Competition of simple and complex adoption on interdependent networks
CCompetition of simple and complex adoption on interdependent networks
Agnieszka Czaplicka, Raul Toral, Maxi San Miguel
Instituto de F´ısica Interdisciplinar y Sistemas Complejos,IFISC (CSIC-UIB), Campus UIB, 07122 Palma de Mallorca, Spain (Dated: October 11, 2018)We consider the competition of two mechanisms for adoption processes: a so-called complexthreshold dynamics and a simple Susceptible-Infected-Susceptible (SIS) model. Separately, thesemechanisms lead, respectively, to first order and continuous transitions between non-adoption andadoption phases. We consider two interconnected layers. While all nodes on the first layer followthe complex adoption process, all nodes on the second layer follow the simple adoption process.Coupling between the two adoption processes occurs as a result of the inclusion of some additionalinterconnections between layers. We find that the transition points and also the nature of the tran-sitions are modified in the coupled dynamics. In the complex adoption layer, the critical thresholdrequired for extension of adoption increases with interlayer connectivity whereas in the case of anisolated single network it would decrease with average connectivity. In addition, the transition canbecome continuous depending on the detailed inter and intralayer connectivities. In the SIS layer,any interlayer connectivity leads to the extension of the adopter phase. Besides, a new transitionappears as as sudden drop of the fraction of adopters in the SIS layer. The main numerical findingsare described by a mean-field type analytical approach appropriately developed for the threshold-SIScoupled system.
I. INTRODUCTION
Dynamical collective phenomena emerging from inter-acting units show nontrivial dependencies on the topol-ogy and other characteristics of the network of interac-tions. Examples of this dependency in complex networksoccur in synchronization phenomena [1–3], in orderingdynamics, where coarsening only occurs below a criticaleffective dimension[4], or in the appearance of new typesof phase transitions, such as explosive percolation or ex-plosive synchronization [5–10]. In these phenomena, anadditional important aspect is the multilayering of thenetwork or the existence of underlying interdependentnetworks [11–14]. In general, dynamical processes in thistype of networks are non-reducible to dynamics in a singleeffective network [15, 16] and new forms of phase transi-tions and changes in the order of the transition have beenfound in such multilayared/interdependent networks [17–23]. While most studies of dynamics in complex net-works isolate a single dynamical process, many real sit-uations involve the coupling of two processes, a situa-tion most naturally described in the context of multi-layer/interdependent networks [11, 24–26]. In this pa-per we consider the situation of interdependent networkswhere nodes in each layer are distinct entities which fol-low only one type of contagion mechanism, but due tointerconnections are influenced by the nodes from bothlayers. We address the question of the effect of couplingtwo dynamical processes featuring, respectively, a discon-tinuous and a continuous phase transition.As a specific illustration we focus on contagion pro-cesses that describe adoption phenomena. The adoptionof an innovation or a new technology can follow from pro-cesses of simple or complex contagion [27–29]. In simplecontagion a node in the network adopts by interactingwith a single neighbor who has already adopted, in the same way than in an infection process. On the contrary,in complex contagion, adoption requires simultaneous ex-posure to multiple neighboring nodes that have alreadyadopted, so that adoption depends on the global state ofthe neighborhood. A prototype model for simple conta-gion is the SIS (Susceptible-Infected-Susceptible) model,known to have a continuous transition to the adoptionphase for a critical value of the infection rate [30, 31].Complex contagion is described by a threshold model [32]which has a discontinuous transition [33] to the adop-tion phase at a critical value of the fraction of neighborsrequired for individual adoption. A number of recentstudies [34–36] consider different aspects of this model,including comparison with available data from online in-teractions [28, 37–39]. While coupling of two simple [40–47] or two complex [48–50] contagion processes in mul-tilayared/interdependent networks has been discussed indifferent situations, our goal here is to consider the cou-pling of a SIS and a threshold contagion models, eachof them running in one of two interdependent networklayers. The proposed set-up is motivated by the study ofthe adoption of a given innovation by two interdependentpopulations each of which follows a different contagionmechanism, or alternatively the coupled adoption of twodifferent innovations each one associated with a differ-ent contagion mechanism. Concerning the fundamentalquestion of the coupling of a continuous and a discontinu-ous phase transition, we find that not only the transitionpoints and the nature of the transitions can change, butalso new transitions appear. These changes turn out todepend on the interlayer connectivity and on the asym-metry between the average intralayer connectivity of thetwo layers.This paper is organized as follows. In section II wedefine the model and give precise dynamical rules for theevolution of the state variables. In section III we in- a r X i v : . [ phy s i c s . s o c - ph ] N ov troduce a mean-field approximation which allows us toderive evolution equations for the density of adopters ineach layer and study the fixed points and their stability,finding the phase diagram in the uncoupled and coupledcases. The more technical details of the derivation ofthe evolution equations for the density of adopters in thethreshold and SIS layers are given, respectively, in ap-pendices A and B. In section IV we show the results ofnumerical simulations and compare them with the an-alytical predictions of the previous section. In sectionV we discuss the continuous or discontinuous orders ofthe different transitions found in the model. Finally, insection VI we summarize the main conclusions. II. MODEL
We consider two interdependent [11, 17–19] layers.Each layer (cid:96) = 1 , N (cid:96) nodes connected as a ran-dom Erd¨os-R´enyi (ER) topology with average degree k (cid:96) .Additionally, there are M interlinks randomly connectingnodes in both layers. Throughout the paper, we considerequal size layers, N = N ≡ N and denote by m = M/N the average number of interlinks per node. Each nodeholds a binary state variable s (cid:96),i , (cid:96) = 1 , , i = 1 , . . . , N ,taken as s (cid:96),i = 0 (not adopter, neutral) and s (cid:96),i = 1(adopter). Nodes in the two layers are distinct entitiessuch that each individual is subjected to only one typeof contagion mechanism, not to two simultaneously.In the first layer, (cid:96) = 1, nodes change their statesthrough a complex adoption process [28, 29], followinga variant of the threshold model rules [32]: a neutralnode switches to the adoption state when the fraction ofits adopter neighbors is above a threshold value θ . It isassumed that adoption is irreversible and an adopter cannot go back to the neutral state. On a single, uncou-pled, network the threshold model displays a discontinu-ous transition [33] at a critical value θ c which decreasesas 1 /k . Below the critical threshold θ < θ c all nodes inthe system become eventually adopters, while for θ > θ c adoption does not spread and only the initial group ofadopters remains.In the second layer, (cid:96) = 2, nodes evolve by a simple adoption process following a variant of the SIS dynam-ics: adoption spreads by pairwise interactions betweenadopter and non-adopter nodes. The probability that aninteraction between a non-adopter and an adopter nodeleads to adoption in the neutral node is λ . SIS dynamicsdoes allow adopter nodes to become neutral again. Asingle, uncoupled network, displays a continuous phasetransition at a critical value λ c : when λ < λ c all nodesend up in the non-adopter state, while for λ > λ c there isa non-vanishing fraction of nodes that become adopters.For our particular rules, we find λ c = 1 /k , see sectionIII C 1.When the networks are coupled, m >
0, the detaileddynamics is as follows: we start at t = 0 with a smallseed of adopters in the first layer (one randomly chosen FIG. 1. Schematic representation of the model’s dynamicsin two layers forming an interdependent network in which anode can interact with neighbors from both layers. In the up-per layer (Thr) nodes change their states through a complexadoption process following a threshold model rules. In thelower layer (SIS) nodes evolve by a simple adoption processfollowing the SIS type of dynamics. node and all its neighbours in the first layer) and a singleadopter in the second layer. At successive time steps t > (cid:96) and one node i from this layer, and its state is updated according to thefollowing rules (see Fig. 1 for a schematic representationof these rules): • If the node belongs to the first layer and- s ,i ( t ) = 0, then s ,i ( t + 1) = 1 if at least a fraction θ of its neighbours (from both layers) are adopters,- s ,i ( t ) = 1, nothing happens. Adopters cannot be-come neutral. • If the node belongs to the second layer and- s ,i ( t ) = 0, then all neighbors (from both layers)of the node are visited sequentially. Adoption fromany contact arises with probability λ .- s ,i ( t ) = 1, then it goes back to neutral state. III. MEAN-FIELD APPROACH TO SYSTEM’SDYNAMICSA. Threshold layer
Threshold dynamics on a single network had been pre-viously described analytically (e.g. see [33, 48, 51]). Inthese treatments the authors used assumptions that arenot valid in our particular case because of the couplingbetween two different types of dynamics. For instance,we can not assume in the SIS layer that a node can changeits state at most once during the evolution, or it is notconvenient to treat the network as tree-like for large m values. Therefore we develop in this paper an approachbased on a mean-field type approximation in which lo-cal fractions are replaced by global averages. As shownin Appendix A, under this approximation the evolutionequation for the fraction of adopters in threshold layer is β d (cid:104) s ( t ) (cid:105) dt = (1 − (cid:104) s ( t ) (cid:105) ) Prob [ (cid:104) s ( t ) (cid:105) n ≥ θ ] , (1)with β = N N + N and (cid:104) s (cid:105) n is the average fraction ofneighbours which are adopters.Although it is possible to relax this condition, to pro-ceed further we assume that each site i in the first layerhas exactly k neighbors in layer 1 and m neighbors inlayer 2. Within the mean-field approximation it can beassumed that the average fraction of neighbours whichare adopters is a weighted average of the average numberof neighbors in each layer: (cid:104) s (cid:105) n = k (cid:104) s (cid:105) + m (cid:104) s (cid:105) k + m . (2)The probability to find exactly j adopters amongst the k + m neighbors, given that the fraction of adopters is (cid:104) s (cid:105) n , follows a binomial distribution B ( (cid:104) s (cid:105) n , j ) ≡ (cid:18) k + mj (cid:19) ( (cid:104) s (cid:105) n ) j (1 − (cid:104) s (cid:105) n ) k + m − j . (3)To find the probability that a neutral node becomesadopter, we sum all cases when j ≥ (cid:98) ( k + m ) θ (cid:99) , where (cid:98) x (cid:99) denotes the largest integer not greater than x . Thisyields Prob [ (cid:104) s ( t ) (cid:105) n ≥ θ ] = k + m (cid:88) j = (cid:98) ( k + m ) θ (cid:99) B ( (cid:104) s (cid:105) n , j ) . (4)The binomial distribution Eq. (3) can be approximatedby a Gaussian distribution with mean ( k + m ) (cid:104) s (cid:105) n andvariance ( k + m ) (cid:104) s (cid:105) n (1 − (cid:104) s (cid:105) n ), leading toProb [ (cid:104) s ( t ) (cid:105) n ≥ θ ] ≈
12 erfc (cid:34) (cid:98) ( k + m ) θ ) (cid:99) − / − ( k + m ) (cid:104) s (cid:105) n ) (cid:112) k + m ) (cid:104) s (cid:105) n (1 − (cid:104) s (cid:105) n ) (cid:35) , (5)where erfc[ x ] is the complementary error function. Thisturns out to be a good numerical approximation for nottoo large values of k + m (an error small than 0 . θ for k + m = 10). In the limit of large k + m this can be further approximated byProb [ (cid:104) s ( t ) (cid:105) n ≥ θ ] ≈
12 erfc θ − (cid:104) s (cid:105) n (cid:113) k + m (cid:104) s (cid:105) n (1 − (cid:104) s (cid:105) n ) . (6)We still need a final ingredient to derive the evolutionequation for (cid:104) s ( t ) (cid:105) . As we are studying the evolutionin a finite system, if the probability Prob [ (cid:104) s ( t ) (cid:105) n ≥ θ ] issmaller that 1 /N , it means effectively that the conditioncan not be reached in the finite system. Therefore, weintroduce the function G ( x ) = (cid:40) Prob [ x ≥ θ ] , if Prob [ x ≥ θ ] ≥ N , , if Prob [ x ≥ θ ] < N , (7)and then write the evolution equation as: β d (cid:104) s ( t ) (cid:105) dt = (1 − (cid:104) s (cid:105) ) G (cid:18) k (cid:104) s (cid:105) + m (cid:104) s (cid:105) k + m (cid:19) , (8)where we have used Eq. (2). This is the final mean-fieldequation for the evolution of the density of adopters inthe threshold layer. B. SIS layer
In the usual Susceptible-Infected-Susceptible (SIS)model of infection [30, 31], the evolution rules are thata randomly chosen agent which happens to be adopter(“infected” state) can either infect one of its neighbourswith a given probability or can go back to the neutral(“susceptible” state); if the randomly chosen agent is al-ready in the neutral state, nothing happens. Here we usea slightly modified version of the SIS rules that we be-lieve are more appropriate to model adoption processes.In this version it is the neutral node the one adoptingfrom its adjacent neighbors adopters. The way of inter-action remains the same, pairwise interactions betweennodes are considered, but the direction of interaction hasbeen changed from outgoing (original SIS for infection)to ingoing (adoption process). In this way we keep thesymmetry in the interaction between the threshold andthe SIS layers and we implement two-way influence be-tween complex and simple adoption layers. By a similarreasoning to the one developed before for the thresholdlayer, we can derive the mean-field equation for the frac-tion of adopters in the SIS layer (details of the derivationare given in Appendix B)(1 − β ) d (cid:104) s ( t ) (cid:105) dt = −(cid:104) s (cid:105) +(1 − (cid:104) s (cid:105) ) (cid:16) − e − λ ( k (cid:104) s (cid:105) + m (cid:104) s (cid:105) ) (cid:17) . (9)Equations (8) and (9) are the starting point of our an-alytical treatment. In the next section we discuss thefixed points and their stability. For the sake of brevitywe adopt henceforth the notation x ≡ (cid:104) s (cid:105) , x ≡ (cid:104) s (cid:105) . C. Fixed points
1. Independent layers
We first analyze the fixed points in the absence of cou-pling between the layers, m = 0.For the first, threshold, layer, Eq. (8) has always thestable fixed point x ∗ = 1. New fixed points appear assolution of the equation G ( x ∗ ) = 0 . (10)If (cid:98) k θ (cid:99) = 0, the sum in Eq. (1) is always equal to 1and there are no new fixed points. They appear when (cid:98) k θ (cid:99) = 1, or θ = ∆ θ ≡ /k , as now the sum Eq. (1)misses the term with j = 0 and hence it is equal to1 − (1 − x ∗ ) k . According to its definition, G ( x ∗ ) = 0if 1 − (1 − x ∗ ) k < /N or x ∗ < − (cid:18) − N (cid:19) /k ≈ ( N k ) − , for large N . When θ = 2∆ θ the sum Eq. (1)misses two terms and the interval of fixed points is givenby the condition 1 − (1 − x ∗ ) k − k x (1 − x ∗ ) k − < /N ,or x ∗ (cid:46) (cid:114) N k ( k −
1) , for large N . The appearance ofan enlarged interval of fixed points continues until θ = 1,where the only term in the sum in Eq. (1) is ( x ∗ ) k andthe interval of fixed points is x ∗ < N − /k <
1. The com-plete phase diagram for the uncoupled threshold layer isplotted in Fig. 2, left panel. If the initial condition x (0)falls inside the shaded area, then it remains there. If, oth-erwise, the initial condition is outside the shaded area,then the dynamics leads to the only stable stationarysolution, x ∗ = 1, corresponding to global adoption. Asthe initial condition can not be smaller that 1 /N (onesingle adopter) one must consider adoption possible onlywhenever x ∗ > /N . In our particular version of thethreshold model for a wide range of reasonable values of N and k this occurs for θ = θ c = 2 /k . A more precisetreatment was presented in reference [33]. It was foundthere that the condition for θ c for threshold dynamics onER graphs is k Q ( K ∗ − , k ) = 1 , (11)where Q ( a, x ) is the incomplete gamma function and K ∗ = (cid:98) /θ c (cid:99) . When the initial group of adopters is suf-ficiently small (three orders of magnitude less than thenumber of nodes) an approximation for θ c ≈ /k can beused. Both in this more detailed calculation, and in oursimple treatment, it is found that θ c varies as the inverseof the number of neighbors k .For the second, SIS, layer, Eq. (9) always possessesthe solution x ∗ = 0. This is stable up to λ ≤ λ c ≡ /k .A transcritical bifurcation leads to a new, stable, fixed point x ∗ ∈ [0 , /
2] for λ > /k appearing as a solutionof e − λk x ∗ = 1 − x ∗ − x ∗ . (12)The complete phase diagram for the uncoupled thresh-old layer is plotted in Fig. 2, right panel. For λ ≤ λ c ,and independently on the initial condition, the systemtends to the only stable fixed point x ∗ = 0. For λ > λ c the dynamics leads to the stable solution x ∗ > x ∗ ≤ / λ c = 1 /k and above this value the fraction of adopters grows as x ∗ = 1 − λk (see the dashed line in the right panel ofFig. 2).
2. Coupled layers
We now consider the case m > x ∗ = 1 is still a fixed point for Eq. (8). The corre-sponding solution x ∗ is obtained from Eq. (9) e − λk x ∗ = e λm − x ∗ − x ∗ . (13)It is easy to show graphically that this solution exists forall values of λ and it belongs to the interval x ∗ ∈ [0 , / x ∗ has to be found numerically as a afunction of λk and λm . The stability of the fixed point( x , x ) = (1 , x ∗ ) is analyzed by means of the eigenvaluesof the matrix of first derivatives: ∂ ˙ x ∂x ∂ ˙ x ∂x ∂ ˙ x ∂x ∂ ˙ x ∂x (14)evaluated at (1 , x ∗ ). The two eigenvalues µ , are µ = − β − G (1 , x ∗ ) , (15) µ = − k λ (1 − x ∗ )(1 − x ∗ )(1 − β )(1 − x ∗ ) . (16)While it is clear that µ <
0, a graphical analysis showsthat the second eigenvalue µ is always negative as welland the fixed point (1 , x ∗ ) is, hence, stable.Other fixed points might appear as simultaneous solu-tions of the equations: G (cid:18) k x + mx k + m (cid:19) = 0 , (17) e − λ ( k x + mx ) = 1 − x − x , (18)satisfying the conditions x ∈ [0 , , x ∈ [0 , x θ x λ FIG. 2. Phase diagram in the independent layer case m = 0. Left-side panel shows the case of the threshold layer for k = 10, N = 1000. Here the dashed area and the line x = 1 is the set of stable fixed points. The right panel displays the steady statefraction of adopters for the SIS model k = 10, N = 1000. Dashed line presents results for usual model of infection, solid linepresents our model of simple adoption. Replacing x from the second equation into the firstwe arrive at the single condition G k λ log (cid:18) − x − x (cid:19) + ( m − k k ) x m ( k + m ) = 0 . (19)An analysis similar to the one performed in the case m = 0 shows that this equation is satisfied in a rangeof values x ∈ [0 , x ∗ ] only when θ > ∆ θ ≡ / ( k + m )(otherwise the function G is identically equal to 1). How-ever, it might occur that x ∗ is such that the correspond-ing value for x ∗ obtained from Eq.(18) does not belongto the interval [0 , θ until thecondition x ∗ ∈ [0 ,
1] is satisfied. Furthermore, as we dis-cussed above, we must require that x ∗ is greater that1 /N . When these conditions are met, besides findingthe fixed points x ∗ , x ∗ , one finds the critical value θ c asa function of λ and the other parameters, k , k , m, N of the model. The resulting line θ c ( λ ) is plotted as asolid line in Fig. 3 and it has been plotted on top of thenumerical results in Fig. 4.To prove that x ∗ is different from zero whenever λ > λ = 0 and replace x ∗ = 1 toobtain x ∗ = mλ + O ( λ ). As a way of example, the wholedependence of x ∗ ( m ) displayed in Fig. 8 is obtained by anumerical solution of Eq.(18) fixing λ = 0 . x ∗ = 1.When analyzing the values of x ∗ and x ∗ correspondingto the fixed points, it turns out that if λ < /k bothvalues x ∗ and x ∗ are close to zero, but when λ > /k , x ∗ is still close to zero, but x ∗ takes a value larger thanthe solution of Eq.(12). This means that the number ofadopters in the SIS layer is always larger in the coupledcase than in the uncoupled layers.In summary, for m >
0, the structure of the fixedpoints of Eqs.(8,9) is as follows: I: x ∗ = 1, x ∗ ≈ .
5, for θ ≤ θ c . IIa: x ∗ (cid:38) x ∗ (cid:38)
0, for θ > θ c and λ < /k , IIb: x ∗ (cid:38) x ∗ >
0, for θ > θ c and λ > /k .These three regimes have been identified in Fig. 3 for aparticular value of the system parameters. In the nextsections we will compare the results of our analyticaltreatment with those obtained in computer simulations. IV. NUMERICAL RESULTS COMPARED WITHANALYTICAL FINDINGS
We have performed numerical simulations of the dy-namical rules of the model. We run the dynamics untilthe steady state is reached (absorbing state in thresholdlayer but still active in SIS) and then measure the frac-tion of adopters in each layer s (cid:96) = N (cid:96) (cid:80) N (cid:96) i =1 s (cid:96),i and itsaverage (cid:104) s (cid:96) (cid:105) over many realizations and network config-urations. The results of numerical simulations shown inFig. 4 evidence that the main effect of the number of in-terconnections m is to facilitate adoption in the coupledlayer system and that there is great correlation betweenthe adoption areas in both layers.Our numerical findings are generally well described bythe mean-field type analysis described in Section III. Theanalytical approach is able to predict the main trends ob-served in Fig. 4, namely the splitting of the parameterspace in distinct regions: region I characterized by thevalues (cid:104) s (cid:105) ≈ (cid:104) s (cid:105) ≈ .
5; and region II character-ized by a low value (cid:104) s (cid:105) (cid:38)
0, further splitted in regions
IIa : very small (cid:104) s (cid:105) (cid:38)
0, and
IIb : larger but still small (cid:104) s (cid:105) >
0. The border between regions
IIa and
IIb cor-responds to λ = 1 /k , the critical value in absence ofcoupling between the layers (see Section III C 1). We now I IIbIIa I IIb
IIa
FIG. 3. Phase diagram in the coupled layer cases k = 12, k = 3 (left panel) and k = 3, k = 12 (right panel). In both casesit is m = 10, N = N = 1000. describe in details the main features of the different tran-sitions that occur between these regions and whose exactnature depend on the intralayer ( k , k ) and interlayer m connectivities.We first focus on the variation of the adoption regionin the threshold layer. As observed in Fig. 4 for a partic-ular value m = 10 (although results are qualitatively thesame for other m values), in the coupled layer system (cid:104) s (cid:105) still experiments a transition from the adopter I to theneutral II phase. Evidence of this transition is providedin the left panels of Fig. 5 where we plot, for m = 10 anddifferent values of the intralayer connectivities ( k , k ),the variation of (cid:104) s (cid:105) as we cross (i) from region I to re-gion IIa (left top panel) varying θ , (ii) from region I toregion IIb varying θ (left middle panel), and (iii) fromregion II to region I varying λ (left bottom panel).For fixed λ the transition between regions I and II occurs at a critical threshold θ c ( λ, m ) that varies with m and λ in an intricate way. As λ increases towards λ = 1, θ c tends to a constant value that depends both on theintralayer ( k , k ) and interlayer ( m ) connectivities. Onthe one hand, θ c ( λ = 0 , m >
0) is necessarily smallerthan the critical value θ c ( m = 0) for the uncoupled case:neighbors in the second layer can not become adopters,hence decreasing the fraction of adopter neighbors of anode in the first layer and making adoption more difficult.On the other hand, the increase of θ c ( λ > , m ) with m seems to go against intuition, since in a single layer θ c varies as the inverse of the number of neighbors, whichnow is k + m . However, the number of adopters in thesecond layer can increase due to its own dynamics and,through the interlayer connections, favor the spread inthe first layer. The combined effect leads to an increasein the critical threshold value θ c for λ, m > θ c as a function of m for a particular value λ = 0 . λ ). There appear to be cleardifferences between the cases k > k and k ≤ k . When I IIbIIa
FIG. 4. (Color online) Fractions of adopters in threshold layer (cid:104) s (cid:105) (left column), and in SIS layer (cid:104) s (cid:105) (right column) as afunction of θ and λ for m = 10 and k = 3 and k = 12 (toppanel), k = k = 6 (middle panel), k = 12 and k = 3(bottom panel). Results come after computer simulationsperformed for a network of N = N = 1000 nodes, whereboth layers are ER random networks, and are averaged over500 realizations of networks and 500 realizations of the dy-namics for each network configuration. The continuous linesdisplayed in the SIS column come from the full stability anal-ysis of the mean-field dynamical equations; the horizontal lineis λ = 1 /k . The arrows indicate transitions between the re-gions that will be discussed in other figures. k > k , the critical threshold changes visibly only when m > k , i.e. when the number of interlayer connectionsovercomes the number of connections inside the thresholdlayer. It then grows steadily towards the limiting value θ c = 0 .
5. When k ≤ k , θ c increases with the numberof interconnections up to a maximum at m ≈
10 andthen decreases until the limiting value of θ c = 0 .
5. Itis interesting to note that in a single network it is notpossible to exceed the value θ c = 0 . θ = 0 . FIG. 5. (Color online) Fractions of adopters in threshold layer(left column) and in SIS layer (right column) as a function of θ (resp. λ ) when λ (resp. θ ) is fixed and for m = 10. Top panelshows results when λ = 0 .
05, middle panel when λ = 0 .
5, andbottom panel when θ = 0 .
25. Red circles correspond to thecase k = k = 6, green triangles to k = 12 > k = 3, andblue squares to k = 3 < k = 12. Symbols present the resultsfrom computer simulations and solid lines with correspondingcolors stand for analytical solutions. On the SIS layer the most remarkable feature appear-ing when m > θ and m > (cid:104) s (cid:105) is always larger than zero for λ >
0, or, in other words λ c ( m >
0) = 0. This occurs asthere are always adopters in the first layer (at least theinitial group of adopters remain) and adoption alwaysspreads from them to the second layer due to the in-terlayer connections. The second noticeable effect of theinterlayer connectivity on the SIS layer is the appearanceof a new transition when crossing from region I to region -2 -1 θ c m k =k =6k =12, k =3k =3, k =12 FIG. 6. (Color online) Critical value of the threshold θ c as afunction of the interconnectivity m for λ = 0 .
5. Same symbolsand line meanings than in figure 5. II . This shows up as a drop in the fraction of adopters (cid:104) s (cid:105) exactly at the same values of ( θ, λ ) for which the firstlayer experiments its transition from adopter to neutralglobal states, as shown in Figs. 4 and 5. V. CHARACTERIZING THE ORDER OFTRANSITIONS
A detailed look at Fig. 5 suggests that the order of thetransition in the threshold layer (discontinuous in the ab-sence of coupling to the SIS layer) might now depend onthe connectivities and, most remarkably, on the exactway some transition lines are crossed. Quite generally alltransitions between the different regions are discontinu-ous, except for the transition I to IIb occurring increas-ing θ at fixed λ for k ≤ k which becomes continuous forsufficiently large m . The change from discontinuous tocontinuous transition in that case is evident from Fig. 7where we plot the jump of (cid:104) s (cid:105) at the transition point.Note, however, that the transition II to I occurringincreasing λ at fixed θ is always discontinuous.When both layers are coupled the transition in theSIS layer remains continuous but moves to λ = 0. Wehave already shown that for small number of interlayerconnections m and adoption probability λ the fractionof adopters in the SIS layer grows linearly as x ∗ ≈ mλ ,so proving the continuous nature of the transition, seeFig. 8 where we plot the stationary solution of equation(18) which fits very well the results obtained by computersimulations. Interestingly we observe also a new, secondtransition in the SIS layer. It appears for the same set ofparameters ( θ, λ ) than a transition in the threshold layeris observed.It appears from the numerical results that in thethreshold layer the order of the transition between the I and IIb regions depends, for large m and k ≤ k ,on whether the transition line is crossed vertically (atconstant θ ) or horizontally (at constant λ ). In fact, itchanges from discontinuous to continuous when λ is fixedand we increase θ going from phase I to IIb (for suffi-ciently large m and when k ≤ k ). However, our analyt-ical calculations do not predict this change in the orderof the transitions in the threshold layer.To provide further numerical evidence of the order ofthe transitions for different connectivities k and k anddifferent ways of crossing the transition lines we havestudied the probability distribution P ( s ) of the numberof adopters in the first layer. In Figs. 9 and 10 we plotthe location of the maxima of this distribution for differ-ent parameter and connectivity values. Panels (a), (b),(c) in Fig. 9 and Fig. 10 show the results when the tran-sition line is crossed horizontally (varying θ at fix λ ) whilepanel (d) in Fig. 9 focuses on a vertical crossing (varying λ at fix θ ) for different values of the inter and intralayerconnectivities. As shown in those figures, the transitionin the threshold layer crossing horizontally from I to IIa occurring at λ < /k remains discontinuous for all k , k cases –blue circles in panels (a), (b) and (c)– as sodoes the analogous transition from I to IIb occurringat λ > /k when k > k –green triangles in panel (b).This discontinuous nature of the transition is clearly ev-idenced by the coexistence of the maxima at s = 0 and s = 1 for a range of values of θ . In fact, in the numericalsimulations, one can even observe the typical hysteresisbehavior typical of a discontinuous transition. However,the same transition from I to IIb in the case k ≤ k and for a sufficiently large number m of interlink connec-tions becomes continuous: only a single maximum of thedistribution, varying continuously from s = 1 to s = 0as a function of θ , is observed –green triangles in panels(a) and (c). To show that our observation does not de-pend on the specific choice of k and k values we plotin Fig. 10 the maxima of the distribution P ( s ) when k = 6, m = 10 and for different values of k . We seethat the order of the transition changes from discontin-uous to continuous when increasing intraconnectivity inSIS layer, k . Only when k (cid:29) k the transition remainsdiscontinuous. For this particular example, the order ofthe transition changes to continuous for k = 4. Fig. 9(d) shows evidence of the discontinuous character of thetransition when one crosses instead the transition linevertically (at constant θ ) from I to II . As mentionedbefore, although the second transition in the SIS layer iscaused by the transition in the threshold layer, the or-der of the transitions in both layers agree only when thetransition line is crossed horizontally ( λ constant), butdiffers for vertical crossing ( θ constant). In the lattercase the transition in the threshold layer is discontinuousbut at the same time the SIS layer experiences a con-tinuous transition with a big slope exhibiting substantialjumps in the average (cid:104) s (cid:105) . -1 ∆ < s > mk =k =6k =12, k =3k =3, k =12 FIG. 7. (Color online) Jump of the fraction of adopters inthreshold layer ∆ (cid:104) s (cid:105) at the transition point θ c as a functionof m and for λ = 0 .
5. Values of the θ c ( m ) are shown in Fig.6.Same symbols and line meanings than in figure 5. -4 -3 -2 -1 -2 -1 < s ( λ = . ) > m k =k =6k =12, k =3k =3, k =12 FIG. 8. (Color online) Fraction of adopters right above λ c = 0as a function of m . Different colors represent different valuesof k and k . Red circles correspond to the case k = k ,green triangles - k > k , and blue squares - k < k . Sym-bols present the results from computer simulations for a net-work of N + N = 2000 nodes, where both layers are ERrandom networks. Results are averaged over 500 realizationsof networks and 500 realizations of dynamics for each networkconfiguration. We took θ = λ in computer simulations. Solidlines with corresponding colors stand for analytical solutions. VI. CONCLUSIONS
In summary, we have considered the competition be-tween two different, simple and complex, adoption pro-cesses as a specific case of competition of a continuousand a first order transition on interdependent networks,and we have developed a mean-field approach appro-priate to describe this situation. We have found thatwith the presence of interlayer connections the system re-veals a wider range of parameters where global adoption
0 0.5 1 0.05 0.51 (a) θ λ
0 0.5 1 0.05 0.51 (b) θ λ
0 0.5 1 0.05 0.51 (c) θ λ
0 0.5 1(cid:9)(cid:9) k =3, k =12(cid:9)(cid:9)(cid:9)(cid:9) k =k =6 (cid:9)(cid:9) k =12 k =31 (d) λ FIG. 9. Maxima of the fraction of adopters in threshold layer (cid:104) s (cid:105) max as a function of θ –for fixed values of λ , different panelspresent results for different intralayer connectivities, i.e. (a) k = k = 6, (b) k = 12 and k = 3, (c) k = 3 and k = 12– andas a function of λ –for fixed θ , panel (d). In all cases the number of interlinks is set to m = 10.
0 0.5 1 1 2 3 4567912181 θ k FIG. 10. (Color online) Maxima of the fraction of adoptersin threshold layer (cid:104) s (cid:105) max as a function of θ when k = 6, m = 10 and λ = 0 . k . takes place. Furthermore, both threshold and SIS layerschange their behavior quantitatively and qualitatively. In the threshold layer the critical value θ c increases withthe interlayer connectivity m , whereas in the case of anisolated single network it would decrease with averageconnectivity. The transition remains discontinuous ex-cept in the case of asymmetric intralayer connectivities k ≤ k and large intralayer connectivity m , when it be-comes continuous. We also find that the critical thresholdreaches a local maximum, θ c > .
5, located at intermedi-ate values of m . In the SIS layer the original transitionremains continuous but it moves to λ c ( m ) = 0 for any m (cid:54) = 0, signaling the disappearance of the neutral state.A new transition in SIS layer between regions of low andlarge number of adopters appears caused by the inter-layer coupling. This new transition can be continuous ordiscontinuous according to the particular values of theinter and intralayer connectivities. Remarkably the na-ture of the transitions in both layers might depend onthe direction in which the transition lines are crossed.Our results indicate that interconnection can result innew transitions and modifications of the nature of pre-existing transitions, opening the way to further researchon universal characteristics of the coupling of networktransitions of different order and their dependence on in-0ter and intralayer connectivities. ACKNOWLEDGMENTS
This work was supported by FEDER (EU) andMINECO (Spain) under Grant ESOTECOS FIS2015-63628-C2-R.
APPENDIX A: MEAN-FIELD APPROACH FORCOMPLEX ADOPTION
We develop in this appendix an approach based on amean-field type approximation in which local fractionsare replaced by global averages.Let us first introduce the general notation. We de-note by s ,i ( t ), i = 1 , . . . , N the states of agents in thefirst layer and s ,i ( t ), i = 1 , . . . , N those in the secondlayer. An agent i in layer (cid:96) is said to be in the adopterstate at time t if s (cid:96),i ( t ) = 1; otherwise, it is in the neu-tral (non-adopter) state when s (cid:96),i ( t ) = 0. We will use (cid:104) s ,i ( t ) (cid:105) n for the fraction of neighbors of agent (1 , i ) whoare adopters, i.e. (cid:104) s ,i ( t ) (cid:105) n = n ,i (cid:80) ( (cid:96),j ) ∈ n ,i s (cid:96),j ( t ), be-ing n ,i the set of neighbors of (1 , i ) in both layers and s (cid:96),j the value of the state of such a neighbor which mightbelong to the first layer, s ,j , or to the second layer, s ,j .Once selected, the state s ,i ( t ) updates according to thefollowing dynamical rule: if (cid:104) s ,i ( t ) (cid:105) n is smaller than thethreshold θ nothing happens; otherwise, it becomes anadopter. This can be written as s ,i ( t + τ ) = (cid:40) s ,i ( t ) , if (cid:104) s ,i ( t ) (cid:105) n < θ, , if (cid:104) s ,i ( t ) (cid:105) n ≥ θ. (20)Note that, according to this rule, once a node becomes anadopter it cannot go back to the neutral state. Therefore,with the course of time the fraction of adopters in thesystem can either increase or stay unchanged, but neverdecrease.We now aim at deriving an approximate equation forthe evolution of the fraction of the number of adopters inlayer 1, s ( t ) = N (cid:80) N i =1 s ,i ( t ). We follow closely [53, 54]in the derivation. The ensemble average (cid:104) s ( t ) (cid:105) evolvesaccording to the general, exact, relation: N (cid:104) s ( t + τ ) (cid:105) = N (cid:104) s ( t ) (cid:105) + (cid:104) s ,i ( t + τ ) − s ,i ( t ) |{ s ( t ) }(cid:105) , (21)where { s ( t ) } = ( s , ( t ) , ..., s ,N ( t ) , s , ( t ) , . . . , s ,N ( t ))denotes the particular realization of the state variablesand (cid:104)· · · | · · · (cid:105) means a conditional average. Considering that time (as measured in Monte Caro units) increasesby τ = 1 / ( N + N ) after one individual update, we canwrite Eq. (21) in the form β (cid:104) s ( t + τ ) (cid:105) − (cid:104) s ( t ) (cid:105) τ = (cid:104) s ,i ( t + τ ) − s ,i ( t ) |{ s ( t ) }(cid:105) == −(cid:104) s ( t ) (cid:105) + (cid:104) s ,i ( t + τ ) |{ s ( t ) }(cid:105) (22)with β = N N + N We now make a mean-field type ap-proximation and consider that the fraction of neighborswhich are adopters (cid:104) s ,i ( t ) (cid:105) n is independent of the site i . Hence, the probability that the fraction of adoptersin the neighbourhood of the randomly chosen node i is at least θ is approximated by Prob [ (cid:104) s ,i ( t ) (cid:105) n ≥ θ ] ≈ Prob [ (cid:104) s ( t ) (cid:105) n ≥ θ ], being (cid:104) s ( t ) (cid:105) n the average value of (cid:104) s ,i ( t ) (cid:105) n over all sites i = 1 , . . . , N . Using the dynami-cal rules described in Eq. (20) we derive: (cid:104) s ,i ( t + τ ) |{ s ( t ) }(cid:105) = (1 − Prob [ (cid:104) s ( t ) (cid:105) n ≥ θ ]) × (cid:104) s ( t ) (cid:105) +Prob [ (cid:104) s ( t ) (cid:105) n ≥ θ ] × (cid:104) s ( t ) (cid:105) + (1 − (cid:104) s ( t ) (cid:105) )Prob [ (cid:104) s ( t ) (cid:105) n ≥ θ ] . (23)Replacing in Eq. (22) and treating the left hand sideas a time derivative we obtain β d (cid:104) s ( t ) (cid:105) dt = (1 − (cid:104) s ( t ) (cid:105) ) Prob [ (cid:104) s ( t ) (cid:105) n ≥ θ ] , (24)which is Eq.(1) in the main text. APPENDIX B: MEAN-FIELD APPROACH FORSIMPLE ADOPTION
We recall that the rules of the adoption process in theSIS layer are the following: at time t an agent from layer2 is randomly selected, let s ,i ( t ) be the state of thisagent. If s ,i ( t ) = 1 (adopter) it goes back to the neutralstate s ,i ( t + τ ) = 0. If s ,i ( t ) = 0 (neutral) then it visitssequentially all its neighbors, having a probability λ ofbecoming an adopter from the interaction with anyoneof them (of course, if it becomes adopter in a given in-teraction, it is not necessary to continue the sequence ofinteractions with the neighbors). Namely, s ,i ( t + τ ) = , if s ,i ( t ) = 0 and adoption fromany neighbour happens,0 , if s ,i ( t ) = 0 and adoption doesnot happen,0 , if s ,i ( t ) = 1 . (25)By a similar reasoning to the one developed beforefor the threshold layer, we can derive an exact evolu-tion equation for the ensemble average of the fraction ofadopters in the SIS layer s ( t ) = N (cid:80) N i =1 s ,i ( t )(1 − β ) (cid:104) s ( t + τ ) (cid:105) − (cid:104) s ( t ) (cid:105) τ = (cid:104) s ,i ( t + τ ) − s ,i ( t ) |{ s ( t ) }(cid:105) = −(cid:104) s ( t ) (cid:105) + (cid:104) s ,i ( t + τ ) |{ s ( t ) }(cid:105) (26)1According to the dynamical rules Eq. (25), the condi-tional average is (cid:104) s ,i ( t + τ ) |{ s ( t ) }(cid:105) = (1 − (cid:104) s ,i (cid:105) ) × Prob[A] , (27)where Prob[A]=Prob[Adoption occurs from anyneighbour]=1 − Prob[Adoption does not occur from anyneighbour]. If κ i is the number of adopter neighbours inany layer of site (2 , i ) the probability that adoption doesnot occur for that site is (1 − λ ) κ i . In the mean-fieldapproximation we will replace this probability by theaverage probability (cid:104) (1 − λ ) κ i (cid:105) over all nodes. We willfurther assume that the number of adjacent adoptersis given by a Poisson distribution (as in ER networks), P ( κ ) = (cid:104) κ (cid:105) κ e −(cid:104) κ (cid:105) κ ! , leading to (cid:104) (1 − λ ) κ i (cid:105) = ∞ (cid:88) κ =0 (cid:104) κ (cid:105) κ e −(cid:104) κ (cid:105) κ ! (1 − λ ) κ = e − λ (cid:104) κ (cid:105) . (28) We replace (cid:104) κ (cid:105) = k (cid:104) s (cid:105) + m (cid:104) s (cid:105) . Thus probability thatadoption happens isProb[Adoption] = 1 − e − λ ( k (cid:104) s (cid:105) + m (cid:104) s (cid:105) ) . (29)Replacing in Eq. (26) and identifying the left side as atime derivative we obtain the mean-field equation for thefraction of adopters in the SIS layer(1 − β ) d (cid:104) s ( t ) (cid:105) dt = −(cid:104) s (cid:105) +(1 − (cid:104) s (cid:105) ) (cid:16) − e − λ ( k (cid:104) s (cid:105) + m (cid:104) s (cid:105) ) (cid:17) , (30)which is Eq.(9) in the main text. [1] T. Nishikawa, A. E. Motter, Y.-Ch. Lai, and F. C. Hop-pensteadt, Phys. Rev. Lett. , 014101 (2003).[2] A. Arenas, A. D´ıaz-Guilera, and C. J. Perez-Vicente,Phys. Rev. Lett. , 114102 (2006).[3] A. Arenas, A. D´ıaz-Guilera, J. Kurths, Y. Moreno, andCh. Zhou, Physics Reports , 93 (2008).[4] K. Suchecki, V. M. Eguiluz, and M. San Miguel, Phys.Rev. E , 036132 (2005).[5] D. Achlioptas, R. M. D’Souza, and J. Spencer, Science , 1453 (2009).[6] F. Radicchi and S. Fortunato, Phys. Rev. Lett. ,168701 (2009).[7] P. Grassberger, C. Christensen, G. Bizhani, S.-W. Son,and M. Paczuski, Phys. Rev. Lett. , 225701 (2011).[8] J. G´omez-Garde˜nes, S. Gomez, A. Arenas, and Y.Moreno, Phys. Rev. Lett. , 128701 (2011).[9] R. M. D’Souza and J. Nagler, Nat. Phys. , 3378 (2015).[10] O. Riordan, and L. Warnke, Science 333 , 322-324 (2011).[11] S.V. Buldyrev, R. Parshani, G. Paul, H.E. Stanley andS. Havlin, Nature , 1025 (2010).[12] M. Kivela, A. Arenas, M. Barthelemy, J.P. Gleeson, Y.Moreno, and M. A. Porter, Journal of Complex Networks , 203 (2014).[13] M. De Domenico, A. Sole-Ribalta, E. Cozzo, M. Kivela,Y. Moreno, M. A. Porter, S. Gomez, and A. Arenas,Phys. Rev. X , 041022 (2013).[14] S. Boccaletti, G. Bianconi, R. Criado, C. del Genio,J. G´omez-Garde˜nes, M. Romance, I. Sendina-Nadal, Z.Wang, and M. Zanin, Physics Reports , 1 (2014).[15] M. De Domenico, V. Nicosia, A. Arenas, and V. Latora,Nat. Commun. , 6864 (2014).[16] M. Diakonova, V. Nicosia, V. Latora, and M. San Miguel,New J. Phys. , 023010 (2016).[17] J. Gao, S.V. Buldyrev, H.E. Stanley, and S. Havlin, Na-ture Phys. , 40 (2012).[18] R. Parshani, S.V. Buldyrev, and S. Havlin, Phys. Rev.Lett. , 048701 (2010).[19] Y. Hu, B. Ksherim, R. Cohen, and S. Havlin, Phys. Rev.E , 066116 (2011).[20] D. Zhou, A. Bashan, R. Cohen, Y. Berezin, N. Shnerb,and S. Havlin, Phys. Rev. E , 012803 (2014). [21] F. Radicchi, Phys. Rev. X , 021014 (2014).[22] F. Radicchi, and A. Arenas, Nature Phys. , 2761 (2013).[23] K. Zhao, and G. Bianconi, J Stat Phys, 152(6), pp.1069-1083 (2013).[24] S. Skardal, V. Nicosia, V. Latora, and A. Arenas,arXiv:1405.5855.[25] C. Granell, S. Gomez, and A. Arenas, Phys. Rev. Lett. , 128701 (2013).[26] Q. Guo, X. Jiang, Y. Lei, M. Li, Y. Ma, and Z. Zheng,Phys. Rev. E , 012822 (2015).[27] Virality L. Weng, F. Menczer, and Y.-Y. Ahn, Sci. Rep. , 2522 (2013).[28] D. Centola, Science , 1194 (2010).[29] D. Centola, V. M. Egu´ıluz, and M. W. Macy, Physica A , 449 (2007).[30] R. M. May, and R. M. Anderson, Infectious Diseasesof Humans: Dynamics and Control (Oxford UniversityPress, Oxford, UK, 1991).[31] R. Pastor-Satorras, and A. Vespignani, Phys. Rev. E ,066117 (2001).[32] M. Granovetter, Am. J. Sociol. , 1420 (1978).[33] D. J. Watts, PNAS , 5766-5771 (2002).[34] J. C. Gonzalez-Avella, V.M. Egu´ıluz, M. Marsili, F.Vega-Redondo, and M. San Miguel, PLoS ONE (5),e20207 (2011).[35] P. Singh, S. Sreenivasan, B.K. Szymanski, and G.Korniss,Sci. Rep. , 2330 (2013).[36] Z. Ruan, G. I˜niguez, M. Karsai, J. Kert´esz, Phys. Rev.Lett. , 218702 (2015).[37] S. Gonzalez-Bailon, J. Borge-Holthoefer, A. Rivero, andY. Moreno, Sci. Rep. , 197 (2011).[38] M. Karsai. G. Iniguez, K. Kaski, J. Kertesz, J. R. Soc.Interface , 20140694 (2014).[39] M. Karsai, G. I˜niguez, R. Kikas, K. Kaski, and J. Kert´esz,Sci. Rep. 6, 27178 (2016).[40] F. Vazquez, M.A. Serrano, and M. San Miguel, Sci. Rep.6, 29342 (2016).[41] A. Saumell-Mendiola, M. A. Serrano, and M. Boguna,Phys. Rev. E , 026106 (2012).[42] J. Sanz, Ch.-Y. Xia, S. Meloni, and Y. Moreno, Phys.Rev. X , 041005 (2014). [43] M. Dickison, S. Havlin, and H.E. Stanley, Phys. Rev. E , 066109 (2012).[44] S. Shai, and S. Dobson, Phys. Rev. E , 042812 (2013).[45] J. G´omez-Garde˜nes, A. S. de Barros, S. T. R. Pinho, andR. F. S. Andrade, EPL , 58006 (2015).[46] C. Buono, L. G. Alvarez-Zuzek, P. A. Macri, and L. A.Braunstein, PLoS One, (3), e92200 (2014).[47] F. Darabi Sahneh and C. Scoglio, Phys. Rev. E ,062817 (2014).[48] Ch. D. Brummitt, K.-M. Lee, and K.-I. Goh, Phys. Rev.E , 045102 (2012). [49] K.-M. Lee, Ch. D. Brummitt, and K.-I. Goh, Phys. Rev.E , 062816 (2014).[50] A. R. Akhmetzhanov, L. Worden, and J. Dushoff, Phys.Rev. E , 012816 (2013).[51] J.P. Gleeson, Phys. Rev. X , 021004 (2013).[52] S. Morris, Rev. Econ. Stud.
57 (2000).[53] T. Vaz Martins, R. Toral, and M. A. Santos, Eur. Phys.J. B , 329 (2009).[54] C. J. Tessone, and R. Toral, Eur. Phys. J. B71