aa r X i v : . [ phy s i c s . s o c - ph ] A ug Homophilic networks evolving by mimesis
Jos´e Manuel Rodr´ıguez CaballeroInstitute of Computer ScienceUniversity of TartuTartu, EstoniaAugust 14, 2020
Abstract
We provide a mathematical model for networks based on similar-ities (homophily) and evolving by mutual imitation (mimesis). Weshow that such social networks will converge to a state of segregation,where the in-group interactions will be maximal and there will be noout-group flow of information. We establish some connections betweenour model and the Wolfram model for fundamental physics.
There is in [people’s] naturalpropensity, from childhoodonward, to engage in mimeticactivities.And this distinguishes [people]from other creatures, that [theyare] thoroughly mimetic andthrough mimesis [take they] firststeps in understandingAristotle (The Poetics)
The mimetic behavior in human activity is a well-established scientific fact[Gar11]. The origins of mimetic theory can be found in Plato’s
Republic and1ristotle’s
Poetics [Law18]. R. Girald [Gir14, Gir98] developed the theory of d´esir mim´etique , in which mimesis of the wishes is the main driving force ofhuman social behavior. His main postulate is L’homme d´esire toujours selon le d´esir de l’Autre.The theory of d´esir mim´etique , which is mainly philosophical and literary[DD82, Law18], may be also related to well-studied neurological structuresknown as mirror neurons [FR14, FBF09, Hey10, Iac09, WWSP01]. A mirrorneuron in a human is a neuron that is activated by observing the behaviorof other humans. Nevertheless, the connection between mirror neurons andimitation is still a subject of debate among scientists [Hic09].Mimetic human behavior is present in the theory of social laser [Khr16],where people willing to imitate each other are described in a similar way tothe bosons from particle physics. In this approach, mimesis is modeled by asocial version of Bose-Einstein statistics.The tendency of humans to interact more frequently with other humanssharing similarities is known as homophily [MSLC01, BCJ +
12, KA17]. Thistendency is also present in other animals and it is likely to play some evolu-tionary role [FNCF12]. The homophilic network [Mei18, page 87] is a rep-resentation of social connections among people are determined by proximityof interest, status, wealth, ethnicity, position in a hierarchy, etc. Severalmathematical techniques, mostly from graph theory, have been developed inorder to study this structure [New10, section 7.13].The aim of the present paper is to introduce a mathematical model forhomophilic networks (Definition 1) evolving by mimesis (equation (1)). Ourmodel is a generalization of the model for diffusion in graphs [New10, section6.13.1]. Diffusion equations (equation (2)) have already been used in orderto model social phenomena[YOMV18]. Nevertheless, we have not found ourequation (1) in the existing literature, and the continuous update of the time-dependent network t G ε ( ψ ( t )) seems to be a new feature in the subjectof diffusion models.Making an analogy with fundamental physics via the Wolfram model[Wol20, Gor20b, Gor20a], we will show (Theorem 1) that the evolution of anetwork, in our model, will converge to a state that is the social analogousof what cosmologist call the Black Hole Era, i.e., society will be fragmented A literary translation is: “Man always desires according to the desire of the Other”.Of course, “Man” here stands for human, including any gender.
Our standpoint is the following discrete structure, modeling homophily.
Definition 1.
Fix a positive integer n (size of the population) and an ex-tended nonnegative real number ε ∈ [0 , + ∞ ] (tolerance threshold). Con-sider a function that assigns an undirected simple graph G ε ( ψ ) to any ψ =( ψ , ..., ψ n ) ∈ R n . The vertices of G ε ( ψ ) are the numbers 1 , , ..., n . In G ε ( ψ ),there is an edge between i and j if and only if i = j and | ψ i − ψ j | ≤ ε . Wewill call G ε ( ψ ) the homophilic network of the population ψ at tolerance level ε . In the next definition we provide a mathematical model for the evolutionby mimesis of a homophilic network. We will use the notation L [ G ] for theLaplacian matrix [New10, section 6.13.1] of the graph G . Definition 2.
Fix a positive integer n , a positive real number C (diffusionconstant), a real number t (initial time) and a vector ψ ∈ R n (initialmicrostate). The homophilic-mimetic model for the evolution of a networkis given by the matrix integral equation ψ ( t ) + C Z tt L [ G ε ( ψ ( τ ))] ψ ( τ ) dτ = ψ , (1)defined for t ∈ ( t , + ∞ ), where ψ : [ t , + ∞ ] −→ R n is a continuous function .We call ψ ( t ) the microstate of the network at time t . The graph G ε ( ψ ( t ))will be called the macrostate of the network at time t . We will call (1) the homophilic-mimetic equation .The deduction of the equation (1) as a description of a sociological phe-nomenon is given by the assumption that the mimetic human behavior in A number that is either a nonnegative real number or the positive infinite. Notice that ψ L [ G ε ( ψ )] is piecewise constant. So, the integral is well-defined. The existence of ψ (+ ∞ ) is a consequence of Theorem 1 and the convergence of adiffusion process described by (2). Newton’s law ofcooling . Furthermore, we let the network of human interactions be updatedevery instant in order to guarantee the exchange of information among peoplesharing similarities and destroying connections among people who are differ-ent enough. In order to avoid discontinuities, we express the resulting setof differential equations (defined of different intervals where the network isconstant), as an integral. We do not pretend to give an accurate descriptionof social reality with this equation, but just a toy model of social evolutionunder the assumptions of homophily and mimesis.The macrostate of society G ε ( ψ ( t )) can be easily measured using meta-data from internet, e.g., online social networks, or even polls. On the otherhand, the measurement of the microstate of society ψ ( t ), and its unit ofmeasurement, is an extremely complicated problem and it will not be dis-cussed in the present paper. It is natural to guess that the microstate maybe related to wealth, ethnicity, sexual attractiveness, social status, academicand military hierarchy, etc. Nevertheless, the choice of a random value pro-ducing an observed macrostate could to be the best strategy for simulationsof real-life situations. Following S. Wolfram [Wol20] and J. Gorard [Gor20b, Gor20a], we will in-terpret the evolution of the time-dependent network t G ε ( ψ ( t )) as theevolution of the spatial graph , i.e., the evolution of space in the Wolframmodel. In this framework it is natural to import concepts from fundamentalphysics, e.g., event horizon and black hole, to graph theory. Definition 3.
Consider a time-dependent network t
7→ N ( t ), having thesame set of vertices V for all values of t . Let W ⊆ V be a subset of vertices ofthe network . We say that W will produce an event horizon of t
7→ N ( t ) if forall t large enough, given two vertices, w ∈ W and z ∈ V \ W (complement of W ), there is no path between w and z in N ( t ). Furthermore, if the induced The Wolfram model is really about hypergraphs , but we will focus on the particularcase of graphs. In our model, N ( t ) = G ε ( ψ ( t )). of W in N ( t ) is a complete graph for all t large enough, we saythat W will end as a black hole . If there is a partition of V such that eachpart will end as a black hole, then we say that the network t
7→ N ( t ) willend as a Black Hole Era .The motivation behind this graph-theoretical definition of event horizonis that in physics it is a boundary in spacetime preventing any exchange ofinformation between both sides of the boundary. In our model, the flow ofinformation is interpreted as the paths in the graph.The analogy between black holes and complete graphs is given by the factthat complete graphs are the densest graphs whereas black holes are thedensest forms of organized matter [Sus08].The following theorem states that, in our model, networks will alwaysend as a Black Hole Era. Theorem 1.
Fix a positive integer n , an extended nonnegative real number ε , a positive real number C , a real number t and a vector ψ ∈ R n . Thetime-dependent network t G ε ( ψ ( t )) determined by (1) will end as a BlackHole Era, i.e., there is a graph Ω , which is a union of complete graphs andsatisfies G ε ( ψ ( t )) = Ω for all t large enough. As a preliminary, we need the following lemmas.
Lemma 1.
Fix a positive integer n and a graph G having n vertices. For anypair of real numbers t < t , any positive real number C , any vector ψ ∈ R n and any continuous function, differentiable on ( t , t ) , ψ : [ t , t ] −→ R n satisfying the matrix differential equation d ψ dt + C L [ G ] ψ = 0 , (2) on the interval ( t , t ) and the initial condition ψ ( t ) = ψ , the inequality | ψ ( t ) | ≤ | ψ ( t ) | (3) holds for t ≤ t ≤ t . Let’s recall that the induced subgraph of a set of vertices W is the graph obtained byrestricting the graph to the vertices of W and the edges to the edges having vertices of W at both extremes. This implies that W will evolve to an event horizon. This name came from cosmology [AL16]. Let’s recall that the density of a graph is equal to the number of edges divided by themaximum possible number of edges. roof. Let v , v , ..., v n be an orthonormal system of eigenvectors of L [ G ],associated to the eigenvalues 0 = λ ≤ λ ≤ ... ≤ λ n , respectively. Theexplicit solution of (2) satisfying the initial condition and continuous on[ t , t ], is ψ ( t ) = n X i =1 h ψ , v i i e − Cλ i ( t − t ) v i , (4)for t ≤ t ≤ t .Applying the Pythagorean theorem, | ψ ( t ) | = n X i =1 |h ψ , v i i| e − Cλ i ( t − t ) . (5)Also, using the fact that t
7→ |h ψ , v i i| e − Cλ i ( t − t ) is monotonically de-creasing, n X i =1 |h ψ , v i i| e − Cλ i ( t − t ) ≤ n X i =1 |h ψ , v i i| . Notice that P ni =1 |h ψ , v i i| = | ψ | . Finally, by transitivity we get | ψ ( t ) | ≤ | ψ | . Therefore, | ψ ( t ) | ≤ | ψ | . Lemma 2.
Fix a positive integer n and a graph G having n vertices. Forany pair of real numbers t < t , any positive real number C , any vector ψ ∈ R n , any connected component H of G and any continuous function,differentiable on ( t , t ) , ψ : [ t , t ] −→ R n satisfying the matrix differentialequation (2) on the interval ( t , t ) and the initial condition ψ ( t ) = ψ , theinequality max i ∈ H ψ i ( t ) − min j ∈ H ψ j ( t ) ≤ n − k G ) | ψ | e − Cλ G ( t − t ) , (6) holds for all t ≤ t ≤ t , where ψ = ( ψ , ..., ψ n ) , k G is the number of con-nected components of G and λ G is the minimum nonzero eigenvalue of L [ G ] .We used the notation i ∈ H to express that i is a vertex in H .Proof. Let v , v , ..., v n and λ , λ , ..., λ n be as in the proof of Lemma 1.Using the fact that k G is the dimension of the kernel of L [ G ], defining ψ ∞ = P k G i =1 h ψ , v i i v i , we have ψ ( t ) − ψ ∞ = n X i = k G +1 h ψ , v i i e − Cλ i ( t − t ) v i , (7)6or t ≤ t ≤ t .Applying the triangle inequality, the Cauchy-Schwarz inequality and thefact that v , v , ..., v n are normalized, we obtain the inequality | ψ ( t ) − ψ ∞ | ≤ ( n − k G ) | ψ | e − Cλ G ( t − t ) (8)for all t ≤ t ≤ t . Let H , H , ..., H k G be the connected components of G . Up to rearrangement of the indices, for 1 ≤ r ≤ k we have v r =( H r ) − / ( v r, , v r, , ..., v r,n ), with v r,i = 1 if i is a vertex of H r and v r,i = 0otherwise. Hence, for any connected component H r , the inequality | ψ i ( t ) − h ψ , v r i| ≤ ( n − k G ) | ψ | e − Cλ G ( t − t ) (9)holds for all i which are vertices of H r . Finally, applying triangle inequalitymax i ∈ H r ψ i ( t ) − min j ∈ H r ψ j ( t )= (cid:12)(cid:12)(cid:12)(cid:12) max i ∈ H r ψ i ( t ) − min j ∈ H r ψ j ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) max i ∈ H r ψ i ( t ) − h ψ , v r i (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) h ψ , v r i − min j ∈ H r ψ j ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ n − k G ) | ψ | e − Cλ G ( t − t ) . Now, we proceed to the proof of our main result.
Proof. (of Theorem 1) Let t < t < t < ... be all the instants such that G ε ( ψ ( t i +1 )) is not equal to G ε ( ψ ( t i )). In virtue of Lemma 2, there arepositive real numbers M G and λ G such that, for any connected component H k of G ε ( ψ ( t k )) the inequality max i ∈ H k ψ i ( t ) − min j ∈ H k ψ j ( t ) ≤ M G | ψ k | e − Cλ G ( t − t ) (10)holds for t k ≤ t < t k +1 , where G = G ε ( ψ ( t k )). Also, the inequality M G | ψ k | e − Cλ G t ≤ M | ψ k | e − Cλ ( t − t ) (11) H r is the number of vertices in the connected component H r . The notation i ∈ H k was explained in the statement of Lemma 2. M is the maximum of M G for all G having n vertices, and λ isthe minimum of λ G for all G having n vertices. In virtue of Lemma 1, M | ψ k | e − Cλ ( t − t ) ≤ M | ψ | e − Cλ ( t − t ) . (12)Finally, we getmax i ∈ H k ψ i ( t ) − min j ∈ H k ψ j ( t ) ≤ M | ψ | e − Cλ ( t − t ) , (13)for all t k ≤ t < t k +1 .Using the fact that the right hand side of the equation above does notdepend on k , we conclude that, for all t ≥ t , and any connected component H t of G ε ( ψ ( t )), the inequalitymax i ∈ H t ψ i ( t ) − min j ∈ H t ψ j ( t ) ≤ M | ψ | e − Cλ ( t − t ) . (14)holds. So, for all t large enough and for any connected component H t of G ε ( ψ ( t )) we have max i ∈ H t ψ i ( t ) − min j ∈ H t ψ j ( t ) ≤ ε. (15)According to Definition 1 and the inequality above, any connected com-ponent of G ε ( ψ ( t )) will be a complete graph and this structure will notchange in the future. Therefore, there is a graph Ω, which is the union ofcomplete graphs, such that, for all t large enough, G ε ( ψ ( t )) = Ω. Needless to say that our model is not meant to make numerical predictionsabout society. What we developed was an idealization. The applicationsof our model could be in order to explore some sociological theories usingthe intuition from our model, i.e., to compare how claims about the humanbehavior in society agree or disagree with the properties of our model. Atthe moment of writing this paper, it is not clear to what extent our modelis related to actual statistical data of real-life social networks. Also, it maybe interesting to look for applications of our model in physics, chemistry,biology and computer science. 8 cknowledgments and disclaimers
The author is an external affiliate of the Wolfram Physics Project but is notengaged by any formal agreement in any activity constituting a competitionwith his Employer (University of Tartu). The production of the present pa-per, done during the free time of the author, was only the result of intellectualcuriosity and it was not funded by any company or institution.The author would like to thank Matthew Szudzik for the suggestionsin order to improve the mathematica notebook “A Wolfram-like model oflanguage secessionism”.
References [AL16] Fred C Adams and Greg Laughlin.
The five ages of the universe:inside the physics of eternity . Simon and Schuster, 2016.[BCJ +
12] Yann Bramoull´e, Sergio Currarini, Matthew O Jackson, PaoloPin, and Brian W Rogers. Homophily and long-run integrationin social networks.
Journal of Economic Theory , 147(5):1754–1786, 2012.[DD82] Michel Deguy and Jean-Pierre Dupuy.
Ren´e Girard et leprobl`eme du mal . Grasset, 1982.[FBF09] PF Ferrari, L Bonini, and L Fogassi. From monkey mirror neu-rons to primate behaviours: possible directand indirect path-ways.
Philosophical Transactions of the Royal Society B: Biolog-ical Sciences , 364(1528):2311–2323, 2009.[FNCF12] Feng Fu, Martin A Nowak, Nicholas A Christakis, and James HFowler. The evolution of homophily.
Scientific reports , 2:845,2012.[FR14] Pier Francesco Ferrari and Giacomo Rizzolatti. Mirror neuronresearch: the past and the future, 2014.[Gar11] Scott R Garrels.
Mimesis and Science: Empirical Research onImitation and the Mimetic Theory of Culture and Religion . MSUPress, 2011. 9Gir98] Ren´e Girard. Mimesis and violence.
Herm`es, La Revue ,1(22):47–52, 1998.[Gir14] Ren´e Girard.
Mensonge romantique et v´erit´e romanesque . Gras-set, 2014.[Gor20a] Jonathan Gorard. Some Quantum Mechanical Properties of theWolfram Model, 2020.[Gor20b] Jonathan Gorard. Some Relativistic and Gravitational Proper-ties of the Wolfram Model. arXiv preprint arXiv:2004.14810 ,2020.[Hey10] Cecilia Heyes. Where do mirror neurons come from?
Neuro-science & Biobehavioral Reviews , 34(4):575–583, 2010.[Hic09] Gregory Hickok. Eight problems for the mirror neuron theoryof action understanding in monkeys and humans.
Journal ofcognitive neuroscience , 21(7):1229–1243, 2009.[Iac09] Marco Iacoboni. Imitation, empathy, and mirror neurons.
An-nual review of psychology , 60:653–670, 2009.[KA17] Kibae Kim and J¨orn Altmann. Effect of homophily on networkformation.
Communications in Nonlinear Science and NumericalSimulation , 44:482–494, 2017.[Khr16] Andrei Khrennikov. Social laser: action amplification by stim-ulated emission of social energy.
Philosophical Transactions ofthe Royal Society A: Mathematical, Physical and EngineeringSciences , 374(2058):20150094, 2016.[Law18] Nidesh Lawtoo. Violence and the mimetic unconscious (partone): The cathartic hypothesis: Aristotle, freud, girard.
Conta-gion , 25:159–192, 2018.[Mei18] Reshef Meir. Strategic voting.
Synthesis Lectures on ArtificialIntelligence and Machine Learning , 13(1):1–167, 2018.[MSLC01] Miller McPherson, Lynn Smith-Lovin, and James M Cook. Birdsof a feather: Homophily in social networks.
Annual review ofsociology , 27(1):415–444, 2001.10New10] Mark E. J. Newman.
Networks: An Introduction . Oxford Uni-vertity Press, 2010.[Sus08] Leonard Susskind.
The black hole war: My battle with StephenHawking to make the world safe for quantum mechanics . Ha-chette UK, 2008.[Wol20] Stephen Wolfram. A Class of Models with the Potential to Rep-resent Fundamental Physics. arXiv , pages arXiv–2004, 2020.[WWSP01] Justin HG Williams, Andrew Whiten, Thomas Suddendorf, andDavid I Perrett. Imitation, mirror neurons and autism.
Neuro-science & Biobehavioral Reviews , 25(4):287–295, 2001.[YOMV18] Petukhov Alexander Yurevich, Malkhanov Alexey Olegovich,Sandalov Vladimir Mikhailovich, and Petukhov Yuri Vasilievich.Modeling conflict in a social system using diffusion equations.