A generalized linear threshold model for an improved description of the spreading dynamics
AA generalized linear threshold model
A generalized linear threshold model for an improved description of thespreading dynamics
Yijun Ran, Xiaomin Deng, Xiaomeng Wang, and Tao Jia a) College of Computer and Information Science, Southwest University, Beibei, Chongqing,400715 P. R. China (Dated: 18 August 2020)
Many spreading processes in our real-life can be considered as a complex contagion, and the linear threshold (LT)model is often applied as a very representative model for this mechanism. Despite its intensive usage, the LT modelsuffers several limitations in describing the time evolution of the spreading. First, the discrete-time step that capturesthe speed of the spreading is vaguely defined. Second, the synchronous updating rule makes the nodes infected inbatches, which can not take individual differences into account. Finally, the LT model is incompatible with existingmodels for the simple contagion. Here we consider a generalized linear threshold (GLT) model for the continuous-timestochastic complex contagion process that can be efficiently implemented by the Gillespie algorithm. The time in thismodel has a clear mathematical definition and the updating order is rigidly defined. We find that the traditional LTmodel systematically underestimates the spreading speed and the randomness in the spreading sequence order. We alsoshow that the GLT model works seamlessly with the susceptible-infected (SI) or susceptible-infected-recovered (SIR)model. One can easily combine them to model a hybrid spreading process in which simple contagion accumulates thecritical mass for the complex contagion that leads to the global cascades. Overall, the GLT model we proposed can bea useful tool to study complex contagion, especially when studying the time evolution of the spreading.
The linear threshold (LT) model is a typical model for thecomplex contagion process. However, it systematically un-derestimates the spreading speed and the randomness inthe spreading sequence order. To cope with this issue,we propose a generalized linear threshold (GLT) model,where the time evolution is controlled by the continuous-time stochastic process. The GLT model can be effi-ciently implemented by the Gillespie algorithm, providinga useful tool to investigate and simulate more complicatedspreading processes, especially when the time evolution isthe focus.
I. INTRODUCTION
The process of adoption such as the adoption ofinnovations , commercial products and socialbehavior , and the process of diffusion such as thespread of rumors , opinions and knowledge canall be described as a kind of contagion process . In theseprocesses, things like information or ideas pass from oneperson to another through the association between the twoindividuals, analogous to the infection of diseases. This kindof contagion process is of particular interest when it occurs insparsely connected networks, where the topology of the net-work has a big impact on the outcome of the spreading ,giving rise to a set of interesting phenomena .The underlying mechanisms generally fall into two cate-gories: simple contagion and complex contagion . The sim-ple contagion is based on disease spreading. An individual, orequivalently a node of a network, has a non-zero probability a) Electronic mail: [email protected]. to be infected if one of the connected neighbors is infected.The infection probability also increases monotonically withthe number of infected neighbors. The complex contagion isinspired by collective behaviors in social systems. It assumesthat the infection will occur only when some critical mass hasreached , which can be either the number of contacts orthe number of infected neighbors . Correspondingly, the in-fection probability is non-monotonic, typically captured by astep function that goes directly from 0 to 1 when the criticalmass has been reached.The linear threshold (LT) model is widely used to studycomplex contagion . In the model, a node will defi-nitely become infected if the fraction of its infected neigh-boring nodes goes beyond a threshold value. Previous worksusing the LT model usually focus on the final consequenceof the spreading, such as when the global cascading couldoccur or how to select effective seed nodes to maximizethe spreading . When it comes to the spreading dynam-ics, however, the LT model suffers three limitations. First, theevolution in LT model is controlled by discrete-time steps thatlack a proper definition, which gives rise to issues when thespeed of the spreading needs to be investigated. Second, thestatus of a node is updated in a synchronous manner. At eachtime step, all nodes currently satisfying the spreading thresh-old will turn into the infected state. This can be an issue inapplication such as machine learning where the order of in-fection can be important information . Finally, the LT modelis not very flexible. It is both theoretically and practicallychallenging if one plans to combine the LT model and othersimple contagion model to model some complicated hybridspreading processes.To overcome these limitations, we consider a generalizedlinear threshold (GLT) model for the continuous-time stochas-tic spreading process that can be efficiently implemented bythe Gillespie algorithm . The evolutionary time in the newmodel has a physical meaning, which is associated with the a r X i v : . [ phy s i c s . s o c - ph ] A ug generalized linear threshold model 2rate of the underlying stochastic process. We find that com-pared with the GLT model, the traditional LT model tends tounderestimate the spreading speed. The order of nodes beinginfected is properly defined in the GLT model, allowing us tobetter generate synthetic spreading node sequence to modelthe spreading in real systems. Finally, the GLT model iscompatible with the susceptible-infected (SI) or susceptible-infected-recovered (SIR) model , because they are definedunder the same mathematical framework. One can easily builda hybrid spreading by combing both simple and complex con-tagion, or adding the recovery process into the complex con-tagion. The remainder of the paper is structured as follows.We first give a brief description of the classical LT model withboth the synchronous and asynchronous updating rules. Wethen propose the GLT model and show how to model it effi-ciently with the Gillespie algorithm. To further shed light onthis model, we compare the spreading results from the GLTand LT model. Finally, we show how the GLT model can becombined with other spreading models. II. RESULTSA. The linear threshold (LT) model
The LT model was first introduced in the field of socialscience to analyze the effects of social reinforcement by as-suming that each adoption requires a certain fraction of expo-sures. The community of network science may be more fa-miliar with the work by Duncan Watts where the LT modelis used to study the condition for global spreading. The samemodel was also applied in the community of computer scienceto find the optimal initiator set that maximizes the spreadingoutcome . In the LT model, each node is in one of the twopossible states: 0 (inactive, susceptible, etc. ) or 1 (active, in-fected, etc. ). A node i in the network can switch only fromstate 0 to state 1. The transition probability depends on thefraction of its neighbors that are on state 1, denoted by φ i , as p ( φ i ) = (cid:40) φ i < φ ∗ i φ i ≥ φ ∗ i , (1)where φ ∗ i is the threshold value of node i , which can be chosenfrom a probability distribution or stay fixed for all nodes.Note that there are other variations for the choice of threshold,such as the number of contacts or the number of infectedneighbors . In this paper, we adopt the model by Watts thatuses the fraction of infected neighbors.The evolution of the system is characterized by discrete-time steps in the LT model. At each time step, we go throughthe network and calculate the transition probability of eachnode according to Eq.(1). All nodes that can change the stateare updated synchronously in that time step. The process is re-peated until no more nodes can change the state. The time stepthat characterizes the system evolution, however, is never ex-plicitly defined. This may be because that initial studies that proposed the model mainly focused on the outcome of thespreading, which does not depend on the choice of time step or how the system actually evolves with time. Nevertheless,the time step needs a proper definition when the spreading dy-namics are concerned. Indeed, while the discrete-time step isused in both the LT and SI model, they are inherently differ-ent with distinct physical meanings. In the SI model, the timestep is associated with the probability that the disease is trans-ferred from one node to another, or the chance that a nodegets infected from an infected neighbor. If the time step isequivalent to a longer period of real time, the infection proba-bility would be tuned larger, which eventually gives the samespreading dynamics. As an example, the infection probabilityin disease spreading would be different if the time step refersto an hour or a day. In the LT model, however, the time step isassociated with a node’s status updating, which is independentof the transmission probability. Its physical meaning, relatedto why every node updates its status within one time step, isnot clearly interpreted.Another issue is the order of the infection. Under the syn-chronous updating rule, all nodes satisfying the threshold con-dition change the state together in one time step. Consideringthe case that node i changes the state from 0 to 1, which makesits neighboring node j reach the threshold. While the transi-tion condition is satisfied, node j can not change the state inthat time step. In other words, node j ’s state is frozen till allnodes in the same batch of node i complete the transition. Interms of the infection order, node j always ranks behind them.Note that the infection order is important in tasks such astracking the spreading source or learning the embedding ofthe underlying network . The simplification of LT modelmay limit its application in generating the synthetic spreadingnode sequence in real systems. A simple fix of this issue isto use the asynchronous updating rule. One option is that ateach time step, we randomly pick only one node from thosewhose threshold is reached and update the node’s state. Inthis way, the spreading order would be more realistic. How-ever, the spreading dynamics would become unrealistic as thenumber of infected nodes increases linearly with time steps.An alternative option is to randomly pick an arbitrary node ateach time step regardless of its threshold condition and up-date its states according to its p ( φ i ) . This actually becomesa Monte Carlo simulation . But the computational com-plexity raised to O ( N ) where N is the number of nodes ina network. More importantly, even though the asynchronousupdating rule can fix the order, it is very difficult to modela system with individual differences. For example, if we as-sume that some nodes are more active and would change thestates faster than others, it would be very difficult to imple-ment this feature in the model.Finally, the LT model is not very flexible. This is partiallyrelated with the vague definition of the discrete-time step. Ifwe want to model a system with both simple and complexcontagion, we need to define two types of time steps. Onetype of time step is for the deterministic infection in the LTmodel and the other for the probabilistic infection in the SImodel. The conversion between the two types of time stepcan be an interesting interplay, which, however, lacks a properdefinition and brings challenges for theoretical interpretation.Because the node status is updated synchronous at the end generalized linear threshold model 3of LT time step, the spreading curve will not be smooth butcontaining multiple bursts separated by fixed time intervals.We also need to propose a rule to decide which action shouldoccur first when the two types of time step coincides. All thesedifficulties increase when more dynamics are involved, suchas adding a recovery process to have the susceptible?infected-susceptible (SIS) or SIR model in the system. Therefore, itis challenging to apply the traditional LT model for complexspreading process. B. A generalized linear threshold model and its stochasticsimulation
To cope with the issues mentioned, we consider a sim-ple variation of the original LT model and generalize it tocontinuous-time stochastic process. In the generalized linearthreshold (GLT) model, a node i has a certain rate to transferfrom state 0 to state 1, which is given by β i ( φ i ) = (cid:40) φ i < φ ∗ i k i if φ i ≥ φ ∗ i . (2)Eq.(2) is similar to Eq.(1), both capturing a threshold dy-namic. When φ i is below the threshold, the transition (or in-fection) can not occur. When φ i is above the threshold, thetransition will occur in certain. The extra information givenby Eq.(2) is the rate k i , which controls the speed of the transi-tion and to what extent node i would be infected ahead of othernodes. By assigning different k i value to different nodes, theindividual differences on the transition are well characterized.To efficiently simulate the GLT model, we apply the Gille-spie algorithm . It is an efficient simulation method forthe stochastic process and was heavily used to investigate theinteractions of molecules in chemical systems or the cel-lular growth and division in biological systems . It canalso be used to simulate the epidemic spreading such as SIand SIR model . Indeed, though it is not explicitly spec-ified, when we use a rate k to quantify a dynamic process, weimply that it is a Poisson process with a rate k . The inter-eventtime or waiting time τ is random and follows an exponentialdistribution with a rate k . This property can be generalized tocases when multiple Poisson processes coexist. Assume thatthere are N nodes in the network, each has a transition rate β i . The inter-event time τ for the occurrence of next transitionfollows an exponential distribution with rate ˜ β = ∑ Ni = β i . Inpractice, τ can be efficiently calculated from a random num-ber r uniformed picked from the interval (0,1) as τ = − ln r ˜ β . (3)The probability that the transition takes place on node j lin-early depends on its transition rate as p = β j ˜ β . (4) Algorithm 1
The generalized linear threshold model based onthe Gillespie algorithm
Input:
Network G , infection rate β , threshold φ ∗ , initial seeds ρ Output:
Time series list T, susceptible number list S, infected num-ber list I function GLT ( G , β , φ ∗ , ρ ) T , S , I ← [ ] , [ | G | − ρ ] , [ ρ ] where the | G | is the number ofnodes nodes ← nodes in the G in f ected _ nodes ← random.sample( nodes , ρ ) risk _ nodes ← the susceptible neighbors of the in f ected _ nodes susceptible _ nodes ← [] for u in nodes do in f ected _ rate [ u ] ← β end for τ ← for n in risk _ nodes do num [ n ] ← the number of infected neighbors of n degree [ n ] ← the number of neighbors of n if num [ n ] degree [ n ] ≥ φ ∗ then add n into susceptible _ nodes remove n from risk _ nodes end if end for total _ rate ← ∑ n ∈ susceptible _ nodes in f ected _ rate [ n ] while total _ rate > do n = random . choice ( susceptible _ nodes ) (cid:46) If each node has different rate β i in a network, please see belowfor an optimization. remove n from susceptible _ nodes add n into in f ected _ nodes τ ← τ − ln ( random . uni f orm ( . , . )) total _ rate Update T , S , I susceptible _ neighbors ← the susceptible neighbors ofthe n for u in susceptible _ neighbors do if u not in susceptible _ nodes then risk _ nodes ← u end if end for for n in risk _ nodes do num [ n ] ← the number of infected neighbors of n degree [ n ] ← the number of neighbors of n if num [ n ] degree [ n ] ≥ φ ∗ then add n into susceptible _ nodes remove n from risk _ nodes end if end for total _ rate ← ∑ n ∈ susceptible _ nodes in f ected _ rate [ n ] end while return T , S , I end function The Gillespie algorithm takes this property of the stochasticprocess. At each simulation step, it decides, in a randommanner, which event would occur and when it would occur.The procedure can be summarized as follows:1. At the time t , find all events that may occur (with a positiverate) and get the sum of the rate ˜ β .2. Generate a random variable τ from an exponential distri- generalized linear threshold model 4bution with rate ˜ β .3. Randomly draw an event according to the probability of p in Eq.(4)4. Update the system according to the event drawn. Updatethe time from t to t + τ .5. Repeat from step 1.To illustrate the simulation of the GLT model, we providethe pseudocode in Algorithm 1. At each simulation step, weneed to determine which action would occur from the rates ofall actions. When the k i is the same for all nodes, or thereare only a few choice of k i values, we can do a random selec-tion of actions to simplify this process, which takes only O ( ) complexity. In comparison with the Monte Carlo version ofthe LT model with complexity O ( N ) , the Gillespie al-gorithm significantly reduces the computation cost. When allnodes have different k i values, the Monte Carlo version of theLT model would fail because it assumes that all nodes arepicked to update the states with equal probability . TheGillespie algorithm can handle this situation by deciding theprocess that happens according the rate k i . The selection is atypical fitness proportionate selection, also known as roulette-wheel selection . The complexity is usually O ( N ) becausewe need to calculate β j / ˜ β for every node at each simulationstep. However, using a recently proposed optimization, thecomplexity can be reduced to O ( ) type . Taken togetherwith the N nodes in the system, the complexity to simulatethe whole evolution is roughly O ( N ) . C. The application of the GLT model
To show features of the GLT model, we compare its timeevolution with that of the LT model. Because the transitionrate k i can be any value, we have to first adjust the continuousrate and the discrete-time step to make the continuous-timeand discrete-time model comparable. Unlike SI model wherethe relationship among the rate, the infection probability andthe discrete-time step is known , there is no method yet tohandle the parameter conversion in the threshold model. Tocope with this issue, we consider re-scaling the spreading timewindow. We choose average cascade size S = .
98 as our ref-erence point and record the time (either discrete-time steps orcontinuous time) takes from the beginning of the spreadingto S = .
98 as the time window. The discrete-time steps andthe continuous time are then re-scaled such that the spreadingtime window is the same. We consider S = .
98 instead of S = as the baseline, where an arbitrary node is pickedat random at each time step regardless of its threshold condi-tion. The state of the node is then updated according to Eq.(1).This baseline is compared with the spreading generated by the GLT model and the traditional LT model with synchronousupdating rule. The dynamics by the GLT model matches withthe baseline, but the dynamics of the LT model is different,where the infected size grows slower than both the GLT modeland the baseline (Fig. 1(b)). This indicates that the LT modelunderestimates the speed of spreading, supporting our initialstatement of the LT model’s limitations. (a) (b) S t
SI(continuous-time)
SI(discrete-time)
S t LT GLT
LT(Monte Carlo)
FIG. 1: Time evolution of the average cascade size S (usually known as infectedfraction in epidemic researches) on Erdös-Rényi network with the size N = (cid:104) k (cid:105) =
4. (a) The continuous-time and discrete-time SI model. Thecontinuous-time SI model is implemented by the Gillespie algorithm in which eachnode has the same infection rate β =
1. In the discrete-time SI model, the infectionprobability is p = .
01 and nodes’ states are updated synchronously. (b) The GLT andLT model in which each node has the same threshold value φ ∗ = .
16. In the GLTmodel, each node has the same rate β =
1. All curves in (a) and (b) are based on theaverage over 10 runs of simulation. In each run, we choose 1 same node as initiator. The LT model underestimates the spreading dynamics dueto its coarse-grained description of the time evolution. Thespreading speed is not instantaneously updated according tothe number of nodes satisfying the threshold condition. Asan example, let us assume there are 10 nodes satisfying thethreshold in the LT model. Naturally, these 10 nodes will beinfected in the next time step. If we assume one time stepcorresponds to a continuous time T , the infection of each ofthe 10 nodes takes T /
10 time on average. In the GLT model,if there are 10 nodes satisfying the threshold, the infection rateof the first node will be 10 × k (assuming k i = k for all nodes).The average waiting time to infect the first node is T /
10 if k is set as k = / T , which is the same as that in the LT model.However, the infection of one node will activate more nodesduring the spreading. Therefore, after the infection of thisnode, there will be more than 10 nodes in the system satisfyingthe threshold. The rate for the next infection to occur is greaterthan 10 × k and the average waiting time takes less than T / k i of the node selected, giving it amuch higher priority to change the state, its rank is still very generalized linear threshold model 5 P rank (a) (b)
P rank (c)
P rank
FIG. 2: The distribution of the rank of an infected node in spreading sequences on an Erdös-Rényi network with N = (cid:104) k (cid:105) =
4. The threshold valueis φ ∗ = .
16. We fix the node in all models and check when it will be infected. (a) The LT model, (b) the GLT model in which every node has the same infection rate β =
1, (c) theGLT model in which the node selected has the infection rate β =
10 and other nodes have infection rate β =
1. The results are based on 10 runs of simulation. We select 1 node asthe initiator and fix it in each run. S t
Hybrid
GLT
S t
SI(continuous-time) Hybrid GLT (a) (b)
FIG. 3: Time evolution of the average cascade size S for among the continuous-time SImodel (the blue diamond), the hybrid model (the black square) and the GLT model (thered circle) on the Erdös-Rényi network with N = (cid:104) k (cid:105) =
4. (a) The threshold value φ ∗ = .
25 and every node has the same infection rate β = φ ∗ = .
16 and every nodehas the same infection rate β =
4. The fraction of nodes that follow the GLT model andthe SI model is 1:1 in the hybrid model. We select 1 node as the initiator and fix it ineach run. The curve is based on 10 runs of simulation. randomly distributed (Fig. 2(c)). Therefore, when using theLT model to generate synthetic spreading data, we may under-estimate the complexity of spreading brought by the inherentrandomness.Finally, because the GLT model is based on thecontinuous-time stochastic process, it is compatible with othercontinuous-time stochastic processes. We only need to addmore reactions in the queue when multiple processes coexist.As an example, we apply the GLT model as a tool to simulatethe hybrid spreading process. There are works in epidemicsassuming that the disease infection rate can be different underdifferent conditions . Hence there will be two infectionrates in the system. This feature is hard to implement in theLT model but can be easily added by the GLT model. Here weconsider a more interesting situation. In the threshold spread-ing, to have a global cascade triggered by a single initiator, thethreshold value φ ∗ needs to be small ( φ ∗ ≤ / (cid:104) k (cid:105) ) . Thisbrings questions on how a social spreading, which is usuallybelieved to be the complex contagion, could occur since thethreshold of a real social system may not be that small. Oneexplanation is that there can be multiple initiators . Al-ternatively, we may also assume that both simple and complexcontagion are active . Here we analyze co-evolutionary con-tagion which is a hybrid model combining the SI and GLTmodel. In the model, we assume that there are two types of nodes, one evolves according to the SI model and the otherto the GLT model. The result shows that complex contagionalone can not take place, but the global cascade could occurwhen simple contagion co-exist (Fig. 3(a)). The simulationresult demonstrates the model’s capability in combining otherspreading mechanisms.We further find that the dynamic of the hybrid model alwayslocates between the SI model and the GLT model whatever theinfected rate β is when the threshold φ ∗ is smaller than the1 / (cid:104) k (cid:105) (Fig. 3(b)). This shows that the simple contagion accu-mulates the critical mass for the complex contagion, which al-lows other nodes to be infected earlier than when the complexcontagion alone takes place. It also implies that the simpleand complex contagion may demonstrate identical spreadingdynamics under certain parameters. III. CONCLUSION AND DISCUSSION
To summarize, we propose a GLT model for thecontinuous-time complex contagion process. It overcomesthe limitations of the LT model in studying the system evo-lution. The GLT model can be efficiently implemented by theGillespie Algorithm. We find that the traditional LT modeltends to underestimate the speed of spreading and the random-ness of the spreading sequence, compared with cases whenthe dynamics are more properly defined. We show that theGLT model can be very efficient to simulate more compli-cated spreading. Taken together, the GLT model we proposedcan be a useful tool to study complex contagion, especiallywhen the time evolution of the spreading is the focus. Ourresult not only sheds light on a series of important questionsthat were not emphasized previously, but also brings insightinto the modeling process of real spreading data. Previous re-search shows that real spreading process is usually more com-plex. There are examples of combining multiple spreadingmechanisms . More importantly, the recovery process isincluded in real spreading . This urges us to combine the lin-ear threshold model with SIR model, which is readily doablewith the GLT model proposed in this paper. These more so-phisticated models together with real spreading data would generalized linear threshold model 6definitely help us understand the underlying patterns in infor-mation spreading.Our model also has some shortcomings, the events basedon the spreading dynamics are described as a Poisson ran-dom process for the GLT model, which may not deal with thereal information spreading well. In the GLT model, whethera node becomes active depends only on the number of cur-rent exposures from its neighbors, without memory effects.The previous records, however, could impact the informationspreading in current time in the real data . Miller stud-ied the equivalence between the generalized epidemic processand the LT model through the percolation theory. They findthat the generalized epidemic process is completely equiva-lent to the LT model. Using the GLT model, we can extendthe analyses to the continuous-time dynamics. In the future,we can study the equivalence based on the temporal dynamicsbetween the GLT model and the simple contagion. In addi-tion, we can study under what circumstances the two modelscan be distinguished, and factors that make the two modelsequivalent. ACKNOWLEDGMENTS
This research is supported by the Chongqing Graduate Re-search and Innovation Project (Grant No. CYB18080), andthe S-Tech Internet Communication Academic Support Plan.
DATA AVAILABILITY STATEMENT
Data sharing is not applicable to this article as data weregenerated by the theoretical model.
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