Asynchronous semi-anonymous dynamics over large-scale networks
Chiara Ravazzi, Giacomo Como, Michele Garetto, Emilio Leonardi, Alberto Tarable
AAsynchronous semi-anonymous dynamics overlarge-scale networks
C. Ravazzi, G. Como, M. Garetto, E. Leonardi, A. Tarable
Abstract
We analyze a class of stochastic processes, referred to as asynchronous and semi-anonymous dynamics (ASD), over directed labeled random net-works. These processes are a natural tool to describe general best-responseand noisy best-response dynamics in network games where each agent, atrandom times governed by independent Poisson clocks, can choose amonga finite set of actions . The payoff is determined by the relative popular-ity of different actions among neighbors, while being independent of thespecific identities of neighbors .Using a mean-field approach, we prove that, under certain conditionson the network and initial node configuration, the evolution of ASD canbe approximated, in the limit of large network sizes, by the solution ofa system of non-linear ordinary differential equations. Our framework isvery general and applies to a large class of graph ensembles for which thetypical random graph locally behaves like a tree. In particular, we willfocus on labeled configuration-model random graphs, a generalization ofthe traditional configuration model which allows different classes of nodesto be mixed together in the network, permitting us, for example, to in-corporate a community structure in the system. Our analysis also appliesto configuration-model graphs having a power-law degree distribution, anessential feature of many real systems. To demonstrate the power andflexibility of our framework, we consider several examples of dynamicsbelonging to our class of stochastic processes. Moreover, we illustrateby simulation the applicability of our analysis to realistic scenarios byrunning our example dynamics over a real social network graph.
Many complex systems arising in different domains exhibit cascading phenom-ena that spread through networks of local interactions. Examples of such cas-cades include, but are not limited to, infrastructure failures [39], adoption ofinnovations, conventions and technologies [33, 36, 29], diffusion of beliefs, opin-ions, fake news [37], memes, and the like [38]. These phenomena can have pro-found effects on politics [14], social norms [4], financial networks [20], marketingcampaigns [21]. 1 a r X i v : . [ c s . S I] F e b he standard mathematical approach to modeling cascading processes is toconsider a graph (finite or infinite) in which nodes stand for individuals thatcan be in one of several (discrete or continuous) states, and edges (directed orundirected, possibly weighted) represent interactions with neighboring nodes.Individuals are supposed to repeatedly update their state over (discrete or con-tinuous) time, depending on the current state of their neighbors [35, 8, 17].Simple epidemic models in which nodes can change their state as conse-quence of a single contact with a neighboring node [31] turn out to be toosimplistic to describe systems in which individuals tend to react to the jointstates of their neighbors. To represent such combined effect, one of the mostcommonly used models in the literature is the linear threshold model, origi-nally introduced by Granovetter [20] and widely investigated in several variants[40, 15]. The general idea behind such models is to assume that a node adoptsa given state if the fraction of neighbors (possibly weighted by edges) currentlyadopting that state exceeds a certain threshold. More in general, researchershave considered so-called networked coordinated games, in which nodes adoptthe best-response (according to some payoff matrix) in reaction to the strategiesadopted by neighbors [10].Some fundamental distinctions in this wide class of models are the following.In progressive processes, state transitions are irreversible: once a node joinsa given state, it keeps such state indefinitely, irrespective of what happens toneighbors [22, 11, 1, 27, 4]. In non-progressive processes transitions are, instead,reversible, since nodes still remain under the influence of their neighbors afterthe adoption of a given state [2, 30, 34]. Another crucial distinction concernsthe update rule of the nodes: does the future state of a node also depend on thecurrent state of the node, in addition to the states of neighbors, or is it uniquelydetermined by the neighbors? Indeed the above distinctions, combined to thenature of edges (i.e., directed or undirected), lead to models of widely differentnature and analytical tractability.In this paper we analyze a class of cascading processes, referred to as Asyn-chronous and Semi-anonymous Dynamics (ASD), with the following character-istics: i) nodes update their (discrete) state over continuous time, accordingto independent Poisson clocks, following an arbitrary rule that depends on thenumber (or relative fraction) of neighbors in each possible state, but not onthe specific identities of neighbors (which are order-independent); ii) edges aredirected; iii) state transitions are reversible (non-progressive model).In this paper we extends previous work in [34], where authors analyze atwo-state, deterministic linear threshold model with synchronous node update,adopting a mean-field approach. Here, we seek to understand how this approachcan be pushed to its greatest generality, extending the class of networks andunderlying node dynamics to which it can be applied. We mention, however,that in [34] authors consider also the progressive variant of their model, whilehere we focus only on the non-progressive model. The interested reader canrefer to [18] for new methods to approximate the dynamics on large networkssampled from a graphon. Although this framework is flexible, it works well ondense network formation models, such as stochastic block models, but not on2parse networks.One stream of related work [7, 30, 24] analyzes the possible equilibria of abinary-decision game in the case of undirected graphs and synchronous update.Similarly to these works, we study a game where players’payoffs depend on theactions taken by their neighbors in the network but not on the specific identitiesof these neighbors.Another huge stream of related work is concerned with the algorithmic as-pects of influence maximization [22, 28], where the goal is to find the initial nodeconfiguration that maximizes the final size of the cascade. In contrast to suchstream of work, here we assume the initial node configuration to be randomlyselected according to a given node statistics. In our analysis we will make use of a mean-field approximation. This is apowerful and standard tool for studying complex systems that are constitutedby a large number of interacting units [9]. The core idea in the mean-fieldtheory is to describe a complex stochastic system using a simpler deterministicdynamical system [23, 9], assuming that each unit interacts with an average ofother units. When the states can take values in a finite or countable space, theevolution of the fraction of units in a given state can be described by the solutionof a system of non-linear ordinary differential equations. The main goal is thento quantify and estimate the accuracy of this approximation. We anticipate thatour results guarantee that the discrepancy between the behavior of the actualsystem and the solution given by the approximation can be made small whenthe network size grows and the time horizon scales only logarithmically withthe network size.In Section 3 we provide the complete analysis of ASD over a labeled branch-ing process, i.e., an infinite ensemble of labeled graphs with a rooted tree struc-ture. In this case it can be easily shown that the evolution of the fractionof nodes in a given state indeed corresponds to the solution of some ordinarydifferential equations (see Proposition 1).In Section 4 we turn our attention to general graph ensembles. The coremessage is the following: if the graph exhibits a local tree structure, then theanalysis on a suitably chosen labeled branching process provides a good approx-imation of the expected fraction of nodes in a given state (see Proposition 2).A property of Local Weak Convergence is the key feature that provides thislink and formalizes the idea that, for large n , the local structure of the graphnear a vertex chosen uniformly at random is approximately a branching process.Finally, the analysis of the concentration around the expectation allows us toderive the accuracy of the above approximation.We will then focus in Section 5 on the labeled configuration model, a generalmix of heterogeneous nodes with class-specific node statistics, which includes theclassic configuration model (CM) as special case. We will explore conditions forLocal Weak Convergence (see Theorem 1) and the concentration of the ASDevolution around the expectation (see Theorem 2). In Section 5.3 we will show3hat the sequence of node degrees is allowed to follow a power-law distributionscaling with the network size. This is particularly important for applications tosocial network graphs.As a second example, in Section 5.4 we will consider a labeled configurationmodel with a community structure, which is another fundamental feature foundin many real systems. Indeed, by considering as label of a node its membershipto a given community, we can represent graphs with a general distribution ofin/out degrees among nodes belonging to the same or different communities.This allows us to describe, for example, “assortative” graphs, in which intra-community edges are denser than inter-community edges.In Section 6 and 7 we present numerical results of ASD previously introducedin large but finite networks, considering both synthetically generated graphsand real-world social networks. More precisely, we compare the solutions of thedifferential equations derived through mean field approximation with resultsobtained by running Monte-Carlo simulations. Throughout this paper, we use the following notational conventions. Let N , Z + , R be the set of natural, non-negative integers and real numbers, respectively.Given n ∈ N we use the notation [ n ] = { , . . . , n } .The symbol | · | denotes theabsolute value if applied to a scalar value and the cardinality if applied to a set.We denote the indicator function of set A with the notation A . Given a finiteset V , R V denotes the space of real vectors with components labelled by elementsof V . If x ∈ R n , we denote the (cid:96) -th entry by x (cid:96) and x [ (cid:96) ] is the projection of x onthe sub-space generated by the first (cid:96) elements, i.e. x [ (cid:96) ] = ( x , x , . . . , x (cid:96) ) ∈ R (cid:96) .This paper makes frequent use of Landau symbols. The notation “ f ( N ) = O ( g ( N )) when N → ∞ ” means that positive constants c and N exist, so that f ( N ) ≤ cg ( N ) for all N > N . The expression “ f ( N ) = o ( g ( N )) when N → ∞ ”means that lim N →∞ f ( N ) g ( N ) = 0.A labeled directed multigraph is a 6-ple ( V , E , A , λ, σ, τ ), where V , E , and A are the sets of nodes, links, and labels, respectively, all finite; λ : V → A is themap giving the label λ ( v ) of a node v ∈ V ; and σ, τ : E → V are the maps givingthe tail node σ ( e ) and head node τ ( e ) of a link e ∈ E so that e is directed from σ ( e ) to τ ( e ). The set of in-neighbors and out-neighbors of a node v ∈ V aredefined as N − v = { w ∈ V\{ v } : ( w, v ) ∈ E} and N + v = { w ∈ V\{ v } : ( v, w ) ∈ E} ,respectively, and the corresponding in-degree and out-degree as d v = |N − v | and k v = |N + v | . We define its out-degree vector k v ∈ Z A + as the vector whosecomponent a ∈ A represents the number of out-neighbors of v belonging toclass a . Similarly, we define for node v the in-degree vector d v ∈ Z A + .A path from a vertex u ∈ V to a vertex v ∈ V (i.e. a path u → v ) is a finitesequence of edges ( u i , v i ) ≤ i ≤ L with u = u , v L = v , v i = u i +1 . If there is atleast a path from u to v , we say that u is connected to v , and the graph distancefrom u to v is then defined as the minimum length of a path from u to v . If all(ordered) vertex pairs are connected, the graph G is said strongly connected. If,4nstead, for any pair ( u, v ) either a path u → v or v → u exists, we say that thegraph is weakly connected. We define a simple path as a path along which allvertices are distinct. A directed tree is a weakly connected graph in which nomore than one path exists between every pair of vertices ( u, v ). Let us consider a finite population of n agents interacting in a connected net-work, which we map onto the nodes of a labeled directed multigraph G =( V , E , A , σ, τ, λ ), whereby a link e ∈ E represents a direct influence of its headnode τ ( e ) on its tail node σ ( e ) and each class of nodes V a = { v ∈ V : λ ( v ) = a } ,for a ∈ A may have a different behavior, thus allowing to account for hetero-geneity.Let each agent v ∈ V be endowed with a time-varying state Z v ( t ) takingvalues from a finite set X for every t ≥
0. We shall denote the vector of all agents’states by Z ( t ) = ( Z v ( t )) v ∈V and refer to it as the network configuration at time t . We shall consider asynchronous and semi-anonymous dynamics (ASD) wherethe state of each agent is updated at random activation times by choosing a newstate in response to the state of its out-neighbors, according to a conditionalprobability distribution that is invariant with respect to permutations of suchout-neighbors.Formally, let Z ( t ) be a continuous-time Markov chain with finite state spaceequal to the set of configurations Z = X V and the structure illustrated below. Definition 1 (Asynchronous semi-anonymous dynamics) . Let P = { θ ∈ R X + : (cid:48) θ = 1 } be the simplex of probability vectors over X . For every label a ∈ A , let Θ ( a ) : Z A×X + → P (1) be a stochastic kernel, and, for every node v ∈ V , let Υ G v : Z → Z A×X + be defined by (cid:0) Υ G v ( z ) (cid:1) ax = (cid:12)(cid:12)(cid:8) e ∈ E : σ ( e ) = v, λ ( τ ( e )) = a, z τ ( e ) = x (cid:9)(cid:12)(cid:12) , a ∈ A , x ∈ X . Then, Z ( t ) evolves as a continuous-time Markov chain on Z with transitionrates: Λ z , z + = (cid:40) γ Θ ( λ ( v )) z + v (Υ G v ( z )) if z and z + differ in the v -th entry only if z and z + differ in more than one entrywhere γ denotes the Poisson rate at which node v updates its state. The formulation above in Definition 1 is very general. Some remarks are inorder. 5 emark 1.
Classes can describe heterogeneous nodes in a variety of ways. Inour examples, we will consider the following three cases: i) classes describingdifferent update rules of the nodes; ii) classes describing nodes with differentdegree distributions; iii) classes describing node membership to different ‘com-munities’. In the most general scenario, a class might represent nodes belongingto a specific community, with a given update rule and a particular degree distri-bution.
Remark 2.
We emphasize that the new state of an agent, when it gets updated,does not need to be a deterministic function of its neighborhood. Indeed, weexplicitly allow for a stochastic rule of adopting a certain state. This allows usto model noisy or mixed-strategy best-response dynamics in networked games.
Our main interest in this paper is to track the evolution of some macroscopicfeatures, e.g., the evolution of the fraction of nodes belonging to a specific classthat are in a given state at time t . We will demonstrate that a mean-fieldapproximation can yield insight into this analysis for a large class of randomnetworks. To clarify the general formulation introduced above, we provide three examplesof ASD dynamics that will later be studied in more details in our numericalillustration section (see Section 7). Since in our examples the node update ruledepends only on the total number of neighbors in a given state, and not on theirlabel, we simplify the general notation introduced before and define: ξ x ( z ) = (cid:88) a ∈A (cid:0) Υ G v ( z ) (cid:1) ax , x ∈ X . (2)The explicit dependence on the Markov-chain state z will be omitted in thefollowing, whenever possible. Let X = {− , , } be the set of admissible states and let G = ( V , E , A , λ, σ, τ )be a labeled multigraph. The label a v = ( a + v , a − v ) of node v determines twogiven (in general, asymmetric) thresholds a + v , a − v , which trigger the transitionto state 1 and −
1, respectively. Specifically, when activated, the update of node v is given by Z v ( t ) = (cid:80) j ∈N + v Z j ( t − ) ≥ a + v (cid:80) j ∈N + v Z j ( t − ) ∈ ( − a − v , a + v ) − (cid:80) j ∈N + v Z j ( t − ) ≤ − a − v Z v ( t − ) = lim x ↑ t Z v ( x ) . The above rule can be encoded in our generalformulation by considering the functions:Θ ( a )1 ( ξ , ξ − , ξ ) = { ξ − ξ − ≥ a + } Θ ( a )0 ( ξ , ξ − , ξ ) = { ξ − ξ − ∈ ( − a − ,a + ) } Θ ( a ) − ( ξ , ξ − , ξ ) = { ξ − ξ − ≤− a − } which depend only on the numbers ξ , ξ − and ξ of out-neighbors in state 1, − Inspired by the model in [32], we consider a network game where each agentcan choose between two actions in X = {− , } . The network consists of twoclasses of nodes, i.e. A = { + , −} and V = V + ∪ V − . We assume that V + and V − represent agents following the majority (i.e., coordinating) or the minority(i.e., anti-coordinating) of their out-neighbors, respectively.Specifically, an agent is updated according to the following rule: if v ∈ V + then Z v ( t ) = (cid:80) j ∈N + v Z j ( t − ) > − (cid:80) j ∈N + v Z j ( t − ) < ± (cid:80) j ∈N + v Z j ( t − ) = 0and if v ∈ V − then Z v ( t ) = (cid:80) j ∈N + v Z j ( t − ) < − (cid:80) j ∈N + v Z j ( t − ) > ± (cid:80) j ∈N + v Z j ( t − ) = 0 . In essence, when a node is updated, it counts the number of neighbors in state − v ∈ V + , or itadopts the state of the minority of its neighbors if v ∈ V − . In the case of a tie,it chooses uniformly at random between states 1 and − (+)1 ( ξ , ξ − ) = { ξ >ξ − } + 12 { ξ = ξ − } Θ (+) − ( ξ , ξ − ) = 1 − Θ (+)1 ( ξ , ξ − )Θ ( − )1 ( ξ , ξ − ) = { ξ <ξ − } + 12 { ξ = ξ − } Θ ( − ) − ( ξ , ξ − ) = 1 − Θ ( − )1 ( ξ , ξ − )which depend only on the number ξ and ξ − of neighboors in state 1 and − .2.3 Evolutionary Roshambo Game (ERG) In this example, the best response of an agent follows the same rationale of thepopular rock-paper-scissors game. Specifically, we assume that nodes have threepossible states, i.e. X = { R, P, S } . When an agent is updated, it performs thefollowing computation:i) for each out-neighbor, it determines its best pairwise response accordingto the two-player game ϕ R P SR b c P c b
0S 0 c b with c > b , i.e. R wins over S, S wins over P, P wins over R. In Section 7we will consider the case with b = c/ Z v ( t ) = argmax ω ∈X (cid:88) j ∈N + v ϕ ( ω, Z j ( t − )) . When the maximizing set is not unique, the new state is selected uniformlyat random among the maximizing alternatives.In this case we have (ignoring ties for simplicity)Θ ( R ) ( ξ R , ξ P , ξ S ) = { bξ R + cξ S > max { cξ R + bξ P ,cξ P + bξ S }} Θ ( P ) ( ξ R , ξ P , ξ S ) = { cξ R + bξ P > max { bξ R + cξ S ,cξ P + bξ S }} Θ ( S ) ( ξ R , ξ P , ξ S ) = { cξ P + bξ S > max { bξ R + cξ S ,cξ R + bξ P }} . In this section we consider a labeled branching process, i.e. a particular ensembleof infinite labeled directed graphs with rooted tree structure, and then analyzeASD on it. As already said, the reason why we introduce this special graph isthat the analysis of ASD on it provides fundamental hints for the analysis ofASD on a general locally tree-like ensemble of graphs.More precisely, we will consider a labeled branching process completely de-scribed by probabilities distributions p k ,a = p k | a p a and q b k | a . The first is thejoint probability distribution that characterizes the root, i.e. the probabilitythat the root has label a ∈ A and out-degree vector k ∈ Z A + . We recall that thecomponent k b represents the number of out-neighbors belonging to class b ∈ A .The latter is the vectorial out-degree distribution for a non-root node with label8 , whose parent has label b . In next sections we will show that probability distri-butions p k ,a and q b k | a specifying the “approximating” labeled branching process,will be chosen so to exactly match statistics’ of the network under investigation.For this reason, we inform the reader that the same notation will be adoptedto denote statistics on both the network and the associated labelled branchingprocess. Recall that in our notation k v ∈ Z A + denotes the the out-degree vector of ver-tex v , whose component a ∈ A represents the number of out-neighbors of v belonging to class a ∈ A .We will call labeled branching process T with node set V = { v , v , ... } andlabel set A the rooted tree built through the following procedure: • Step 0: Start with a root node v and assign to it a random label A ∈ A and a random out-degree vector K (0) ∈ Z A + with joint probability distri-bution P ( A = a, K (0) = k ) = p k ,a . For every a ∈ A , add K (0) a out-edges with label ( A , a ) to the root v anddeclare all these edges active. Note that an edge label is defined as theordered pair of the labels associated to adjacent nodes.Then, for h = 1 , , . . . • Step h : If there are no active edges, stop. Otherwise, take any active edge e , let ( a, b ) be its label and declare the edge inactive. Assign to edge e ahead node τ ( e ) = v h with label λ ( v h ) = b and generate a random vector K ( h ) = k in Z A + with conditional probability distribution q a k | b , then forevery label c ∈ A add K ( h ) c new active outgoing edges to v h with label( b, c ). Let us now consider the ASD process over the graph T built above. In thefollowing matrix notation, vectors are meant to be column vectors, unless oth-erwise specified. Proposition 1.
Let Z ( t ) , for t ≥ , be the state vector of the ASD on T . Then,for every fixed time t ≥ , the following facts hold1. For every i ∈ V , the states { Z τ ( e ) ( t ) | e ∈ E : σ ( e ) = i } of the offsprings j of i in T are independent and identically distributed random variableswith ζ ω | a,b ( t ) = P ( Z j ( t ) = ω | A j = a, A i = b ) , ω ∈ X , a, b ∈ A satisfying d ζ ω | a,b ( t )d t = γ (cid:0) φ ω | a,b ( ζ ( t )) − ζ ω | a,b ( t ) (cid:1) , (3)9 here φ ω | a,b ( ζ ) = (cid:88) k ∈ Z A + ϕ ( k ,a ) ω ( ζ ) q b k | a and ϕ ( k ,a ) ω ( ζ ) = (cid:88) ξ ∈ Z A×X + : ξ = k Θ ( a ) ω ( ξ ) (cid:18) k ξ (cid:19) (cid:89) c ∈A (cid:89) g ∈X [ ζ g | c,a ] ξ cg . where (cid:18) k ξ (cid:19) = (cid:89) c (cid:18) k c ξ c (cid:19) , ξ c is the c-th row of matrix ξ and ξ cg denotes the ( c, g ) -th element ofmatrix ξ .2. The state Z v ( t ) of the root node v is a random variable with y ω | a ( t ) = P ( Z v = ω | A = a ) satisfying d y ω | a ( t )d t = γ (cid:0) ψ ω | a ( ζ ( t )) − y ω | a ( t ) (cid:1) , (4) with ψ ω | a ( ζ ) = (cid:88) k ∈ Z A + ϕ ( k ,a ) ω ( ζ ) p k | a Proof.
1. Let v be the root of T . Then, For every i ∈ V , the states Z j ( t ) :( i, j ) ∈ E of the offsprings of v i in T are independent and identicallydistributed Bernoulli random variables. Define ζ ω | a,b ( t ) = P [ Z j ( t ) = ω | A j = a, A i = b ], j ∈ V \ { v } , where v i is the father of v j , we have ζ ω | a,b ( t + ∆ t )= e − γ ∆ t ζ ω | a,b ( t ) + (cid:0) − e − γ ∆ t (cid:1) ×× (cid:88) k ∈ Z A + P [ Z j ( t ) = ω | K j = k , A j = a, A i = b ] P [ K j = k | A j = a, A i = b ]= ( γ ∆ t + o (∆ t )) (cid:88) k ∈ Z A + ϕ ( k ,a ) ω ( ζ ( t )) q b k | a + (1 − γ ∆ t + o (∆ t )) ζ ω | a,b ( t ) + o (∆ t )= ( γ ∆ t + o (∆ t )) φ ω | a,b ( ζ ( t )) + (1 − γ ∆ t + o (∆ t )) ζ ω | a,b ( t ) , from which we concluded ζ ω | a,b ( t )d t = lim ∆ t → ζ ω | a,b ( t + ∆ t ) − ζ ω | a,b ( t )∆ t = γ ( φ ω | a,b ( ζ ( t )) − ζ ω | a,b ( t )) .
10. Define y ( t ) = P [ Z v ( t ) = ω | A = a ], then with the same arguments wehave d y ω | a d t = γ ( ψ ω | a ( ζ ( t )) − y ω | a ( t )) . with ψ ω | a ( ζ ( t )) = (cid:88) k ∈ Z A + ϕ ( k ,a ) ω ( ζ ( t )) p k | a . Remark 3.
Whenever labels of neighbor nodes are independent, p k | a and q b k | a depend on a, b only through k = (cid:80) i k i , and things become simpler. Indeed, if wedefine ζ ω ( t ) = P [ Z j ( t ) = ω ] = (cid:80) a,b ∈A ζ ω | a,b ( t ) p a p b , then, it can be easily shownthat, similarly to (3) , we can derive an ODE for ζ ω ( t ) in the form d ζ ω ( t )d t = γ ( φ ω ( ζ ( t )) − ζ ω ( t )) , (5) where φ ω ( ζ ( t )) = (cid:88) a,b ∈A (cid:88) k ∈ Z + ϕ ( k,a ) ω ( ζ ( t )) q k | a,b p a p b ,q k | a,b = (cid:88) k ∈ Z A + : k T = k q b k | a , and ϕ ( k,a ) ω ( ζ ( t )) = (cid:88) ξ ∈ Z X + : ξ T = k Θ ( a ) ω ( ξ ) (cid:18) k ξ (cid:19) (cid:89) g ∈X [ ζ g ( t )] ξ g . Analogously, defining y ω ( t ) = P [ Z ( t ) = ω ] = (cid:80) a ∈A y ω | a ( t ) p a , the ODEreplacing (4) can be written as d y ω ( t )d t = γ ( ψ ω ( ζ ( t )) − y ω ( t )) , (6) where ψ ω ( ζ ( t )) = (cid:88) a ∈A (cid:88) k ∈ Z + ϕ ( k,a ) ω ( ζ ( t )) p k | a p a ,p k | a = (cid:88) k ∈ Z A + : k T = k p k | a . Remark 4.
The above analysis of ASD over the ensemble T can be easilyextended to the case in which the rate of activation of a vertex depends on itslabel a ∈ A . Without loss of generality we will assume γ = 1 in the following. ASD on labeled random networks
In this section we consider the evolution of ASD process over a multigraph G taken from a general ensemble of labeled directed graphs E ( n ) of size n . Inparticular, we show that, under certain conditions on the ensemble and on theinitial node configuration, the ASD process over G can be well approximatedby the same process over a labeled branching process T . The ensemble E ( n ) isdescribed by the ‘node statistics’ p d , k ,a,s , which provides the probability that anode picked at random has in-degree vector d , out-degree vector k , label a ∈ A and initial state s ∈ X . We shall assume that p d , k ,a,s factorizes as p d , k ,a,s = p d , k ,a p s | a . The above node statistics clearly provides all information needed to computeany marginal or conditional distribution we might be interested in. For example, p d , k ,a = (cid:80) s ∈X p d , k ,a,s is the distribution of in-degree vector, out-degree vectorand label of a generic node. As another example, p k ,a = (cid:80) d p d , k ,a providesthe distribution of out-degree vector and label of a generic node. We denote p a = (cid:80) k p k ,a the probability for a node to be associated with label a . Withintuitive notation, p k | a = p k ,a /p a denotes the distribution of out-degree vectorof a node with label a , and so on.As it always happens in graphs with heterogeneous degrees, we will needto distinguish the probability law of k v for a generic node v picked uniformlyat random, and the probability law of k v for a node v reached by traversingan edge. This because, in general, we could have correlation between in-degreeand out-degree. Moreover, when we reach a node by following a certain edge, itis also important to distinguish the label of the node originating the traversededge picked uniformly at random. To account for the above generality, we needto introduce some additional notation. Specifically, we define: q a d , k | b = d a p d , k ,b (cid:80) d , k d a p d , k ,b , (7)which is the distribution of in-degree vector d and out-degree vector k of a nodewith label b , reached by traversing an edge from a node with label a . Similarly, q a k | b = (cid:80) d q a d , k | b is the marginal distribution of out-degree vector of a node withlabel b , reached by traversing an edge from a node with label a . t We first observe that, since the process evolves through local interactions, thestate of a generic node v on a multigraph G = ( V , E , A , λ, σ, τ ) at time t isdetermined only by the structure and state of a relatively small neighborhoodaround v . Given a generic node v ∈ V , we define the relevant neighborhood N t of v as the subgraph induced by the set of all nodes in V having an impacton Z v ( t ), i.e., on the state of v at time t . Similarly, we define the relevantneighborhood T t as the subtree induced by the set of all nodes in T having animpact on Z v ( t ), where v is the root node of T .12he relevant neighborhood can be built by looking backward in time, iden-tifying dependencies between neighboring nodes. First, observe that the stateof v at time t depends on its out-neighbors v (cid:48) (one-hop away nodes) if and onlyif v has updated its state in [0 , t ] at least once, i.e., we can find an updatetime of v , ϑ v ( t ) ≤ t . The state of node v depends on a two-hop away node v (cid:48)(cid:48) , if and only if we can find a common neighbor v (cid:48) of v and v (cid:48)(cid:48) , such that ϑ v (cid:48) ( t ) < ϑ v ( t ) ≤ t . Similarly the state of v depends on a three-hops away node v (cid:48)(cid:48)(cid:48) only if we can find two nodes v (cid:48) and v (cid:48)(cid:48) along a directed path from v to v (cid:48)(cid:48)(cid:48) such that ϑ v (cid:48)(cid:48) ( t ) < ϑ v (cid:48) ( t ) < ϑ v ( t ) ≤ t , and so on.Due to the fact that update times of each node form independent Poissonprocesses with rate γ = 1, we can exploit well-known properties of the Poissonprocess (time-reversibility, memoryless property) to obtain N t (or T t ) as theresult of a process evolving forward in time, and exploring progressively theneighborhood of v by adding an exponentially distributed delay (of mean 1) oneach explored node, up to time t .More precisely, the relevant neighborhood of v is obtained by the follow-ing process. Vertices can be active, neutral or inactive. Initially, the relevantneighborhood is empty.1. The process starts by activating node v at time t = 0. All of the othernodes are set neutral.2. Upon the activation of a node, a random timer is associated to it, takenfrom an exponential distribution of mean γ = 1. Moreover, the node isadded to the relevant neighborhood, together with the outgoing edges.3. Upon expiration of its associated timer: i) an active node is set inactive;ii) all of its neutral out-neighbors are set active and added to the relevantneighborhood, together with outgoing edges.For t ≥
0, we can stop the above exploration process at time t (i.e., weno longer add nodes to the relevant neighborhood after time t ), obtaining atruncated version N t of G , composed of all the nodes that have been activated.Similarly, we obtain a truncated version T t of T . Let G = ( V , E , A , λ, σ, τ ) be a multigraph sampled from a given labeled networkensemble E ( n ) of size n . For t ≥
0, let N t be the relevant neighborhood at time t of a node v chosen uniformly at random from V , and let µ N t be its distributionon the multigraph space. Let T t be a labeled branching process as defined inSec. 3.1, truncated at time t , and let µ T t be its distribution.Proposition 2 identifies some sufficient conditions to guarantee that the ASDprocess over a network is well approximated by the solution of the differentialequation in (4). Proposition 2.
For t ≥ , let Z ( t ) be the state vector of the ASD at time t on G . Let z ω ( t ) = n |{ v ∈ V : Z v ( t ) = ω }| be the fraction of state- ω adopters t time t , and z ω ( t ) = E [ z ω ( t )] be its expectation over the ensemble. For any (cid:15) > , P ( | z ω ( t ) − y ω ( t ) | ≥ (cid:15) ) ≤ P ( | z ω ( t ) − z ω ( t ) | ≥ (cid:15) − (cid:107) µ N t − µ T t (cid:107) TV ) where y ω ( t ) is the solution of (4) .Proof. Notice that P ( | z ω ( t ) − y ω ( t ) | ≥ (cid:15) ) ≤ P ( | z ω ( t ) − z ω ( t ) | + | z ω ( t ) − y ω ( t ) | ≥ (cid:15) ) (8)We prove now that | z ω ( t ) − y ω ( t ) | ≤ (cid:107) µ N t − µ T t (cid:107) TV from which we get theresult.Observe that, by definition, the state Z v ( t ) of node v depends exclusively onthe initial states Z j (0) = σ j of the agents belonging to the relevant neighborhood N t of node v at time t , i.e., P ( Z v ( t ) = ω ) = χ ω ( N t ) where χ ω is a function inthe range [0,1].We thus have z ω ( t ) = E [ z ω ( t )] = 1 n (cid:88) v ∈V P ( Z v ( t ) = ω ) = (cid:90) χ ω ( g )d µ N t ( g ) . On the other hand, considering the state of the root in the labeled branchingprocess T , the output of the ODE (4) satisfies y ω ( t ) = (cid:90) χ ω ( g )d µ T t ( g ) . It then follows | z ω ( t ) − y ω ( t ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) χ ω ( g )d µ N t ( g ) − (cid:90) χ ω ( g )d µ T t ( g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:18) χ ω ( g ) − (cid:19) d µ N t ( g ) − (cid:90) (cid:18) χ ω ( g ) − (cid:19) d µ T t ( g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:107) µ N t − µ T t (cid:107) TV . From Proposition 2 we deduce that the evolution of the ASD process iswell approximated by the solution of the differential equation in (4) for graphensembles enjoying the following two fundamental properties:(a) Topological Property: Local Weak Convergence is required, in the sensethat (cid:107) µ N t − µ T t (cid:107) TV can be made arbitrarily small by increasing the graphsize.(b) Concentration Property: for large graph size, the fraction of state- ω adoptersin the ASD process must concentrate around its expectation with proba-bility close to one. 14 ASD over labeled configuration model
Considering the ensable E ( n ) of all labeled networks with given size n and statis-tics p d , k ,a,s , we define the corresponding labeled configuration model ensamble C n,p , on which we will restrict our investigation in the rest of the paper. Inparticular, we provide general bounds for ASD evolution over C n,p .Next we consider two specific examples of labeled configuration model, whichwe believe are particularly interesting, and apply to them the general boundsabove, showing asymptotic convergence to the ODE solution as the network sizegrows large. We first explicitly describe the construnction of the labeled configuration model C n,p . For each v ∈ V , denote with κ v = ( κ av ) a ∈A and δ v = ( δ av ) a ∈A the out-degree and in-degree vectors, respectively, such that there is exactly a fraction p d , k ,a of nodes v ∈ V with ( δ v , κ v , a v ) = ( d , k , a ). Denote with L a,a (cid:48) a set ofstubs, and define arbitrary maps ν a,a (cid:48) , γ a,a (cid:48) : L a,a (cid:48) → V , satisfying the property: | ν − a,a (cid:48) ( v ) | = δ av for nodes v with label λ ( v ) = a (cid:48) and | γ − a,a (cid:48) ( v ) | = κ a (cid:48) v with λ ( v ) = a . For all a, a (cid:48) ∈ A , let π a,a (cid:48) be chosen uniformly at random among all permuta-tions of L a,a (cid:48) and define multigraph G = ( V , E , A , λ, σ, τ ) with set of nodes V and E = (cid:83) ( a,a (cid:48) ) ∈A×A E a,a (cid:48) , where E a,a (cid:48) = ( γ a,a (cid:48) ( h ) , ν a,a (cid:48) ( π a,a (cid:48) ( h )) : h ∈ L a,a (cid:48) , and σ ( γ a,a (cid:48) ( h ) , ν a,a (cid:48) ( π a,a (cid:48) ( h ))) = γ a,a (cid:48) ( h ) and τ ( γ a,a (cid:48) ( h ) , ν a,a (cid:48) ( π a,a (cid:48) ( h ))) = ν a,a (cid:48) ( π a,a (cid:48) ( h )).Denote with l a,a (cid:48) = |L a,a (cid:48) | the total number of edges incoming to nodes withlabel a (cid:48) , originating from nodes with label a , so that: l a,a (cid:48) = n (cid:88) d , k d a p d , k ,a (cid:48) = n (cid:88) d , k k a (cid:48) p d , k ,a The total number of edges in the graph is l = (cid:80) a,a (cid:48) l a,a (cid:48) . The average in-degreeof a node, which is equal to the average out-degree, will be denoted by ¯ d = l/n .We repeat here for readers’ ease the expression of the fraction of nodes withlabel a (cid:48) , reached from a node with label a , having in-degree vector d and out-degree vector k : q a d , k | a (cid:48) = d a p d , k ,a (cid:48) (cid:80) d , k d a p d , k ,a (cid:48) (9)We summarize the notations in Table 1. In Figure 1, we show an example ofnotation use for a simple case in which A = { + , −} .We emphasize that the graph ensemble defined above extends the classicalconfiguration model, which can be recovered as a particular case by setting |A| = 1 . ordered set of labels a v = λ ( v ) label of node v V a = { v ∈ V : a v = a } δ v = ( δ av ) a ∈A in-degree of node v κ v = ( κ av ) a ∈A out-degree of node v L a,a (cid:48) set of stubs from nodes with label a to nodes with label a (cid:48) l a,a (cid:48) = |L a,a (cid:48) | number of edges from nodes with label a to nodes with label a (cid:48) π a,a (cid:48) permutation of L a,a (cid:48) ν a,a (cid:48) : L a,a (cid:48) → V map with the property | ν − a,a (cid:48) ( v ) | = δ av for all v ∈ V a (cid:48) γ a,a (cid:48) : L a,a (cid:48) → V map with the property | γ − a,a (cid:48) ( v ) | = κ a (cid:48) v for all v ∈ V a E a,a (cid:48) = ( γ a,a (cid:48) ( h ) , ν a,a (cid:48) ( π a,a (cid:48) ( h )) : h ∈ L a,a (cid:48) } set of edges from nodeswith label a to nodes with label a (cid:48) E = (cid:83) a,a (cid:48) E a,a (cid:48) Table 1: Notations+ + + + + − − − v + + + − − − − − a v = + κ v = ( κ + v , κ − v ) = (3 , δ v = ( δ + v , δ − v ) = (5 , ν − , + ( v ) ν − − , + ( v ) γ − , + ( v ) γ − , − ( v )Figure 1: Labeled configuration model with two classes A = { + , −} In order to apply the general approximation result stated in Proposition 2 tothe labeled configuration model defined above, we need to prove both LocalWeak Convergence and Concentration Property of ASD. The following theoremsactually provide the main results of our paper: • Theorem 1 is a topological result and is related exclusively to the proper-ties of the labeled configuration model. More precisely, it provides a usefulbound on the total variation distance between the relevant neighborhood16f a graph drawn uniformly at random from the labeled configurationmodel ensemble and the labeled branching process described in Section 3.The proof is rather technical and is postponed to Appendix A.1. • Theorem 2 is related to the ASD evolution and to the specific propertiesof the labeled configuration model. It provides a bound on the distancebetween the fraction of state- ω adopters and its expectation. The proofcan be found in Appendix A.2.Let G = ( V , E , A , λ, σ, τ ) be a multigraph sampled from the ensemble C n,p .For t ≥
0, let N t be the relevant neighborhood at time t of a node v chosenuniformly at random from V , and let µ N t be its distribution. Let T t be thelabeled branching process truncated at time t and let µ T t be its distribution.Moreover, let W b,at be the number of edges in T t from nodes with label b ∈ A to nodes with label a ∈ A and let F W b,at ( x b,a ) = P ( W b,at > x b,a ). Theorem 1 (Topological Property) . We have (cid:107) µ T t − µ N t (cid:107) TV ≤ P ( T t (cid:54) = N t ) ≤ inf X ∈ ( R + ) A×A (cid:88) a,b ∈A (cid:34) F W b,at ( x b,a ) + x b,a ( x b,a + 1)2 (cid:80) d , k d b q b d , k | a l b,a (10)+ (cid:88) b (cid:48) (cid:54) = b x b,a x b (cid:48) ,a (cid:80) d , k d b q b (cid:48) d , k | a l b,a (11) Example 1 (Topological Property for the classical configuration model) . If |A| = 1 then the model ensemble C n,p boils down to the classical configurationmodel and the bound derived in (10) reduces to (cid:107) µ T t − µ N t (cid:107) TV ≤ P ( T t (cid:54) = N t ) ≤ inf x> (cid:20) F (cid:102) W t ( x ) + (cid:80) d,k dq d,k x ( x + 1)2 nd (cid:21) (12) where (cid:102) W t is the number of nodes in T t and F (cid:102) W t ( x ) = P ( (cid:102) W t > x ) . We next introduce the concentration result that allows us to estimate towhat extent the fraction of state- ω adopters in the ASD process concentratesaround its expectation. Theorem 2 (Concentration Property) . Let G be a multigraph sampled fromthe ensemble C n,p . We denote with N vt the relevant neighborhood at time t ofa node v , sampled with a probability proportional to its in-degree, and with V vt the number of nodes in it. For t ≥ , let Z ( t ) be the state vector of the ASDdynamics on G , b ( t ) = |{ v ∈ V : Z v ( t ) = ω }| be the number of state- ω adopters t time t conditioned to G . For any (cid:15) > , η > , x > , s ≥ we have P ( | b ( t ) − E [ b ( t )] | > ηn ) ≤ − η n (cid:15) ) tx + (cid:18) η (cid:19) (1 + (cid:15) ) tn s E v [ | V vt | s ] x s + 2e − nt(cid:15) (cid:15) ) + 2e − η n t + η/ + (cid:18) η (cid:19) s x s (cid:88) w ∈V | δ w | [ E [ | V wt | s ] + 2e − η n dx Remark 5.
We emphasize that the bounds presented in Theorem 1 and Theorem2 represent an important step forward with respect to results already known inliterature. In particular, Lemma 5 and Proposition 2 in [34] states a similarresult for a different, simplified version of our system dynamics in which: i)node updates are synchronized, being triggered by a common discrete time step;ii) the update rule is restricted to be the deterministic linear threshold model; iii)both the maximum in-degree d max and the maximum out-degree k max of nodesare supposed to be finite. In contrast to [34], we introduce a much more generalresult along three different directions:1. We consider asynchronous dynamics (each node is updated by an inde-pendent Poisson clock). Hence the neighborhood exploration process in theproof has to take into account this new source of randomness. Specifically,the estimation of the total variation (10) is split into two terms, which areobtained by conditioning on the number of nodes in T t . The necessity ofthis refined analysis will be clear in the next section.2. We consider arbitrary semi-anonymous dynamics, with possible random(noisy) response to the state of neighbors. Moreover, we define our dy-namics on a much more general ensemble of labelled random graphs, whichallows us to differentiate the distribution of incoming/outgoing edges foreach pair of classes.3. We allow the maximum in- and out-degree of nodes to possibly scale with n (under some technical constraints). This is crucial for applications tosocial networks and many other complex systems in which the degree dis-tribution has often been observed to follow a power law. But notice thateven in the case of the classic Erd¨os-R´enyi random graph G ( n, p ) , d max or k max of course are not independent of n . Note that, in the case of finite d max , k max , by taking x = k t max our bound in (12) leads to (cid:107) µ N t − µ T t (cid:107) TV ≤ d max k max ( k t max + 1)2 nd . recovering the result in [34] (Lemma 5). Remark 6.
We emphasize that our results could be also extended to weighteddirected networks, under the assumption that weights on edges are described by .i.d. random variables. For example, one could consider the relative weightedpopularity of a given state among the neighbors, to determine the next stateof an agent. It is obvious that in this case the dynamics are not purely semi-anonymous since different neighbors exert different influence strengths. How-ever, the randomness of the weights and i.i.d. assumption make our approachstill valid. In the next section, we show how the bounds derived above for Local WeakConvergence and concentration property can be used to study asymptotic be-havior of ASD on a labeled configuration model with power-law degree distri-bution. More precisely, we will consider a sequence of labeled graphs with size n and described by distributions p ( n ) k | a , q ( n ) k | a such that p ( n ) k | a n →∞ −→ p k | a , q ( n ) k | a n →∞ −→ q k | a . Then we will consider the labeled branching process obtained by the construc-tion above with asymptotic distributions. The following proposition quantifiesthe distance between the solution corresponding to the differential equation withdistribution p ( n ) k | a , q ( n ) k | a and the solution corresponding to the differential equationwith asymptotic distribution p k | a , q k | a . Proposition 3.
Let • ζ ( n ) ( t ) be the solution of (3) with q ( n ) k | a and initial condition ζ ( n )0 ; • ζ ( t ) be the solution of (3) with q k | a and initial condition ζ . • y ( n ) ( t ) be the solution of (3) with p ( n ) k | a and initial condition y ( n )0 ; • y ( t ) be the solution of (3) with p k | a and initial condition y .In addition let φ ( z ) − z and ψ ( z ) be Lipschitz continuous in [0 , |A| , and let L and M > be the Lipschitz constants corresponding to infinity norm. Then sup t ∈ [0 ,m ∆] (cid:107) ζ ( n ) ( t ) − ζ ( t ) (cid:107) ∞ ≤ (cid:107) ζ ( n )0 − ζ (cid:107) ∞ (1 − ∆ L ) m + 1 L (cid:18) − ∆ L ) m − (cid:19) (cid:107) q ( n ) k | a − q k | a (cid:107) TV with ∆ < /L and sup t ∈ [0 ,m ∆] (cid:107) y ( n ) ( t ) − y ( t ) (cid:107) ∞ ≤ (cid:107) y ( n )0 − y (cid:107) ∞ (1 − ∆) m + (cid:18) − ∆) m − (cid:19) (cid:34) M sup t ∈ [0 ,m ∆] (cid:107) ζ ( n ) ( t ) − ζ ( t ) (cid:107) ∞ + (cid:107) p ( n ) k | a − p k | a (cid:107) TV (cid:35) . The proof can be found in Appendix B.In particular, if p ( n ) k | a is a truncated version of p k | a , we can apply to theprevious bound the following statement of immediate verification:19 roposition 4. Consider a generic distribution p k | a and its truncated version p ( n ) k | a , i.e. p ( n ) k | a = p k | a k ∈B n (cid:80) k ∈B n p k | a for a generic compact set B n ∈ N |A| , then we have: (cid:107) p ( n ) k | a − p k | a (cid:107) TV = 1 − (cid:88) k ∈B n p k | a . In this section we consider the classical configuration model with a truncatedpower-law degree distribution , which is a particular case of labeled configurationmodel with |A| = 1. We simplify the notation: let p ( n ) d,k be the fraction of nodeswith in-degree d and out-degree k , where we have highlighted the number ofnodes n , and let d = (cid:80) d,k dp ( n ) d,k be the average degree. (cid:80) d,k dq ( n ) d,k = Θ( n δ ) 0 ≤ δ < / q ( n ) k ∝ k − β β > k max = Θ( n ζ ) ζ < min (cid:110) − δ , β − (cid:111) Table 2: Assumptions on power-law degree distribution
Assumption 1.
Let us assume that (cid:88) d,k dq ( n ) d,k = (cid:88) d,k d d p ( n ) d,k = Θ( n δ ) with ≤ δ < / . This means that we allow the average in-degree of a node,reached by an edge selected uniformly at random, to possibly scale with n . More-over, we will assume that q ( n ) k = 1 d (cid:88) d dp ( n ) d,k = O ( k − β ) follows a power-law of exponent β > and maximum value k max = Θ( n ζ ) with ζ < min (cid:110) − δ , β − (cid:111) . Let µ s be the s -th moment of q = { q ( n ) k } k ≥ . From Assumption 1 we have µ s = (cid:40) Θ(1) if β > s + 1Θ( n ζ ( s +1 − β ) ) if 2 < β < s + 1 . (13)20otice that, being β > µ is always finite and, therefore, does not scale with n. In order to guarantee that the ASD over a network drawn uniformly atrandom from the configuration model ensemble is well approximated by thesolution of ODE, it is sufficient that the terms in the upper bounds derivedin (12) (see Example 1), in Theorem 2, and in Proposition 3 go to zero when n → ∞ . In the following, let N ( n ) = ( V ( n ) , E ( n ) ) be a sequence of networks,each one sampled from the corresponding model ensemble C n,p ( n ) , where { p ( n ) } n is a sequence of truncated versions of a power law distribution of p satisfyingAssumption 1. For t ≥
0, let N ( n ) t be the relevant neighborhood of a node v chosen uniformly at random from the node set V ( n ) . Moreover, let T ( n ) t bethe sequence of truncated Galton-Watson (GW) processes (see [16]) for whichthe root offspring follows distribution p ( n ) , while the degree of non-root nodesfollow law q ( n ) . Finally, let p ( n ) n →∞ → p and q ( n ) n →∞ → q . We summarize themain assumptions and notations in Table 2.Before presenting the topological result for the configuration model withpower-law degree distribution, we present two technical results, whose proofsare postponed to Appendix C. Lemma 1 (Bound on the number of nodes/edges in T t ) . Consider the GWprocess T in which the offspring distribution of the root follows law p , while thedegree of remaining nodes follows law q . Let T t be the corresponding randomtree obtained by truncating T at time t , and (cid:102) W t be the number of nodes in T t .Let h n = c log n for some c > and t = o ( h n ) as n → ∞ , then for any s > we have F (cid:102) W t ( x n ) ≤ E [ N sh n ] x sn + o (1 /n ) n → ∞ where { N h } h ∈ N is the number of nodes in a truncated version of T with maximalwidth h . Lemma 2.
Let { N h } h ≥ be a supercritical GW process, in which the offspringdistribution of the root follows law p , while the degree of remaining nodes followslaw q . We have: E [ N sh ] = O ( µ s · µ s ( h − ) , ∀ β > , where µ j is the j -th momentof q . Theorem 3 (Topological result for configuration model with power-law degreedistribution) . With the above definitions, let µ N ( n ) t and µ T ( n ) t be the distributionsof N ( n ) t and T ( n ) t , respectively. Under Assumption 1, for t = o (log n ) , we have (cid:107) µ T ( n ) t − µ N ( n ) t (cid:107) TV ≤ P ( T t (cid:54) = N t ) = o (1) when n → ∞ .Proof. From inequality (12), we have (cid:107) µ T t − µ N t (cid:107) TV ≤ P ( T t (cid:54) = N t ) ≤ inf x> (cid:20) F (cid:102) W t ( x ) + (cid:80) d,k dq d,k x ( x + 1)2 nd (cid:21) (14)21here (cid:102) W t is the number of nodes in T t . Let x n = n (1 − δ ) / − γ for some γ > h n = c log n for some c > (cid:107) µ T t − µ N t (cid:107) TV ≤ (cid:80) d dq d,k x n ( x n + 1)2 nd + F (cid:102) W t ( x n ) n → ∞≤ (cid:80) d dq d,k x n ( x n + 1)2 nd + E [ N sh n ] x sn + o (1 /n )where the last inequality holds for any s > { N h } h ∈ N is (the number ofnodes of) a truncated GW process of maximal width h , in which the offspringdistribution of the root follows law p , while the degree of remaining nodes followlaw q (see Lemma 1).Under Assumption 1 we prove that there exists s such that E [ N shn ] x sn = o (1 /n )as n → ∞ and we conclude that for some γ > (cid:107) µ T t − µ N t (cid:107) TV ≤ dn γ + o (1 /n ) = o (1) n → ∞ . To find a suitable value of s , we distinguish two cases.(i) If β > (cid:98) − δ (cid:99) + 2 we can simply choose s = (cid:98) − δ (cid:99) + 1. By so doing,we stay in the case β > s + 1, and from (13) and Lemma 2 we get: E [ N sh n ] = Θ( n cs log µ ) and E [ N sh n ] x sn = Θ( n − s ( − δ − γ − c log µ )) = o (1 /n ) n → ∞ (ii) If β ≤ (cid:98) − δ (cid:99) + 2, we choose instead a sufficiently large value of s , fallingin the case s > β − µ s scales with n as in (13). In particular,from Lemma 2 we have E [ N sh n ] = Θ( n ζ ( s +1 − β )+ cs log µ ) . Thus E [ N sh n ] /x sn = Θ (cid:16) n ζ ( s +1 − β )+ cs log µ − s ( − δ − γ ) (cid:17) . We now observe that if there exists s ∈ N such that s (cid:18) − δ − ζ − c log µ − γ (cid:19) > − ζ ( β −
1) (15)then E [ N sh n ] /x sn = o (1 /n ) for n → ∞ . Since ζ < / ( β − ζ < min { − δ } , we can always find twosufficiently small constants γ and c such that (cid:0) − δ − ζ − c log µ − γ (cid:1) isalso positive. Therefore, there exists an integer s large enough such thatboth s > β − orollary 1. For t ≥ , let Z ( t ) be the state vector of the ASD dynamics on N ( n ) and z ( n ) ω ( t ) = n |{ v ∈ V : Z v ( t ) = ω }| be the fraction of state- ω adoptersat time t . Under Assumptions 1 for t = o (log n ) , for any η > P ( | z ( n ) ω ( t ) − y ω ( t ) | > η ) = o (1) for n → ∞ where y ω ( t ) is the solution of (4) over a GW tree T t with the asymptotic degreestatistics p and q .Proof. Let T ( n ) t be the continuous-time branching process truncated up to time t , µ T ( n ) t be its distribution. Denote by y ( n ) ( t ) the solution of (4) with p ( n ) k | a andinitial condition y ( n )0 . We have P ( | z ( n ) ω ( t ) − y ω ( t ) | ≥ η ) ≤ P ( | z ( n ) ω ( t ) − y ( n ) ω ( t ) | ≥ η/
2) + P ( | y ( n ) ω ( t ) − y ω ( t ) | ≥ η/ t = o (log n ) as n → ∞ we have (cid:107) µ T ( n ) t − µ N ( n ) t (cid:107) TV = o (1) when n → ∞ . We conclude that for any η > n such that, if n ≥ n , then (cid:107) µ N ( n ) t − µ T ( n ) t (cid:107) TV ≤ P ( T t (cid:54) = N t ) ≤ η. Using Proposition 2, it follows that for any η > n : P ( | z ( n ) ω ( t ) − y ω ( t ) | ≥ η ) ≤ P ( | z ( n ) ω ( t ) − z ( n ) ω ( t ) | ≥ η/
4) + P ( | y ( n ) ω ( t ) − y ω ( t ) | ≥ η/ x = n / , s = 3, and (cid:15) >
0, we get P ( | z ( n ) ω ( t ) − y ω ( t ) | ≥ η ) ≤ P ( | y ( n ) ω ( t ) − y ω ( t ) | ≥ η/
2) + o (1)where, we have applying jointly Corollary 4 and Corollary 5 in Appendix C tobound the third moment of V vt . Finally Propositions 3 and 4 guarantee that | y ( n ) ω ( t ) − y ω ( t ) | → P ( | y ( n ) ω ( t ) − y ω ( t ) | ≥ η/
2) = 0 for large enough n . Remark 7 (Relation to Theorem 1 in [34]) . In [34] a similar approximationresult was proved in the specific case of binary synchronous LTM. More preciselyTheorem 1 in [34] implies that, for sequences of networks whose network statis-tics converge to a given limit as the network size grows large, the concentrationresult is guaranteed on the configuration model for finite values of t , providedthat the maximum in- and out-degrees remain bounded. It is worth remarkingthat Theorem 1, proved in this paper, is much more general and the result appliesto: (a) asynchronous semi-anonymous dynamics; (b) networks with maximumin- and out-degrees growing as a function of network size n ; (c) networks withpower-law degree distribution. Remark 8 (Transient behavior versus Asymptotic behavior) . Theorem 1 guar-antees the approximation result for values of t growing at most as o (log n ) .However, using techniques devised in [6] for the exchange of limits in t and n ,it can be shown that with high probability, as the network size grows large, theasymptotic fraction of ω -adopters concentrates on the set of all stationary points f the ODE in (1) . This implies that, when the ODE in (1) admits multiplestationary points, then from our result it is not possible to predict the asymptoticlimit of the system. However, if the system in (1) admits a unique (globally at-tractive) stationary point, convergence is guaranteed in that point for all initialconditions. In this section, we apply the bounds derived for Local Weak Convergence andConcentration Property in Section 4 to study the asymptotic ASD on a labeledconfiguration model with community structure, which is a key feature of manyreal systems. In particular, we consider a Configuration Block Model (CBM)with K communities of sizes { n i } Ki =1 , which are mapped into correspondingclasses with labels { a i } Ki =1 .When the maximum in/out degree of nodes is finite both Local Weak Con-vergence and Concentration property can be easily proven by taking the simpleworst-case in which all nodes have in/out degree equal to the maximum in/outdegree, so we will consider here a more challenging case, where in/out degree ofnodes is allowed to scale with n .However, to simplify the analysis, we assume no correlation between in-degree and out-degree of a node. As a consequence, the law of p is the same asthe law of q , and ζ ω | a,a (cid:48) ( t ) = y ω | a ( t ) (see Proposition 1).Moreover, we assume that the number of edges established from a node ofcommunity i towards nodes of community j is independent for any pair ( i, j ),including the special case i = j , i.e., intra-community edges. Therefore, p d , k | a factorizes into: p d , k | a = (cid:89) i ∈A p in i,a [ d i ] (cid:89) j ∈A p out a,j [ k j ]We will require that in/out degree sequences of the nodes, although possiblydependent on the network size n , generate empirical distributions p in i,a [ d ] and p out a,j [ k ] with a light tail, for any pair ( i, a ) or ( a, j ), thus having finite momentsof any order.In order to guarantee that the ASD over a network drawn uniformly atrandom from the CBM ensemble is well approximated by the solution of theODE, it is sufficient that the terms in the upper bound derived in (10) and inTheorem 2 go to zero when n → ∞ . In the following, let N ( n ) = ( V ( n ) , E ( n ) ) be asequence of networks, each one sampled from the corresponding CBM ensemble G ( n ) = G ( { n i } Ki =1 , p ( n ) d , k ,a ), and, for t ≥
0, let N ( n ) t be the relevant neighborhoodof a node v chosen uniformly at random from the node set V ( n ) . Moreover, let T ( n ) t be the sequence of GW processes with offspring distribution following law p ( n ) . Finally, let p ( n ) n →∞ → p . Theorem 4.
Under above definitions and assumptions on the CBM ensemble,let µ N ( n ) t and µ T ( n ) t be the distributions of N ( n ) t and T ( n ) t , respectively. For t = o (log n ) , we have (cid:107) µ T ( n ) t − µ N ( n ) t (cid:107) TV = o (1) when n → ∞ . Corollary 2.
Under above definitions and assumptions on the CBM ensemble,for t ≥ , let Z ( t ) be the state vector of the ASD dynamics on N ( n ) and z ( n ) ω ( t ) = n |{ v ∈ V : Z v ( t ) = ω }| be the fraction of state- ω adopters at time t . For t = o (log n ) and any η > : P ( | z ( n ) ω ( t ) − y ω ( t ) | > η ) = o (1) for n → ∞ where y ω ( t ) is the solution of (4) over a GW tree T t with the asymptotic degreestatistics p and q . The proof follows exactly the same lines of Corollary 1.
Remark 9.
Note that the approach followed in the proof of Theorem 4 canbe extended to a significantly more general class of labeled configuration graphs C n,p . The key step of the approach pursued in Theorem 4 is to find a distributionmeeting the following two constraints: i) it stochastically dominates the out-degree distribution of every class; ii) its tail is not too heavy, so that we caneffectively bound the moments on the number of nodes in the corresponding GWtruncated tree by exploiting (43) (see Appendix C). Under such conditions wecan conclude that (cid:107) µ T t − µ N t (cid:107) TV = o (1) when n → ∞ . In particular, theprevious considerations apply every time the dominating distribution is a powerlaw meeting the constraints in Assumption 1. In the case of regular random graphs, where all nodes have the same out-degree,we can analytically derive some interesting properties of the ODEs describingthe temporal evolution of the system (see Proposition 1), for each of the threeexamples of ASD introduced in Section 2.2. In particular, we can characterizethe equilibrium points of the system and their stability.
We assume that all agents have the same out-degree k and symmetric thresholds,i.e., k v = k and a ± v = r for all v ∈ V . In this case, there is a single class a = r for which p k | r = q k | r = 1, andΘ ( r )1 ( ξ , ξ − , k − ξ − ξ − ) = { ξ − ξ − ≥ r } Θ ( r )0 ( ξ , ξ − , k − ξ − ξ − ) = { ξ − ξ − ∈ ( − r,r ) } Θ ( r ) − ( ξ , ξ − , k − ξ − ξ − ) = { ξ − ξ − ≤− r } y (0) , y − (0)),the dynamics over the continuous-time branching process is described by (cid:40) d y d t = φ ( k,r )+ ( y ( t ) , y − ( t )) − y ( t ) , d y − d t = φ ( k,r ) − ( y ( t ) , y − ( t )) − y − ( t ) , (17)and y = 1 − y − y − , where φ ( k,r )+ ( x, z ) = k (cid:88) u = r ( k − u ) ∧ ( u − r ) (cid:88) v =0 (cid:18) ku (cid:19)(cid:18) k − uv (cid:19) x u z v (1 − x − z ) k − u − v = (cid:98) k − r (cid:99) (cid:88) v =0 k − v (cid:88) u = v + r (cid:18) ku (cid:19)(cid:18) k − uv (cid:19) x u z v (1 − x − z ) k − u − v (18)and φ ( k,r ) − ( x, z ) = φ ( k,r )+ ( z, x ).Some analytical properties of the dynamical system can be deduced fromthe analysis of φ ( k,r )+ ( x, z ) and φ ( k,r ) − ( x, z ). In particular, we are interested infinding stationary points, i.e. those points satisfying the following equations (cid:40) φ ( k,r )+ ( x, z ) = xφ ( k,r ) − ( x, z ) = z, x = z ≤ Lemma 3.
Let φ ( k,r )+ ( x, z ) be as defined in (18) . Then the following propertieshold:1. φ ( k,r )+ ( x, z ) is non decreasing in x and strictly increasing if < r ≤ k ;2. φ ( k,r )+ ( x, z ) is non increasing in z and strictly decreasing if ≤ r < k ;3. φ ( k,r )+ (0 , z ) = 0 for < r ≤ k , φ ( k, ( x,
0) = 1 , φ ( k,r )+ (1 ,
0) = 1 , for ≤ r ≤ k ;4. ∇ x φ ( k,r )+ = (cid:80) (cid:98) k − r (cid:99) v =0 χ ( v + r − , v ) x v + r − z v (1 − x − z ) k − v − r with χ ( u, v ) = (cid:18) ku (cid:19)(cid:18) k − uv (cid:19) ( k − u − v ) ∇ z φ ( k,r )+ = − (cid:80) ku = r χ ( u, v (cid:48) ) x u z v (cid:48) (1 − x − z ) k − u − v (cid:48) − with v (cid:48) = ( k − u ) ∧ ( u − r ) . The proof is trivial and we omit it for brevity.26 roposition 5. If ≤ r < k , then1. the equation φ ( k,r )+ ( x,
0) = x has exactly three solutions { , x (cid:63) , } with x (cid:63) ∈ (0 , ;2. for every x ∈ ( x (cid:63) , ¯ x ) with < x (cid:63) < ¯ x ≤ there exists a unique value z ( x ) such that φ ( k,r )+ ( x, z ( x )) = x.
3. the equation φ ( k,r ) − (0 , z ) = z has exactly three solutions { , z (cid:63) , } with z (cid:63) ∈ (0 , ;4. for every z ∈ ( z (cid:63) , ¯ z ) with < z (cid:63) < ¯ z ≤ , there exists a unique value x ( z ) such that φ ( k,r ) − ( x ( z ) , z ) = z. Proof.
From definition (18) we have φ ( k,r )+ ( x,
0) = k (cid:88) u = r (cid:18) ku (cid:19) x u (1 − x ) k − u If 2 ≤ r < k , the function φ ( k,r )+ ( x,
0) has a lazy-S-shaped graph, i.e., it isincreasing, with a unique inflection point at (cid:101) x = ( r − / ( k − (cid:101) x and concave on the right-hand side of (cid:101) x (seeLemma 4 in [34]). From this fact and the observation that φ ( k,r )+ (0 ,
0) = 0and φ ( k,r )+ (1 ,
0) = 1 we get the assertion at Point 1. It can also be provedthat x (cid:63) ∈ [( r − /k, r/k ]. Denoting F ( x, z ) = φ ( k,r )+ ( x, z ) − x and observingthat F ( x (cid:63) ,
0) = φ ( k,r )+ ( x (cid:63) , − x (cid:63) = 0 and ∇ z F ( x (cid:63) , (cid:54) = 0 (see expression inPoint 3 of Lemma 3), the statement in Point 2. is obtained by the implicitfunction theorem [5]. Point 3. and 4. are straightforward from the relation φ ( k,r ) − ( x, z ) = φ ( k,r )+ ( z, x ).In Figure 2 the functions z ( x ) such that φ ( k,r )+ ( x, z ( x )) = x and x ( z ) suchthat φ ( k,r ) − ( x ( z ) , z ) = z are depicted for threshold values r = 2 (left) and r = 3(right) and degree k = 10. In addition to stationary points { (0 , , ( x (cid:63) , , (1 , , (0 , z (cid:63) ) , (0 , } , as derived in Lemma 3, extra stationary points are placed at intersections be-tween curves z ( x ) and x ( z ). In the specific case with k = 10 it can be noticedthat if r = 2 then there are two additional stationary points. The followingproposition gives sufficient conditions guaranteeing that the set of stationarypoints only contains the trivial points { (0 , , ( x (cid:63) , , (1 , , (0 , z (cid:63) ) , (0 , } . Proposition 6. If r ≥ ( k + 1) / , { (0 , , ( x (cid:63) , , (1 , , (0 , z (cid:63) ) , (0 , } are theonly fixed points of the system. x y φ − ( x,y ) > yy ⋆ x ⋆ φ + ( x,y ) > x x y φ − ( x,y ) > yy ⋆ x ⋆ φ + ( x,y ) > x Figure 2: k = 10 , r = 2 (left), k = 10 , r = 3 (right). Proof.
Notice that φ ( k,r )+ ( x, x ) = φ ( k,r ) − ( x, x ) = k (cid:88) u = r ( k − u ) ∧ ( u − r ) (cid:88) v =0 (cid:18) ku (cid:19)(cid:18) k − uv (cid:19) x u + v (1 − x ) k − u − v ≤ k (cid:88) u = r (cid:18) ku (cid:19) x u (1 − x ) k − u = ϕ ( k,r ) ( x ) . If r ≥ ( k + 1) /
2, we have ϕ ( k,r ) ( x ) < x for all x < / φ ( k,r )+ ( x, x ) < x from which we conclude the assertion.The following corollary can be proved by linearization. Corollary 3.
The following properties hold1. (0 , is a locally stable stationary point for ≤ r ≤ k , unstable if r = 1 .2. (0 , and (1 , are locally stable stationary points for ≤ r < k , unstableotherwise.3. ( x (cid:63) , and (0 , z (cid:63) ) are unstable stationary points.4. The set of points { ( x, z ) ∈ R : x = z } is invariant. The basins of attraction for the ODE in (17) are shown in Figure 3 for degree k = 10 and threshold r ∈ { , } . Basins are evaluated numerically, by solvingthe ODE system for a wide set of initial conditions. More specifically, for eachinitial condition x , z the color in the picture represents the asymptoticallystable equilibrium point (yellow for (0,1), green for (0,0) and blue for state(1 , , ,
1) and (1 ,
0) are locally stable stationary points.28 x y x y Figure 3: k = 10 and r = 2 (left), r = 3 (right) We again consider the homogenous case in which all nodes have out-degree k .For simplicity, in the following, we will suppose that k is odd, in order to avoidties, although the extension to the case of even k is straightforward. Let α bethe fraction of coordinating nodes. From Proposition 1, the evolution of thenode states is governed by the following ODE:d y ( t )d t = αφ ( y ( t )) + (1 − α ) (1 − φ ( y ( t ))) − y ( t ) , (19)where φ ( y ( t )) is the probability with which a coordinating node enters state1, given by φ ( y ) = k (cid:88) k = (cid:100) k/ (cid:101) (cid:18) kk (cid:19) y k (1 − y ) k − k . (20)The above ODE derives from the fact that the probability of stepping to state1 for an anti-coordinating node is equal to the probability of stepping to state-1 for a coordinating node. Let us define (cid:96) ( y ( t )) the RHS of (19). It is easilyseen that, since φ (1 /
2) = 1 / (cid:96) (1 /
2) = 0, so that y = 1 / α .The derivative of (cid:96) with respect to y is given by:d (cid:96) d y = (2 α −
1) d φ d y − α − k (cid:18) k − (cid:98) k (cid:99) (cid:19) ( y (1 − y )) (cid:98) k (cid:99) − α ≤ α th or α > α th , with α th = 12 k − k (cid:0) k − (cid:98) k (cid:99) (cid:1) (23) • If α ≤ α th , then d (cid:96) d y ≤ < y <
1. It then turns out that the onlystationary point is y = 1 /
2, which is stable and has a basin of attractionequal to [0 , • If α > α th , then d (cid:96) d y has two zeros, symmetric with respect to z = 1 / y (1 − y ) = 1 (cid:98) k (cid:99) (cid:113) (2 α − k (cid:0) k − (cid:98) k (cid:99) (cid:1) Because of that, there are three stationary points, out of which the one in z = 1 / z = 1 /
2, i.e., y ± = 1 / ± (cid:15) , and stable, with basins of attraction [0 , / / , (cid:15) as a function of k and α . α ε k = 5k = 15k = 25k = 35k = 45k = 55 Figure 4: Position of the stable stationary points as a function of the fractionof cooperative nodes and the common node degree.
In this section, we consider the ERG dynamics on a regular graph with k v = k .30et y ω , ω ∈ { R, P, S } be the fraction of nodes in state ω . Moreover, let π ω , ω ∈ { R, P, S } be the probability that a given node, when activated, switches tostate ω thanks to its neighbors’ states. More precisely, for ω = R : π R = P (cid:26) ξ S > k , ξ P < k (cid:27) + 12 P (cid:26) ξ S = k , ξ P < k (cid:27) + 12 P (cid:26) ξ S > k , ξ P = k (cid:27) + 13 P (cid:26) ξ S = k , ξ P = k (cid:27) (24)and analogously for π S and π P .When it is activated, a node in state ω changes state with probability 1 − π ω ,while a node not in state ω switches to that state with probability π ω , bydefinition. We thus have: dy ω dt = (1 − y ω ) π ω − y ω (1 − π ω ) , ω ∈ { R, P, S } (25)At an equilibrium point, dy ω dt = 0, so that y ω = π ω , ω ∈ { R, P, S } . Proposition 7.
The only equilibrium point for the ERG dynamics on a regulargraph is ( y R , y P , y S ) = (1 / , / , / .Proof. We limit the proof to the case in which k is not divisible by 3, to deal witha simpler expression for π ω . The extension to the general case is straightforward.The point ( y R , y P , y S ) = (1 / , / , /
3) is clearly of equilibrium since y ω = π ω = 1 / ω ∈ { R, P, S } .We prove that there cannot be other equilibria by first considering the regionfor which y S < y P < y R and showing that, in this region, π P > π R . By writingdown the expression of the probabilities, we have π P − π R = k (cid:88) u = (cid:100) k/ (cid:101) (cid:98) k/ (cid:99) (cid:88) v =0 (cid:18) ku, v (cid:19) (cid:0) y uR y vS y k − u − vP − y uS y vP y k − u − vR (cid:1) (26)Out of the terms in the above sum, those for which k − u − v ≤ u are positivebecause of the rearrangement inequality. Instead, those for which k − u − v > u can be negative. However, for a given v , the sum of the pair of terms in which k − u − v and u take the pair of values u and w is given by (cid:18) ku, v (cid:19) [ y uR ( y vS y wP − y wS y vP ) + y wR ( y vS y uP − y uS y vP )]which is positive, again for the rearrangement inequality. Thus, for y S < y P
In this section we present a few interesting cases in which we run our examplesof ASD in large but finite networks, considering both synthetically generatedgraphs and a real-world social network. Results are obtained either by numeri-cally solving the differential equation in (4), or by running detailed Monte-Carlosimulations using an ad-hoc event-driven simulator.In particular, we will focus on the online social network Epinions, for whicha popular snapshot is publicly available at the Stanford Large Network DatasetCollection [25]. The online social network Epinions.com is a consumer reviewwebsite where the users can review different kind of items with the purpose ofrating hundred thousand products and ranking the reviewers to be trusted. Theavailable dataset contains the who-trust-whom relationships of all the members,operating from 1999 until 2014. The network consists of |V| = 75879 nodesand |E| = 508837 directed edges, it is highly connected and contains cycles.The average clustering coefficient is 0.1378. The maximum in-degree is 3035,maximum out-degree is 1801, the average in/out-degree is 6.7. In and outdegrees follow an approximate power law distribution with exponent 1.7.
We assume that all nodes have symmetric threshold a ± v = 2. In Figure 5, weshow on the left the loci of the stationary solution of (3) in the plane ( ζ − , ζ ),and on the right an arrow plot representing the gradient of the system of the twoODEs in each possible point for which ζ − + ζ ≤
1. As it can be seen from thecombinations of the two plots, there are three stable stationary points, whichare located at ( ζ − , ζ ) = (0 , ζ − , ζ ) = (¯ ζ,
0) and ( ζ − , ζ ) = (0 , ¯ ζ ), with¯ ζ (cid:39) .
91 while there are four unstable stationary points at values ( ζ − , ζ ) =( (cid:101) ζ, ζ − , ζ ) = (0 , (cid:101) ζ ), ( ζ − , ζ ) = ( (cid:101) ζ , (cid:101) ζ ) and ( ζ − , ζ ) = ( (cid:101) ζ , (cid:101) ζ ), where (cid:101) ζ (cid:39) . × − , (cid:101) ζ (cid:39) . × − and (cid:101) ζ (cid:39) . ζ − = ζ for (cid:101) ζ ≤ ζ ≤ / ζ ω ( t ) and y ω ( t ), ω ∈ {− , , } , obtained through the numerical solution of ODEs in equations(3)-(4). In particular, y ω ( t ) represents the fraction of nodes in state ω at time t , while ζ ω ( t ) represents the fraction of edges connected to a node in state ω at time t . At time t = 0, the fraction of nodes of any degree in states − , , . , . , .
2, respectively. As it can be seen, the fraction of nodes in state 1decreases exponentially with time (the curve of ζ ( t ) is superimposed on that of y ( t )). Instead, the curve for ζ − ( t ) reaches the fixed point ζ = 0 .
91, as predictedby Figure 5. The fixed point for y − ( t ) is lower, at about 0 .
39, implying thatthe fraction of nodes with higher degree that asymptotically reach state − - Locus of stationary points for state -1Locus of stationary points for state 1 -3 -3 - Figure 5: TLTM in the Epinions social network with symmetric thresholds r ± = 2.: (a) Stationary solution of ODEs in (3). (b) Gradient of ODE systemin (3) . In this subsection, we show results for a network with n = 10 nodes dividedinto two equal-size communities (classes), with size n/ · . We considerthe TLTM with symmetric thresholds a ± v = 2. We put 10 seeds in state 1in community 1, and 10 seeds in state -1 in community 2. The out-degreedistribution of the first class is given by p ( k ,k ) | = p ( k ) p ( k ) where p ( k ) = (cid:18) n/ − k (cid:19) p k A (1 − p A ) − k and p ( k ) = (cid:18) n/ k (cid:19) p k B (1 − p B ) − k p A and p B being chosen so that the average degree toward community 1 is 20,while the average degree toward community 2 is 6. Analogously, the out-degreedistribution of the second class is given by p ( k ,k ) | = p ( k ) p ( k ) where p ( k ) = (cid:18) n/ k (cid:19) p k C (1 − p C ) − k and p ( k ) = (cid:18) n/ − k (cid:19) p k A (1 − p A ) − k p C being chosen so that the average degree toward community 1 is 5. Thenetwork shows a slight asymmetry, since nodes in community 1 have slightlymore edges directed towards nodes of community 2 than viceversa.33 time, t y -1 (t)y (t)y (t) -1 (t) (t) (t) Figure 6: TLTM in the Epinions social network with symmetric thresholds a ± v = 2 : Transient solution of ODEs in (3)-(4).Figure 7 compares the fraction of nodes in state 1 and -1 in either community,averaged across 100 simulation runs, against analytic results. We observe verygood agreement between analysis (thin curves) and simulation (thick curves).Interestingly, in the beginning we have two weakly interfering percolation pro-cesses in the two communities, producing a significant increase of nodes in state-1 in community 2, and a significant increase of nodes in state 1 in community 1.However, the percolation process in community 2 grows faster, because nodes incommunity 2 receive less influence from nodes in community 1 than viceversa.As a consequence, the percolation process of nodes in state -1 eventually invadesalso community 1, while nodes in state 1 vanish to zero throughout the network. Here we consider a simple regular graph where all nodes have fixed out-degree k = 21 and fixed in-degree d = 21. Note that, since the out-degree is odd,the best response is always deterministic (i.e., there are no ties). We performa single simulation run with n = 10 , with the following initial configuration:30,000 coordinating nodes in state 1, 10,000 coordinating nodes in state -1,40,000 anti-coordinating nodes in state 1, 20,000 anti-coordinating nodes instate -1.In Figures 8 and 9 we show the fraction of nodes in each of the possiblestates as function of time, according to simulation and analysis, respectively.We notice a perfect agreement between analytical prediction and simulation.Small fluctuations around the equlibrium configuration appear on the (single)simulation sample path.To assess the degree of concentration of the process around its average, wecarried out the following experiment: we performed 400 runs of the system,34 .0000100.0001000.0010000.0100000.1000001.000000 0 1 2 3 4 5 F r ac ti on o f nod e s time, t-1 in community 21 in community 1-1 in community 11 in community 2 Figure 7: Evolution over time of the fraction of nodes in states 1 and -1 in thetwo communities, according to analysis (thin curves) and simulation (averageof 100 runs) (thick curves).where the variability across runs is due to multiple reasons: i) the networktopology generated by the configuration model; ii) the initial selection of nodesin the various states; iii) the temporal dynamics of the process (Poisson clocks).We then sampled the system with time granularity ∆ t = 0 .
01, and at each timeinstant we evaluated the average, the minimum, and the maximum fraction ofnodes in each state, across the 400 runs. In Figure 10 we show the results ofthe above experiment for the fraction of nodes in state -1 (either coordinatingor anti-coordinating), with n = 10 (top-left), n = 10 (top-right), n = 10 (bottom-left), n = 10 (bottom-right).Thin curves above and below the thicker line (denoting the average), corre-spond to maximum and minimum values. Results for the fraction of nodes instate 1 are not shown, and they exhibit a similar variability. We observe thatresults become more concentrated passing from n = 10 to n = 10 nodes. We now investigate BRCA dynamics on the Epinions graph with a fraction ofcoordinating nodes equal to α = 0 .
7, evenly distributed among nodes of anydegree. Our main goal in this section will be to understand better the origin ofpossible discrepancies between analysis and simulation.A numerical analysis of (3)-(4) shows that, similarly to the regular case, thestationary points for the Epinions degree distribution are three, out of whichthe one in ζ = 1 / ζ (cid:39) . ζ (cid:39) .
67 and are stable. Such stationary points correspond to fractions ofnodes in state 1 given by y (cid:39) .
426 and y (cid:39) . F r ac ti on o f nod e s i n g i v e n s t a t e time, tstate 1 - coordinatingstate -1 - coordinatingstate 1 - anti-coordinatingstate -1 - anti-coordinating Figure 8: Evolution over time of thefraction of nodes in the ABRD game,according to a single simulation run. F r ac ti on o f nod e s i n g i v e n s t a t e time, tstate 1 - coordinatingstate -1 - coordinatingstate 1 - anti-coordinatingstate -1 - anti-coordinating Figure 9: Evolution over time of thefraction of nodes in the ABRD game,according to analysis. F r ac ti on o f nod e s i n s t a t e - time, taverage (coordinating)min/max (coordinating)average (anti-coordinating)min/max (anti-coordinating) F r ac ti on o f nod e s i n s t a t e - time, taverage (coordinating)min/max (coordinating)average (anti-coordinating)min/max (anti-coordinating) Figure 10: Evolution over time of the fraction of nodes in state -1 in the ABRDgame, across 400 simulation runs.We consider the following initial condition: 42% of coordinating nodes in state1, 28% of coordinating nodes in state -1, 10% of anti-coordinating nodes in state1, 20% of anti-coordinating nodes in state -1.The left plot in Figure 11 shows the fraction of nodes in each possible state asfunction of time, comparing simulation (thick curves) and analysis (thin curves).For simulations, we have plotted the average of 400 runs, where in each run we36 F r ac ti on o f nod e s i n g i v e n s t a t e time, t 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 1 2 3 4 5 6 7 8 9 10 F r ac ti on o f nod e s i n g i v e n s t a t e time, t 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 1 2 3 4 5 6 7 8 9 10 F r ac ti on o f nod e s i n g i v e n s t a t e time, t Figure 11: BRCA dynamics: evolution over time of the fraction of nodes in eachpossible state, according to simulation (thick curves) and analysis (thin curves).From top curve to bottom curve: coordinating nodes in state 1, coordinatingnodes in state -1, anti-coordinating nodes in state 1, anti-coordinating nodes instate -1. From left to right plot: original Epinions graph (left); configurationmodel matched to the Epinions graph (middle); configuration model matchedto the Epinions graph, with 10 n nodes (right).randomly select a different seed set. We observe that, after a very similar initialbehavior, simulations results tend to a different equilibrium point with respectto analysis, as one can see by looking at the fraction of nodes at time t = 10.We have identified two main reasons for the observed discrepancies: i) thefirst one due to the fact that the structure of the Epinions graph is not capturedby the configuration model; ii) the second one due to the fact that the networksize is not large enough to converge to a unique equilibrium point across differentruns, due to random effects.To separate out the impact of the above two reasons, we have performed thefollowing experiments. First, we have run simulations in which, in each run,we randomly reshuffle the edges while maintaining the same node statistics.Note that, by so doing, we generate graph according to the configuration modelmatched to the Epinions graph. The results of this experiment are shown in themiddle plot of Fig. 11. As expected, now we observe a much better agreementbetween analysis and simulation. Still, there are non negligible discrepancies inthe final fraction of nodes in each possible state.An in-depth inspection of simulation results revealed that about 5% of sim-ulation runs tend to a completely different equilibrium than the remaining 95%of the runs. This fact is illustrated by the middle plot in Figure 12, where wehave put a mark for each of the 400 runs, showing on the x axes the fractionof coordinating nodes in state 1, sampled at time t = 10. For each run, wealso computed the fraction of coordinating nodes that would transit to state 137 F Fraction at time 10 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0 0.1 0.2 0.3 0.4 0.5 0.6 F Fraction at time 10 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0 0.1 0.2 0.3 0.4 0.5 0.6 F Fraction at time 10
Figure 12: F metric vs fraction of coordinating node in state 1 (sampled attime t = 10): original Epinions graph (left); configuration model matched to theEpinions graph (middle); configuration model matched to the Epinions graph,with 10 n nodes (right).if their clock would fire at time t = 0. This fraction, denoted as F , is plottedon the y axes, and it is meant to capture a possible initial bias towards enteringstate 1, due to the initial network condition, which is especially dependent onthe random seed allocation.We can observe that simulations converge to two main equilibria, and thatsimulations runs where the final fraction of coordinating nodes in state 1 issmaller (around 0.2) have smaller values of the F metric specified above, sug-gesting that the initial seed allocation is the main responsible for driving thesystem into a different configuration. We emphasize that a similar behaviorcan be observed also on the original Epinions graph, for which an analogousinvestigation of single simulation runs produced the left plot in Fig. 12.To remove the bi-stable outcome of simulations, we performed the followingadditional experiment: we considered again the node statistics of the originalEpinions graph, but this time we generated graphs of size ten times larger thanthe original one (i.e., with n = 758790 nodes), using the configuration model.The right plot in Fig. 12 shows that simulation results are now much moreconcentrated around a unique equilibrium. Moreover, in the right plot of Fig.11 we observe an almost perfect agreement between analysis and simulation forthe evolution over time of the fraction of nodes in each possible state (in thisplot thick and thin curves are essentially overlapped and thus indistinguishable).We conclude that, when our analytic approach is used to predict the be-havior of ASD dynamics on realistic (finite) graph, one must be aware of twomain sources of errors: one due to the fact that real graphs are not completelydescribed by the configuration model; the other due to the fact that, in finitegraphs, randomness can possibly drive the system to different equilibria, espe-38ially when initial conditions are close to the border of the attraction basin ofthe expected equilibrium. However, our experiments with the Epinions graphsuggest that our approach has remarkable accuracy even in realistic graphs.Moreover, when the number of nodes exceeds, say, one million, results are suf-ficiently concentrated around their average to justify our mean-field approach,at least for the types of ASD dynamics that we have examined so far. Here, we show numerical results for the ERG dynamics on a simple regulargraph where all nodes have fixed out-degree k = 50 and fixed in-degree d = 50,and all nodes starts in the same state (the rock state). In simulation, we take1 million nodes, and perform a single run of the system. In Figure 13 we showthe fraction of nodes in each of the three states as function of time, accordingto simulation. In Figure 14, in the plane representing the fraction of nodes inthe paper and rock states, we show the loci of the stationary solutions for (4)with ω ∈ { R, P } . As shown in Section 6.3, the only stationary solution for theRoshambo game is y ω = 1 / ω ∈ { R, P, S } , which is the only point lying inthe intersection of the curves shown in Figure 14. Starting from y R = 1, as itis shown in Figure 13, there is a rapid increase in the fraction of nodes in thepaper state, since it is the best response. When the node populations in the rockand paper states become of equal size, then nodes in the scissors state start toappear (in the plane, the trajectory deviates from the boundary y R + y P = 1).The three populations then tend to the equilibrium point in which they havethe same size. fr ac ti on o f nod e s i n d i ff e r e n t s t a t e s time, t rockpaperscissors Figure 13: Evolution over time of thefraction of nodes in states rock, paper,scissors, according to a single simula-tion run in a regular graph. Fraction of nodes in Rock state F r a c t i on o f node s i n P ape r s t a t e Figure 14: Stationary solution of ODEsin (4) for the ERG dynamics in a regu-lar graph.39
Conclusions
In this paper we have proposed a mathematical framework showing that gen-eral semi-anonynoums dynamics in large scale random graphs converge to thesolution of ordinary differential equations, allowing fast numerical prediction ofthe transient behavior of many cascading processes in complex systems and,in some cases, analytical estimation of their points of equilibrium. With re-spect to existing literature, we have extended the above mean-field approxi-mation along several directions: i) asynchronous node activation; ii) arbitrarysemi-anonymous dynamics, including noisy best-response and class-dependentbehavior; iii) general random graph exhibiting a local tree-structure, possiblymixing heterogeneous nodes and unbounded in/out degrees. Our main contri-bution is a rigorous mathematical proof of convergence, which requires a carefulcombination of many independent results related to the different frameworkcomponents. Despite the generality of our approach, we have not consideredimportant variations of semi-anonynoums dynamics such as non-reversible tran-sitions. Moreover, it remains still largely open how to analytically characterizein a tractable way the behavior of ASD in undirected network.
A Proofs of Section 5.2
A.1 Topological result: Proof of Theorem 1
Let N t be the relevant neighborhood of a node chosen uniformly at randomfrom V , respectively. Moreover, Let T t be the truncated branching process attime t . In order to compare the distributions µ N t and µ T t of these two randomvariables, we define a coupling between them. Definition of coupling
As a starting point, we define two different sequences of random variables. For a, a (cid:48) ∈ A , let ( L a,a (cid:48) h ) h ∈ N be a sequence of i.i.d. random variables distributedaccording to a uniform distribution on the finite set L a,a (cid:48) . Let ( M a,a (cid:48) h ) h ∈L a,a (cid:48) be a finite sequence of random variables such that P ( M a,a (cid:48) h = L a,a (cid:48) h | L a,a (cid:48) h / ∈ { M a,a (cid:48) , . . . , M a,a (cid:48) h − } ) = 1while M a,a (cid:48) h is uniformly distributed on the set L a,a (cid:48) \ { M a,a (cid:48) , ..., M a,a (cid:48) h − } , if L a,a (cid:48) h ∈ { M a,a (cid:48) , . . . , M a,a (cid:48) h − } . Notice that the marginal distribution of the se-quences ( L a,a (cid:48) h ) h ∈ N and ( M a,a (cid:48) h ) h ∈L a,a (cid:48) are equivalent to sampling with replace-ment and sampling without replacement, respectively, from the set L a,a (cid:48) . Wethus have P ( L a,a (cid:48) h +1 (cid:54) = M a,a (cid:48) h +1 | ( L a,a (cid:48) , . . . , L a,a (cid:48) h ) = ( M a,a (cid:48) , . . . , M a,a (cid:48) h )) = hl a,a (cid:48) . (27)40e build the neighborhood N t with a dynamic exploration procedure drivenby variables ( M a,a (cid:48) h ) h ∈ N . Our procedure starts adding to N t the root v alone(chosen uniformly at random over V ), then N t grows through the addition ofnew edges/nodes, according to the following mechanism. Upon insertion, anewly added node is declared unexplored and its out-degree is initialized to 0.Then some of the unexplored nodes in N t are sequentially explored at randomtimes (smaller than t ). Upon its exploration, a node acquires a non-null out-degree and its out-neighbors are introduced in the network (if not already inthe network).Before describing in detail the algorithm, we introduce the following vari-ables: • i denotes the iteration, which is determined by the number of explorednodes in the network; • h a,a (cid:48) i denotes the number of ( a, a (cid:48) ) edges at iteration i ; • h i denotes the total number of edges in N t at iteration i , • S i denotes the set of nodes (except for the root v ) in N t at iteration i ; • S (cid:48) i denotes the set of unexplored nodes in N t at iteration i ; • T i denotes the time at which the i -th node activates (i.e., it is explored). • Γ i denotes the time lag between the ( i − i -th exploration. • V i denotes the identity of the i -th explored node. • A i denotes the class of the i -th explored node.Then N t is generated according to the following procedure:1. Set i = 0. Start from the root v (chosen uniformly at random from V ).Assign it label A = λ ( v ) and out-degree vector J = , and declare itunexplored. Set h a,a (cid:48) = 0, for all a, a (cid:48) ∈ A , h = 0, S = S (cid:48) = ∅ andextract T ∼ Exp(1).2. If T > t , stop the process. If T ≤ t : declare v explored, change itsout-degree vector to J = K v , and add J = (cid:80) a (cid:48) ∈A J a (cid:48) new edges to N t . In particular, for all a (cid:48) ∈ A , add exactly J a (cid:48) out-going edges M A ,a (cid:48) h (cid:48) , h (cid:48) ∈ { , . . . J a (cid:48) } , from v , pointing to nodes ν A ,a (cid:48) ( M A ,a (cid:48) h (cid:48) ). Note thatsuch nodes are not necessarily different from each other. Newly introducednodes are declared unexplored and their out-degree is set to 0.Then, counters are updated as: i = 1, h A ,a (cid:48) = J a (cid:48) for all a (cid:48) ∈ A and h a,a (cid:48) = 0 for a (cid:54) = A , h = (cid:80) a,a (cid:48) h a,a (cid:48) , S = ∪ a (cid:48) ,h (cid:48) { ν A ,a (cid:48) ( M A ,a (cid:48) h (cid:48) }} and S (cid:48) = S . 41. Let us define T i = T i − + Γ i and letΓ i ∼ Exp ( |S (cid:48) i | )where |S (cid:48) i | represents the number of unexplored nodes in N t at iteration i ; and let V i be chosen uniformly at random from S (cid:48) i .4. If T i > t , stop the process. If T i ≤ t : declare the node V i explored,assign it a degree J i = K V i and add J i = (cid:80) a (cid:48) ∈A J a (cid:48) i new edges to N t . Inparticular, add for all a (cid:48) ∈ A exactly J a (cid:48) i out-going edges M A i ,a (cid:48) h (cid:48) from V i pointing to nodes ν A i ,a (cid:48) ( M A i ,a (cid:48) h (cid:48) ) with h (cid:48) ∈ { h A i ,a (cid:48) i + 1 , . . . , h A i ,a (cid:48) i + J a (cid:48) i } .Note that such nodes are not necessarily all distinct, nor they are distinctfrom other nodes already inserted in N t . Among them, those that are notalready in N t are added and declared unexplored and their out-degree isset to 0.Then, h A i ,a (cid:48) i +1 = h A i ,a (cid:48) i + J a (cid:48) i , and, for all a (cid:54) = A i , h a,a (cid:48) i +1 = h a,a (cid:48) i . Then h i +1 = h i + (cid:80) a (cid:48) J a (cid:48) i , S i +1 = S i (cid:83) {∪ a (cid:48) ,h (cid:48) { ν A i ,a (cid:48) ( M A i ,a (cid:48) h (cid:48) ) } and S (cid:48) i +1 = S (cid:48) i \{ V i } (cid:83) ( ∪ a (cid:48) ,h (cid:48) { ν A i ,a (cid:48) ( M A i ,a (cid:48) h (cid:48) ) } . Finally, increment i and go back toPoint 3.Note that by construction, Point 4 is repeated for all i ≤ i M where i M =max { i ≥ | T i ≤ t } . Observe that the random network N t generated in this wayhas, by construction, the same structure and desired distribution µ N t , of therelevant neighborhood of a random node in G . Let E a,a (cid:48) t be the total numberof edges in N t from nodes with label a to nodes with label a (cid:48) . Notice that E a,a (cid:48) t = h a,a (cid:48) i M .Now we build the tree T t with a similar dynamic exploration procedure drivenby variables ( L a,a (cid:48) h ) h ∈ N , which starts from a root (cid:101) v alone and sequentially addsnew edges/nodes (cid:101) v h , h = 1 , , . . . , to the tree. Nodes { (cid:101) v h } h ≥ are assumed tobe pairwise different. However a correspondence between node { (cid:101) v h } h in the treeand nodes in the graph G is dynamically established. We emphasize that thiscorrespondence is, in general, non bijective: the same node in the network maycorrespond to several distinct nodes of T t , which are replicas of it.More in detail, first we define the following variables: • i denotes the iteration, i.e., the number of activated nodes in the tree; • (cid:101) h a,a (cid:48) i denotes the number of ( a, a (cid:48) ) edges at iteration i ; • (cid:101) h i denotes the total number of edges in the tree at iteration i , which, byconstruction, equals the total number of non-root nodes S i in the tree; • (cid:101) S i denotes the set of nodes (except for the root (cid:101) v ) in T t at iteration i ; • (cid:101) S (cid:48) i denotes the set of unexplored nodes in T t at iteration i ; • (cid:101) T i denotes the time at which the i -th node activates.42 (cid:101) Γ i denotes the time lag between the ( i − i -th exploration. • (cid:101) V i denotes the identity of the i -th explored node. • W ( · ) represents the function that maps the nodes of the tree into V . • (cid:101) A i is the class of the i -th activated node.Then T t is generated according to the following procedure:1. Set i = 0. Start from the root (cid:101) v . Establish a correspondence between (cid:101) v and v (i.e., W ( (cid:101) v ) = v ). Assign it label (cid:101) A = λ ( v ) and out-degreevector (cid:101) J = . Set (cid:101) h a,a (cid:48) = (cid:101) h = 0, for all a, a (cid:48) ∈ A , (cid:101) S = ∅ and (cid:101) T = T .2. If (cid:101) T > t , stop the process. If (cid:101) T ≤ t : change the out-degree vector of (cid:101) v to (cid:101) J = K v , and add (cid:101) J = (cid:80) a (cid:48) ∈A (cid:101) J a (cid:48) new nodes to T t (as childrenof (cid:101) v ). In particular, for all a (cid:48) ∈ A , add exactly (cid:101) J a (cid:48) out-going edgesfrom (cid:101) v , pointing to different nodes (cid:101) v h , h ∈ { , . . . , (cid:80) a (cid:48) (cid:101) J a (cid:48) } . Establisha correspondence between newly inserted tree nodes and graph nodes as W ( (cid:101) v h ) = ν A ,a (cid:48) ( L A ,a (cid:48) h (cid:48) ), for h = h (cid:48) + (cid:80) a (cid:48) t , stop the process. If (cid:101) T i ≤ t : declare the node (cid:101) V i explored, assignit a degree (cid:101) J i = K W ( (cid:101) V i ) and add (cid:101) J i = (cid:80) a (cid:48) ∈A (cid:101) J a (cid:48) i new nodes to T t (aschildren of (cid:101) V i ). In particular, for all a (cid:48) ∈ A , add exactly (cid:101) J a (cid:48) i out-goingedges from (cid:101) V i , pointing to different nodes v (cid:101) h i + h , h ∈ { , . . . , (cid:80) a (cid:48) (cid:101) J a (cid:48) i } .Establish a correspondence between newly inserted tree nodes and graphnodes as W ( (cid:101) v (cid:101) h i + h ) = ν A i ,a (cid:48) ( L A i ,a (cid:48) (cid:101) h Ai,a (cid:48) i + h (cid:48) ), for h = h (cid:48) + (cid:80) a (cid:48)
Using the coupling inequality (see Proposition 4.7 in [26]) we have (cid:107) µ T t − µ N t (cid:107) TV ≤ P ( N t (cid:54) = T t ) . (28)Define the two events B = (cid:92) a,a (cid:48) ∈A (cid:26) ( L a,a (cid:48) , L a,a (cid:48) , . . . , L a,a (cid:48) (cid:101) E a,a (cid:48) t ) = ( M a,a (cid:48) , M a,a (cid:48) , . . . , M a,a (cid:48) (cid:101) E a,a (cid:48) t ) (cid:27) and B = (cid:92) a ∈A (cid:92) ( b,h ) (cid:54) =( b (cid:48) ,h (cid:48) ): b,b (cid:48) ∈A ,h ∈{ ,..., (cid:101) E b,at } ,h (cid:48) ∈{ ,..., (cid:101) E b (cid:48) ,at } { ν b,a ( L b (cid:48) ,ah )) (cid:54) = ν b (cid:48) ,a ( L b,ah )) } (cid:92)(cid:92) (cid:92) b ∈A (cid:92) h ∈{ ,..., (cid:101) E b,a t } { ν b,A ( W ( (cid:101) v h )) (cid:54) = v } , v uniformly at random from V , and set its out-degree J = .Set i := 0. Set h a,a (cid:48) := 0 for all a, a (cid:48) ∈A . Set V = v . Let (cid:101) v = v be the tree root and set itsout-degree (cid:101) J = .Set i := 0. Set (cid:101) h a,a (cid:48) := 0 for all a, a (cid:48) ∈A . Set (cid:101) V = (cid:101) v .Extract T ∼ Exp(1). Set (cid:101) T = T .For i = 0 , , , . . . For i = 0 , , , . . . IF T i ≤ t - assign V i the out-degree J i = K V i andthe class A i = λ ( V i );- for all a ∈ A , add J ai out-going edges from V i , pointing to nodes ν A i ,a ( M A i ,ah Ai,ai + j ), with j ∈ { , . . . , J ai } ;- assign all newly inserted nodes an out-degree equal to ;- update h A i ,a (cid:48) i +1 := h A i ,a (cid:48) i + J a (cid:48) i , andfor all a (cid:54) = A i h a,a (cid:48) i +1 := h a,a (cid:48) i , S i +1 = S i (cid:83) {∪ a (cid:48) ,h (cid:48) { ν A i ,a (cid:48) ( M A i ,a (cid:48) h (cid:48) ) } and S (cid:48) i +1 = S (cid:48) i \ V i (cid:83) ( ∪ a (cid:48) ,h (cid:48) { ν A i ,a (cid:48) ( M A i ,a (cid:48) h (cid:48) ) } .ELSE end. IF (cid:101) T i ≤ t - assign (cid:101) V i the out-degree (cid:101) J i = K W ( (cid:101) V i ) and the class (cid:101) A i = λ ( W ( (cid:101) V i ));- for all a ∈ A , add (cid:101) J ai out-going edgesfrom (cid:101) V i , pointing to (cid:101) J ai new nodes,which are replicas of ν A i ,a ( L A i ,a (cid:101) h Ai,ai + j ),with j ∈ { , . . . , (cid:101) J ai } ;- assign all newly inserted nodes an out-degree equal to ;- update (cid:101) h A i ,a (cid:48) i +1 := (cid:101) h A i ,a (cid:48) i + (cid:101) J a (cid:48) i , andfor all a (cid:54) = A i (cid:101) h a,a (cid:48) i +1 := (cid:101) h a,a (cid:48) i , (cid:101) S i +1 = (cid:101) S i (cid:83) ( ∪ h { (cid:101) v (cid:101) h i + h } and (cid:101) S (cid:48) i +1 = (cid:101) S (cid:48) i \ (cid:101) V (cid:48) i (cid:83) ( ∪ h { (cid:101) v (cid:101) h i + h } .ELSE end.Generate T i +1 = T i + Γ i +1 , whereΓ i +1 ∼ Exp (cid:0) |S (cid:48) i +1 | (cid:1) Sample node V i +1 uniformly at randomfrom S (cid:48) i +1 . Generate (cid:101) T i +1 = (cid:101) T i + (cid:101) Γ i +1 , where (cid:101) Γ i +1 = Γ i +1 , if | (cid:101) S (cid:48) i +1 | = |S (cid:48) i +1 | and (cid:101) Γ i +1 ∼ Exp (cid:16) | (cid:101) S (cid:48) i +1 | (cid:17) otherwise.Sample node (cid:101) V i +1 as follows. If (cid:101) S (cid:48) i +1 = S (cid:48) i +1 then (cid:101) V i +1 = V i +1 , else (cid:101) V i +1 istaken uniformly at random from (cid:101) S (cid:48) i +1 .Table 3: Coupling between the generation processes of N t (left column) and T t (right column) 45hich are, in words, the event that there are no repeated edges in T t and thatthe map W ( · ) is bijective (i.e., just a single node in T t corresponds to everynode in N t ).We are going to show that {B ∩ B } ⊆ {N t = T t } . The assertion, indeed,can be easily checked by induction over the iteration i . First observe that at theend of iteration 0, by construction, under B and B the structure of T t and N t are necessarily equal. Indeed by construction they can be different only if eithersome L A ,a (cid:48) h (cid:48) (cid:54) = , M A ,a (cid:48) h (cid:48) for h (cid:48) ≤ J A ,a (cid:48) = ˜ J A ,a (cid:48) or there are h (cid:48) and h (cid:48)(cid:48) such that ν A ,a ( L A ,a (cid:48) h (cid:48) ) = ν A ,a (cid:48) ( M A ,a (cid:48) h (cid:48) ) = ν A ,a (cid:48) ( M A ,a (cid:48) h (cid:48)(cid:48) ) = ν A ,a (cid:48) ( L A ,a (cid:48) h (cid:48)(cid:48) ). Now supposethat the structure of N t is equal to the structure of T t at the end of iteration i − i ≥ S i = (cid:101) S i and S (cid:48) i = (cid:101) S (cid:48) i h a,a (cid:48) i = (cid:101) h a,a (cid:48) i ;therefore V i = (cid:101) V i and Γ i = (cid:101) Γ i and A i = (cid:101) A i J A i ,a (cid:48) i = (cid:101) J A i ,a (cid:48) i . During iteration i we add to N t nodes ν A i ,a (cid:48) ( M A i ,a (cid:48) h Ai,a (cid:48) i + h (cid:48) ) = ν A i ,a (cid:48) ( L A i ,a (cid:48) h Ai,a (cid:48) i + h (cid:48) ) for h (cid:48) ∈ { , . . . , J a (cid:48) i } (where the equality descends from B ), which, from B , are all different anddifferent from nodes already in N t . In T t we add brand-new replicas of nodes ν A i ,a (cid:48) ( L A i ,a (cid:48) h Ai,a (cid:48) i + h (cid:48) ) = ν A i ,a (cid:48) ( M A i ,a (cid:48) h Ai,a (cid:48) i + h (cid:48) ). Therefore the structures of N t and T t are still equal at the end of iteration i .Thus: P ( N t (cid:54) = T t ) ≤ P B C ∪ B C (cid:12)(cid:12)(cid:12)(cid:12) (cid:92) b,a (cid:110) (cid:101) E b,at ≤ x b,a (cid:111) + P (cid:91) b,a (cid:110) (cid:101) E b,at > x b,a (cid:111) ≤ P B C (cid:12)(cid:12)(cid:12)(cid:12) (cid:92) b,a (cid:110) (cid:101) E b,at ≤ x b,a (cid:111) + P ˜ B C (cid:12)(cid:12)(cid:12)(cid:12) ˜ B , (cid:92) b,a (cid:110) (cid:101) E b,at ≤ x b,a (cid:111) + (cid:88) b,a P (cid:16)(cid:110) (cid:101) E b,at > x b,a (cid:111)(cid:17) (29)where in the first inequality we have let out the probability that the number ofnodes exceeds a fixed threshold.Define the event E b,ah = (cid:110) ( L b,a , . . . , L b,ah ) = ( M b,a , . . . , M b,ah )) (cid:111) (30)46he first term of (29) is upper bounded by P B C (cid:12)(cid:12)(cid:12)(cid:12) (cid:92) b,a (cid:110) (cid:101) E b,at ≤ x b,a (cid:111) = P (cid:91) b,a ∈A (cid:16) E b,a (cid:101) E b,at (cid:17) c (cid:12)(cid:12)(cid:12)(cid:12) (cid:92) b,a (cid:110) (cid:101) E b,at ≤ x b,a (cid:111) ≤ (cid:88) b,a ∈A P (cid:18)(cid:16) E b,a (cid:101) E b,at (cid:17) c (cid:12)(cid:12)(cid:12)(cid:12) (cid:101) E b,at ≤ x b,a (cid:19) ≤ (cid:88) b,a ∈A x b,a − (cid:88) h b,a =0 P ( L b,ah b,a +1 (cid:54) = M b,ah b,a +1 |E b,ah b,a )= (cid:88) b,a ∈A x b,a − (cid:88) h b,a =0 h b,a l b,a (31)where the first inequlity is the union bound, the second inequality is the chainrule, while the last equality comes from (27). Using the same arguments, thesecond term of (29) becomes P B C (cid:12)(cid:12)(cid:12)(cid:12) B , (cid:92) b,a (cid:110) (cid:101) E b,at ≤ x b,a (cid:111) ≤ (cid:88) a,b ∈A x b,a (cid:88) h b,a =1 P ( ν b,a ( M b,ah b,a ) ∈ { w , ν b,a ( M b,a ) , ..., ν b,a ( M b,ah b,a − ) }|E b,ah b,a )+ (cid:88) a,b ∈A (cid:88) b (cid:48) (cid:54) = b x b,a (cid:88) h b,a =1 x b (cid:48) ,a (cid:88) h b (cid:48) ,a =1 P (cid:18) ν b,a ( M b,ah b,a ) = ν b (cid:48) ,a ( M b (cid:48) ,ah b (cid:48) ,a ) (cid:12)(cid:12)(cid:12)(cid:12) E b,ah b,a , E b (cid:48) ,ah b (cid:48) ,a (cid:19) (32)where the first term above gives the probability that two edges from the sameclass point to the same node or that a given edge points to the root, while thesecond term computes the probability that two edges from two different classespoint to the same node. From the definition in (9), we have that the first termis upper bounded by (cid:88) a,b ∈A x b,a (cid:88) h b,a =1 P ( ν b,a ( M b,ah b,a ) ∈ { w , ν b,a ( M b,a ) , ..., ν b,a ( M b,ah b,a − ) }|E b,ah b,a ) ≤ (cid:88) a,b ∈A x b,a (cid:88) h b,a =1 ( h b,a − (cid:80) d , k ( d b − q b d , k | a + (cid:80) d , k d b q b d , k | a l b,a ≤ (cid:88) a,b ∈A x b,a ( x b,a + 1)2 (cid:80) d , k d b q b d , k | a l b,a − (cid:88) a,b ∈A x b,a − (cid:88) h =0 hl b,a . (33)47sing similar arguments, we get (cid:88) a,b ∈A (cid:88) b (cid:48) (cid:54) = b x b,a (cid:88) h b,a =1 x b (cid:48) ,a (cid:88) h b (cid:48) ,a =1 P (cid:18) ν b,a ( M b,ah b,a ) = ν b (cid:48) ,a ( M b (cid:48) ,ah b (cid:48) ,a ) (cid:12)(cid:12)(cid:12)(cid:12) E b,ah b,a , E b (cid:48) ,ah b (cid:48) ,a (cid:19) ≤ (cid:88) a,b ∈A (cid:88) b (cid:48) (cid:54) = b x b,a x b (cid:48) ,a (cid:80) d , k d b q b (cid:48) d , k | a l b,a . (34)Combining these bounds and inequalities in (28) and (31) we conclude thethesis. A.2 Concentration property: Proof of Theorem 2 Before presenting the proof of the main result we fix some notations and westate some preliminary lemmas. First, we recall a simple variant of Azuma’sinequality which will be useful in our arguments. Let { Y k : k = 0 , , , , . . . } bea martingale. The classical Azuma’s inequality [3, Theorem 7.2.1] states that if | Y k − Y k − | ≤ c k with probability one, then P ( | Y N − Y | ≥ η ) ≤ − η (cid:80) N(cid:96) =0 c (cid:96) . The following martingale concentration result generalizes the Azuma inequalityto the case in which | Y k − Y k − | is not bounded. Lemma 4 (Lemma 1 in [12]) . Let { Y k : k = 0 , , , , . . . } be a martingale.Then for all sequences of positive numbers ( c (cid:96) ) and η > , we have the followinginequality P ( | Y N − Y | ≥ η ) ≤ − η (cid:80) N(cid:96) =1 c (cid:96) + (cid:18) (cid:63) η (cid:19) n (cid:88) (cid:96) =1 P ( | Y (cid:96) − Y (cid:96) − | ≥ c (cid:96) ) , with ∆ (cid:63) = sup i | Y i − Y i − | . We recall that we consider three sources of randomness: the dynamics de-fined by Θ in (1), the activation process and the labeled network. The con-centration property is proved in two steps. First, we study concentration bysequentially unveiling the edges in the labeled network (Lemma 5) and then weconsider the other sources of randomness for a fixed graph (see Lemma 7). A.2.1 Unveiling the network We recall that for any (cid:96) ∈ N we use the notation [ (cid:96) ] = { , . . . , (cid:96) } . Let Π a,a (cid:48) be the set of all permutations of L a,a (cid:48) = { , . . . , l a,a (cid:48) } for any a, a (cid:48) ∈ A anddenote by Π = × a,a (cid:48) ∈A Π a,a (cid:48) . Since each of permutation π a,a (cid:48) ∈ Π a,a (cid:48) defines aspecific pairing of out-links from nodes with label a and in-links of nodes with48abel a (cid:48) , there are exactly (cid:81) a,a (cid:48) ∈A l a,a (cid:48) ! distinct elements in Π. We define thefollowing cylinder sets: C (cid:96) ( π [ (cid:96) ] ) = { υ ∈ Π : υ [ (cid:96) ] = π [ (cid:96) ] } , ∀ π [ (cid:96) ] (35)We notice that C (cid:96) ( · ) are disjoint and exhaustive events, i.e., C (cid:96) ( π [ (cid:96) ] ) ∩C (cid:96) ( π (cid:48) [ (cid:96) ] ) = ∅ if π [ (cid:96) ] (cid:54) = π (cid:48) [ (cid:96) ] and ∪ π [ (cid:96) ] C (cid:96) ( π [ (cid:96) ] ) = Π. Lemma 5 (Unveiling network) . Let N = (( V , E , A , λ, σ, τ )) be a network sam-pled from the model ensemble C n,p of all labeled networks with given size n and statistics p . We denote the induced graph obtained in the exploration pro-cess of the neighborhood of a node v by N vt , and with V vt the number of nodesin it. For t ≥ , let Z ( t ) be the state vector of the ASD dynamics on N , b ( t ) = |{ v ∈ V : Z v ( t ) = ω }| be the number of state- ω adopters at time t . Wedenote the expectation over the ensemble of labeled graphs by (cid:101) b ( t ) . For any s ≥ we have P (cid:16) | b ( t ) − (cid:101) b ( t ) | ≥ ηn (cid:17) ≤ inf x> (cid:40) − η n dx + (cid:18) η (cid:19) s x s (cid:88) v ∈V | δ v | [ E [ | V vt | s ] (cid:41) Proof. Let π ∈ Π be the random element of Π (uniformly extracted by Π) whichdescribes the network N = (( V , E , A , λ, σ, τ )). We denote with F (cid:96) the naturalfiltration generated by π [ (cid:96) ] , with F equal to the trivial σ -algebra. Let π (cid:96) ∈ Π,for any given (cid:96) ∈ { · · · |E|} , a random element in Π satisfying the followingproperties: i) π (cid:96) [ (cid:96) ] = π [ (cid:96) ] ; ii) π (cid:96)(cid:96) +1 and π (cid:96) +1 are conditionally independent given π [ (cid:96) ] ; iii) for any i > π (cid:96)(cid:96) + i = π (cid:96) + i if π (cid:96) + i (cid:54) = π (cid:96)(cid:96) +1 and π (cid:96)(cid:96) + i = π (cid:96) +1 if π (cid:96) + i = π (cid:96)(cid:96) +1 . Observe that by construction π (cid:96) is extracted uniformly from Π, as well.Furthermore the conditional law of both π and π (cid:96) , given π [ (cid:96) ] with π [ (cid:96) ] = π (cid:96) [ (cid:96) ] isuniform in C (cid:96) ( π [ (cid:96) ] ). Let G (cid:96)(cid:96) (cid:48) be the natural filtration induced by π (cid:96) [ (cid:96) (cid:48) ] . Observethat by construction G (cid:96)(cid:96) (cid:48) = F (cid:96) for (cid:96) (cid:48) ≤ (cid:96) . At last let ˆ F (cid:96) +1 and ˆ G (cid:96)(cid:96) +1 the naturalfiltration induced by π (cid:96) +1 and π (cid:96)(cid:96) +1 . Of course F (cid:96) +1 = σ ( F (cid:96) ∪ ˆ F (cid:96) +1 ) and G (cid:96)(cid:96) +1 = σ ( G (cid:96)(cid:96) ∪ ˆ G (cid:96)(cid:96) +1 ) . We have P ( | b π ( t ) − (cid:101) b ( t ) | ≥ ηn ) = P (cid:0)(cid:12)(cid:12) E [ b π ( t ) |F |E| ] − E [ b π ( t ) |F ] (cid:12)(cid:12) ≥ ηn (cid:1) . Let us emphasize the dependence of the number of ω -adopters b ( t ) on a specificgraph π ∈ Π with notation b π ( t ) and define A (cid:96) = E [ b π ( t ) |F (cid:96) ]. Note, indeed,that { A (cid:96) } (cid:96) is a martingale.In order to estimate the above probability we apply Lemma 4. First, wecompute P ( | A (cid:96) +1 − A (cid:96) | ≥ c (cid:96) ) = P ( | A (cid:96) +1 − A (cid:96) | s ≥ c s(cid:96) ) ≤ E [ | A (cid:96) +1 − A (cid:96) | s ] c s(cid:96) Notice that by construction A (cid:96) = E [ b π ( t ) |F (cid:96) ] = E [ b π (cid:96) ( t ) |F (cid:96) ] = E [ b π (cid:96) ( t ) |F (cid:96) +1 ]49here the first equation holds because π and π (cid:96) are both uniform on C (cid:96) ( π [ (cid:96) ] ),and last equation descends from the fact that ˆ G (cid:96)(cid:96) +1 and ˆ F (cid:96)(cid:96) +1 are conditionallyindependent given F (cid:96) = G (cid:96)(cid:96) . Furthermore we have: A (cid:96) +1 = E [ b π ( t ) |F (cid:96) +1 ] . Therefore A (cid:96) +1 − A (cid:96) = E [ b π ( t ) | F (cid:96) +1 ] − E [ b π ( t ) | F (cid:96) ] , = E [ b π ( t ) | F (cid:96) +1 ] − E [ b π (cid:96) ( t ) | F (cid:96) +1 ]= E [ b π ( t ) − b π (cid:96) ( t ) | F (cid:96) +1 ] (36)Now observing that by construction π and π (cid:96) differ in at most two positions,hence we have: E [ b π ( t ) − b π (cid:96) ( t ) | F (cid:96) +1 ] ≤ E [ |N v( π (cid:96) +1 ) t | | F (cid:96) +1 ]and E [ | A (cid:96) +1 − A (cid:96) | s ] ≤ s E (cid:104)(cid:16) E [ | N v( π (cid:96) +1 ) t | |F (cid:96) +1 ] (cid:17) s (cid:105) ≤ s E (cid:104) | V v( π (cid:96) +1) t | s (cid:105) where v( π (cid:96) ) is the in-node of edge π (cid:96) . We conclude that P ( | A (cid:96) +1 − A (cid:96) | ≥ c (cid:96) ) ≤ s (cid:104) E [ | V v( π (cid:96) +1 ) t | s (cid:105) c s(cid:96) For any x > c (cid:96) = x for all (cid:96) . Then, by applying Lemma 4 and observingthat ∆ (cid:63) ≤ n , we obtain P (cid:16) | b ( t ) − (cid:101) b ( t ) | ≥ ηn (cid:17) ≤ − η n dx + (cid:18) η (cid:19) s x s dn (cid:88) (cid:96) =1 (cid:104) E [ | V v ( π (cid:96) ) t | s (cid:105) = 2e − η n dx + (cid:18) η (cid:19) s x s (cid:88) v ∈V | δ v | [ E [ | V vt | s ]from which the thesis. Remark 10. The approach followed in Lemma 5 can be potentially extendedto more general classes of random graphs, with a variable number of edges, aslong as: i) the number of edges in the graph is sufficiently concentrated aroundits expectation; ii) random variables associated to the presence of different edgesin the graph are sufficiently weakly correlated, so that we can effectively bound E [ | A (cid:96) +1 − A (cid:96) | ] , through a coupling argument similar to the one established be-tween π and π (cid:96) in Lemma 5. Let N = (( V , E , A , λ, σ, τ )) be a labeled graph. We denote the random timesat which the opinion update occurs, random node sequence activated, and therandom state, by { T (cid:96) } (cid:96) ∈ N , { w (cid:96) } (cid:96) ∈ N , { z (cid:96) } (cid:96) ∈ N , respectively. For t ≥ 0, let Z ( t ) bethe state vector of the ASD dynamics on N and b ( t ) = |{ v ∈ V : Z v ( t ) = ω }| be the number of state- ω adopters at time t .50 emma 6. Let { T (cid:96) } (cid:96) ∈ N be the random times at which the opinion update occurs.For t > define the random variable ι ( t ) = sup { k ∈ N : T k ≤ t } . Then for any (cid:15) > and ∆ n < tn the following bounds hold P ( ι ( t ) ≥ (1 + (cid:15) ) tn ) ≤ e − nt(cid:15) (cid:15) ) P ( | ι ( t ) − tn | ≥ ∆ n ) ≤ − ∆2 n tn +∆ n ) . Proof. This is a straightforward consequence of Chernoff bound [19]. Lemma 7 (Unveiling dynamics) . Let N = (( V , E , A , λ, σ, τ )) be a labeled graph.Let { T (cid:96) } (cid:96) ∈ N , { w (cid:96) } (cid:96) ∈ N , { z (cid:96) } (cid:96) ∈ N be the random times at which the opinion updateoccurs, random node sequence activated, and random state of activated sequence,respectively. We denote the size of the induced graph obtained in the explorationprocess of the neighborhood of a node v with V vt at time t . For t ≥ , let Z ( t ) be the state vector of the ASD dynamics on N , b ( t ) = |{ v ∈ V : Z v ( t ) = ω }| be the number of state- ω adopters at time t conditioned to N . We denote theexpectation over the activation process by b ( t ) = E [ b ( t ) |N ] . For any (cid:15) > wehave P ( | b ( t ) − b ( t ) | > ηn ) ≤ x> (cid:26) − η n (cid:15) ) tx + (cid:18) η (cid:19) (1 + (cid:15) ) tn s E v [ | V vt | s ] x s (cid:27) + 2e − nt(cid:15) (cid:15) ) + 2e − η n t + η/ with v chosen uniformly at random in V .Proof. For t ∈ R + let ι ( t ) = sup { k ∈ N : T k ≤ t } , w and z be the randomsequences of activated nodes and the corresponding random state. We recallthat for any (cid:96) > w [ (cid:96) ] is uniformly distributed over V [ (cid:96) ] . We denoteby F (cid:96),s the natural filtration generated by w [ (cid:96) ] and z [ s ] . Given ι ( t ), let w be arandom vector uniformly distributed in V [ ι ( t )] (let w (cid:96) +1 = v ) and ˆ w (cid:96) be a randomvector in V [ ι ( t )] which is obtained by choosing some v (cid:48) uniformly at random fromthe set of nodes V and putting ˆ w (cid:96)(cid:96) +1 = v (cid:48) and ˆ w (cid:96)i = w i for all i ∈ [ ι ( t )] \ { (cid:96) + 1 } .It should be noticed that w (cid:96) +1 and ˆ w (cid:96)(cid:96) +1 are conditionally independent given w [ (cid:96) ] . Furthermore, by construction ˆ w (cid:96) is uniformly distributed over V [ ι ( t )] .In an analogous way, recall that z s = Z w s is a random variable distributedas defined in Definition 1. Given ι ( t ), let z be a vector of length ι ( t ), whosecomponents are independent with the (cid:96) -component distributed as Θ ( λ ( w (cid:96) )) inDefinition 1 (let z s +1 = ω ) and, let ˜ z s be a random vector which is obtained bychoosing some ω (cid:48) according to Θ ( λ ( w (cid:96) )) in Definition 1 and putting ˜ z ss +1 = ω (cid:48) and ˜ z si = z i for all i ∈ [ ι ( t )] \ { s + 1 } . 51et us emphasize the dependence of the number of ω -adopters b ( t ) on aspecific sequence of activated nodes w and states z with notation b w,z ( t ). Given ι ( t ), we define for any ( (cid:96), s ) ∈ [ ι ( t )] × [ ι ( t )] B ι ( t ) , N (cid:96),s = E [ b ( t ) | ι ( t ) , F (cid:96),s , N ] , then P (cid:0) | b w,z ( t ) − b ( t ) | ≥ ηn (cid:1) ≤ P (cid:16)(cid:12)(cid:12) E [ b w,z ( t ) | ι ( t ) , F ι ( t ) ,ι ( t ) , N ] − E [ b w,z ( t ) | ι ( t ) , F ι ( t ) , , N ] (cid:12)(cid:12) ≥ ηn (cid:17) + P (cid:16)(cid:12)(cid:12) E [ b w,z ( t ) | ι ( t ) , F ι ( t ) , , N ] − E [ b w,z ( t ) | ι ( t ) , F , , N ] (cid:12)(cid:12) ≥ ηn (cid:17) + P (cid:16) | E [ b w,z ( t ) | ι ( t ) , N ] − E [ b w,z ( t ) |F , , N ] | ≥ ηn (cid:17) = P (cid:16) | B ι ( t ) , N ι ( t ) ,ι ( t ) − B ι ( t ) , N ι ( t ) , | > ηn (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) (T1) + P (cid:16) | B ι ( t ) , N ι ( t ) , − B ι ( t ) , N , | > ηn (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) (T2) + P (cid:16)(cid:12)(cid:12)(cid:12) B ι ( t ) , N , − E [ b w,z ( t ) |F , , N ] (cid:12)(cid:12)(cid:12) ≥ ηn (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) (T3) We now evaluate (T1) and (T2) by applying Lemma 4 and (T3) using simplearguments. • In order to estimate (T1) we first consider B ι ( t ) , N (cid:96) +1 , − B ι ( t ) , N (cid:96), = E [ b w,z ( t ) | ι ( t ) , F (cid:96) +1 , , N ] − E [ b w,z ( t ) | ι ( t ) , F (cid:96), , N ]= E [ b w,z ( t ) | ι ( t ) , F (cid:96) +1 , , N ] − E [ b ˆ w (cid:96) +1 ,z ( t ) | ι ( t ) , F (cid:96) +1 , , N ]= E [ b w,z ( t ) − b ˆ w (cid:96) ,z ( t ) | ι ( t ) , F (cid:96) +1 , , N ] . By observing that, by construction, w and ˆ w (cid:96) − differ in at most oneposition, we get E [ b w,z ( t ) − b ˆ w (cid:96) ,z ( t ) | ι ( t ) , F (cid:96) +1 , , N ] ≤ E [ | V vt ||F (cid:96), ]where v is chosen uniformly at random in V . We thus have for any x > P (cid:16) | B ι ( t ) , N (cid:96) +1 , − B ι ( t ) , N (cid:96), | > x (cid:17) = P (cid:16) | B ι ( t ) , N (cid:96) +1 , − B ι ( t ) , N (cid:96), | m > x m (cid:17) ≤ E (cid:104) | B ι ( t ) , N (cid:96) +1 , − B ι ( t ) , N (cid:96), | m (cid:105) x m ≤ m E [( E [ | V vt ||F (cid:96) +1 , ]) m ] x m ≤ m E [ | V vt | m ] x m v is chosen uniformly at random. We thus have for any (cid:15) > P (cid:16) | B ι ( t ) , N ι ( t ) , − B ι ( t ) , N , | > ηn (cid:17) ≤ P (cid:16) | B ι ( t ) , N ι ( t ) , − B ι ( t ) , N , | > ηn (cid:12)(cid:12) ι ( t ) < (1 + (cid:15) ) tn (cid:17) + P ( ι ( t ) ≥ (1 + (cid:15) ) tn ) ≤ inf x> (cid:26) − η n (cid:15) ) tx + (cid:18) η (cid:19) (1 + (cid:15) ) tn m E [ | V vt | m ] x m (cid:27) + e − nt(cid:15) (cid:15) ) (37)with v chosen uniformly at random in V . • (T2): Following the same arguments used in the previous point and ob-serving that ˜ z ss differs from z [ ι ( t )] only at position s + 1, we get P (cid:16) | B ι ( t ) , N ι ( t ) ,ι ( t ) − B ι ( t ) , N ι ( t ) , | > ηn (cid:17) ≤ inf x> (cid:26) − η n (cid:15) ) tx + (cid:18) η (cid:19) (1 + (cid:15) ) tn m E [ | V vt | m ] x m (cid:27) + e − nt(cid:15) (cid:15) ) . (38) • (T3): Let now ˆ ι ( t ) a random variable taking values in Z + , distributed as ι ( t ), and independent of it; let w [ ι ( t )] ∈ V ι ( t ) and ˆ w [ˆ ι ( t )] ∈ V ˆ ι ( t ) randomuniform sequences of activation of length ι ( t ) and ˆ ι ( t ) respectively and b ( t ) and ˆ b (cid:96) ( t ) the corresponding the numbers of state- ω adopters at time t on the network N :For any ∆ n we have P (cid:16) | E [ b ( t ) | ι ( t ) , F , , N ] − E [ b ( t ) |F , , N ] | ≥ ηn (cid:17) (39)= P (cid:16)(cid:12)(cid:12)(cid:12) E [ b ( t ) | ι ( t ) , F , , N ] − E [ˆ b (cid:96) ( t ) | ι ( t ) , F , , N ] (cid:12)(cid:12)(cid:12) ≥ ηn (cid:17) ≤ P (cid:16) | ι ( t ) − ˆ ι ( t ) | ≥ ηn (cid:17) ≤ P (cid:16) | ι ( t ) − ˆ ι ( t ) | ≥ ηn (cid:12)(cid:12)(cid:12) | ˆ ι ( t ) − tn | ≤ ∆ n , | ι ( t ) − tn | ≤ ∆ n (cid:17) + 2 P ( | ι ( t ) − tn | > ∆ n ) ≤ − ∆2 n tn +∆ n ) (40)where the second inequality descend from the fact that we can establish acoupling between the sequences of activation w [ ι ( t )] and ˆ w [ˆ ι ( t )] by forcingthem to have common initial part of length min( ι ( t ) , ˆ ι ( t )) while the lastinequality follows from Lemma 6. Choosing ∆ n = ηn we get P (cid:16) | ι ( t ) − ˆ ι ( t ) | ≥ ηn (cid:12)(cid:12)(cid:12) | ˆ ι ( t ) − tn | ≤ ∆ n , | ι ( t ) − tn | ≤ ∆ n (cid:17) = 0and we conclude the proof combining with (38).53 roof of Theorem 2 For any (cid:15) > P ( | b ( t ) − E [ b ( t )] | > ηn ) ≤ P ( | b ( t ) − E [ b ( t ) |N ] | > ηn/ 2) + P ( | E [ b ( t ) |N ] − E [ b ( t )] | > ηn/ v is sampled with a probability proportional with its in-degree. CombiningLemma 7 with Lemma 5 we get that for any (cid:15) > η > x > P ( | b ( t ) − E [ b ( t )] | > ηn ) ≤ − η n (cid:15) ) tx + (cid:18) η (cid:19) (1 + (cid:15) ) tn s E v [ | V vt | s ] x s + 2e − nt(cid:15) (cid:15) ) + 2e − η n t + η/ + (cid:18) η (cid:19) s x s (cid:88) w ∈V | δ w | [ E [ | V wt | s ] + 2e − η n dx . B Convergence to ODE solution with asymp-totic degree distribution Proof of Proposition 3. Let f ( n ) ( z ) = φ ( n ) ( z ) − z and f ( z ) = φ ( z ) − z . Forany ∆ > t ∈ [0 , m ∆] we have ζ ( n ) ( t ) = ζ ( n ) (0) + (cid:90) ( m − f ( n ) ( ζ ( n ) ( s ))d s + (cid:90) t ( m − f ( n ) ( ζ ( n ) ( s ))d s ζ ( t ) = ζ (0) + (cid:90) ( m − f ( ζ ( s ))d s + (cid:90) t ( m − f ( ζ ( s ))d s from which (cid:107) ζ ( n ) ( t ) − ζ ( t ) (cid:107) m ∆ : = sup t ∈ [0 ,m ∆] (cid:107) ζ ( n ) ( t ) − ζ ( t ) (cid:107) ∞ ≤ sup t ∈ [0 , ( m − (cid:107) ζ ( n ) ( t ) − ζ ( t ) (cid:107) ∞ + ∆ (cid:13)(cid:13)(cid:13) f ( n ) − f (cid:13)(cid:13)(cid:13) ∞ + sup t ∈ [( m − ,m ∆] (cid:90) t ( m − (cid:13)(cid:13)(cid:13) f ( ζ ( n ) ( s )) − f ( ζ ( s )) (cid:13)(cid:13)(cid:13) ∞ d s. Since f is Lipschitz on a compact and invariant set B = [0 , |A| , then thereexists L > ζ , ζ ∈ B it holds that (cid:107) f ( ζ ) − f ( ζ ) (cid:107) ∞ ≤ (cid:107) ζ − ζ (cid:107) ∞ . Then for any ∆ < /L we have (cid:107) ζ ( n ) ( t ) − ζ ( t ) (cid:107) m ∆ ≤ (cid:107) ζ ( n ) ( t ) − ζ ( t ) (cid:107) ( m − − L ∆ + ∆ (cid:13)(cid:13)(cid:13) f ( n ) − f (cid:13)(cid:13)(cid:13) ∞ − L ∆ ≤ (cid:107) ζ ( n ) (0) − ζ (0) (cid:107) ∞ (1 − L ∆) m + 11 − L ∆ m − (cid:88) j =0 ∆ (cid:13)(cid:13)(cid:13) f ( n ) − f (cid:13)(cid:13)(cid:13) ∞ (1 − L ∆) j = (cid:107) ζ ( n ) (0) − ζ (0) (cid:107) ∞ (1 − L ∆) m + (cid:13)(cid:13)(cid:13) f ( n ) − f (cid:13)(cid:13)(cid:13) ∞ L (cid:18) − ∆ L ) m − (cid:19) ≤ (cid:107) ζ ( n )0 − ζ (cid:107) ∞ (1 − L ∆) m + (cid:107) q ( n ) k | a − q k | a (cid:107) TV L (cid:18) − ∆ L ) m − (cid:19) . with a similar procedure we obtain the bound on (cid:107) y ( n ) ( t ) − y ( t ) (cid:107) . C Proofs of Sections 5.3 and 5.4 Proof of Lemma 1 Let d( w , w ) be the geodesic distance (i.e. the number ofedges in a shortest path) between nodes w and w . We denote the maximumnumber of hops traversed from the root v to a node w in T t with H v ( t ) =max w ∈T t d( v, w ) . Equivalently, H v ( t ) is the depth of the tree T t . Let us fix h n = c log n with c > F (cid:102) W t ( x n ) ≤ P ( (cid:102) W t > x n | H v ( t ) < h n ) + P ( H v ( t ) ≥ h n )= P ( (cid:102) W t > x n | H v ( t ) < h n ) + P (cid:18) max w ∈T t d( v, w ) ≥ h n (cid:19) = P ( (cid:102) W t > x n | H v ( t ) < h n ) + P ( ∃ w ∈ T t : d( v, w ) = h n ) ≤ P ( N h n > x n ) + n P ( (cid:101) P ( t ) ≥ h n )where { N h } h ∈ N is a truncated GW process of maximum depth h , in which theoffspring distribution of the root follows law p , while the degree of remain-ing nodes follow law q , and (cid:101) P ( t ) is a variable representing the number of pointsfalling in [0 , t ) according to a homogeneous Poisson process with constant param-eter γ = 1. Note that n P ( (cid:101) P ( t ) ≥ h n ) represents an obvious upper-bound to theprobability that the depth of T t exceeds h n , since by conduction P ( (cid:101) P ( t ) ≥ h n )is equal to the probability that a given branch of T t has depth larger than h n .55e have P ( (cid:101) P ( t ) ≥ h n ) ≤ ∞ (cid:88) h = h n e − t t h h ! = e − t t h n h n ! (cid:88) h ≥ h n t h − h n h ( h − . . . ( h n + 1)= e − t t h n h n ! (cid:88) s ≥ t s ( h n + s )( h n + s − . . . ( h n + 1) ≤ e − t t h n h n ! (cid:88) s ≥ (cid:18) th n (cid:19) s = e − t t h n h n ! (cid:18) − th n (cid:19) − for h n → ∞ where the last inequality follows from t/h n < t = o ( h n ) for h n → ∞ . Using Stirling’s approximation [19] h n ! ≥ √ πh h n +1 / n e − h n we obtain P ( (cid:101) P ( t ) ≥ h n ) ≤ e − t + h n log t − h n (log h n )+ h n − log(2 πh n ) − log(1 − t/h n ) . (41)Using bound in (41), we obtain for any s > F (cid:102) W t ( x n ) ≤ P ( N h n > x n ) + n e − h n log h n + o ( h n log h n ) ≤ E [ N sh n ] x sn + o (1 /n ) n → ∞ , where the second last inequality follows from the Markov inequality [19]. Atlast, we emphasize that the an analogous bound holds for the number W t ofedges, since (cid:102) W t = W t + 1. Lemma 8. Let N = ( V , E ) be a network sampled from the configuration modelensemble C n,p of compatible size n and statistics p and q , N w h be the inducedgraph obtained by the exploration process of the h -depth neighborhood of a node w chosen uniformly at random from the node set V . Let ˙ q be the distribu-tion defined as follows: (cid:80) kh =0 ˙ q h = min( (cid:80) kh =0 p h , (cid:80) kh =0 q h ) , ∀ k . Note that ˙ q stochastically dominates both p and q . Moreover, let (cid:98) q k be the distribution re-lated to ˙ q k as follows: (cid:98) q k + k = ˙ q k , with k = min k : (cid:80) kj =1 ˙ q j > (cid:15) . Let N w h bethe number of nodes in N w h . We have that for every x n ≤ (cid:98) (cid:15)n (cid:99) : P ( N w h > x n ) ≤ P ( N w h > x n ) where N w h is the total number of nodes over a tree of depth h , in which thedegree of all the nodes follow law (cid:98) q .Proof. First note that, as long as k is bounded, the order of magnitude ofthe moments of ˙ q and (cid:98) q is the same. We prove the assertion through couplingarguments. First we show that N w h ≤ st (cid:98) N w h where (cid:98) N w h is the number of nodesin a tree in which the root has a degree distributed as p and the degree of theother nodes are obtained by extractions without repetitions from an empirical56istribution matching q . Then we show that P ( (cid:98) N w h > x n ) ≤ P ( N w h > x n )under the assumption that x n ≤ (cid:98) (cid:15)n (cid:99) .To show that N w h ≤ st (cid:98) N w h , we start performing a breadth-first explorationof N h and we denote with (cid:101) T h the induced spanning tree of N h . Note that thisspanning tree is obtained as result of the exploration of the edges { M i,j } i,j of N t (we use a double index, where i represents the level, i.e., the distance fromthe root of the tree, while j is an ordinal number induced by the breadth-firstexploration among edges of the same level) by retaining only those M i,j satisfy-ing the following condition: { ν ( M i,j ) (cid:54) = w , ν ( M i,j ) (cid:54) = ν ( M k,q ) , ∀ ( k, q ) with k
For any h and for any s ≥ we have: E [( N w h ) s ] = O ( E [( N w h ) s ])57 roof. note that by construction: N w h { N w h < (cid:98) (cid:15)n (cid:99)} + (cid:98) (cid:15)n (cid:99) { N w h ≥(cid:98) (cid:15)n (cid:99)} ≤ N w h ≤ N w h { N w h < (cid:98) (cid:15)n (cid:99)} + n { N w h ≥(cid:98) (cid:15)n (cid:99)} Moreover note that from Lemma 8 we have: N w h ≤ st N w h { N w h < (cid:98) (cid:15)n (cid:99)} + n { N w h ≥(cid:98) (cid:15)n (cid:99)} The assertion follows immediately.Now we introduce a technical result that characterizes the moments of thetotal number of nodes generated in a GW process in which the offspring distri-bution follows a generic law (cid:98) q . Such result will later on be used to prove morespecific results valid when (cid:98) q either follows a truncated power law distribution(Corollary 5) or it has all finite moments (Corollary 6). Lemma 9. Let { N h } h ≥ be a supercritical GW process with power-law degreedistribution (cid:98) q = { (cid:98) q k } k ≥ (with (cid:80) ∞ k =0 k (cid:98) q k > ). Let n h be the number of nodesat depth h , and N h be the total number of nodes generated up to generation h .These quantities are defined recursively as follows: n h +1 = n h (cid:88) i =1 D i ; N h +1 = N h + n h +1 where D i are i.i.d. according to (cid:98) q , and we start with n = N = 1 . Let (cid:101) µ j = E [( D + 1) j ] , with D distributed according to (cid:98) q . We have E [ N sh ] = O (cid:88) ( k ,...,k s ) ∈K s (cid:101) µ k (cid:101) µ k (cid:101) µ k . . . (cid:101) µ k s s (cid:101) µ s ( h − . where k = (cid:80) sj =1 k j , and the summation is over the following set K s = { ( k , . . . , k s ) : s (cid:88) j =1 jk j = s } . Proof. We first show through a coupling argument that the total number ofnodes N h is stochastically dominated by the number Z h of nodes at depth h in a modified branching process in which the outdegree is augmented by one.Indeed, in such modified branching process we have, recursively: Z h +1 = Z h (cid:88) i =1 ( D i + 1)starting again from Z = 1. We prove by induction that N h ≤ Z h . For h = 0we have N = 1 + D = Z , and the assertion is verified. We prove now that,58f the assertion is true for h , then it is true for h + 1. By inductive hypothesis N h ≤ Z h and N h +1 = N h + n h +1 = N h + n h (cid:88) i =1 D i ≤ N h + N h (cid:88) i =1 D i ≤ Z h + Z h (cid:88) i =1 D i = Z h +1 and thus the assertion is verified. Therefore we will analyze in the following howthe number Z h of nodes at depth h in the modified branching process dependson the moments (cid:101) µ j = E [( D + 1) j ] with D ∼ (cid:98) q .Denote by Φ h ( t ) = E [ t Z h ] the probability generating function of the randomvariable Z h , and by Ψ (cid:101) q the probability generating function of the modified off-spring distribution (i.e., of random variable D +1). Then the following recursionholds: Φ ( t ) = t andΦ h ( t ) = Φ h − (Ψ (cid:101) q ( t )) = Ψ (cid:101) q (Φ h − ( t )) ∀ h ≥ . (42)Let F sh = E [ Z h ( Z h − · · · ( Z h − s + 1)] be the factorial moments of Z h . F sh is obtained by evaluating at t = 1 the s -th derivative of Φ h ( t ), i.e., we have F sh = Φ ( s ) h (1).We thus have F h = Ψ (1) (cid:101) q (Φ h − (1))Φ (1) h − (1) = (cid:101) µ F h − Since F = (cid:101) µ , we obtain by induction F h = (cid:101) µ h , hence the result is true forthe first moment of Z h . To show that the result holds for a generic moment s ,we first show that it is true for the factorial moments of Z h . First, notice that F s = O ( (cid:101) µ s ), ∀ s ≥ 1. We use the strong induction on h , showing that, if therelation holds for F sh − , it holds also for F sh .To differentiate s -times Φ h ( t ), we apply Fa`a di Bruno formula [13] to (42)Φ ( s ) h ( t ) = (cid:88) k =( k ,...,k s ) ∈K s b k Ψ ( k + k ... + k s ) (cid:101) q ( t ) s (cid:89) j =1 (cid:16) Φ ( j ) h − ( t ) (cid:17) k j where b k = s ! k ! k ! · · · k s !1! k · · · s ! k s and the summation is over the following set K s = { ( k , . . . , k s ) : s (cid:88) j =1 jk j = s } . Neglecting factorial terms, which are just some constants for any finite s ,and using the inductive step F sh − = O ( (cid:101) µ s · (cid:101) µ s ( h − ), we obtain F sh = O (cid:88) ( k ,...,k s ) ∈K s (cid:101) µ k + ... + k s · ( (cid:101) µ (cid:101) µ h − ) k · ( (cid:101) µ (cid:101) µ h − ) k · . . . · ( (cid:101) µ s (cid:101) µ s ( h − ) k s = O (cid:88) ( k ,...,k s ) ∈K s (cid:101) µ k (cid:101) µ k (cid:101) µ k . . . (cid:101) µ k s s (cid:101) µ s ( h − . (43)59here k = (cid:80) sj =1 k j . At last we observe that an analogous result holds for themoments E [ N sh ], whose scaling order is the same as F sh . Corollary 5 ( Proof of Lemma 2 ) . Let { N h } h ≥ be a supercritical GW processwith power-law degree distribution (cid:98) q = { (cid:98) q k } k ≥ of exponent β > , truncated at (cid:98) k max = Θ( n ζ ) , ζ > . We have: E [ N sh ] = O ( (cid:98) µ s · (cid:98) µ s ( h − ) , ∀ β > , where (cid:98) µ j isthe j -th moment of q .Proof. Let’s first consider the extreme case in which all moments (cid:98) µ j of (cid:98) q areinfinite, including the first one, which happens for 1 < β < 2. From (13) wehave (cid:101) µ j = Θ( (cid:98) µ j ) = Θ( n ζ ( j +1 − β ) ), ∀ j ≥ 1. Plugging the above expression of (cid:101) µ s into (43), we obtain: F sh = O (cid:88) ( k ,...,k s ) ∈K s µ k n ζ ( k (1+1 − β )+ k (2+1 − β )+ ...k s ( s +1 − β )) µ s ( h − = O µ s ( h − (cid:88) ( k ,...,k s ) ∈K s n ζ [ s +1 − β + k (2 − β )] = O (cid:98) µ s (cid:98) µ s ( h − (cid:88) ( k ,...,k s ) ∈K s n ζk (2 − β ) = O (cid:16)(cid:98) µ s (cid:98) µ s ( h − n ζs (2 − β ) (cid:17) = O (cid:16)(cid:98) µ s (cid:98) µ s ( h − (cid:98) µ s (cid:17) = O (cid:16)(cid:98) µ s (cid:98) µ s ( h − (cid:17) (44)where we have used the fact that, since we are assuming β < 2, the dominantterm in the summation is the one associated to the largest possible value of k ,obtained when k = s , while all others k i = 0, i > j − 1, whereas moments of order j or higher are infinite. This happenswhen β > j . Repeating the same passages as before, adding and subtractingthe ‘missing’ terms corresponding to finite moments, we get: F sh = O (cid:98) µ s (cid:98) µ s ( h − (cid:88) ( k ,...,k s ) ∈K s n ζ (( k − k )(2 − β ) − k (3 − β ) − ... − k j − ( j − β )) = O (cid:16)(cid:98) µ s (cid:98) µ s ( h − (cid:17) = O (cid:16)(cid:98) µ s (cid:98) µ s ( h − (cid:17) (45)where we have used the fact that, since β > j , the dominant term is obtainedagain by choosing k = k = s , while all others k i = 0, i > Remark 11. In our application to the single-class configuration-model with(truncated) power law distribution (Section 5.3), we are only interested to thecase β > (so that the average degree is finite), for which we could obtain thestricter bound F sh = O (cid:16)(cid:98) µ s (cid:98) µ s ( h − (cid:17) . However, since we take h = c log n , we arenot penalized by using looser bound stated in Corollary 5. emark 12. To apply Corollary 5 to the single-class configuration-model with(truncated) power law distribution (Section 5.3), one should also consider thefact that the first generation of nodes in the GW process follows law p ( n ) k , whilethe following generations follow law q ( n ) k . However, by Assumption 1, we havethat p ( n ) k and q ( n ) k are both O ( k − β ) , hence p ( n ) k and q ( n ) k are both stochasticallydominated by a power law distribution (cid:98) q of exponent β , which allows us to applyCorollary 5 and obtain a valid bound for our configuration-model. In particularwe can define (cid:98) q as follows: (cid:80) kh =0 (cid:98) q h = min( (cid:80) kh =0 p h , (cid:80) kh =0 q h ) . Corollary 6. Let { N h } h ≥ be a supercritical GW process with degree distri-bution (cid:98) q = { (cid:98) q k } k ≥ (with (cid:80) ∞ k =0 k (cid:98) q k > ) having all finite moments. Let N h be the total number of nodes generated up to generation h . We have E [ N sh ] = O ( (cid:98) µ s ( h − ) , where (cid:98) µ is the first moment of (cid:98) q .Proof. By assumption we have (cid:101) µ j = Θ( (cid:98) µ j ) = Θ(1) for any j ≥ 1. As directapplication of Lemma 9 we get E [ N sh ] = O (cid:16)(cid:98) µ s ( h − (cid:17) . (46) Proof of Theorem 4 First observe that the number of edges W b,at in T t between any pair of classes( a, b ) con be upper bounded by the total number of edges in T t , which itself canbe bounded by the total number of nodes in T t .Recall that the number of edges between a node of community a and nodesof community j conforms to the empirical distribution p out a,j [ k ].Let P out a,j be the cumulant of p out a,j , and define P out ∗ [ k ] = min a,j P out a,j [ k ]. Then,let p out ∗ [ k ] be the distribution whose cumulant is P out ∗ [ k ].The outgoing degree of a generic node is then stochastically dominated by ar.v. distributed as p out ∗ which has, by construction, all finite moments. Indeednote that, since all p out a,j have, by assumption, an exponential tail, p out ∗ has anexponential tail as well.Therefore, we can bound the number of edges in T t with those in a truncatedGW tree T t in which the outgoing degree distribution of nodes is p out ∗ .Then, setting x b,a = x n = n − γ , for any ( a, b ) ∈ A × A , from (10) andLemma 1 we have, for any s > (cid:107) µ T t − µ N t (cid:107) TV ≤ S x n ( x n + 1)2 n ¯ d + |A| E [ N sh n ] x sn + o (1 /n ) n → ∞ (47)where N sh n is the number of nodes in a truncated GW process of maximumdepth h n , in which the offspring distribution of every node is p out ∗ . 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