys. Rev. Lett. 125, 078302]. Here we show that this statement is overly restrictive. Within a Hamiltonian framework that minimizes individuals' social stress, we demonstrate that stationary, balanced, but fragmented states can be reached for any number of attributes, if, in addition to homophily, individuals take into account a significant fraction, q, of their triadic relations. Above a critical value qc, balanced states result. This result also holds for sparse realistic social networks. Finally, in the limit of small q, our result agrees with that of [Phys. Rev. Lett. 125, 078302]. The concept of so-called filter bubbles captures the frag- itive or negative). Stress arises from a homophily-related P mentation of society into isolated groups of people who term, − (i;j) Jijsisj, or a term reflecting social balance, trust each other, but clearly distinguish themselves from P − (i;j;k) JijJjkJki. The former is similar in structure to \other". Opinions tend to align within groups and di- the Edwards-Anderson spin-glass model [41], the latter to verge between them. Interest in this process of social the Baxter-Wu three-spin interaction model [42]. These disintegration, started by Durkheim [1], has experienced two terms have been studied separately, homophily in a recent boost, fuelled by the availability of modern com- [43{46] and Heider balance in [47{49]; only recently were munication technologies. The extent to which societies both terms integrated into a single Hamiltonian [35]. fragment depends largely on the interplay of two ba- The issue of social cohesion and organization was re- sic mechanisms that drive social interactions: homophily cently studied in an attribute-based local triad dynam- and structural balance. Homophily is based on the \prin- ics model (ABLTD) [50]. There, each agent has binary ciple" that individuals with similar opinions tend to be- opinions on G attributes. If two agents agree on more come friends (\similarity breeds connection" [2]); indeed, attributes than they disagree on, they become friends like-minded individuals often form homogeneous struc- (positive link). Agents tend to change their attributes tures in society [3]. Structural balance, first described by to reduce stress in unbalanced triads. The paper showed Heider [4], is the tendency of balanced triads to be over- that given a system of N agents, as N ! 1, the so- represented in societies. A triad of individuals is balanced called \paradise state", where all agents are friends of if all three individuals are mutual friends (friend of my each other, is never reached unless the number of at- friend is my friend) or if two friends have a mutual enemy tributes G scales as O(N γ ), for γ ≥ 2. Instead, the (enemy of my enemy is my friend). Structural balance society remains in a stationary unbalanced state with an has been investigated by social scientists for a long time equal number of balanced and unbalanced triads. This [5{7] and, more recently, by physicists and network scien- means that for realistic situations, where N is large and tists [8{18]. Recent contributions study the dynamics on arXiv:2012.11221v1 [physics.soc-ph] 21 Dec 2020 the number of attributes G remains relatively small, it balanced networks [19{21], the co-evolution of opinions is impossible to reach social balance, let alone the par- and signed networks [22{35], and generalizations of the adise state. This statement is to some extent, contrary concept of structural balance [36]. For an overview, see to empirical findings that societies are balanced to a high [37, 38]. A general survey of statistical physics methods degree; see e.g. recent work on large scale studies [16, 17]. applied to opinion dynamics is found in [39, 40]. In this letter, motivated by the ABLTD, we propose Among the various approaches towards social fragmen- an individual-stress-based model that takes into account tation, the minimization of social stress captured by a the homophily effect between adjacent individuals and so-called social stress Hamiltonian has become particu- structural balance within a local neighborhood. The lat- larly relevant. Here the opinion of individual i is de- ter consists of the subset of the most relevant triads to noted by si and the relation between i and j by Jij (pos- an individual, i.e. those that involve its closest friends 2 and greatest enemies. The ratio of relevant triads to the 1 Ak = −1 total number of triangles the individuals belong to de- [ ] −1 termines whether society fragments or remains cohesive. k k With the help of simulations on a regular network, we flip A1 show that there exists a critical size of the local neighbor- 1 i Aj = −1 hood above which society fragments, yet stays balanced. [ ] 1 j i j i We discuss the relation of the presented model to both, 1 −1 the ABLDT model [50] and the social stress Hamiltonian Ai = 1 Ai = 1 [ ] [ ] approach [35]. For appropriate parameter choices, both 1 1 models can be shown to correspond to special cases of l m l m i i the presented model. H( ) = − 5/3 H( ) = − 3 Local social stress model. Consider a society of N in- dividuals. Each individual i has binary opinions on G FIG. 1. Co-evolutionary interplay of opinions and links. Red ` (blue) links denote positive (negative) relationships. Among issues, characterized by an attribute vector, Ai = fai g, ` the three triads in the graph, the chosen one, (i; j; k) is cir- where ai 2 {−1; +1g; ` 2 1;:::;G. Further, i has rela- cled. We consider the case Q = 1. As agent i flips one of its tions to k other individuals in a social network. Net- 1 i attributes, Ai , this triad becomes balanced and i decreases work topology does not change over time. Following its individual stress from −5=3 to −3. The opinion vectors [50], the relation between two agents i and j is deter- of l and m are Al = Ai and Am = −Ai, respectively (not 1 mined by the sign of their distance in attribute space: shown in the figure). For Q = 2, the same flip of Ai leads to (i) Jij = sign(Ai · Aj), where the dot denotes the scalar an increase of ∆HQ=2 = 2=3, if one of the two chosen triads (i) product. Jij = 1 indicates friendship, Jij = −1 enmity. contains k, or ∆HQ=2 = 14=3, if none of them contains k. For Each agent i has a social stress level, H(i), defined as (i) Q = 3, ∆HQ=3 = 8=3. The higher Q, the more important becomes the role of social balance. 1 X X H(i)(A) = − J A · A − J J J : (1) G ij i j ij jk ki j (j;k) Qi triads, the weights of the two links adjacent to i are ~ ~ ~ The first sum extends over all ki neighbours of i, while recomputed as Jij = sign(Ai · Aj). J is the new ∆ 1 the second is restricted to Qi (out of Ni possible ) triads matrix. that node i belongs to. The notation (j; k)Qi means to (iii) Compute the new stress H~ using J~. The sum over all pairs of j and k which, together with i, form change in stress is ∆H(i) ≡ H~ − H. the Qi triads. These are chosen at each step of the dy- ~ ~ namics (see below). Q represents the number of triads i (iv) Update the system Ai ! Ai and Jij ! Jij i n (i) o would like to have socially balanced { i 's relevant neigh- with probability, min e−∆H ; 1 , otherwise leave borhood. The factor 1=G ensures that contributions from it unchanged. This stochastic rule means that any link towards H(i) do not diverge in the limit G ! 1. agents are not always rational and might choose Assuming agents try to minimize their individual social to increase their stress. stress over time, we implement the following dynamics2: 3. Continue with the next timestep. 1. Initialize. Each node is assigned an opinion vector, Figure 1 illustrates an update where by changing one Ai, whose components are randomly chosen to be 1 or −1 with equal probability. Every node has the opinion, agent i becomes an enemy of j, but the chosen triad, (ijk), becomes balanced. If Q = 1, this decreases same degree, ki = K, and is connected to its neigh- (i) (i) bours in a regular way, forming a ring topology. i's social stress from H = −5=3 to H = −3.
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