Mean field dynamics of graphs I: Evolution of probabilistic cellular automata for random and small-world graphs

Mean Field Dynamics of Graphs I: Ev olution of Probabilistic Cellular Automata for Random and Small-W orld Graphs Lourens J. W aldorp 1 , ∗ and Jolanda J. Kossako wski 1 1 Dep artment of Psycholo gy University of Amster dam Nieuwe A chter gr acht 129-B 1018 XE Amster dam The Netherlands (Dated: Marc h 2, 2022) It w as recently sho wn ho w graphs can be used to provide descriptions of psyc hopathologies, where symptoms of, say , depression, affect each other and certain configurations determine whether some- one could fall in to a sudden depression. T o analyse changes ov er time and c haracterise possible future behaviour is rather difficult for large graphs. W e describ e the dynamics of netw orks using one-dimensional discrete time dynamical systems theory obtained from a mean field approac h to (elemen tary) probabilistic cellular automata (PCA). Often the mean field approach is used on a regular graph (a grid or torus) where eac h no de has the same num ber of edges and the same probabilit y of b ecoming active. W e show that we can use v ariations of the mean field of the grid to describ e the dynamics of the PCA on a random and small-world graph. Bifurcation diagrams for the mean field of the grid, random, and small-world graphs indicate p ossible phase transitions for certain parameter settings. Extensiv e simulations indicate for different graph sizes (num ber of no des) that the mean field approximation is accurate. The mean field approac h allows us to provide p ossible explanations of ‘jumping’ behaviour in depression. P ACS n um b ers: 64.60.aq en 05.45.-a Keywords: cellular automata, discrete dynamical system, nonlinear dynamics, bifurcation I. INTR ODUCTION In psyc hopathology sudden c hanges from ’normal’ to depressed mo ods can o ccur. Suc h ’discon tinuities’ can b e the result of a rela- tiv ely small c hange in the en vironment or p er- son. Recen tly , mental disorders hav e b een describ ed as a net work of in teracting symp- toms [1, 2], which pro vides a framework where an explanation for suc h sudden changes in mo od could b e found. A mental disorder can b e viewed as a netw ork of symptoms, eac h symptom influencing other symptoms. F or in- stance, lack of sleep during the night could lead to p o or concentration during the da y , whic h in turn could lead to lack of sleep again b y worrying that your job may b e on the line. Here we use this idea and mo del the dynamics of psyc hopathology netw orks as probabilistic cellular automata. Then to analyse the dy- namics we use a mean field approac h where ∗ Corresponding author: waldorp@uv a.nl eac h no de is similar in b eha viour to all others. W e extend known results for the mean field ap- proac h in this context to other t yp es of graphs (random and small-world graphs), where the mean field can b e interpreted as a w eighted a verage of all nodes in the graph. W e also give concen tration inequalities for approximations using v ariations of the grid mean field. Cellular automata are discrete dynamical systems that ha v e deterministic, lo cal rules to mo ve from one generation to the next [3, 4]. In tro duced b y V on Neumann, the most famous v ersion is Conw a ys game of life, popularised b y Gardner, and has found man y applications from computer science [7] to neuronal popu- lation mo delling [8] to ep demiology [9]. In a cellular automaton eac h cell or no de in a finite grid (usually a subset of Z 2 ) can b e ’active’ or ’inactiv e’ (1 or 0) and if, for instance, t wo (di- rect) neighbours are active, then the no de will b ecome active at the next time step. Another example of a cellular automaton is b ootstrap p ercolation, where eac h no de can only b ecome activ e and cannot b e inactiv ated by its neigh- 2 b ours, and the ob jective is to determine the initial configuration of active no des that re- sult in all no des b eing active [10]. In general, a new generation in a cellular automaton is determined b y a lo cal and homogeneous up- date rule φ . F or each no de x in the graph this induces a sequence of states, an orbit. A (ran- dom) configuration at time 0 then determines whether all no des in the net work will be active, inactiv e, or whether the netw ork will demon- strate p erio dic b eha viour. A generalisation of a cellular automaton is to introduce a proba- bilit y p φ to decide whether or not a no de will b ecome active or not determined b y a no de’s neigh b ours. One such rule is the ma jorit y rule whic h gives the probability to switch dep end- ing on whether the ma jority of its neighbours are activ e. Suc h a system is called a probabilis- tic cellular automaton (PCA). Here we will in vestigate the dynamic b eha viour of the pro- p ortion of activ e no des (density) for PCA with a ma jorit y rule that are defined on toroidal, random and small-w orld graphs. Man y v ersions of PCA exist and of partic- ular interest are those that b eha v e similar to the Ising net w ork. The reason is that the Ising net work is often used to mo del realistic phe- nomena, like magnetisation [11, 12] or psy- c hopathologies [2]. W e ha v e in mind the appli- cation to psyc hopathology here. In suc h sys- tems the symptoms of disorders are the no des in the graph and edges b et ween the symptoms are estimated from data using the Ising mdoel [2] or from verbal accounts. In our companion pap er in this issue we elaborate on real data analysis using results from discrete Mark ov c hains. W atts sho wed that a one-dimensional, large- scale cellular automaton (deterministic) where the connectivit y b et w een no des was arranged as a small-world, could perform the densit y (all zero es) and synchronisation (alternating all zeroes and all ones) tasks. Newman and W atts gav e approximations for path length and clustering on a small-w orld, to obtain an analytic solution to the threshold ab o ve whic h a large num b er of no des are active. Call- a wa y et al. also studied p ercolation in differ- en t graph topologies in deter, fo cussing on the consequences of (randomly) deleting no des. Here again the ob jectiv e w as to concen trate on stable solutions of the graphs. In a proba- bilistic version, T omassini et al. inv estigated a one-dimensional PCA on a regular and small- w orld graph in terms of its p erformance on the density and sync hronisation tasks. They determined by using evolutionary algorithms that a small-world top ology is most efficient to solving b oth tasks, corresponding to the re- sults of W atts in a deterministic v ersion. Their ob jective was different from ours in that here w e are in terested in all types of dynamic b e- ha viour (stable or not), and sp ecifically rep- resen ting this b eha viour for the PCA by the mean field. Our starting p oin t is the work b y Balis- ter et al. and Kozma et al. where a tw o- dimensional (toroidal) grid on a PCA is de- fined. The mean field is then used to deter- mine the unconditional probability distribu- tion of the densit y (relative num b er of activ e no des). Balister et al. show that the mean field mo del predicts a bifurcation for small v alues of the probability of a no de switch- ing to another state and determine its criti- cal p oin t for a neighbourho od of size five [see also 8, 18]. This is of particular in terest in our case as it ma y explain mo od disorders (e.g., de- pression or manic-depression) from symptoms and their connectivit y . T o apply these results to random and small-w orld graphs w e deter- mine the marginal distribution across the p os- sible degree probabilities giv en the topology of the random or small-world graph. Extend- ing results of homogeneous graphs has b een applied to so cial netw orks [19] and to cellular automata [20]. W e first in tro duce probabilistic cellular au- tomata in Section I I. Then in Section I II we sho w ho w the traditional version of a PCA on a grid can b e reduced to a single discrete time dynamical system, called the mean field. In Section I II B w e show that for the random graph we can use a v ariation on the form u- lation for the grid of the dynamical system to describ e dynamics. W e use these results on the random graph to show in Section I II C that w e can obtain a similar appro ximation for the small-w orld graph, again using the form u- lation for the grid. Ha ving sho wn that these appro ximations are appropriate, we see in Sec- tion IV what the dynamics of the pro cess is 3 for the differen t topologies. W e follo w these theoretical results by extensive sim ulations to v erify the accuracy of the mean field in Section V. I I. PR OBABILISTIC CELLULAR A UTOMA T A A cellular automaton is a dynamical system of nodes in a fixed, finite grid where directly connected no des determine the state of a no de at each subsequent time step [7]. Each no de x in a no de set V = { 1 , 2 , . . . , n } is at time t in one of the states of a finite alphab et Σ. The nearest neighbours in the graph G = ( V , E ) are given by the edges in E . Often the graph G is the square lattice Z 2 , where eac h node has exactly four neighbours [see e.g., 21]. A lo cal rule determines based on the direct neighbours what the v alue of the alphab et of no de x ∈ V will b e at time t + 1. Let the neighbourho od of x b e the set of no des that are directly con- nected to x , Γ( x ) = { y ∈ V : ( x, y ) ∈ E } . A local rule φ : Γ → Σ assigns for eac h con- figuration of the neigh b ourhoo d of x a v alue a ∈ Σ. If we additionally in tro duce a proba- bilit y for eac h lo cal configuration φ , we obtain a probabilistic cellular automaton (PCA). The probabilit y is a function p a : Σ | Γ | × Σ → [0 , 1] suc h that a probability is assigned to each no de x to ha ve label a for a configuration φ dep enden t on the neigh b ourhoo d Γ( x ), with P a ∈ Σ p a = 1. The local rule φ is applied it- erativ ely to each result, and hence induces a sto c hastic pro cess with sequence Φ t : V → Σ for eac h time step t . F or no de x ∈ V we write Φ t ( x ) = φ t ( x 0 ), where x 0 is the v alue at time t = 0, and Φ t ( V ) is the image for all no des simultaneously . Eac h no de there- fore has an orbit [22], whic h is the sequence (Φ t ( x ) , t ≥ 0) = ( x 0 , φ ( x 0 ) , φ ( φ ( x 0 )) , . . . ). Example 1 (Ma jority rule on 0/1) Let the al- phab et b e Σ = { 0 , 1 } and tak e a finite sub- set of the square lattice V ⊂ Z 2 . In this lat- tice V eac h node has 4 (nearest) neighbours. W e use the ma jorit y rule, which sa ys that if | φ − 1 x (1) | = |{ y ∈ Γ( x ) : Φ t ( y ) = 1 }| is greater than 2, then the no de x will b e 1 with proba- bilit y 1 − p , and otherwise 0 with probability p . The sequence (Φ t ( x ) , t ≥ 0) is any orbit of 0s and 1s (0 , 1 , 1 , . . . ). In Example 1 the probability p determined b y the neighbourho od Γ( x ) is indep enden t of the state of the no de x itself. Suc h a model is called totalisitc [17]. Additionally , the proba- bilities for 0 and 1 were defined by the same parameter p , which is then called symmetric , i.e., p 1 = 1 − p 0 . Here w e fo cus on totalis- tic and symmetric models with size 2 alphabet Σ = { 0 , 1 } . I II. MEAN FIELD APPR OXIMA TION ON GRAPHS The key ingredient of the mean field approxi- mation, shown b y Balister et al. , is that the prop erties of interest are uniform ov er the graph. F or the (toroidal) grid top ology this is easy to see: An y no de x ∈ V has the same n umber of neighbours | Γ | = 4, where eac h no de in the neighbourho od b ecomes 0 or 1 by the same lo cal rule. It follows that an y four no des in the grid could serv e as part of the neigh b ourhoo d for x . In a probabilistic au- tomaton, therefore, the lo cal rule depends only on the num ber of 1s in any random dra w of 4 no des from all no des V . W e first consider the case for a grid and then mo ve onto the random and small-world graph. A. Mean field on a grid Let the graph G grid ( n, Γ) b e a grid with n no des and b oundary conditions suc h that eac h no de has exactly four neighbours. W e consider the densit y ρ t defined b y | Φ − 1 t (1) | /n , where the set of no des that are 1 is Φ − 1 t (1) = { y ∈ V : φ t ( y ) = 1 } . It follo ws that we require the probability of { Φ t ( x ) = 1 } given a certain n umber of no des in state 1 at the previous time p oin t. The probabilit y of switching to state 1, ξ | Γ | ( r ), is conditional on r of the neigh b ours that are 1. Let | φ − 1 (1) | = |{ y ∈ Γ : φ ( y ) = 1 }| b e the num b er of 1s in the neighbourho od of x . Then w e define the probability of state 1 4 giv en that r neighbours are 1 as ξ | Γ | ( r ) = P (Φ t ( x ) = 1 | | φ − 1 (1) | = r ) (1) One of the p ossibilities to define ξ | Γ | is the ma- jorit y rule: if the ma jority of neigh b ours in Γ are 1, then the node will b e 1 with probability 1 − p at t + 1. The ma jority rule is defined as ξ | Γ | ( r ) = ( p if r ≤ | Γ | / 2 1 − p if r > | Γ | / 2 (2) T o obtain the probability of Φ t ( x ) = 1, the state of x b eing in state 1, we need to deter- mine the probability of a neighbourho od hav- ing r active no des. In the mean field we hav e that the probabilit y of a 1 is homogeneous and so we obtain a binomial distribution for the n umber of 1s in the neighbourho od Γ. The probabilit y that the n um b er of nodes in the neigh b ourhoo d equals r is in the mean field the same as | Γ | Bernoulli trials each with proba- bilit y of success ρ t . Hence P ( | φ − 1 (1) | = r | ρ t ) =  | Γ | r  ρ r t (1 − ρ t ) | Γ |− r (3) where r = 0 , 1 , . . . , | Γ | . Then the probability of the ev ent { Φ t ( x ) = 1 | ρ t } , that no de x will ha ve state 1 at time t + 1 giv en the density ρ t at time t , is p grid ( ρ t ) = | Γ | X r =0 ξ | Γ | ( r )  | Γ | r  ρ r t (1 − ρ t ) | Γ |− r . (4) Let q | Γ | / 2 ( ρ t ) = | Γ | / 2 X r =0  | Γ | r  ρ r t (1 − ρ t ) | Γ |− r Com bining this probabilit y with the ma jority rule (2) giv es p grid ( ρ t ) = pq | Γ | / 2 ( ρ t ) + (1 − p )  1 − q | Γ | / 2 ( ρ t )  whic h is used in Kozma et al. . This mean field result follows directly from Theorem 2.1 in Balister et al. . Let B ( n, p ) denote a bino- mial random v ariable with n Bernoulli trials eac h with success probability p . Lemma 2 [17, The or em 2.1] L et G grid ( n, Γ) b e a grid with a PCA as define d ab ove with ξ | Γ | ( r ) ac c or ding to the majority rule in (2). Then the evolution of the numb er of active no des nρ t is nρ t +1 = B ( n, p grid ( ρ t )) . (5) The me an and varianc e for the density ρ t , r e- sp e ctively, µ grid = p grid and σ 2 grid = p grid (1 − p grid ) /n . It follo ws that the conditional probability of nρ t +1 activ e no des in the grid on { nρ t = k } at the previous time step is P ( nρ t +1 = r | nρ t = k ) =  n r  p grid ( ρ t ) r (1 − p grid ( ρ t )) n − r (6) It is easily seen that this is a discrete time Mark ov process on a finite state space of size n , since the probability of nρ t +1 dep ends only on ρ t . Equation (6) is the transition probability of the discrete time Marko v pro cess for the n umber of active nodes in the graph. Because φ t ( x ) is Bernoulli distributed B (1 , p grid ( ρ t )) for all no des x ∈ V , and the n umber of active nodes | Φ − 1 (1) | is the sum of these Bernoulli trials, we can apply the law of large num b ers so that for large n , ρ t +1 is close to µ t := µ ( ρ t ) with high probabilit y . Indeed, w e can use Chernov’s b ound to suggest that using p grid is go od enough for large graphs. Lemma 3 (A c cur acy b ound of density) L et nρ t = P x ∈ V φ t ( x ) b e the sum of n Bernoul li trials given by (5), with me an of the density p grid ( ρ t ) . F or every 0 < ε < min { p grid , 1 − p grid } , let δ = 2 exp( − ε 2 / 2 σ 2 grid ) . We then have with pr ob ability at le ast 1 − δ | ρ − p grid | ≤ r p grid (1 − p grid ) n 2 log (2 /δ ) (7) A pro of is in the App endix. So we can use the mean field p grid ( ρ t ) for grids of large size n . With δ = 0 . 05, we obtain the inter- v al with probabilit y at least 0.95 of [ µ grid − 2 . 72 σ grid , µ grid − 2 . 72 σ grid ]. Another interv al 5 can b e obtained from the DeMoivre-Laplace cen tral limit theorem. This theorem tells us that for large enough n , z grid = ( ρ t − p grid ) /σ grid is distributed as N (0 , 1). In fact, if the third order momen t of z grid is c < ∞ , the Berry-Esseen theorem says that the or- der of approximation of the distribution of ρ t to the normal distribution is O (3 c/ √ n ) [23]. This pro vides an interv al for ρ t +1 as a mea- sure of accuracy with [ µ grid − 1 . 96 σ grid , µ grid + 1 . 96 σ grid ] with probability 0 . 95. Clearly , in b oth limit la ws the size of the netw ork n de- termines the accuracy of the approximation. B. Mean field on a random graph In the original setting of a grid (with b ound- ary conditions, so a torus) the num b er of neigh b ours is fixed and it w as seen that the mean field appro ximation p grid w as accurate for the density b ecause each no de is identi- cal with resp ect to a change dep ending on its neighbours. Here w e introduce the neigh- b ourhoo d size | Γ | as a random v ariable and then determine the probabilit y of Φ t ( x ) = 1 giv en ρ t b y av eraging o v er all possible sizes of neigh b ourhoo ds weigh ted b y its probabil- it y for neigh b ourhoo d size. This is done in a random graph where eac h no de has a bino- mial n umber of neighbours. Let G rg ( n, p e ) b e a random graph with n no des and (constan t) probabilit y p e of an edge b eing presen t [24, 25]. Let the size of the neighbourho od | Γ | b e a bi- nomial random v ariable with maximal v alue n − 1 neigh b ours and probability p e , that is, B ( n − 1 , p e ) and P p e ( | Γ | = k ) =  n − 1 k  p k e (1 − p e ) n − k − 1 . Then the probabilit y of obtaining an activ e no de can b e defined conditionally on the even t {| Γ | = k } , the neighbourho o d ha ving size k . Then marginalising o v er the p ossible the neigh b ourhoo d size, we obtain p rg for the probabilit y of a no de being active in the bi- nomial pro cess. Proofs can be found in the App endix. Lemma 4 (Pr ob ability on a r andom gr aph) Consider a PCA on a r andom gr aph G rg ( n, p e ) with e dge pr ob ability p e and a lo c al rule φ de- termine d by the majority rule for the neigh- b ourho o d | Γ | = k ac c or ding to ξ k as in (4). Then the pr ob ability of obtaining an active no de at time t + 1 is p rg ( ρ t ) = n − 1 X k =0 k X r =0 ξ k ( r )  k r  ρ r t (1 − ρ t ) k − r P p e ( | Γ | = k ) . (8) In tuitively we w ould exp ect that if w e restrict the neighbourho ods in (8) to the exp ected neigh b ourhoo d size p e ( n − 1) for each node, then the approximation should b e reasonably close. This would make it p ossible to analyt- ically determine fixed p oin ts more easily and simplify computation for large graphs consid- erably . W e next sho w that this is a reasonable approac h. Lemma 5 (Simple pr ob ability on a r andom gr aph) If the neighb ourho o d in G rg ( n, p e ) of e ach no de in a PCA as in L emma 4 is fixe d with the exp e cte d numb er of no des under the r andom gr aph ν = b p e ( n − 1) c such that k = ν in p grid , then the pr ob ability p rg r e duc es to p ν grid ( ρ t ) = ν X r =0 ξ ν ( r )  ν r  ρ r t (1 − ρ t ) ν − r . (9) The appr oximation err or is | p rg − p ν grid | ≤ | p − 1 / 2 | × 2 exp( − ( n − 1) ε 2 /p e (1 − p e ) + log( n )) , as ε de cr e ases to 0 and with 0 < p e < 1 . This implies of course that the n umber of ac- tiv e nodes nρ t in the random graph with prob- abilit y p rg in (8) and the n umber of active no des with probability p ν grid in (9) conv erge in probabilit y with exp onen tial rate with graph size n . Remark T o retain a probability of an edge in p ν grid leads to a larger approximation error, i.e., 6 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ µ ( ρ ) rg grid n = 25 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ µ ( ρ ) rg grid n = 100 FIG. 1: The exp ectation p rg (blue, solid curv e) of equation (8) and p rmr and (red, dotted curv e) of equation (9) with p = 0 . 1 and p e = 0 . 3. Left panel shows the curves for a graph of size n = 25, showing a clear difference b et w een the curves, and the right panel for graph size n = 100. Note that the difference b et w een the curves at the crossings with the 45 ◦ line is small. using p n − 1 grid ( ρ t ) = n − 1 X r =0 ξ ν ( r )  n − 1 r  ( ρ t p e ) r (1 − ρ t p e ) n − r − 1 (10) leads to an error of at most | p − 1 / 2 | , which mak es it non ignorable (see the App endix). W e now hav e expression (9) similar to (4) for a random graph with the probability of an active no de at time t determined by both the density ρ t and an edge b eing present p e in the size of the neighbourho od. F rom (8) in Lemma 4 and Corollary 2 and Lemma 5 the evolution equation for the random graph follows. Theorem 7 (Evolution on a r andom gr aph) L et a PCA with lo c al rule φ b e define d on a r andom gr aph G rg ( n, p e ) , and let the pr ob abil- ity of obtaining an active no de in G rg ( n, p e ) b e p ν grid as define d in (9). Then the evolution e quation for the r andom gr aph with lar ge n is nρ t +1 = B ( n, p ν grid ( ρ t )) . (11) The me an and varianc e of the density ρ t r e- sp e ctively, µ ν grid = p ν grid and σ 2 grid ,ν = p ν grid (1 − p ν grid ) /n . W e immediately ha v e that the probabilit y p ν grid is close to the density ρ for eac h time p oin t t for large graph size n . In fact, we find b y the triangle inequality | ρ − p ν grid | ≤ | ρ − p rg | + | p rg − p ν grid | , and b oth terms con verge to 0. The first term | ρ − p rg | conv erges to 0 b y Lemma 3 with p rg and using Lemma 4, and | p rg − p ν grid | conv erges to 0 b y Lemma 5. The pro cess nρ t on a random graph is also a discrete time Marko v pro cess, as b efore, and has transition probabilit y (6) with p ν grid . W e can then apply a similar analysis of dynamics to µ ν grid = p ν grid as b efore. Note that we require for obvious reasons that the graph is connected. It follows that w e need a minim um probabilit y p e suc h that the graph is connected. The probability that a random graph G rg is connected is exp( − exp( − λ )), where p e = (log n + λ + o (1)) /n with λ fixed [24, Theorem 7.3]. F or instance, if w e c ho ose the probabilit y of G rg 7 b eing connected to be 0.99 and we use n = 50, then we obtain λ = 4 . 6 and hence p e = 0 . 17. W e can therefore not go b elo w 0.17 for a graph with n = 50 no des. C. Small-w orld graph A small-w orld graph is one which has high av erage clustering and low a verage path length, relativ e to a random graph with the same num b er of nodes and edges. These graphs ha ve b een shown to model realistic net- w orks lik e those of working relations b et w een actors and the nerve cells in the worm C. el- egans [26], and subsequen tly the small-w orld has b een sho wn to apply to man y different net- w orks, lik e the (parcellated) brain [27]. And most recen tly , the net work of symptoms as de- fined by the diagnostic statistical manual (a comp endium to diagnose patients) has b een found to be a small-w orld. This finding is a p ossible explanation for the correlations b e- t ween pairs of symptoms found in different subp opulations [1]. Here we use the mo dified Newman-W atts (NW) small-world of Newman and W atts [14], where for a giv en grid structure where each no de has neighbourho od Γ, a set of ( n − 1) p w edges is on av erage indep enden tly added to the graph, where p w is the probability of tw o no des being wired. Suc h a graph is denoted b y G sw ( n, Γ , p w ). The same idea as with the random graph, where the probability for an ac- tiv e no de was corrected b y the probability of the degree of a no de, a v eraged o v er all p ossible neigh b ourhoo d sizes, can b e used for the ran- dom part in the NW small-world. In the NW small-w orld w e start with a grid with neigh- b ourhoo d size | Γ | , which is fixed, and augmen t the graph randomly with edges according to a binomial v ariable with probability p w . Let P p w ( | Γ | = k ) =  n − 1 k  p k w (1 − p w ) n − k − 1 W e then obtain p sw ( ρ t ) = n − 1 X k = | Γ | k −| Γ | X r =0 ξ k ( r )  k r  ρ r t (1 − ρ t ) k − r P p w ( | Γ | = k ) . (12) W e could define the small-world probabilit y using this definition. But w e can split up p sw in tw o terms, one inv olving the fixed neigh- b ourhoo d Γ of the grid, and one random neigh- b ourhoo d consisting of the p ossible shortcuts. W e therefore start with the probability in a grid p grid corrected b y the (1 − p w ) n −| Γ | requir- ing that no possible randomly added edges are presen t, i.e., we obtain p sw grid = p grid (1 − p w ) n −| Γ | (13) for the first part of the fixed grid. Then, in accordance with the random part of the NW small-w orld, a probability is added to emulate the p ossible additional neigh b ours in the ran- dom part of the graph, ignoring the first | Γ | neigh b ours from the grid. Define the proba- bilit y p rg , Γ ( ρ t ) = n − 1 X k = | Γ | +1 k −| Γ | X r =0 ξ k ( r )  k r  ρ r t (1 − ρ t ) k − r P p w ( | Γ | = k ) , (14) where the first | Γ | neigh b ours are ignored since they were included already as neighbours in the grid structure in p sw grid . Then we can write the small-world probability as p sw = p grid (1 − p w ) n −| Γ | + p rg , Γ . F or the second part, ho wev er, w e hav e the approximation as b efore from the random graph, lea ving out the first Γ nodes from the grid. This leads to the simplification using the grid probability only p ν grid , Γ ( ρ t ) = ν X r = | Γ | +1 ξ ν ( r )  ν r  ( ρ t ) r (1 − ρ t ) ν − r , (15) where ν = b p w ( n − | Γ | ) c . The error of appro xi- mation using p ν grid , Γ instead of p rg , Γ follo ws im- mediately from Lemma 5 for fixed grid neigh- b ourhoo d Γ, except that the first | Γ | no des in the grid are taken out. 8 Corollary 8 L et G sw ( n, Γ , p w ) b e the NW smal l-world gr aph of size n with | Γ | no des in the fixe d neighb ourho o d for e ach no de. F ur- thermor e, let 0 < p w < 1 b e the wiring pr ob a- bility and ν = b p w ( n − | Γ | ) c . Then the appr ox- imation err or for the pr ob ability using the grid structur e p ν grid in (15) in the r andom p art is | p rg , Γ − p ν grid , Γ | ≤ | p − 1 / 2 | × 2 exp( − ( n − | Γ | ) ε 2 /p w (1 − p w ) + log( n − | Γ | + 1)) , for ε > 0 . Equations (12) to (15) and Corollary 8 prov e the following equation for the evolution on an NW small-world. Theorem 9 (Evolution on a smal l-world) Define a PCA on a Newman-Watts smal l- world gr aph G sw of size n with | Γ | neighb ours for the initial gr aph and wiring pr ob ability p w . Then with pr ob abilities p sw grid in (13) and p ν grid , Γ in (15), the evolution of the numb er of active no des is nρ t +1 = B ( | Γ | , p sw grid ( ρ t )) + B ( n − | Γ | , p ν grid , Γ ( ρ t )) (16) with me an and varianc e for the density ρ t , r e- sp e ctively, µ sw = | Γ | n p sw grid + n − | Γ | n p ν grid , Γ (17) and σ 2 sw = | Γ | n 2 p sw grid (1 − p sw grid ) + n − | Γ | n 2 p ν grid , Γ (1 − p ν grid , Γ ) . (18) W e write p ν sw = p sw grid ( | Γ | /n ) + p ν grid , Γ (( n − | Γ | ) /n ) for the NW small-world probabilit y based on the approximation with the grid. Figure 2 shows tw o examples of the approx- imation p ν grid , Γ for the random part in the NW small-w orld. It is clear from the corollary that con vergence is a bit slo w for small graphs since the difference of nodes in the fixed neighbour- ho od Γ and in the exp ected p w ( n − | Γ | ) neigh- b ours in the random part, determines the rate. Again, w e can use p ν sw to determine the dy- namics of the mean field. IV. D YNAMICS OF THE MEAN FIELD T o inv estigate the dynamics we treat the mean field function p grid for the grid, p ν grid for the random graph, and p sw for the NW small- w orld as a discrete dynamical system. W e can then determine in principle the fixed p oin ts and describe its b eha viour in the long term. In general, how ever, obtaining the fixed points is not trivial. Indeed, Balister et al. [17] pro- vide an analytical solution for the fixed p oin ts for a sp ecific case in G grid , but men tion that a general solution is difficult. Janson et al. [20] giv e analytical solutions for the fixed points when lea ving out the ma jorit y rule, making it a deterministic system. Here we keep the ma jority rule sacrificing the p ossibilit y of de- termining the critical p oin ts analytically . W e therefore describ e the qualitative b eha viour of p grid , p ν grid , and p sw . A. Dynamics of the mean field in a grid The dynamics of the mean field in the grid G grid from (4) hav e been describ ed in Balister et al. [17] and Kozma et al. [8] for a neighbour- ho od size of | Γ | = 5. The function µ grid = p grid is con tinuous and since [0 , 1] is closed and b ounded, we find that µ grid has at least one fixed point in [0 , 1] [22, 28]. A fixed p oin t is one where we find p grid ( ρ t ) = ρ t . Finding the fixed p oin ts for p grid in general is not trivial. Balister et al. [17] sho w ed that if | Γ | = 5 in the finite grid, then p = 7 / 30 ≈ 0 . 233 is a critical point, such that if p is in [7 / 30 , 1 / 2] then there is a stable fixed p oin t at ρ = 0 . 5, but when p < 7 / 30 then ρ = 0 . 5 is unstable and there are t w o other stable fixed p oin ts. This can b e seen in Figure 3, whic h shows t wo bifurcation plots, where for each v alue of 0 < p ≤ 0 . 5 the function µ grid = p grid is iterativ ely applied for ab out 1000 steps, and 9 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ µ ( ρ ) sw-rg sw-grid n = 25 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ µ ( ρ ) sw-rg sw-grid n = 100 FIG. 2: The exp ectation p sw (blue, solid curv e) of equation ( ?? ) and p ν sw (red, dotted curv e) of equation (9) with p = 0 . 1 and p w = 0 . 3. Left panel shows the curves for a graph of size n = 25, sho wing a clear difference b et w een the curves, and the right panel for graph size n = 100. Note that the difference b et w een the curves at the crossings with the 45 ◦ line is small. only the last 50 are plotted. Figure 3 sho ws that for | Γ | = 5 neighbours the fixed p oin t at is stable for p ∈ [0 , 7 / 30) and bistable for p ∈ [7 / 30 , 0 . 5], as predicted. Since p grid is con tinuous, stability can be c hec k ed by con- sidering the deriv ativ e ∂ µ grid /∂ ρ = ˙ µ grid . If | ˙ µ grid | is b ounded b y 1, then the fixed p oin t ρ is attractive, otherwise it is rep ellen t. The deriv ative with resp ect to ρ t is ˙ µ grid ( ρ t ) = | Γ | X r =0 ξ | Γ | ( r )  | Γ | r  ( r − ρ t | Γ | ) ρ r − 1 t (1 − ρ t ) | Γ |− r − 1 F or example, the deriv ativ e for p = 0 . 15 is not b ounded b y 1 for all v alues of ρ ; the fixed p oin t ρ = 0 . 5 is rep ellen t since at this p oin t ˙ µ (0 . 5) ≈ 1 . 359, and so iteration of µ will lead a wa y from 0.5. The deriv ativ e for p = 0 . 35 is smaller than 1 (0.672) and so in this case ρ = 0 . 5 is an attractive fixed p oin t. It can b e seen that for | Γ | = 5 the critical p oin t is at 0.233, as predicted by theory [17]. It can also b e seen that increasing the neigh b ourhoo d size to | Γ | = 15 (righ t panel) increases the criti- cal point. This increase in critical p oin t cor- resp onds to the simulations in Kozma et al. [8] where (’long range’) edges w ere added to the no des, which increased the neigh b ourhoo d size. B. Dynamics of the mean field in a random graph The dynamics of p ν grid in the random graph G rg are similar to that of the grid. The main difference is that the critical point of the bi- furcation is closer to p = 1 / 2. As is clear from the definition of p ν grid in (9), the only difference with that of the grid is the neighbourho o d size whic h is increased to ν = b p e ( n − 1) c . Figure 4 shows the result for a graph with n = 25 no des (left panel) and for a graph with n = 100 no des. The approximation of p ν grid is quite accurate, also for the lo cation of the critical p oin t. With a graph of size n = 100 the accu- racy is suc h that p rg and p ν grid are nearly indis- tinguishable, which corresp onds to the result in Lemma 5. 10 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 p µ ( ρ ) | Γ | = 5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 p µ ( ρ ) | Γ | = 15 FIG. 3: Bifurcation plots of p grid in a graph of size n = 100 for | Γ | = 5 and | Γ | = 15 neighbours. 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 p µ ( ρ ) rg g ri d n = 2 5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 p µ ( ρ ) rg g ri d n = 1 0 0 FIG. 4: Plots of µ rg (red) and µ ν grid (blue) as a function of p . In the left panel the bifurcation plots are giv en for a random graph of size n = 25 and in the right panel for a random graph of size n = 100. All plots are obtained with edge probabilit y p e = 0 . 4. C. Dynamics of the mean field in a small-w orld The dynamic b eha viour of p sw is sho wn in Figure 5. Generally , the b eha viour is similar to that on the random graph. In Figure 5 the left panel sho ws a bifurcation plot of p sw and and p ν sw on G sw (49 , 0 . 4). The accuracy of p ν sw impro ves greatly for larger n , as seen in the righ t panel of Figure 5 for G sw (100 , 0 . 4). F or lo w v alues of new edges in the NW small-w orld p w , the probability p ν sw is smaller than in the grid. This is b ecause the probability p sw grid = p grid (1 − p w ) n −| Γ | is corrected by the num ber of edges not added to the graph. 11 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 p µ ( ρ ) sw -rg sw -g ri d n = 4 9 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 p µ ( ρ ) sw -g ri d sw -rg n = 1 0 0 FIG. 5: Bifurcation plots of the small-world mean field µ ν sw ( ρ ) based on the grid with ν = b p e ( n − 1) c neigh b ours (blue) and the mean field µ sw ( ρ ) based on the all p ossible neigh b ours in the random graph (red). In the left panel a small-world of n = 25 no des and in the right panel a graph of n = 100 no des; all graphs are obtained with the probability of wiring (adding edges) in the NW small-world of p w = 0 . 4. V. NUMERICAL EV ALUA TION OF THE MEAN FIELD T o ev aluate the accuracy of the predictions of the mean field in the grid, random, and small-w orld graph, w e sim ulated netw orks of differen t sizes in the topology of a grid, a ran- dom graph, and a small-world graph. F or each sim ulation run 0/1 data were generated ac- cording to one of the three types of graph using the R pac k age IsingSampler [2]. The in com- bination with the ma jority rule the PCA was run for a certain duration T and the a v erage states of the last section of the time series was determined to see if it matc hes that of the pre- dictions of the mean field. T o determine the accuracy w e used b oth 90% and 95% confi- dence in terv als obtained from the cen tral limit theorem for each of the three different graphs (see Section I II A). F or eac h combination of parameters, 100 graphs w ere sim ulated in the top ology of an un weigh ted grid, a random graph, and a small- w orld graph. W e v aried the size of the net- w ork n ∈ { 16 , 25 , 49 , 100 } , the num b er of time p oin ts T ∈ { 50 , 100 , 200 , 500 , 5000 } , and the probabilit y of an activ e no de in the ma jority rule p ∈ { 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 } , see (2). W e also v aried the probability of an edge in the random graph p e ∈ { 0 . 1 , 0 . 2 , . . . , 0 . 9 } , and the probabilit y of wiring in the small-w orld graph p w ∈ { 0 . 1 , 0 . 2 , . . . , 0 . 9 } . F or t = 0, a random n umber of no des w as set to activ e b y using the R pac k age IsingSampler version 0.2 [29]. Figures 6 and 7 show the top ology of some sim ulated random graphs (6) and small-world graphs (7) arranged in a square. All sim u- lated data, figures, as well as the used R-co de are publicly a v ailable at the OSF [30]. Re- sults in bifurcation diagrams with 90% and 95% confidence interv als are sho wn in Figure 8. Mean densit y estimates, visualized with red dots, w ere calculated b y dividing eac h sim ula- tion in to snipp ets in which the distance b e- t ween density estimates ( δ ) did not exceed 0 . 4. Other v alues for δ w ere considered, but it was found that c hanging δ to 0 . 3 or 0 . 5 re- sulted in similar percentages. W e calculated for eac h snipp et the p ercen tage of mean den- sit y estimates that fell within a 90% and 95% confidence interv al. T able I sho ws the mean p ercen tages, marginalized ov er the n umber of time p oin ts T , the v ariation in p e , p w and the n umber of no des n obtained from the central 12 n = 16 p e = 0 . 1 n = 25 n = 49 n = 100 p e = 0 . 2 p e = 0 . 3 p e = 0 . 4 p e = 0 . 5 FIG. 6: Examples of simulated random graphs with netw ork size n ∈ { 16 , 25 , 49 , 100 } and p e ∈ { 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 } . limit theorem with b ounds ˆ ρ ± 1 . 96 for the 95% confidence interv al, and ˆ ρ ± 1 . 64 for the 90% confidence interv al. Figure 9 gives a 3- dimensional represen tation of the p ercen tage of mean density estimates that fall within a 95% confidence in terv al. Results from the 90% confidence in terv al are not presented, as these w ere similar to the results from the 95% con- fidence interv al. It can b e seen that the mean field appro ximation accurately estimates the densit y of the netw ork structures across v ari- ous simulation conditions. As seen in Figure 9, a lo cal dip o ccurs for all net w ork structures at p = 0 . 3 and b ecomes more extreme as n increases in size. Also sho wn in Figure 8, the mean density estimates at p = 0 . 3 fall less often in the 95% confi- dence interv al in comparison to the mean den- 13 n = 16 p w = 0 . 1 n = 25 n = 49 n = 100 p w = 0 . 2 p w = 0 . 3 p w = 0 . 4 p w = 0 . 5 FIG. 7: Examples of simulated small world graphs with netw ork size n ∈ { 16 , 25 , 49 , 100 } and p w ∈ { 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 } . sit y estimates at other v alues for p . This is partly due to the fact that the standard er- ror, a parameter needed for the calculation of the confidence in terv als, dep ends on the net- w ork size; as the netw ork size n increases, the standard error b ecomes smaller as well as the resulting confidence in terv al. F urthermore, as this phenomenon o ccurs in all simulated net- w ork structures, w e believe that this behaviour results from the fact that the mean field ap- pro ximation has a bit of trouble adjusting to the one-phase stability , after b eing in an area where phase transitions may occur. All in all, results sho w that the mean field appro xima- tion also p erforms w ell when non-regular net- w ork structures are under consideration. Figure 10 visualizes the ev olution of selected sim ulation conditions. Phase transitions were 14 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p t = 50 n = 16 T orus 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p t = 500 n = 49 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p t = 5000 n = 100 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p Random graph 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p Small- w orld graph 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p 0.1 0.2 0.3 0.4 0.5 −0.25 0.25 0.75 1.25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ ρ p FIG. 8: Bifurcation diagrams of a torus (upp er panel), a random graph (middle panel; p e = 0 . 5) and a small w orld graph (low er panel; p w = 0 . 5). Grey solid area = 90% confidence in terv al around bifurcation. Dashed grey lines = 95% confidence interv al around bifurcation. Red dots = mean densit y estimates at different v alues of p . obtained for the random graph, specifically at p e = { 0 . 3 , 0 . 4 } and p = { 0 . 3 , 0 . 4 } . W e also ob- tained phase transitions for the small world graph at p w = { 0 . 9 } and p = { 0 . 1 } . F or the duration in our simulations, we did not obtain phase transitions in an y other sim ula- tions. How ev er, as w e observed stable chains throughout the simulation study , w e exp ect to obtain phase transitions in all conditions when t approaches infinit y . VI. CONCLUSIONS AND DISCUSSION T o model the complex dynamics of large- scale net w orks (graphs) is in general diffi- cult. This is because there are man y differ- 15 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 p % in interval n = 16 T orus 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 p % in interval n = 25 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 p % in interval n = 49 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 p % in interval n = 100 p p(e) % in interval Random graph p p(e) % in interval p p(e) % in interval p p(e) % in interval p p(w) % in interval Small- w orld graph p p(w) % in interval p p(w) % in interval p p(w) % in interval FIG. 9: Visualization of the p ercen tage of mean density estimates that fall within a 95% confidence interv al in 2D (torus; left column) and 3D (random graph; middle column, and small world graph; right column) at t = 500. en t ’agents’ that operate within the graph. In particular, if the no des in the graph repre- sen t symptoms and the edges represen t their m utual influence, then the interacting symp- toms sho w complex b eha viour on a macro- scopic scale, e.g., at the lev el of the n umber of activ e symptoms. Here we show ed that the mean field model for a probabilistic cel- lular automaton with ma jorit y rule, can serve as an accurate approximation to such large- scale graphs, and can simplify analysis of the dynamics of such systems. Sp ecifically , we sho wed that av eraging across the differen t pos- sible degrees for a random and small-world graph, results in approximations that lie with high probability close to a generalised version of the mean field on a torus. These theoret- ical results were confirmed by extensiv e sim- ulations, sho wing correspondence b et w een the mean field and simulated graph activit y for dif- feren t size graphs. Our approximation is based on the formula- tion of the grid (torus) where a relatively sim- ple sum ov er p ossible active no des determines the probabilit y of a randomly selected no de in the graph b eing active. W e sho wed that for large graphs this approximation is accu- rate. This simplification could serv e to obtain a more extensiv e analysis of the dynamics suc h as that presented in Janson et al. [20]. There the ma jority rule (the probabilistic element) w as remov ed from the model, to obtain ex- act fixed p oin ts for the mo del. Here w e chose not to remo v e the probabilistic element since w e aim to introduce different rules for up dates than the ma jority rule, like a conditional Ising probabilit y . Our initial motiv ation for these results w as to obtain a model where we could assess the risk of a single p erson based on the estimate of the graph and the corresp onding probabil- it y of an activ e no de p , to determine the risk of that p erson ’jumping’ from one state into another. This risk assessmen t might b e useful 16 T ABLE I: Percen tages of densit y estimates with a 90% or a 95% confidence in terv al. p e = edge probabilit y (random graph). p w = rewiring probabilit y (small-world graph). Structure T N p e / p w p 90%CI 95%CI T orus 50 16 - 0.1 0.75 0.80 0.2 0.99 0.99 0.3 0.92 0.99 0.4 0.94 0.99 0.5 0.96 0.99 500 49 - 0.1 0.29 0.67 0.2 0.97 0.99 0.3 0.63 0.75 0.4 0.75 0.83 0.5 0.81 0.87 5000 100 - 0.1 0.12 0.55 0.2 1.00 1.00 0.3 0.50 0.50 0.4 0.75 0.83 0.5 0.75 0.75 Random graph 50 16 0.5 0.1 0.97 0.97 0.2 1.00 1.00 0.3 0.55 0.76 0.4 0.78 0.88 0.5 0.85 0.92 500 49 0.5 0.1 1.00 1.00 0.2 1.00 1.00 0.3 0.98 0.98 0.4 0.92 0.97 0.5 0.94 0.98 5000 100 0.5 0.1 1.00 1.00 0.2 1.00 1.00 0.3 0.87 0.91 0.4 0.98 1.00 0.5 0.99 1.00 Small-w orld graph 50 16 0.5 0.1 0.34 0.76 0.2 0.20 0.43 0.3 0.38 0.59 0.4 0.54 0.69 0.5 0.63 0.75 500 49 0.5 0.1 1.00 1.00 0.2 1.00 1.00 0.3 1.00 1.00 0.4 0.98 0.99 0.5 0.99 1.00 5000 100 0.5 0.1 1.00 1.00 0.2 1.00 1.00 0.3 0.99 0.99 0.4 1.00 1.00 0.5 1.00 1.00 in a clinical setting where a decision in a par- ticular t yp e of in terv ention is required. This idea is pursued in the companion pap er in this issue. APPENDIX Pro of (Lemma 3) Let the Kullbac k-Leibler div ergence b et ween p + ε and p b e defined as 17 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 µ ρ t 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 µ ρ t 0 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 µ ρ t FIG. 10: Examples of the evolution of a torus (upp er panel; p = 0 . 1), a random graph (middle panel; p = 0 . 3 , p e = 0 . 6) and a small w orld graph (low er panel; p = 0 . 1 , p w = 0 . 9). a function of 0 ≤ ε ≤ p h + ( ε ) = ( p + ε ) log p + ε p + (1 − p − ε ) log 1 − p − ε 1 − p and similarly , define h − ( ε ) = h + ( − ε ). Then Cherno v’s b ound [23, 31] for the density ρ t of a grid with n no des and its mean at time t , p grid ( ρ t ) defined in (4), for 0 < ε < min { p grid , 1 − p grid } immediately giv es P ( | ρ t − p grid ( ρ t ) | > ε ) ≤ exp( − nh + ( ε )) + exp( − nh − ( ε )) The Kullback-Leibler divergence can b e ap- pro ximated quadratically by h + ( ε ) = ε 2 2 p (1 − p ) + O ( ε 3 ) as ε → 0 18 Similarly for h − ( ε ) gives P ( | ρ t − p grid ( ρ t ) | > ε ) ≤ 2 exp( − ε 2 / 2 σ 2 grid ( ρ t )) (19) where σ 2 grid = p grid (1 − p grid ) /n . Let δ = 2 exp( − ε 2 / 2 σ 2 grid ( ρ t )) such that ε = q 2 σ 2 grid log(2 /δ ). Then w e obtain the result with probability at least 1 − δ . 2 Pro of (Lemma 4) By Theorem 2.1 in Balister et al. [17] we ha ve that the probabilit y of state 1 at time t + 1 on the even t {| Γ | = k } and giv en ρ t is P (Φ t ( x ) = 1 | | Γ | = k , ρ t ) = k X r =0 ξ k ( r )  k r  ρ r t (1 − ρ t ) k − r and in the random graph of Erd¨ os-Ren yi the probabilit y of k neigh b ours for any no de is P p e ( | Γ | = k ) =  n − 1 k  p k e (1 − p e ) n − k − 1 It follows that the marginal P k P (Φ t ( x ) = 1 | | Γ | = k , ρ t ) P ( | Γ | = k ) is p rg ( ρ t ) = n − 1 X k =0 k X r =0 ξ k ( r )  k r  ρ r t (1 − ρ t ) k − r P p e ( | Γ | = k ) as claimed. 2 Pro of (Equation 10) T o obtain (10) assume a fixed v alue ν for all k , ξ k = ξ ν for all k . Then ξ ν ( r ) only dep ends on r . First note that  k r  n − 1 k  =  n − 1 r  n − r − 1 k − r  Second, observ e that the order of the tw o sums in p Φ ,G can be switched since the first sum is for the sequence ( k = 0 , 1 , . . . , n − 1) and the second is ( r = 0 , 1 , . . . , k ), which can b e seen as the lo wer triangular of the t wo- dimensional array for ( k = 0 , 1 , . . . , n − 1) and ( r = 0 , 1 , . . . , n − 1). Up on switching to ( r = 0 , 1 , . . . , n − 1) and ( k = r , r +1 , . . . , n − 1), w e obtain the upp er triangle of the array . Sec- ond, by changing the order of summation and reordering the sums, we get n − 1 X r =0 ξ ν ( r )  n − 1 r  ρ r t p r e n − r − 1 X k = r  n − r − 1 k − r  × ( p e (1 − ρ t )) k − r (1 − p e ) n − k − 1 In the sum on the righ t we can use the bino- mial theorem with m = k − r and N = n − r − 1, whic h gives N X m =0  N m  ( p e (1 − ρ t )) m (1 − p e ) N − m = ( p e (1 − ρ t ) + 1 − p e ) N whic h leads to (10). F or the approximation error, write ξ k ( r ) = p 1 { r ≤ k / 2 } + (1 − p ) 1 { r > k / 2 } and recall that ν is fixed. Then p rg ( ρ t ) − p rand ( ρ t ) = n − 1 X k =0 k X r =0  k r  ρ r t (1 − ρ t ) k − r  n − 1 k  × p k e (1 − p e ) n − k − 1 [ ξ k ( r ) − ξ ν ( r )] Using H¨ older’s inequality with the ` ∞ and ` 1 norms, gives | p rg ( ρ t ) − p rand ( ρ t ) | ≤ n − 1 X k =0 k X r =0  k r  ρ r t (1 − ρ t ) k − r  n − 1 k  × p k e (1 − p e ) n − k − 1 max r,k | ξ k ( r ) − ξ ν ( r ) | The binomial theorem for the first term of the right hand side giv es n − 1 X k =0 k X r =0  k r  ρ r t (1 − ρ t ) k − r  n − 1 k  p k e (1 − p e ) n − k − 1 = n − 1 X r =0  n − 1 r  ( ρ t p e ) r (1 − ρ t p e ) n − r − 1 = 1 . 19 F or each r , k such that r ≤ k w e hav e that ξ k ( r ) − ξ ν ( r ) = p ( 1 { r ≤ k / 2 } − 1 { r ≤ ν / 2 } )+ (1 − p )( 1 { r > k / 2 } − 1 { r > ν / 2 } ) The term | ξ k ( r ) − ξ ν ( r ) | is at most 2 p − 1 if ν < k or 1 − 2 p if ν ≥ k for any r, k , which giv es the size of the error b ound. 2 Pro of (Lemma 5) If we fix ν = b p e ( n − 1) c , the exp ectation of the random v ariable for eac h no de of the p ossible num b er of neigh- b ours B ( n − 1 , p e ), such that each k = ν in the part for the densit y we obtain n − 1 X k =0  n − 1 k  p k e (1 − p e ) n − k − 1 × ν X r =0  ν r  ξ ν ( r ) ρ r t (1 − ρ t ) ν − r ! , from which we obtain p ν grid ( ρ t ). The appro xi- mation error for the probabilities is then | p rg ( ρ t ) − p ν rand ( ρ t ) | =      n − 1 X k =0 ( p k grid ( ρ t ) − p ν grid ( ρ t )) P p e ( | Γ | = k )      . The probabilit y of obtaining a neigh b ourhoo d size k close to the exp ected num b er of neigh- b ours ν = p e ( n − 1) can b e obtained from the Cherno v bound in Lemma 3, giving P ( | k − ν | ≤ t ) ≥ 1 − 2 exp( − ( n − 1) ε 2 /p e (1 − p e )), for ε & 0. This leads to the difference b eing b ound b y | p rg ( ρ t ) − p ν rand ( ρ t ) | ≤     n − 1 X k =0 ( p k grid ( ρ t ) − p ν grid ( ρ t )) × 2 exp( − ( n − 1) ε 2 /p e (1 − p e ))     . Using H¨ older’s inequality with the sup and ` 1 norms, we find that the ab o v e is ≤ max k | p k grid ( ρ t ) − p ν grid ( ρ t ) | × n − 1 X k =0 2 exp( − ( n − 1) ε 2 /p e (1 − p e )) . The difference p k grid ( ρ t ) − p ν grid ( ρ t ) is deter- mined by the mismatc h b et w een k and ν and is at most 2 p − 1 if ν < k and 1 − 2 p if ν ≥ k for any r , k . And so we obtain | p rg ( ρ t ) − p ν grid ( ρ t ) | ≤ | p − 1 / 2 | × 2 exp( − ( n − 1) ε 2 /p e (1 − p e ) + log( n )) , completing the pro of. 2 Pro of (Theorem 7) In the random graph G rg ( n, p e ) each no de has the same num ber n − 1 of p ossible neighbours. Hence, for each no de the lo cal rule φ is determined b y r out of n − 1 p ossible neigh b ours b eing b oth con- nected and 1. Lemma 4 shows that the proba- bilit y of this binomial process is then p rg in (8). Then by Lemma 5 for large n the probability p ν grid con verges to p rg , and b y consequence, the n umber of activ e nodes in b oth pro cesses con- v erges. 2 [1] D. Borsb oom, A. O. J. Cramer, V. D. Sc hmittmann, S. Epsk amp, and L. J. W al- dorp, PLoS ONE 6 , e27407 (2011). [2] C. D. v an Borkulo, D. Borsb oom, S. Ep- sk amp, T. F. Blank en, L. 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