[Physics.Soc-Ph] 6 Mar 2013 Tt.W Loso O Eti Es Ewr Limits Network Dense Certain How Show System Also the We of Evolution Dynamical Expected State

arXiv:1303.1414v1 [physics.soc-ph] 6 Mar 2013 tt.W loso o eti es ewr limits network dense certain how show system also the We of evolution dynamical expected state. of the approxima- describe analysis mean-field that and and tions, exact derivation closely both equations, 9]. is the master [7, it on magnetism and and focus [7], [8] percolation We network of Our boolean models of neighborhood. in to related its type take in a will nodes is node of model the states state the to the response determines function,” “response which a map has a node Each models contagion. by social motivated of dynamics, map networked of model be websites. can network media social which through spread with and ease shared the by viral concept driven of This area marketing, contemporary 6]. important [5, vastly the people underlies among behaviors to the or similar category, fashions, ways many last in This is social contagion, [4]. biological of diseases study stability infectious cellular and [1], and dynamics brain [3], ecosystem the [2], in regulation behavior genetic phenomena. neuron of number include a These for as description recognized natural now most are the networks on Dynamical place physical, settings. taking in processes social ubiquity and and technological, generality biological, their of recognition ∗ ‡ † iyo ahntn etl,W 80 S [email protected] [email protected] ; USA 98103 WA Unive [email protected] Seattle, Mathematics, Washington, Applied of of sity Department address: Current nti ok epeetrslsfravr general very a for results present we work, this In the to due research of area exploding an are Networks opeema-edter;terlso epnefunctions response of roles the theory; mean-field Complete otgo,wihrfr otesraigo ideas, of spreading the to refers which contagion, rps nigareetbtentema-edter n s and theory mean-field imitat the fashio “limited between of this agreement aspects of finding simulations important graphs, extensive capture to to theory attempt com our we the way model stan this into we In off-threshold response example, an the incorporating specific by on a non-conformity depending As chaotic synchronicity. to update state dynamic steady function t response from predicts average range the this dynami show of these version and that networks smoothed show a random for we theory however, field network, mean dense determini large, and random stoc a in of of outcomes amounts different varying find Allowing we responses, neighborhood. its of state aeo ekrHarris, Decker Kameron esuybnr tt yaiso ewr hr ahnd ac node each where network a on dynamics state binary study We .INTRODUCTION I. 1 eateto ahmtc n ttsis emn Advance Vermont Statistics, and Mathematics of Department ycrn;adapiain osca contagion social to applications and synchrony; emn ope ytm etr n opttoa tr La Story Computational and Center, Systems Complex Vermont yaiso nunepoesso networks: on processes influence of Dynamics nvriyo emn,Brigo,V 50 USA 05405 VT Burlington, Vermont, of University 1, ∗ hitpe .Danforth, M. Christopher Dtd ac ,2013) 7, March (Dated: r- where h eut fsmltosadter.Fnly nSec- in further Finally, for directions and theory. research. conclusions present and compare we we VI, simulations social tion case, of many specific results in this the For that expect function we random processes. response imitation Poisson contagion of limited on kind the model specific reflects the a In with consider networks networks. we mean-field random V, generalized a underlying on develop Section the we model IV, the when of Section we model theory In III, the fixed. Section is of network In analysis an variants. provide stochastic and terministic and [12] Schelling of is work model seminal [13]. Our Granovetter the time. to sta- through related are wildly trends closely vary some some and others familiar: not, while do are ble some we and off which take it with trends quantitative, but features be experiments, the to designed meant captures carefully not in destruction is perhaps and model except creation Our the these fashions. behind that of forces Simmel, believed main [1957], indeed, the “Fashion” are trends; essay two classic all these his to that in essential argue imita- are can of One ingredients tendencies non-conformity. competing and to tion representing according Nodes, act [10]. people, model contagion imitation average ited the to dynamics dynamics. the map of function convergence response the to lead nScinI,w en h eea oe n t de- its and model general the define we II, Section In lim- particular a to theory general our apply then We Let adtrsodmdl fsca contagion. social of models threshold dard o otgo”mdlo oso random Poisson on model contagion” ion G V atct nbt h ewr n node and network the both in hasticity .Tu,tebhvo ftesse can system the of behavior the Thus, s. 1, tcvrin ftemdl ntelimit the In model. the of versions stic stend e and set node the is ( = ucin,ntokcnetvt,and connectivity, network functions, a h yaiso h ewr are network the on dynamics the hat sadscea rns ecompare We trends. societal and ns † ohsi simulations. tochastic scicd.W osrc general a construct We coincide. cs eigtnece fiiainand imitation of tendencies peting n ee hrdnDodds Sheridan Peter and ,E V, I EEA MODEL GENERAL II. si epnet h average the to response in ts eantokwith network a be ) optn Core, Computing d onciiy and connectivity, , b, E steeg e.W let We set. edge the is 1, N ‡ = | V | nodes, 2

Rewiring network Fixed network A = A(G ) denote the adjacency matrix; entry Aij is the number of edges from node j to node i. Assign each Probabilistic response P-R P-F node i ∈ V a response function fi : [0, 1] → {0, 1}, and Deterministic response D-R D-F let x(0) ∈{0, 1}N be the vector of initial node states. At time step t, each node i computes the fraction TABLE I. The four different ways we implement the model, corresponding to differing amounts of quenched randomness. N These are the combinations of fixed or rewired networks and j=1 Aij xj (t) φi(t)= N (1) probabilistic or deterministic response functions. In the ther- P j=1 Aij modynamic limit of the rewired versions, where the network G and response functions change every time step, the mean-field of their neighbors in whoP are active and takes the state theory (Sec. IV) is exact.

xi(t +1)= fi (φi(t)) (2) at the next time step. are parameterized solely by two thresholds, φon and φoff , The above defines a deterministic dynamical system so the distribution of response functions is determined given a network and set of response functions. This is by the joint density P (φon, φoff ). Again, the specific net- called a realization of the model [7]. Each node is in ei- work and response functions define a realization of the ther the 0 or 1 state; we refer to these as the off/inactive model. When these are fixed for all time, we have, in and on/active states, respectively. In the context of con- principle, full knowledge of the possible model dynamics. tagion, these would be the susceptible and infected states. Given an initial condition x(0), the dynamics x(t) are With these binary states, our model is a particular kind deterministic and known for all t ≥ 0. As for all finite of Boolean network [7]. Note that each node reacts only Boolean networks [7], the system dynamics are eventually to the fraction of its neighbors who are active, rather periodic, since the state space {0, 1}N is finite. than the absolute number, and the identities of the in- With the introduction of noise, the system is no longer put nodes do not matter. Each node’s input varies from N eventually periodic. Fluctuations at the node level allow 0 to 1 in steps of 1/ki, where ki = j=1 Aij is node i’s a greater exploration of state space, and the behavior is degree (in-degree if G is not a simple graph). P comparable to that of the general class of discrete-time In the rest of this Section, we will describe some vari- maps. Roughly speaking, the mean-field theory we de- ations of the basic model which also differentiate our velop in Section IV becomes more accurate as we intro- model from the Boolean networks extant in the litera- duce more stochasticity. ture. This is mainly due to the response functions, but We introduce randomness in two parts of the model: also the type of random network on which the dynamics the network and response functions. Allowing for the take place, varying amounts of stochasticity introduced network and responses to be either fixed for all time or into the networks and response functions, and the possi- resampled each time step and taking all possible combi- bility of asynchronous updates. nations yields four different designs (see Table I).

A. The networks considered 1. Rewired networks The mean-field analysis in Section IV is applicable to any network which can be characterized by its degree First, the network itself can change every time step. distribution. The vast majority of the theory of ran- This is the rewiring (R), as opposed to fixed (F), net- dom Boolean networks considers only regular random work case. For example, we could draw a new network networks. Fortunately, such theories are easily general- from G(N, kavg/N) every time step. This amounts to ized to other types of networks with independently cho- rewiring the links while keeping the degree distribution sen edges, such as Poisson (Erdös-Rényi) and configura- fixed, and it is alternately known as a mean field, an- tion model random networks [9, 14]. We develop specific nealed, or random mixing variant as opposed to a fixed results for Poisson random networks, and these are the network or quenched model [7]. networks considered in Section V.

B. Stochastic variants 2. Probabilistic responses

The specific network and response functions determine Second, the response functions can change every time exactly which behaviors are possible. These are chosen step. This is the probabilistic (P), as opposed to the de- from some distribution of networks, such as G(N, kavg/N) terministic (D), response function case. For our social (Poisson random networks on N nodes with edge prob- contagion example, there needs to be a well-defined dis- ability kavg/N), and some distribution of response func- tribution P (φon, φoff ) for the thresholds. This amounts to tions. In the example of Section V, the response functions having a single response function, the expected response 3 function random walker), D is the diagonal degree matrix, and N f = (fi) [15]. If α = 1, then x(t) ∈ {0, 1} and we re- ¯ f(φ)= dφon dφoff P (φon, φoff )f(φ; φon, φoff ). (3) cover the fully deterministic response function dynamics Z Z given by (1) and (2). We call f¯ : [0, 1] → [0, 1] the probabilistic response function. Its interpretation is the following. For an updating node with a fraction φ of active neighbors at the current A. Asynchronous limit time step, then, at the next time step, the node assumes the active state with probability f¯(φ) and the inactive Here, we show that when α ≈ 1/N, time is effectively state with probability 1 − f¯(φ). continuous and the dynamics can be described by an ordi- nary differential equation. This is similar to the analysis of Gleeson [16]. Consider Eqn. 5. Subtracting x(t) from 3. Temperature in the system both sides and setting ∆x(t)= x(t+1)−x(t) and ∆t =1 yields In this paper, the network and response functions are ∆x(t) either fixed for all time or resampled every time step. = α (f(T x(t)) − x(t)) . (6) One could tune smoothly between the two extremes by ∆t introducing rates at which these reconfigurations occur. Since α is assumed small, the right hand side is small, These rates are inversely related to quantities that behave and thus ∆x(t) is also small. Making the continuum ap- like temperature (one for the network and another for the proximation dx(t)/dt ≈ ∆x(t)/∆t yields the differential response functions). Holding the network or response equation functions fixed corresponds to zero temperature, since there are no fluctuations. The stochastic and rewired dx = α (f(T x) − x) . (7) cases correspond to high or infinite temperature, because dt reconfigurations occur every time step. The parameter α sets the time scale for the system. Be- low we see that, from their form, similar asynchronous, C. Update synchronicity continuous time limits apply to the dynamical equations in the densely connected case, Eqn. (8), and in the mean- field theory, Eqns. (11) and (12). Finally, we introduce a parameter α for the probability that a given node updates. When α = 1, all nodes update each time step, and the update rule is said to be B. Dense network limit for Poisson random synchronous. When α ≈ 1/N, only one node is expected networks to update with each time step, and the update rule is said to be effectively asynchronous. This is equivalent to The following result is particular to Poisson random a randomly ordered sequential update. For intermediate networks, but similar results are possible for other ran- values, α is the expected fraction of nodes which update dom networks with dense limits. The normalized Lapla- each time step. cian matrix is defined as L ≡ I − D−1/2AD−1/2, where I is the identity [17]. So T = D−1/2(I − L)D1/2. By III. FIXED NETWORKS Oliveira [18], when kavg is Ω(log N) there exists a typical typ 1 1T Laplacian matrix L = IN − N N /N such that the actual L ≈ Ltyp in the induced 2-norm with high proba- Consider the case where the response functions and bility [1N is the length-N column vector of ones]. In this network are fixed (D-F), but the update may be syn- limit, if we approximate the degrees as uniform, ki ≈ kavg chronous or asynchronous. Extend the definition of xi(t) typ T for all i ∈ V , then T ≈ T = 1N 1 /N. So T effectively to now be the probability that node i is in the active state N averages the node states: T x(t) ≈ N x (t)/N ≡ φ(t). at time t. Note that this agrees with our previous defini- i=1 i Without a subscript, φ(t) denotes the active fraction of tion as the state of node i when x (t)=0 or 1. Then the i the network at time t. We make theP above approxima- x follow the master equation i tion in Eqn. 5 and average that equation over all nodes, N finding j=1 Aij xj (t) xi(t +1)= αfi + (1 − α)xi(t), (4) N ¯ P j=1 Aij ! φ(t +1)= αf(φ(t)) + (1 − α)φ(t) ≡ Φ(φ(t); α), (8) which can be written inP matrix-vector notation as where we have assumed that N is large and the aver- N x(t +1)= α f (T x(t)) + (1 − α)x(t). (5) age of nodes’ individual response functions i=1 fi/N converges in a suitable sense to the stochastic response Here T = D−1A is sometimes called the transition prob- function f¯, Eqn. (3). This amounts to assumingP a law ability matrix (since it also occurs in the context of a of large numbers for the response functions, i.e., that the 4 sample average converges to the expected function. Note if all of the nodes update synchronously, the active den- that α tunes between the probabilistic response function sity of stubs at t + 1 will be ¯ Φ(φ;1) = f(φ) and the line Φ(φ;0) = φ. Also, the fixed ∞ ∞ points of Φ are fixed points of f¯, but their stability will kpk g(ρ; pk, f¯)= qkFk ρ; f¯ = Fk ρ; f¯ . (10) depend on α. When the network is dense, it ceases to kavg k=1 k=1 affect the dynamics, since each node sees a large number X X of other nodes. Thus the network is effectively the com- Finally, if each node only updates with probability α, we plete network. In this way we recover the map models of have the following map for the density of active stubs: Granovetter and Soong [19]. ρ(t +1)= αg ρ(t); pk, f¯ + (1 − α)ρ(t) (11) ¯ ≡ G ρ(t); pk, f, α . IV. MEAN-FIELD THEORY By a similar argument, the active density of nodes is given by Making a mean-field calculation refers to replacing the complicated interactions among many particles by φ(t +1)= αh ρ(t); pk, f¯ + (1 − α) φ(t) (12) a single interaction with some effective external field. ≡ H ρ(t), φ(t); p , f,¯ α , There are analogous techniques for understanding net- k work dynamics. Instead of considering the |E| interac- where tions among the N nodes, network mean-field theories ∞ derive self-consistent expressions for the overall behavior h (ρ; pk,f)= pkFk ρ; f¯ . (13) of the network after averaging over large sets of nodes. k=0 These have been fruitful in the study of random Boolean X networks [20] and can work well when networks are non- Note that the edge-oriented state variable ρ contains all random [21]. of the dynamically important information, rather than We derive a mean-field theory, in the thermodynamic the node-oriented variable φ. limit, for the dynamics of the general model by blocking nodes according to their degree class. This is equivalent to nodes retaining their degree but rewiring edges every B. Analysis of the map equation time step. The model is then part of the well-known class of random mixing models with non-uniform con- The function Fk(ρ; f¯) is known in polynomial approx- tact rates. Probabilistic (P-R) and deterministic (D-R) imation theory as the kth Bernstein polynomial (in the response functions result in equivalent behavior for these variable ρ) of f¯ [22]. Bernstein polynomials have im- random mixing models. The important state variables portant applications in computer graphics due to their end up being the active density of stubs, i.e. half-edges “shape-preserving properties” [23]. The Bernstein oper- or node-edge pairs. In an undirected network without ator Bk takes f¯ 7→ Fk. This is a linear, positive operator degree-degree correlations, the state is described by a which preserves convexity for all k and exactly interpo- single variable ρ(t). In the presence of correlations we lates the endpoints f¯(0) p and f¯(1). Immediate conse- must introduce more variables {ρk(t)} to deal with the quences include that each Fk is a smooth function and (k) ¯(k) ¯(k) relevant degree classes. the kth derivatives Fk (x) → f (x) where f (x) exists. For f¯ concave down, such as the tent or logistic maps, then Fk is concave down for all k and Fk increases A. Undirected networks to f¯ (Fk ր f¯) as k → ∞. This convergence is typically slow. Importantly, Fk ր f¯ implies that g ρ; pk, f¯ ≤ f¯ for any degree distribution pk. To derive the mean-field equations in the simplest case—undirected, uncorrelated random networks— In some cases, the dynamics of the undirected mean- consider a degree k node at time t. The probability that field theory given by ρ(t +1) = G (ρ(t)), Eqn. (11), the node is in the active state at time t+1 given a density are effectively those of the map Φ, from the dense limit ρ of active stubs is Eqn. (8). We see that g, Eqn. (10), can be seen as the expectation of a sequence of random functions Fk under k the edge-degree distribution qk. Indeed, this is how it k j k−j Fk(ρ; f¯)= ρ (1 − ρ) f¯(j/k) , (9) was derived. From the convergence of the Fk’s, we ex- j ¯ ¯ j=0 pect that g(ρ; pk, f) ≈ f(ρ) if the average degree kavg X is “large enough” and the edge-degree distribution has where each term in the sum counts the contributions from a “sharp enough” peak about kavg (we will clarify this having 0, 1, . . . , k active neighbors. Now, the probability soon). Then as kavg → ∞, the mean-field coincides with of choosing a random stub which ends at a degree k node the dense network limit we found for Poisson random net- is qk = kpk/kavg in an uncorrelated random network [9]. works, Eqn. 8. Some thought leads to a sufficient condi- This is sometimes called the edge-degree distribution. So tion for this kind of convergence: the standard deviation 5

σ(kavg) of the degree distribution must be o(kavg). In node degrees. We encode correlations of this type by the Appendix A we prove this as Lemma 1. conditional probabilities In general, if the original degree distribution pk is char- 2 (u) k k′ acterized by having mean kavg, variance σ , and skewness pk,k′ ≡ P ( , undirected| ) γ1, then the edge-degree distribution qk will have mean (i) ′ 2 2 2 pk k′ ≡ P (k, incoming|k ) kavg +σ /kavg and variance σ [1+γ1σ/kavg −(σ/kavg) ]. , Considering the behavior as kavg → ∞, we can conclude (o) k k′ pk,k′ ≡ P ( , outgoing| ), that requiring σ = o(kavg) and γ1 = o(1) are sufficient conditions on pk to apply Lemma 1. Poisson degree dis- the probability that an edge starting at a degree k′ node −1/2 tributions (σ = kavg and γ1 = kavg ) fit these criteria. ends at a degree k node and is, respectively, undirected, incoming, or outgoing relative to the destination degree p k node. We introduced this convention in a series of pa- C. Generalized random networks pers [24, 25]. These conditional probabilities can also be defined in terms of the joint distributions of node types In more general random networks, nodes can have both connected by undirected and directed edges. We omit undirected and directed incident edges. We denote node a detailed derivation, since it is similar to that in Sec- degree by a vector k = (k(u), k(i), k(o))T (for undirected, tion IV A and similar to the equations for the time evo- in-, and out-degree) and write the degree distribution lution of a contagion process [24, Eqns. (13–15)] [see also as pk ≡ P (k). There may also be correlations between 26].

The result is a coupled system of equations for the density of active stubs which now may depend on node type (k) and edge type (undirected or directed):

′ ′ (u) (i) k k (u)′ (i)′ (u) (u) (u) k k ρk (t +1)= (1 − α)ρk (t)+ α pk,k′ ′ ju ji k ju=0 ji=0 X X ′X ′ (u) (i) ju (k −ju) ji (k −ji) (u) (u) (i) (i) (14) × ρk′ (t) 1 − ρk′ (t) ρk′ (t) 1 − ρk′ (t) h j i+ hj i h i h i × f¯ u i , (u)′ (i)′ k + k

′ ′ (u) (i) k k (u)′ (i)′ (i) (i) (i) k k ρk (t +1)= (1 − α)ρk (t)+ α pk,k′ ′ ju ji k ju=0 ji=0 X X X′ ′ (u) (i) ju (k −ju) ji (k −ji) (u) (u) (i) (i) (15) × ρk′ (t) 1 − ρk′ (t) ρk′ (t) 1 − ρk′ (t) h j i+ hj i h i h i × f¯ u i . (u)′ (i)′ k + k The active fraction of nodes at a given time is

k(u) k(i) k(u) k(i) φ(t +1)= (1 − α)φ(t)+ α pk ju ji k ju=0 ji=0 X X X (u) (i) ju (k −ju) ji (k −ji) (u) (u) (i) (i) (16) × ρk (t) 1 − ρk (t) ρk (t) 1 − ρk (t) h j i+ jh i h i h i × f¯ u i . k(u) + k(i)

Because these expressions are very similar to the undi- V. LIMITED IMITATION CONTAGION MODEL rected case, we expect similar convergence properties to those in Sec. IV B. However, an explicit investigation of As a motivating example of these networked map dy- this convergence is beyond the scope of the current paper. namics, we study an extension of the classical threshold 6 models of social contagion [such as 12, 13, 27–29, among 1 others]. What diﬀerentiates our limited imitation contagion model from the standard models is that the response function includes an oﬀ-threshold, above which the node 0.8

takes the inactive state. We assign each node i ∈ V ) an on-threshold φon,i and an off-threshold φoff,i, requir- off 0.6 φ ing 0 ≤ φon,i ≤ φoff,i ≤ 1. Node i’s response function , on fi(φi) = fi(φi; φon,i, φoff,i) is 1 if φon,i ≤ φi ≤ φoff,i and φ ; 0.4 0 otherwise. See Figure 1 for an example on-off threshold φ response function. f( 0.2 This is exactly the model of Granovetter and Soong [19], but on a network. We motivate this choice with 0 the following [also see 19]. (1) Imitation: the on state 0 φ 0.5 φ 1 becomes favored as the fraction of active neighbors sur- on φ off passes the on-threshold (bandwagon effect). (2) Non- conformity: the on state is eventually less favorable with FIG. 1. An example on-off threshold response function. the fraction of active neighbors past the off-threshold (re- Here, φon = 0.33 and φoff = 0.85. The node will “activate” if verse bandwagon, snob effect). (3) Simplicity: in the φon ≤ φ ≤ φoff , where φ is the fraction of its neighbors who absence of any raw data of “actual” response functions, are active. Otherwise it takes the “inactive” state. which are surely highly context-dependent and variable, we choose arguably the simplest deterministic functions g(ρ; k ) avg which capture imitation and non-conformity. 1 k =1 avg A crucial difference between our model and many re- 10 0.8 100 lated threshold models is that, in those models, an acti- vated node can never reenter the susceptible state. Glee- son and Cahalane [26] call this the permanently active 0.6 ) property and elaborate on its importance to their anal- ρ ( ¯ ysis. Such models must eventually reach a steady state. f 0.4 When the dynamics are deterministic, this will be some fixed fraction of active nodes. The introduction of the off- threshold builds in a mechanism for node deactivation. 0.2 Because nodes can now recurrently transition between on and off states, the deterministic dynamics can exhibit 0 a chaotic transient (as in random boolean networks [7]), 0 0.2 0.4 0.6 0.8 1 ρ and the long time behavior can be periodic with poten- tially high period. With stochasticity, the dynamics can ¯ be truly chaotic. FIG. 2. The tent map probabilistic response function f(ρ), Eqn. (17), used in the limited imitation contagion model. This is compared to the edge maps g(ρ; kavg) = g(ρ; pk, f¯), The networks we consider are Poisson random net- Eqn. (10), shown for kavg = 1, 10, 100. These pk are Poisson works from G(N, kavg/N). The thresholds φon and φoff distributions with mean kavg . As kavg increases, g(ρ; kavg) are distributed uniformly on [0, 1/2) and [1/2, 1), respec- increases to f¯(ρ). tively. This distribution results in the probabilistic response function (see Figure 2) A. Analysis of the dense limit

When the network is in the dense limit (Section III B), 2φ if 0 ≤ φ< 1/2, the dynamics follow φ(t +1)=Φ(φ(t); α), where f¯(φ)= (17) 2 − 2φ if 1/2 ≤ φ ≤ 1. ( Φ(φ; α)= αf¯(φ)+(1 − α)φ (1 + α)φ if 0 ≤ φ< 1/2, (18) = ( (1 − 3α)φ +2α if 1/2 ≤ φ ≤ 1. The tent map is a well-known chaotic map of the unit interval [30]. We thus expect the limited imitation model Solving for the fixed points of Φ(φ; α), we find one at with this probabilistic response function to exhibit simi- φ = 0 and another at φ = 2/3. When α < 2/3, the larly interesting behavior. nonzero fixed point is attracting for all initial conditions 7

B. Mean-ﬁeld

Here we show how we compute the mean-field maps derived in Section IV. In this specific example, we can write the degree-dependent map Fk(ρ; f¯) in terms of incomplete regularized beta functions Iz(a,b) [31]. Since f¯ is understood to be the tent map, we will write Fk(ρ; f¯)= Fk(ρ). We find that

Fk(ρ)=2ρ − 4ρIρ(M, k − M), (19)

where we have let M = ⌊k/2⌋ for clarity (⌊·⌋ and ⌈·⌉ are the floor and ceiling functions). The details of this derivation are given in Appendix B. The map g(ρ; pk, f¯) is parametrized here by the network parameter kavg, since pk is fixed as a Poisson distribution with mean kavg and f¯ is the tent map, and we write it as simply g(ρ; kavg). To evaluate g(ρ; kavg), we compute Fk(ρ) using Eqn. (19) and constrain the sum FIG. 3. Bifurcation diagram for the dense map Φ(φ; α), in Eqn. (10) to values of k with ⌊kavg − 3 kavg⌋ ≤ Eqn. (18). This was generated by iterating the map at 1000 k ≤ ⌈k +3 k ⌉. This computes contributions to α values between 0 and 1. The iteration was carried out with avg avg p within three standard deviations of the average degree 3 random initial conditions for 10000 time steps each, dis- p carding the first 1000. The φ-axis contains 1000 bins and the in the network, requiring only O( kavg) evaluations of invariant density, shown by the grayscale value, is normalized Eqn. (19). The representation in Eqn. (19) allows for p by the maximum for each α. With α < 2/3, all trajectories quick numerical evaluation of Fk(ρ) for any k, which we go to the fixed point at φ = 2/3. performed in MATLAB. In Figure 2, we show g(ρ; kavg) for kavg = 1, 10, and 100. We confirm the conclusions of Section IVB: g(ρ; kavg) is bounded above by f¯(ρ), and g(ρ; kavg) ր ¯ except φ = 0. When α = 2/3, [1/2, 5/6] is an interval f(ρ) as kavg → ∞. Convergence is slowest at ρ = 1/2, of period-2 centers. Any orbit will eventually land on where the kink exhibited by the tent map has been one of these period-2 orbits. When α > 2/3, this inter- smoothed out by the effect of the Bernstein operator. val of period-2 centers ceases to exist, and more complicated behavior ensues. Figure 3 shows the bifurcation diagram for Φ(φ; α). From the bifurcation diagram, the C. Simulations orbit appears to cover dense subsets of the unit interval when α > 2/3. The bifurcation diagram appears like We performed stochastic simulations of the limited im- that of the tent map (not shown; see [10, 30]) except the itation model for the D-F, P-F, and P-R designs, in the branches to the right of the first bifurcation point are abbreviations of Table I. Unless otherwise noted, N = separated here by the interval of period-2 centers. 104. For all of the bifurcation diagrams, the first 3000 time steps were considered transient and discarded, and the invariant density of ρ was calculated from the following 1000 points. For plotting purposes, the invariant density was normalized by its maximum at those parameters. For example, in Figure 3 we plot P (φ|α)/ maxφ P (φ|α) The effect of conformsits, an aside rather than the raw density P (φ|α). To compare the mean-field theory to those simulations, we numerically iterated the edge map ρ(t +1) = Suppose some fraction c of the population is made up G (ρ(t); kavg, α) for different values of α and kavg. We of individuals without any off-threshold (alternatively, then created bifurcation diagrams of the possible behav- each of their off-thresholds φoff = 1). These individ- ior in the mean-field as was done for the simulations. uals are conformist or “purely pro-social” in the sense that they are perfectly happy being part of the majority. For simplicity, assume α = 1. The map Φ(φ; c)=2φ for D. Results 0 ≤ φ< 1/2and 2−2(1−c)φ for 1/2 ≤ φ ≤ 1. If c> 1/2, then the equilibrium at 2/3 is stable. Pure conformists, To provide a feel for the deterministic dynamics, we then, can have a stabilizing effect on the process. We show the result of running the D-F model on a small net- expect a similar effect when the network is not dense. work in Figure 4. Here, N = 100 and kavg = 17. Starting 8

FIG. 4. Deterministic (D-F) dynamics on a small network. Here, N = 100 and kavg = 17. On the left, we plot the state evolution over time. The upper plot shows individual node states (black = active) sorted by their eventual level of activity, and the lower plot shows the total number of active nodes. We see that the contagion takes oﬀ, followed by a transient period of unstable behavior until time step 80, when the system enters a macroperiod-4 orbit. Note that individual nodes exhibit diﬀerent microperiods (see Sec. V D). On the right, we show the network itself with the initial seed node in black in the lower right.

from a single initially active node at t = 0, the active population grows monotonically over the next 6 time steps. From t =6 to t = 80, the transient time, the active fraction varies in a similar manner to the dynamics in the stochastic and mean-field cases. After the transient, the state collapses into a period-4 orbit. We call the overall period of the system its “macroperiod,” while individual nodes may exhibit different “microperiods.” Note that the macroperiod is the lowest common multiple of the individual nodes’ microperiods. In Figure 4, we observe microperiods 1, 2, and 4 in the timeseries of individual node activity. In other networks, we have observed up to macroperiod 240 [10]. A majority of the nodes end FIG. 5. The 3-dimensional bifurcation diagram computed up frozen in the on or off state, with approximately 20% from the mean-field theory. The axes X = average degree kavg, of the nodes exhibiting cyclical behavior after collapse. Y = update probability α, and Z = active edge fraction ρ. The The focus of this paper has been the analysis of the on- discontinuities of the surface are due to the limited resolution off threshold model, and the D-F case has not been as of our simulations. See Figure 6 for the parameters used. This amenable to analysis as the stochastic cases. We offer a was visualized using Paraview, and a file is available in the deeper examination through simulation of the determin- online Appendix. istic case in [10]. We explore the mean-field dynamics by examining the limiting behavior of the active edge fraction ρ under the The mean-field map dynamics exhibit period-doubling map G (ρ; kavg, α). We simulated the map dynamics for bifurcations in both parameters kavg and α. Visualiz- a mesh of points in the (kavg, α) plane. We plot the 3- ing the bifurcation structure in 3-d (Figure 5) shows in- dimensional (3-d) bifurcation structure of the mean-field terlacing period-doubling cascades in the two parame- theory in Figure 5. We also show 2-d bifurcation plots for ter dimensions. These bifurcations are more clearly re- fixed kavg and α slices through this volume in Figures 6 solved when we take slices of the volume for fixed pa- and 7. For more visualizations of this bifurcation struc- rameter values. The mean-field theory (Figure 6) closely ture, see Appendix C. In all cases, the invariant density matches the P-R simulations (Figure 7). The first deriva- ∂G ∂Φ of ρ is normalized by its maximum for that (kavg, α) pair tive ∂ρ (ρ; kavg, α) < ∂ρ (ρ; α) for any finite kavg, so the and indicated by the grayscale value. bifurcation point α = 2/3 which we found for the dense 9

FIG. 6. Mean-field theory bifurcation diagram slices for various fixed values of kavg and α. The top row (a–c) shows slices for fixed kavg. As kavg → ∞, the kavg -slice bifurcation diagram asymptotically approaches the bifurcation diagram for the dense map, Figure 3. Note that the first bifurcation point, near 2/3, grows steeper with increasing kavg . The bottom row (d–f) shows slices for fixed α. The resolution of the simulations was α = 0.664, 0.665,..., 1, kavg = 1, 1.33,..., 100, and ρ bins were made for 1000 points between 0 and 1.

FIG. 7. Bifurcation diagram from fully stochastic (P-R) simulations, made in the same way as Figure 6. The bifurcation structure of these stochastic simulations matches that of the mean-field theory (Figure 6), albeit with some blurring. 10 map Φ is an upper bound for the first bifurcation point The deterministic case, which we have barely touched of G. The actual location of the first bifurcation point on here, merits further study [see 10]. In particular, depends on kavg, but α = 2/3 becomes more accurate we would like to characterize the distribution of periodic for higher kavg (it is an excellent approximation in Fig- sinks, how the collapse time scales with system size, and ures 6c and 7c, where kavg = 100). When α = 1, the first how similar the transient dynamics are to the mean-field bifurcation point occurs at kavg ≈ 7. dynamics. The bifurcation diagram slices resemble each other and Furthermore, the model should be tested on realistic evidently fall into the same universality class as the lo- networks. These could include power law or small world gistic map [32, 33]. This class contains all 1-d maps with random networks, or real social networks gleaned from a single, locally-quadratic maximum. Due to the prop- data. In a manner similar to Melnik et al. [21], one could erties of the Bernstein polynomials, Fk(ρ; f¯) will univer- evaluate the accuracy of the mean-field theory for real sally have such a quadratic maximum for any concave networks. down, continuous f¯ [22]. So this will also be true for Finally, the ultimate validation of this model would g(ρ; kavg, f¯) with kavg finite, and we see that kavg par- emerge from a better understanding of social dynamics tially determines the amplitude of that maximum in Fig- themselves. Characterization of people’s “real” response ure 2. Thus kavg acts as a bifurcation parameter. The functions is therefore critical [some work has gone in this parameter α tunes between G (ρ; kavg, 1) = g(ρ; kavg, f¯) direction; see 5, 6, 34, 35]. Comparison of model output and G (ρ; kavg, 0) = ρ, so it has a similar effect. Note to large data sets, such as observational data from social that the tent map f¯ and the dense limit map Φ are media or online experiments, is an area for further experi- kinked at their maxima, so their bifurcation diagrams mentation. This might lead to more complicated context- are qualitatively different from those of the mean-field. and history-dependent models. As we collect more data The network, by constraining the interactions among the and refine experiments, the eventual goal of quantifiably population, causes the mean-field behavior to fall into a predicting social behavior, including fashions and trends, different universality class than the individual response seems achievable. function map.

ACKNOWLEDGMENTS VI. CONCLUSIONS We thank Thomas Prellberg, Leon Glass, Joshua Payne and Joshua Bongard for discussions and sugges- We have described a very general class of synchronous or asynchronous, binary state dynamics occuring on net- tions. We are grateful for computational resources pro- vided by the Vermont Advanced Computing Core sup- works. We obtained an exact master equation and showed that, when random networks are sufficiently ported by NASA (NNX 08A096G). KDH was supported by VT-NASA EPSCoR and a Boeing Fellowship; CMD dense, the networked dynamics approach those of the fully-connected case. We developed a mean-field theory was supported by NSF grant DMS-0940271; PSD was supported by NSF CAREER Award #0846668. This and found that it also predicted the same limiting behavior. The convergence of the mean-field map to the aver- work is based on the Master’s Thesis of KDH. age response function is related to the Bernstein polynomials, allowing us to employ many previous results in Appendix A: Proof of Lemma 1 order to analyze the mean-field map equation. We also extended those mean-field equations to correlated ran- Lemma 1. dom networks. For k ≥ 1, let fk be continuous real-valued functions on a compact domain with uni- The general model we describe was motivated by the X fk → f formly. Let be a probability mass function on Z+ the limited imitation model of social contagion. We see pk parametrized by its mean and with standard deviation that including an aversion to total conformity results in µ , assumed to be . Then, more complicated, even chaotic dynamics, as opposed to σ(µ) o(µ) the simple spreading behavior typically seen in the single ∞ threshold case. The theory developed for the general case lim pkfk = f. µ→∞ successfully captured the behavior of the stochastic net- k=0 ! X work dynamics. We have focused on the rich structure of Proof. Suppose 0 ≤ a< 1 and let K = ⌊µ − µa⌋. Then, bifurcations as the two parameters, update synchronicity α and average degree kavg, were varied. We see that the ∞ K ∞ universality class of the dynamics matches those of the g = pkfk = pkfk + pkfk. (A1) logistic map. Using the mean-field theory, we can un- k=0 k=0 k=K+1 X X X derstand this as a result of the smoothing effect of the Since fk → f uniformly as k → ∞, for any ǫ> 0 we can Bernstein polynomials on the tent map average response choose µ large enough that function. However, this universality class will appear for any unimodular, concave down response function. |fk(x) − f(x)| <ǫ (A2) 11 for all k>K and all x ∈ X. Without loss of generality, form of Eqn. (17) to write assume that |fk|≤ 1 for all k. Then, M k 2j F (ρ)= ρj (1 − ρ)k−j k j k j=0 X k k 2j + ρj (1 − ρ)k−j 2 − j k j=XM+1 2 M σ k |g − f|≤ + ǫ. =2 − 2ρ − 2 ρj (1 − ρ)k−j µa j j=0 X 4 M k + ρj (1 − ρ)k−j j. (B1) k j j=0 X We have used the fact that the binomial distribution k ρj(1 − ρ)k−j sums to one and has mean kρ. For a j The σ/µ term is a consequence of the Chebyshev in- n ≤ M, we have the identity equality [14] applied to the first sum in (A1). Since σ grows sublinearly in µ, this term vanishes for some M k 0 ≤ a < 1 when we take the limit µ → ∞. The ǫ term (j) ρj (1−ρ)k−j = ρn(k) I (k−M,M −n+1) n j n 1−ρ comes from using (A2) in the second sum in (A1), and it j=0 X can be made arbitrarily small. (B2) where Ix(a,b) is the regularized incomplete beta function and (k)n = k(k −1) ··· (k −(n−1)) is the falling factorial [31, 36]. This is an expression for the partial (up to M) nth factorial moment of the binomial distribution with parameters k and ρ. Note that when n = 0 we recover the well-known expression for the binomial cumulative distribution function. From Eqns. (B1) and (B2), we arrive at Eqn. (19).

Appendix C: Online material

To better explore the 3-d mean ﬁeld bifurcation structure, we created movies of the the kavg and α slices as the parameters are dialed. We also provide a VTK ﬁle with the 3-d bifurcation data, view- Appendix B: Beta function representation of Fk able in Paraview or other software. Videos of the individual-node dynamics for small networks in the D-F and P-F cases are shown for some parameters which produce interesting behavior. The D-F, P-F, and P-R cases were implemented in Python, and we We now show how, when f¯ is the tent map (17), the provide our code. All of the above is available from degree-k map Fk(ρ; f¯) can be written in terms of incom- http://www.uvm.edu/storylab/share/papers/harris2013a/. plete regularized beta functions. First, use the piecewise

[1] S. Coombes, B. Doiron, K. Josi´c, and E. Shea-Brown, [4] P. Rohani, X. Zhong, and A. King, Science 330, 982 Philosophical Transactions of the Royal Society A: Math- (2010). ematical, Physical and Engineering Sciences 364, 3301 [5] J. Ugander, L. Backstrom, C. Marlow, and J. Kleinberg, (2006). PNAS 109, 5962 (2012). [2] R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, [6] D. Centola, Science 329, 1194 (2010). D. Chklovskii, and U. Alon, Science 298, 824 (2002). [7] M. Aldana, S. Coppersmith, and L. P. Kadanoﬀ, in [3] J. Bascompte, Science 325, 416 (2009). Perspectives and Problems in Nonlinear Science, edited by E. Kaplan, J. E. Marsden, and K. R. Sreeni- 12

vasan (Springer, New York, 2003), chap. 2, pp. 23–90, [23] J. M. Peña, ed., Shape Preserving Representations in http://arxiv.org/abs/nlin/0204062. Computer-Aided Geometric Design (Nova Science Pub- [8] D. Stauffer and A. Aharony, Introduction to Percolation lishers, Inc, 1999). Theory (Taylor and Francis, London, 1994), 2nd ed. [24] J. L. Payne, K. D. Harris, and P. S. Dodds, [9] M. E. J. Newman, SIAM Review 45, 167 (2003). Phys. Rev. E 84, 016110 (2011), URL [10] P. S. Dodds, K. D. Harris, and C. M. Danforth (2013), http://link.aps.org/doi/10.1103/PhysRevE.84.016110. http://arxiv.org/abs/1208.0255. [25] P. S. Dodds, K. D. Harris, and J. L. Payne, [11] G. Simmel, American Journal of Sociology 62, 541 Phys. Rev. E 83, 056122 (2011), URL (1957). http://link.aps.org/doi/10.1103/PhysRevE.83.056122. [12] T. C. Schelling, Journal of Mathematical Sociology 1, [26] J. P. Gleeson and D. J. Cahalane, 143 (1971). Phys. Rev. E 75, 056103 (2007), URL [13] M. Granovetter, American Journal of Sociology 83, 1420 http://link.aps.org/doi/10.1103/PhysRevE.75.056103. (1978). [27] T. C. Schelling, Journal of Conflict Resolution 17, 381 [14] B. Bollobás, Random Graphs (Cambridge University (1973). Press, 2001), 2nd ed. [28] D. J. Watts, PNAS 99, 5766 (2002). [15] When there are isolated nodes, those nodes’ degrees are [29] P. S. Dodds and D. J. Watts, Phys. Rev. Lett. 92, 218701 0, and thus the denominator in the master equation is (2004). zero and D−1 undefined. If the initial network contains [30] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An isolated nodes, we set all entries in the corresponding Introduction to Dynamical Systems (Springer, 1996). rows of T to zero. [31] NIST, Digital Library of Mathematical Functions (2012), [16] J. P. Gleeson, Phys. Rev. E 77, 046117 (2008). http://dlmf.nist.gov. [17] D. B. West, Introduction to Graph Theory (Prentice Hall, [32] M. J. Feigenbaum, Journal of Statistical Physics 19, 25 Upper Saddle River, NJ, 2001), 2nd ed. (1978). [18] R. I. Oliveira (2010), http://arxiv.org/abs/0911.0600, [33] M. J. Feigenbaum, Journal of Statistical Physics 21, 669 0911.0600. (1979). [19] M. Granovetter and R. Soong, Journal of Economic Be- [34] D. Centola, Science 334, 1269 (2011). havior and Organization 7, 83 (1986). [35] D. M. Romero, B. Meeder, and J. Kleinberg, in Proc. 20th [20] B. Derrida and Y. Pomeau, Europhys. Lett. 1, 45 (1986). ACM International World Wide Web Conference (2011). [21] S. Melnik, A. Hackett, M. A. Porter, P. J. Mucha, and [36] R. L. Winkler, G. M. Roodman, and R. R. Britney, Man- J. P. Gleeson, Phys. Rev. E 83, 036112 (2011). agement Science 19, 290 (1972). [22] G. M. Phillips, Interpolation and Approximation by Poly- nomials (Springer, 2003), see chapter 7.