A simple person's approach to understanding the contagion condition for spreading processes on generalized random networks
AA simple person’s approach to understanding the contagion condition for spreadingprocesses on generalized random networks
To appear in “Spreading Dynamics in Social Systems”;Eds. Sune Lehmann and Yong-Yeol Ahn, Springer Nature.
Peter Sheridan Dodds ∗ Vermont Complex Systems Center, Computational Story Lab,the Vermont Advanced Computing Core, Department of Mathematics & Statistics,The University of Vermont, Burlington, VT 05401. (Dated: November 8, 2018)We present derivations of the contagion condition for a range of spreading mechanisms on fam-ilies of generalized random networks and bipartite random networks. We show how the contagioncondition can be broken into three elements, two structural in nature, and the third a meshing ofthe contagion process and the network. The contagion conditions we obtain reflect the spreadingdynamics in a clear, interpretable way. For threshold contagion, we discuss results for all-to-all andrandom network versions of the model, and draw connections between them.
I. INTRODUCTION
Given a local contagion mechanism acting on a ran-dom network, and a seed set of nodes N , we would liketo know the answers to a series of increasingly specificquestions:Q1: Is a global spreading event possible? We’ll definea “global spreading event” as one that reaches anon-zero fraction of a network in the infinite limit.Q2: If a global spreading event is possible, what’s theprobability of one occurring?Q3: What’s the distribution of final sizes for all spread-ing events?Q4: Global or not, how does the spreading from the seedset N unfold in time?Now, if we know the full time course of a spreadingevent (Q4) (see [1]), we evidently will be able to answerquestions 1, 2, and 3. We might be tempted to take ononly the more challenging analytical work and call it day(or appropriate time frame of suffering required). But itturns out to be useful to address each question separatelyWhile we will take on these questions for simple modeldistillations only, their real-world counterparts are someof the most important ones we face. What’s the probabil-ity that a certain fraction of a population will contractinfluenza? Could an ecosystem collapse? Indeed, thebiggest question for many systems is:Q5: If we have limited knowledge of a network and lim-ited control, how do we optimally facilitate or pre-vent spreading [2, 3]? ∗ [email protected] In this chapter, we’ll focus on Q1, determining the con-tagion condition for a range of contagion processes onrandom networks including bipartite ones. We will do soby plainly encoding the course of the spreading processitself into the contagion condition.We will take the basic contagion mechanism to be onefor which there are node states: Susceptible (S) andInfected (I). We will also prevent nodes from recoveringor becoming susceptible; once nodes are infected, theyremain so. In mathematical epidemiology, such mod-els are referred to as SI, where S stands for Susceptibleand I for Infected. Two other commonly studied modelsare SIR and SIRS, where a recovered immune state Ris allowed for both and the possibility of cycling in thelatter.For the most part, we will be considering infiniterandom networks. If needed, we will define such net-works as the limit of a one parameter family of networks(e.g., Erd¨os-R´enyi networks with increasing N and meandegree held constant). As a rough guide for simulations,using around N = 10 nodes is typically sufficient foryield results that visually conform well to theoretical ones(e.g., fractional size of the largest component in Erd¨os-R´enyi networks). II. ELEMENTS OF SIMPLE CONTAGION ONRANDOM NETWORKS
The key feature of random networks for spreading isthat they are locally pure branching structures. Thisremains true for a large number of variations on randomnetworks such as correlated random networks and bipar-tite affiliation graphs. Successful spreading away froma single seed (which could be one of many seeds) canonly occur if nodes are susceptible when just one of theirneighbors is infected (see Fig. 1). We will refer to theseeasily susceptible nodes as critical nodes (called vulner-able nodes in [4]). Denoting a network’s entire node setTypeset by REVTEX a r X i v : . [ phy s i c s . s o c - ph ] A ug FIG. 1. Random networks are locally pure branching struc-tures. For the initial stages of the spread shown, nodes canonly experience the infection from a single neighbor. Forspreading to take off from a simple seed, the network mustcontain a connected macroscopic critical mass network Ω crit of nodes susceptible to a single neighbor becoming infected. as Ω, global spreading will only be possible if there is aconnected subnetwork of critical nodes that forms a giantcomponent, the critical mass network Ω crit .This set of critical nodes behaves in the same way asa critical mass one does for collective action [5–8] butthere is now an internal dynamic. If one node is infectedwithin the critical mass network Ω crit , then spreading tosome fraction of the critical mass network and beyondis possible, depending on the probablistic nature of thecontagion process.There are two other subnetworks that need to be char-acterized to understand spreading on random networks.First, containing the critical mass network and all non-critical nodes connected to the critical mass network isthe triggering component, Ω trig . Knowledge of this struc-ture is required to determine the probability of a globalspreading event [9]. Second, we have Ω final which is theextent of infection realized for any spreading event. Forrandom networks, the distribution of the fractional sizeof Ω final will be either unimodal (the contagion processalways succeeds) or bimodal (initial failure is possible).If Fig. 2, we show how the three subnetworks Ω crit ,Ω trig , and Ω final potentially overlap. A global spreadingevent is only possible if Ω crit takes up a non-zero fractionof the network. Some limiting cases allow for surprisingkinds of robust-yet-fragile contagion, such as Ω crit beingvanishingly small while any successful infection spreadsto the full network [4]. III. THE CONTAGION CONDITION
We would like to devise some kind of general, quick testalgorithm into which we would be able to feed any conta-gion mechanism and any network, whether constructedor real. Such an algorithm would generate what we’ll calla Contagion Condition, and would only be worthwhile ifit avoided simulating all possible spreading events andinstead computed a composite test statistic. Upon run-
FIG. 2. One possible arrangement of the three essential sub-networks for a contagion process on a random network: thecritical mass network Ω crit , the triggering component Ω trig ,and the final extent of a global spreading process, Ω final . Ingeneral, Ω crit ⊂ Ω trig , Ω crit ⊂ Ω final , and Ω trig , Ω final ⊂ Ω . ning a system through our algorithm we would simplyreceive a “Yes” or “No”. Scaling up, we could then testan array of systems in parallel and for the “Yes” respons-es, we would proceed to explore those systems in detail(e.g., those cities which are susceptible to Zombie out-breaks [10]). A. Contagion condition for one-shot spreadingprocesses
For random network models, our test algorithm canbe formulated in a physically-minded way. We willstep through the building of the contagion condition forone-shot, permanent infection spreading on generalized,uncorrelated random networks and then expand fromthere.By one-shot spreading, we mean that each newlyinfected node has one chance in the next time step toinfect its uninfected neighbors. That is, if node i failsto infect a specific neighbor i (cid:48) , then i cannot attemptto infect i (cid:48) again in any following time step. Permanentinfection means that nodes do not recover.For a node i with degree k , we will write i ’s proba-bility of infection given j of its neighbors are infectedas B kj . While our focus on the initial spread on ran-dom networks means we need only consider the proba-bility nodes are infected by one of their neighbors, B k ,we must consider the response to multiple simultaneousinfections for later stages of global spreading on ran-dom networks [1, 11], more complicated contagion mech-anisms, and, more importantly if we care about the realworld, networks with non-zero clustering [12, 13].As is often the case with networks, we open up betterways to understand and explain phenomena if we focuson edges rather than nodes. This is not entirely naturalas for many problems we are ultimately concerned withhow nodes behave and, for contagion especially, we canreadily map ourselves directly onto individual nodes (willmy next movie fail?). But once we lose this anchoringand shift to thinking first about edges with nodes in thebackground, clearer paths emerge.So, instead of framing spreading as rooted in nodeinfection rates, we consider the dynamics of infectededges. For our purposes, an infected edge will be oneemanating from an infected node, and we will have toconsider direction even for undirected networks.We need to determine one number for our system, whatwe’ll call the gain ratio, R [14]. We define R as theexpected number of newly infected edges that will begenerated by a single infected edge leading to an unin-fected node. (In epidemiology, the gain ratio would beequivalent to the reproduction number, R .)For the moment, let’s assume we have computed R fora system. Because sparse random networks are locallypure branching structures (see Fig. 1), the spread ema-nating from a single seed will also be a simple branchingone. Early on, there will be no interactions between anytwo newly infected edges leading to the same uninfectednode.The fraction of newly infected edges at time t , f inf( · ) ( t ),must then follow an elementary evolution: f inf( · ) ( t ) = R f inf( · ) ( t − . (1)The subscript for the count f inf will indicate the edge’stype which for our initial system is irrelevant, hence ( · ).The early growth will therefore be exponential with f inf( · ) ( t ) = R t f inf( · ) (0) , (2)where f inf( · ) (0) equals the degree of the seed node. Wemight guess that we can write down the exact evolutionas f inf( · ) ( t ) = R t f inf( · ) (0) , but the initial step is sneakilydifferent. Well get to this issue later on.Global spreading will evidently be possible only if R > , (3)and this very simple criterion will be our Contagion Con-dition.The above equations maintain the same form if we con-sider not one seed but a random seed set taking up a non-zero fraction of the random network. Writing ρ t as thefraction of edges emanating from newly infected nodes attime t , we have, again for the initial phase of spreading: ρ t = R ρ t − , (4)which leads to ρ t = R t ρ . (5) We now determine the gain ratio R for the simple classof one-shot contagion on random network systems. Indoing so, we show that the Contagion Condition is worth-while beyond being a simple diagnostic as, with the righttreatment, it can be also seen to carry physical intuition.In determining R , there are three (3) pieces to con-sider: two are structural and a function of the network,and the third couples the contagion mechanism to thenetwork.1. We start on an edge that has just become infectedand look toward the uninfected node that has nowbecome exposed. The properly normalized proba-bility that this node has degree k is Q k = kP k (cid:104) k (cid:105) (6)because each degree k node can be reached alongits k edges. This skewing of the degree distributionis a result of some renown as it drives the Simon-like rich-get-richer models of network growth ofPrice [15, 16] and Barab´asi and Albert [17], andalso underlies the friendship paradox and its gener-alizations [18, 19]: Your friends are quite likely tobe different from you, and often in disappointingways such as by having more friends or wealth onaverage.2. Second, we have the action of contagion mecha-nism. As have already defined, with probability B k the node of degree k is infected by the singleincoming infected edge. With probability 1 − B k ,the infection fails.3. Depending on whether or not the infection is suc-cessful, we know that in the next time step thecontagion mechanism will generate either 0 or k − R = ∞ (cid:88) k =0 kP k (cid:104) k (cid:105) (cid:124)(cid:123)(cid:122)(cid:125) prob. ofconnecting toa degree k node • B k (cid:124)(cid:123)(cid:122)(cid:125) Prob. ofinfection • ( k − (cid:124) (cid:123)(cid:122) (cid:125) + ∞ (cid:88) k =0 kP k (cid:104) k (cid:105) (cid:124)(cid:123)(cid:122)(cid:125) prob. ofconnecting toa degree k node • (1 − B k ) (cid:124) (cid:123)(cid:122) (cid:125) Prob. ofno infection • ( 0 ) (cid:124)(cid:123)(cid:122)(cid:125) (7)The second piece evaporates and we have our contagioncondition: R = ∞ (cid:88) k =0 kP k (cid:104) k (cid:105) • B k • ( k − > . (8)Again, the value here is that this structure of R encodesthe contagion mechanism in a clear way. As such, weresist any urge to rearrange the form of Eq. (8) for amore elegant form. As we move to more general systems,the three part form of two pieces for the network and onefor the contagion mechanism will be maintained, and thecriterion of a single number exceeding unity, R >
1, willelevate to being the largest eigenvalue of a gain ratiomatrix exceeding unity.We now move through a few examples of other kinds ofsystems involving contagion mechanisms acting on net-work structures.
B. Contagion condition for multiple-shot spreadingprocesses
We have presumed a one-shot contagion process in ourderivation of Eq. (8). In loosening this restriction tospreading processes that may involve repeated attemptsto infect a node with the possible recovery of the infectednode allowed as well, we can compute B k as the longterm probability of infection. The form of gain ratioremains the same and therefore so does the contagioncondition given in Eq. (8). C. Remorseless spreading and the giant componentcondition
We step back from contagion momentarily to show thatwe can also determine whether or not a random networkhas a giant component. This is now a structural testabsent any processes. A network will have a giant com-ponent if it is, on average, locally expanding. That is, ifwe travel along a randomly chosen edge, we will reach anode which has, on average, more than one other edgeemanating from it. But this is just a remorseless versionof our one-shot contagion mechanism, one where infec-tion always succeeds, i.e., B k = 1.Setting B k = 1 in Eq. (8), we have the giant compo-nent condition: R = ∞ (cid:88) k =0 kP k (cid:104) k (cid:105) • ( k − > , (9)where we have again used the physical sense of a gainratio. D. Simple contagion on generalized randomnetworks If B k = B <
1, A fraction (1- B ) of all edges will nottransmit infection, and the contagion condition becomes R = ∞ (cid:88) k =0 kP k (cid:104) k (cid:105) • B • ( k − > . (10)This is a bond percolation model [20], and Eq. (10) can beseen as a giant component condition for a network with (1- B ) of its edges removed. The resultant network has adegree distribution ˜ P k = B k (cid:80) ∞ i = k (cid:0) ik (cid:1) (1 − B ) i − k P i , andevidently, as B decreases, only increasingly more con-nected networks will be able to facilitate spreading. E. Other routes to determining the contagion andgiant component conditions
There are many other ways to arrive at the contagioncondition in Eq. (8) and the giant component conditionin Eq. (9). The path taken affects the form of the condi-tion and may limit understandability [14]. For example,the giant component condition was determined by Molloyand Reed [21] in 1995 and presented as ∞ (cid:88) k =0 k ( k − P k > . (11)While equivalent to Eq. (9), the framing of local expan-sion is obscured.For a simple spreading mechanism with B k = B ,Newman [13], for example, used generatingfunctionologymethods [22] to first determine the average size of finitecomponents and then find when this quantity diverged.For Granovetter’s social contagion threshold model onrandom networks [6], Watts took the same approach [4].This size divergence is a hallmark of phase transitionsin statistical mechanical systems in general, and whileit can be used to find the critical point, doing so wouldideally be at the level of a consistency check.For the giant component condition, a somewhat moredirect approach using generating functions [23] is basedon the probability distribution that the node at the ran-domly chosen end of a randomly chosen edge has k otheredges is R k = Q k +1 = 1 (cid:104) k (cid:105) ( k + 1) P k +1 . (12)Writing the generating function for the degree distri-bution as F P ( x ) = (cid:80) ∞ k =0 P k x k , we have F R ( x ) = F (cid:48) P ( x ) /F (cid:48) P (1) , where we have used (cid:104) k (cid:105) = F (cid:48) P (1), anelementary result for determining averages with gen-erating functions [22]. The average number of otheredges found at a randomly-arrived-at node is F (cid:48) R (1) = F (cid:48)(cid:48) P (1) /F (cid:48) P ( x ) = (cid:104) k ( k − (cid:105)(cid:104) k (cid:105) . This is exactly our gain ratioand we now have (cid:104) k ( k − (cid:105)(cid:104) k (cid:105) > FIG. 3. For general directed networks, a node has k i incidentedges and k o emanating edges governed by a a joint distribu-tion P k i ,k o .FIG. 4. Nodes in mixed random networks have k u undirectededges, k i incident edges, and k o emanating edges. Node degreeis represented by the vector (cid:126)k = [ k u k i k o ] T and degrees aresampled from a joint distribution P (cid:126)k . F. Simple contagion on generalized directedrandom networks
For purely directed networks, we allow each node tohave an in-degree k i and an out-degree k o with probabil-ity P k i ,k o (see Fig. 3). The same arguments that gave usEq. (8) now end with: R = ∞ (cid:88) k i =0 ∞ (cid:88) k o =0 k i P k i ,k o (cid:104) k i (cid:105) • B k i , • k o > . (14)The three components of the contagion condition havethe same interpretation as before. G. Simple contagion on mixed, correlated randomnetworks
We jump to a more complex possibility of mixed ran-dom networks with a combination of directed and undi-rected (or bidirectional) edges as well as arbitrary degree-degree correlations between nodes, as introduced in [24].Nodes may have three types of edges: k u undirectededges, k i incoming directed edges, and k o outgoing direct-ed edges. The degree distribution is now a function of athree-vector: P (cid:126)k where (cid:126)k = [ k u k i k o ] T . (15)As for directed networks, we require in- and out-degreeaverages to match up: (cid:104) k i (cid:105) = (cid:104) k o (cid:105) . We add two point
FIG. 5. For mixed random networks, node degree corre-lations may be measured along undirected and/or directededges. correlations per [14, 24] through three conditional prob-abilities: • P (u) ( (cid:126)k | (cid:126)k (cid:48) ) = probability that an undirected edgeleaving a degree (cid:126)k (cid:48) nodes arrives at a degree (cid:126)k node. • P (i) ( (cid:126)k | (cid:126)k (cid:48) ) = probability that an edge leaving adegree (cid:126)k (cid:48) nodes arrives at a degree (cid:126)k node is anin-directed edge relative to the destination node. • P (o) ( (cid:126)k | (cid:126)k (cid:48) ) = probability that an edge leaving adegree (cid:126)k (cid:48) nodes arrives at a degree (cid:126)k node is anout-directed edge relative to the destination node.We now require more refined (detailed) bal-ance along both undirected and directed edges(see Fig. 5). Specifically, we must have [14, 24]: P (u) ( (cid:126)k | (cid:126)k (cid:48) ) k (cid:48) u P ( (cid:126)k (cid:48) ) (cid:104) k (cid:48) u (cid:105) = P (u) ( (cid:126)k (cid:48) | (cid:126)k ) k u P ( (cid:126)k ) (cid:104) k u (cid:105) , and P (i) ( (cid:126)k | (cid:126)k (cid:48) ) k (cid:48) o P ( (cid:126)k (cid:48) ) (cid:104) k (cid:48) o (cid:105) = P (o) ( (cid:126)k (cid:48) | (cid:126)k ) k i P ( (cid:126)k ) (cid:104) k i (cid:105) . For all example systems so far, the gain ratio has beena single number. For mixed random networks, infectionsalong directed edges may cause infections along undirect-ed edges and so on. We will need to count undirected anddirected edge infections separately, the growth of infec-tions for a one-shot contagion process will obey the fol-lowing dynamic: (cid:34) f (u) (cid:126)k ( t + 1) f (o) (cid:126)k ( t + 1) (cid:35) = (cid:88) (cid:126)k (cid:48) R (cid:126)k(cid:126)k (cid:48) (cid:34) f (u) (cid:126)k (cid:48) ( t ) f (o) (cid:126)k (cid:48) ( t ) (cid:35) , (16)where we now identify a gain ratio tensor: R (cid:126)k(cid:126)k (cid:48) = (17) (cid:20) P (u) ( (cid:126)k | (cid:126)k (cid:48) ) • B (cid:126)k(cid:126)k (cid:48) • ( k u − P (i) ( (cid:126)k | (cid:126)k (cid:48) ) • B (cid:126)k(cid:126)k (cid:48) • k u P (u) ( (cid:126)k | (cid:126)k (cid:48) ) • B (cid:126)k(cid:126)k (cid:48) • k o P (i) ( (cid:126)k | (cid:126)k (cid:48) ) • B (cid:126)k(cid:126)k (cid:48) • k o (cid:21) . For a gain ratio matrix or tensor, our contagion conditionis now a test of whether or not the largest eigenvalueexceeds 1.
H. Contagion on correlated random networks witharbitrary node and edge types
We make one last step of generalization for correlatedrandom networks [14]. As per Fig. 6, we allow arbitrary
FIG. 6. Element of a general correlated random networkwhere edges and nodes may take on arbitrary characteristics.Node and edge type are specified as α = ( ν, λ ). types of nodes and edges along with arbitrary correlationsbetween node-edge pairs. For multi-shot contagion, wehave f (cid:126)α ( d + 1) = (cid:88) (cid:126)α (cid:48) R (cid:126)α(cid:126)α (cid:48) f (cid:126)α (cid:48) ( d ) (18)where R (cid:126)α(cid:126)α (cid:48) is the gain ratio matrix and has the form: R (cid:126)α(cid:126)α (cid:48) = P (cid:126)α(cid:126)α (cid:48) • k (cid:126)α(cid:126)α (cid:48) • B (cid:126)α(cid:126)α (cid:48) . (19)Here, • P (cid:126)α(cid:126)α (cid:48) = conditional probability that a type λ (cid:48) edgeemanating from a type ν (cid:48) node leads to a type ν node. • k (cid:126)α(cid:126)α (cid:48) = potential number of newly infected edges oftype λ emanating from nodes of type ν . • B (cid:126)α(cid:126)α (cid:48) = probability that a type ν node is eventuallyinfected by a single infected type λ (cid:48) link arrivingfrom a neighboring node of type ν (cid:48) .Finally, we can write down our generalized contagioncondition as: max | µ | : µ ∈ σ ( R ) > , (20)where σ ( R ) denotes the eigenvalue spectrum of R . I. Simple contagion on bipartite random networks
Bipartite networks (or affiliation graphs) connect twopopulations through some association, and induce net-works within each population [23, 25–29]. Bipartitestructures and variants are natural representations ofmany real networked systems with a classic examplebeing boards and directors. The induced distributionsare formed by connecting all pairs of boards that shareat least one director and all pairs of directors that belongto the same board.Base models for real bipartite systems are randombipartite networks which are formed by randomly con-necting two populations with specified degree distribu-tions. Random bipartite networks are able to reproduce induced degree distributions, which may be non-trivialin form [23].To help with our analysis, we’ll consider a randombipartite network between stories and tropes [30]. Eachstory contains one or more trope, and each trope is partof one more stories. Stories sharing tropes are then linkedas are tropes found in the same story. In Fig. 7, we showa small example (center) along with the induced trope-trope and story-story networks.For spreading between stories we may wish to imaginewe’re in the BookWorld of the Thursday Next series [31].We’ll use this notation for our two inter-affiliatedtypes: r for stories and › for tropes.Consider a story-trope system with N r denoting thenumber of stories, N › the number of tropes, and m r , › the number of edges connecting stories and tropes.Let’s have some underlying distributions for numbersof affiliations: P ( r ) k (a story has k tropes) and P ( › ) k (atrope is in k stories).Some bookkeeping arises with balance requirements.Writing (cid:104) k (cid:105) r as the average number of tropes per story,and (cid:104) k (cid:105) › as the average number of stories containing agiven trope, we must have: N r ·(cid:104) k (cid:105) r = m r , › = N › ·(cid:104) k (cid:105) › . Let’s first get to the giant component condition beforetalking about contagion.Just as for random networks, we focus on edges beget-ting edges, and we will need the distributions analogousto Q k , Eq. (6). We randomly select an edge connectinga story r to a trope › . Traveling from the trope to thestory, we have that the probability the story r contains k total tropes is: Q ( r ) k = kP ( r ) k (cid:80) N r j =0 jP ( r ) j = kP ( r ) k (cid:104) k (cid:105) r . (21)Heading instead towards the trope › , we find the proba-bility that the trope › is in k total stories is Q ( › ) k = kP ( › ) k (cid:80) N › j =0 jP ( › ) j = kP ( › ) k (cid:104) k (cid:105) › . (22)To determine the giant component condition for theinduced network of stories (to choose a side), let’s startwith a randomly chosen edge and travel from the story tothe trope. As shown starting on the left of Fig. 8, we hitthe trope and then travel to the other stories contain-ing that trope. This bouncing back and forth betweentropes and stories continues and because the connectionsare random and if the system is large enough, no storyor trope is returned to early on. Just as for random net-works, there are no short loops (technically, finitely manyin the infinite limit).We are thus able to depict the expanding branching inFig. 8 and we can see that the giant component condi-tion will involve the product of the gain ratio for each FIG. 7. Example of a bipartite affiliation network and the induced networks. Center: A small story-trope bipartite graph.The induced trope network and the induced story network are on the left and right. The dashed edge in the bipartite affiliationnetwork indicates an edge added to the system, resulting in the dashed edges being added to the two induced networks.FIG. 8. Spreading on a random bipartite network can beseen as bouncing back and forth between the two connectedpopulations. The gain ratio for simple contagion on a bipar-tite random network is the product of two gain ratios as shownin Eq. (24). distribution. R = R r · R › = (23) (cid:34) ∞ (cid:88) k =0 kP ( r ) k (cid:104) k (cid:105) r • ( k − (cid:35) (cid:34) ∞ (cid:88) k =0 kP ( › ) k (cid:104) k (cid:105) › • ( k − (cid:35) > ∞ (cid:88) k =0 ∞ (cid:88) k (cid:48) =0 kk (cid:48) ( kk (cid:48) − k − k (cid:48) ) P ( r ) k P ( › ) k (cid:48) = 0 , (24)but, again, we have stripped the physics away.Introducing a simple contagion can be done as beforeby allowing tropes to infect other tropes in the same story (with probability B ( › ) k ) and stories to affect other storiesif they share a trope (with probability B ( r ) k ) We adjustEq. (24) to obtain: R = R r · R › = (25) (cid:34) ∞ (cid:88) k =0 kP ( r ) k (cid:104) k (cid:105) r • B ( r ) k • ( k − (cid:35) × (cid:34) ∞ (cid:88) k =0 kP ( › ) k (cid:104) k (cid:105) › • B ( › ) k • ( k − (cid:35) > J. Threshold contagion on generalized randomnetworks
We turn to our last example: threshold contagion, animportant simple model of social contagion [4, 6, 32–37]. In basic threshold contagion models, all individu-als observe the infection status of their neighbors at eachtime step, and become infected if their internal thresholdis exceeded. In the present and following section, we willexplore the contagion condition for threshold models onall-to-all networks and random networks, and examinethe early course of a global spreading event reflecting onthe nature of early adopters.In Granovetter’s mean-field or all-to-all network ver-sion [6], individuals are always aware of the overall frac-tion of the population that is infected. We write thefraction of the population that is infected at time t as a t . If we have a general threshold distribution f ( φ ), thenthe fraction of the population whose threshold will beexceeded at time t and hence be infected at time t + 1 is: φ t +1 = (cid:90) φ t f ( u )d u = F ( u ) | φ t = F ( φ t ) − F (0) (26)where F is the cumulative distribution of f (if F (0) > φ →
0. In this limit, global spreading occursif (1) F (0) > φ = 0 is a fixed point but isunstable (meaning F (0) = 0 and F (cid:48) (0) > φ = 0 isa stable fixed point (meaning F (0) = 0 and F (cid:48) (0) < f ( φ )a degree k node will be part of the critical mass networkwith probability: B k = (cid:90) /k f ( φ )d φ. (27)The gain ratio remains the same as the one givenin Eq. (8).We now link the contagion conditions for the all-to-allnetwork and random network versions of social conta-gion. K. Connecting the contagion condition for all-to-alland random networks for threshold contagion
We make the simple observation that if we examinethe threshold model’s behavior on a random network andallow the average degree (cid:104) k (cid:105) to increase, then the resultswill tend towards what we would observe on an all-to-all network. Since the limiting behavior of the contagionmodel on all-to-all networks is governed by the presenceor absence of fixed points of the cumulative thresholddistribution F , we are therefore able to state what themodel’s behavior on random networks must tend towardsas (cid:104) k (cid:105) increases based solely on the form of F .We consider two examples of threshold distribution f to facilitate our discussion. First, for a general thresholddistribution f , it is useful for us to define a global spread-ing event interval as the range of (cid:104) k (cid:105) for which glob-al spreading events are possible on a random network.A simple example involving a bounded global spreadingevent interval and a non-trivial threshold distribution f is represented in Figs. 9A–B. The main plot of Fig. 9Ashows the cumulative distribution F , and the inset showsthe threshold distribution f . The all-to-all network mod-el, 9A, exhibits a simple kind of critical mass behavior:the infection level approaches unity if the initial activatedfraction φ is above the sole unstable fixed point, or elseit dies away. Thus for all-to-all networks, a small initialinfection level will always fail to yield global infection.For global spreading events to occur on all-to-all net-works, some alternative seeding mechanism (an advertis-ing campaign, perhaps) must precede the word-of-mouthdynamics so as to create a sufficiently large φ .By contrast, global spreading events can arise froma single infected individual in a sparse random net-work with exactly the same distribution of thresholds, a t F ( a t + ) all−to−all networksA 〈 k 〉 〈 S 〉 random networksB a’ a t F ( a t + ) C 〈 k 〉 〈 S 〉 D φ ∗ f ( φ ∗ ) φ ∗ f ( φ ∗ ) FIG. 9. Plots comparing the behavior of the model on all-to-all networks (plots A and C) and random networks (B andD) for two different example threshold distributions. Theinsets to plots A and C show the two underlying thresholddistributions, which are unimodal and bimodal respectively,and the corresponding cumulative distributions are presentedin the main plots of A and C. Plots B and D show globalspreading event intervals for random networks with the samethreshold distributions as A and C respectively. The blacklines in B and D indicate the average size of global spreadingevents that exceed 0 . N , and the dashed lines the averagesize of the largest critical mass network (sizes are normalizedby N ) The threshold distribution in plot A leads to a boundedglobal spreading event interval on random networks while thedistribution in plot C leads to an unbounded one. In plot D,the average size of the largest critical mass network decays to0 as (cid:104) k (cid:105) → ∞ . The results in plots B and D are derived from10 networks with N = 10 and one seed per network. as shown in Fig. 9B. The reason is that when individ-uals are connected to a limited number of alters with-in a population, the fraction of their neighbors that areinfected may now be nonzero and thus may exceed theirthreshold (in infinite all-to-all networks, this fraction isalways 0 for finite seeds). By effectively reducing theknowledge individuals have of the overall population—by increasing their ignorance—global spreading eventsbecome possible. Related observations invoke pluralisticignorance [38, 39] and the importance of small groups infacilitating collective action [5] by circumventing the freerider problem.Thus, when the threshold distribution f is fixed, weobserve a connection between the results for spreadingon all-to-all networks and random networks. Boundedglobal spreading event intervals can only occur when themean-field version exhibits a critical mass property, i.e.,when there exists a stable fixed point at the origin φ = 0(i.e., F (0) = 0 and F (cid:48) (0) < (cid:104) k (cid:105) . Unbounded global spreading event intervals arise whenthere are sufficient individuals who will be vulnerableeven if their degree is very high, i.e., when the thresh-old distribution has enough weight at or near φ = 0. Anexample of an unbounded global spreading event intervalis given in Fig. 9D with the underlying threshold dis-tribution and its cumulative shown in Figs. 9C. Sincesmall seeds always take off in the all-to-all network ver-sion, as network connectivity is increased, global spread-ing events continue to occur and the global spreadingevent interval is unbounded. The size of the largest crit-ical mass network is nonzero for all finite (cid:104) k (cid:105) , though ittends to 0 in the limit (cid:104) k (cid:105) → ∞ . For highly connectedrandom networks, the final size of the global spreadingevent again depends on the fixed points of F . For exam-ple, in Fig. 9B, global spreading events typically reachthe full size of the giant component which correspondsto an upper stable fixed point of F at φ = 1. In Fig. 9D,we see global spreading events only reach half the size ofthe population, corresponding to the stable fixed pointof F at φ = 1 /
2. We thus see that in moving from all-to-all networks torandom networks, the behavior of the threshold modelchanges qualitatively in the sense that there exist thresh-old distributions for which global spreading events start-ed by a small seed cannot occur on an all-to-all network,yet may occur on sparse, random networks.
IV. CONCLUDING REMARKS:
For any parameterized system that may afford glob-al spreading, the contagion condition is a fundamentalcriterion to determine. We have outlined the contagioncondition for a range of contagion mechanisms acting ongeneralized random networks, showing that the conditioncan be derived so as to bear a clear imprint of the mech-anism at work. A similar approach can be used to layout the triggering probability of a global spreading eventin a readable form [9].While generating function approaches provided manyof the first breakthroughs giving the possibility and prob-ability of spreading [23, 40] and have yielded powerfulaccess to many other results, they have tended to obscurethe forms of the simplest ones such as the contagion con-dition. These techniques are also inherently indirect asthey work by avoiding the giant component and charac-terizing only finite ones. Later work focusing on fraction-al seeds was able to go directly into the giant componentand determine not just the final size but full time dynam-ics of global spreading events [1, 11], and we recommendcontinued pursuit of this line of attack going forward. [1] J. P. Gleeson and D. J. Cahalane, Phys. Rev. E ,056103 (2007).[2] P. Lazarsfeld and R. Merton, in Freedom and Controlin Modern Society , edited by M. Berger, T. Abel, andC. Page (Van Nostrand, New York, 1954) pp. 18–66.[3] D. J. Watts and P. S. Dodds, Journal of ConsumerResearch , 441 (2007).[4] D. J. Watts, Proc. Natl. Acad. Sci. , 5766 (2002).[5] M. Olson, The Logic of Collective Action: Public Goodsand the Theory of Groups , Revised ed., Harvard Econom-ic Studies (Harvard University Press, Cambridge, MA,1971).[6] M. Granovetter, Am. J. Sociol. , 1420 (1978).[7] P. E. Oliver, G. Marwell, and R. Teixeira, The AmericanJournal of Sociology , 522 (1985).[8] P. E. Oliver, Annual Review of Sociology , 271 (1993).[9] K. D. Harris, J. L. Payne, and P. S. Dodds, “Direct,physically-motivated derivation of triggering probabili-ties for contagion processes acting on correlated randomnetworks,” (2014), http://arxiv.org/abs/1108.5398.[10] P. Munz, I. Hudea, J. Imad, and R. J. Smith?, in Infectious Disease Modelling Research Progress , edited byJ. M. Tchuenche and C. Chiyaka (Nova Science Publish- ers, Inc., 2009) pp. 133–150.[11] J. P. Gleeson, Phys. Rev. E , 046117 (2008).[12] D. J. Watts and S. J. Strogatz, Nature , 440 (1998).[13] M. E. J. Newman, SIAM Rev. , 167 (2003).[14] P. S. Dodds, K. D. Harris, and J. L. Payne, Phys. Rev.E , 056122 (2011).[15] D. J. de Solla Price, Science , 510 (1965).[16] D. J. de Solla Price, J. Amer. Soc. Inform. Sci. , 292(1976).[17] A.-L. Barab´asi and R. Albert, Science , 509 (1999).[18] Y.-H. Eom and H.-H. Jo, Nature Scientific Reports ,4603 (2014).[19] N. Momeni and M. Rabbat, PloS one , e0143633(2016).[20] D. Stauffer and A. Aharony, Introduction to PercolationTheory , Second ed., patterns (Taylor & Francis, Wash-ington, D.C., 1992).[21] M. Molloy and B. Reed, Random Structures and Algo-rithms , 161 (1995).[22] H. S. Wilf, Generatingfunctionology , 3rd ed., combina-torics (A K Peters, Natick, MA, 2006).[23] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys.Rev. E , 026118 (2001). [24] M. Bogu˜n´a and M. ´Angeles Serrano, Phys. Rev. E ,016106 (2005).[25] Y.-Y. Ahn, S. E. Ahnert, J. P. Bagrow, and A.-L.Barab´asi, Nature Scientific Reports , 196 (2011).[26] C.-Y. Teng, Y.-R. Lin, and L. A. Adamic, in Proceedingsof the 3rd Annual ACM Web Science Conference , WebSci’12 (ACM, New York, NY, USA, 2012) pp. 298–307.[27] C. A. Hidalgo, B. Klinger, A.-L. Barab´asi, and R. Haus-man, Science , 482 (2007).[28] K.-I. Goh, M. E. Cusick, D. Valle, B. Childs, M. Vidal,and A.-L. Barab´asi, Proc. Natl. Acad. Sci. , 8685(2007).[29] L. P. Garc´ıa-P´erez, M. A. Serrano, and M. Bogu˜n´a, “Thecomplex architecture of primes and natural numbers,”(2014), http://arxiv.org/abs/1402.3612.[30] TV Tropes, http://tvtropes.org .[31] J. Fforde,
The Eyre Affair: A Thursday Next Novel (NewEnglish Library, London, 2001).[32] T. C. Schelling, J. Math. Sociol. , 143 (1971).[33] T. C. Schelling, J. Conflict Resolut. , 381 (1973).[34] T. C. Schelling, Micromotives and Macrobehavior , com-plexity (Norton, New York, 1978).[35] M. S. Granovetter and R. Soong, Journal of Mathemati-cal Sociology , 165 (1983).[36] M. S. Granovetter and R. Soong, J. Econ. Behav. Organ. , 83 (1986).[37] M. Granovetter and R. Soong, Sociological Methodology , 69 (1988).[38] T. Kuran, World Politics , 7 (1991).[39] T. Kuran, Private Truths, Public Lies: The Social Conse-quences of Preference Falsification , Reprint ed. (HarvardUniversity Press, Cambridge, MA, 1997).[40] D. J. Watts, P. S. Dodds, and M. E. J. Newman, Science296