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Stochastic Modeling of the Decay Dynamics of

Online Social Networks

Mohammed Abufouda and Katharina A. Zweig

Abstract The dynamics of online social networks (OSNs) involves a complicated mixture of growth

and decay. In the last decade, many online social networks, like MySpace and Orkut, suﬀered from

decay until they were too small to sustain themselves. Thus, understanding this decay process

is crucial for many scenarios that include: (1) Engineering a resilient network, (2) Accelerating

the disruption of malicious network structures, and (3) Predicting users’ leave dynamics. In this

work we are interested in modeling and understanding the decay dynamics in OSNs to handle the

aforementioned three scenarios. Here, we present a probabilistic model that captures the dynamics

of the social decay due to the inactivity of the members in a social network. The model is proved

mathematically to have submodularity property. We provide preliminary results and analyse some

properties of real networks under decay process and compare it to the model’s results. The results

show, at the macro level of the networks, that there is a match between the properties of the

decaying real networks and the model.

1 Introduction

Today’s online social networks represent a main source of communication and information exchange

among people all over the world. Many online social networks have proven their usefulness, like Face-

book, Twitter, and Linkedin, in connecting people and facilitating an exquisite new medium for

sharing news, forming groups of people of the same interests, and eliciting knowledge. The growth

of these networks in terms of user activity shows that these online social networks have become a

vital part in today’s human activities. One well-studied aspect of online social networks dynamics

is the growth dynamics of a network. The work by Barab´asi et al. [6] presented a simple model

for understanding the growth dynamics of a network, namely the Preferential Attachment Model

(PAM), which is a rich-get-richer-model. Jin et al. [15] noticed that the model by Barab´asi et al. [6]

and other similar models, like the work by Dorogovtsev et al. [11] for modeling the growth of ran-

Mohammed Abufouda and Katharina A. Zweig

Computer Science Department, University of Kaiserslautern,

Gottlieb-Daimler-Str. 48, 67663, Kaiserslautern, Germany, e-mail: {abufouda,zweig}@cs.uni-kl.de

1

2 Mohammed Abufouda and Katharina A. Zweig

dom networks, are not suitable to understand the growth dynamics of social networks. Thus, they

provided a model that considered the specialty of social networks without a power law distribution

and with large clustering coeﬃcient [15]. With the availability of the online datasets, Newman [25]

studied empirically the growth of social networks using the scientiﬁc collaboration networks against

the PAM model [6]. Bala et al. [5] provided a non-cooperative game based model for the network

formation. Later, Jackson [14] surveyed the models and methods that were used to capture the net-

work formation process and compared them in terms of stability and eﬃciency. Leskovec et al. [21]

ﬁrst showed on dynamic network data, that networks densify over time and that their diameter is

shrinking. They also provided another growth dynamics model that was able to produce networks

with these properties. The previous work and the availability of rich datasets pushed the research

to an in-depth investigation of the properties of the networks over time. Kumar et al. [20] studied

the growth of a large social network in terms of network component analysis, Kossinets et al. [18]

studied the tie formation process within the social networks that is aﬀected by internal and exter-

nal factors, and Capocci et al. [9] studied the statistical properties of the growth characteristics

of Wikipedia collaboration social networks. Likewise, Backstrom et al. [4] studied empirically how

groups are formed and evolved over time in MySpace social networks , while Mislove et al. [23]

provided a study for the growth of Flicker social network. Even though there are many success-

ful social networks, the evolution of a social network also incorporates decay. In the last decade,

some of the online social networks were closed after a huge loss or inactivity of their members.

Online social networks, like Friendsfeed, Friendster, MySpace, Orkut, and many websites of the

Stack Exchange platform, are now out of service, despite the fact that some of them, e.g., Orkut

and Myspace, showed a tremendous growth [2] just a decade ago. The decay of these networks poses

many questions about the reasons behind their fall down. Garcia et al. [12] and Chhabra et al. [10]

studied the static properties of Friendster and MySpace, respectively, in order to understand the

network-related properties of these networks as an example of a decayed network. Recent studies

by Malliaros et al. [22] and Bhawalkar et al. [8] provided theoretical models for understanding the

social engagement in online social networks with a potential to predict social inactivity. Torkjazi

et al [28] provided an analysis of Myspace online social network and examined the activity and

inactivity of its users with some insights about the reasons behind the fall of MySpace. Similarly,

Ribeiro [26] studied activity and inactivity of the users by providing a model that uses the number

of daily active users as a proxy of the dynamics in the membership based websites. Kairam et

al. [16] provided machine learning prediction models to predict community longevity: how long a

community in an online social network will survive. Another related work done by Asur et al. [3]

discussed the persistence and decay of Twitter tweets. While investigating the reasons behind the

inactivity of members of an online social networks is not in the scope of this work, some recent

studies proposed some answers [27, 17], suggesting that the main reason behind this decay is the

inactivity of the members of the online social networks.

Building a sound understanding of the decay dynamics of networks requires not only studying the

static properties of these networks, but also requires investigating their dynamics and properties

over time, and this is what we are interested in here. As a scenario, we consider the Stack Exchange

websites that were closed after some period of time due to the lack of enough activity required to

keep the website alive. The closed websites are an example of the social network decay, where we

model the members of a website as the nodes of the network and an edge exists between any two

nodes if they post, comment, or answer to the same question in the websites.

While we cannot answer why a person starts losing interest in a social network, we can try to

analyze and model the eﬀect of this behavior on other people. Such a model might in turn hint at

Stochastic Modeling of the Decay Dynamics of Online Social Networks 3

the causes of social decay or at least explain some part of it.

In this work, we provide a probabilistic model for understanding the social decay phenomenon in

online social networks. The model presented here can provide insights regarding the eﬀect of node

leave on the neighborhood nodes. Our contribution in this work is split the following: (1) A longi-

tudinal network analysis of the stack exchange sites showing their decay. (2) A probabilistic model

for social network decay which is a step by step mechanistic model for a node leave and the eﬀect of

its leave. (3) Theoretical proof of the submodularity of the model that leads to viable optimization,

e.g., determining the minimal set of nodes to leave the network for accelerating/decelerating the

decay process. Being submodular renders the maximization problem of the model to be viable.

2 Model and notations

A network G= (V, E) is a tuple of two sets Vand E, where Vis the set of nodes and Eis the set of

edges such that an undirected edge eis deﬁned as e={u, v} ∈ E, where u, v ∈V. As we consider a

dynamic system, the notation Gtis a network at time t. We assume that every node w∈Vhas an

initial Leave Probability πt=0

wwhich denotes the probability of node wleaving the network at time

1, and generally at t+ 1. If a node wdid not leave at t+ 1, i.e., w∈V(Gt+1), then its current leave

probability, πt

w, will be increased depending on its neighbors who left at t−1. The tie strength at

time t−1, representing some possibly dynamic measure of closeness of a relationship, is denoted by

δt−1

v,w and assumed to be ∈(0,1]. The details of this process are described in the following sections.

Deﬁnition 1. A dynamic network Gis called a ”Decaying Network” if |E(G)t−1| ≥ |E(G)t|,

|V(G)t−1|≥|V(G)t|, and V(G)t⊆V(G)t−1,∀t > 0.

t = 0 t = 1 t = 2 t = 3

t = 4 t = 5 t = 6

Fig. 1: An illustration of the model. The color of the nodes represents how likely a node will leave in the future, where

white nodes are very unlikely to leave and the level of grayness correlates with the probability to leave. Whenever a

node leaves the network it is marked as black, all its edges are removed, and all of its neighbors get aﬀected by its

leave by increasing their leave probability. The dotted edges are the removed edges.

We assume the model starts with a Decaying Network, i.e, no further nodes or edges are added to

the network. The main idea of the model is shown in Figure 1.

4 Mohammed Abufouda and Katharina A. Zweig

2.1 Probability Gain

At any point of time twhere t>0, the node’s leave probability changes from πt−1

wto πt

w, by adding

Probability Gain ∆πt

w, that never exceeds the value of 1. Thus, a node wwill leave at time t+ 1

with probability πt+1

wsuch that:

πt+1

w=min{1, πt−1

w+∆πt

w}(1)

If a node wdid not leave the network at time t, then we have two sets: Γt−1

wand Γt−1

w, which are

the sets of w’s neighbors who left and did not leave the network at t−1, respectively.

2.1.1 Probability gain due to one node leave:

We ﬁrst deﬁne the probability gain due to the leave of a single neighbor vof the node wat time

point t−1, and then generalize it to w’s neighbors that left the network: Γt−1

w. Now, the probability

gain that a node wwill get at t+ 1 due to the leave of its neighbor node vat t−1 is deﬁned as:

∆πt+1

w(v)=1−(1 −πt−1

v)(1 −δt−1

v,w ) (2)

where the edge e= (v, w)∈E(G)t−2and e= (v , w)/∈E(G)t−1as v∈Γt−1

wand w∈V(G)t−1.

Thus, the total probability gain produced by the leave of node vto all of its neighbors which did

not leave, see Figure 2 for an illustration, is given by:

∆πt(v) = X

w∈Γt−1

v

1−(1 −πt−1

v)(1 −δt−1

v,w ) (3)

t−2t−1t

Fig. 2: This ﬁgure shows how a node vaﬀects all of its neighbors when it leaves. At t−2, the node vhas a leave probability

πt−2

vwhich was gained by v’s initial leave probability π0

vand possible probability gains caused earlier by leaving neighbors,

i.e., πt−2

v=π0

v+Pt=t−3

t=1 ∆πt

v. At time t−1, the node vleaves the network aﬀecting its neighbors by increasing the leave

probability of nodes 1,2,4,5. Here we assume that the tie strength between vand the nodes 1,2,5 is greater than the tie

strength between vand 4. That is why the nodes 1,2,5 gain more leave probability than node 4, which is represented by a

darker color of nodes 1,2,5.

Stochastic Modeling of the Decay Dynamics of Online Social Networks 5

t−2t−1t

Fig. 3: This ﬁgure shows how a node wis aﬀected by the leave of its neighbors. At t−2, the nodes 1,4 have leave

probabilities πt−2

1and πt−2

4, respectively, which were gained by the nodes’ initial leave probabilities π0

1and π0

4and possible

earlier probability gains. At time t−1, the nodes 1,4 leaves the network aﬀecting their neighbors, here we are interested

in the node w. The leave of nodes 1,4 left node wwith an increased leave probability at t. Note that nodes 2,3,5,6 are

aﬀected also by the leave of 1,4, but for simplicity and for visualization traceability we concentrated on node w.

2.1.2 Probability gain due to multiple nodes leave:

We now generalize the probability gain induced by the leave of a single node to capture the impact

of all neighbors that left, i.e., Γt−1

w.

∆πt

w= 1 −[ (1 −ξt−1

w)

| {z }

Assures leave

(Y

u∈Γt−1

w

(1 −πt−1

u))

| {z }

Leave probabilities effect

(Y

u∈Γt−1

w

(1 −δt−1

u,w ))

| {z }

Tie strength effect

]

= 1 −[(1 −ξt−1

w)( Y

u∈Γt−1

w

(1 −πt−1

u)(1 −δt−1

u,w ))]

(4)

where ξt−1

w=|Γt−1

w|

|Γt−1

w|and the quantity 1−ξt−1

wassures that when all of the neighbors of the node w

leaves, then the node wwill (be forced to) leave too as it will be disconnected. Thus, Equation 1

becomes:

πt

w=min{1, πt−1

w+ 1 −[(1 −ξt−1

w)( Y

u∈Γt−1

w

(1 −πt−1

u)(1 −δt−1

u,w ))]}(5)

3 Monotonicity and submodularity

In this section, we show the monotonicity and submodularity properties of the model’s equations 1.

Deﬁnition 2. Let f: 2V→R≥0, where R≥0={x∈R|x≥0}, be an arbitrary function that

maps the subsets Sand Tto a non-negative real value, where S⊆T⊂V. Then, the function f

is submodular [19] if it satisﬁes the following inequality: f(S∪ {v})−f(S)≥f(T∪ {v})−f(T),

where v∈V\T.

Lemma 1 (Order preserving of the probability gain sum). Let πt={π1, π2,· · · , πn}, where

πi∈πtand πi∈(0,1]. Then we have: P

πi∈πt

πi≤P

πi∈πt+1

πiwhere πt⊆πt+1, and the sets πtand

πt+1 are deﬁned like above.

1Detailed proofs are provided in an earlier technical paper [1].

6 Mohammed Abufouda and Katharina A. Zweig

Lemma 2 (Order preserving of the probability gain product). Let πt={π1, π2,· · · , πn},

where πi∈πtand πi∈(0,1]. Then we have: Q

πi∈πt

πi≥Q

πi∈πt+1

πiwhere πt⊆πt+1, and the sets πt

and πt+1 are deﬁned like above.

Theorem 1. The leave probability gain function, Equation 3, is submodular.

The interpretation of the theorem is that, the more friends a node vhad before leaving, the higher

its total induced leave probability gain.

Theorem 2. The leave probability gain function, Equation 4, is monotone, i.e., for a node wwe

have πt

w≤πt+1

wif the node wdid not leave the network at t+ 1.

Theorem 3. The leave probability gain function, Equation 4, is submodular.

The theorem state that the more of your friends leave, the less important the others become. Sub-

modulariy entails an interesting properties: the minimization problem of submodular function can

be performed in polynomial time [13], and the maximization problem of the submodular function,

which is NP-Hard problem, can be approximated within a factor of α= (1 −1/e) using a greedy

algorithm [24].

4 Results

In this section, we provide the analysis of the decaying stack exchange websites and the results

of the model. Figure 4a shows the distribution of the number of user comments for alive and

decayed websites. The ﬁgure shows that the decayed websites clearly have diﬀerent distribution

characteristics with a low mean and low standard deviation. A similar behavior is found in Figure 4b

and Figure 4c that represents the distribution of users’ total received Reputation and Upvotes,

respectively. These two properties reﬂect the level of knowledge and experience that the members

of a website have. For the decayed websites, it is clear that, on average, the members have much less

reputation and upvotes than those in the alive websites. The three ﬁgures, Figures 4a, 4b, and 4c

show that there is less social activity in the decayed websites, which may be used as an indication

for studying the future of the alive websites. However, understanding the decay dynamics of the

decayed websites requires a deeper investigation and modeling for the nature of the interaction

among the members. Our approach to better understand what happens during the decay process is

to make a network representation of the members’ interactions, like comments, upvotes, and posts,

as networks. Then, we build a network based model for modeling the decay process.

Stochastic Modeling of the Decay Dynamics of Online Social Networks 7

(a) Comments per user. (b) Reputation per user. (c) Upvotes per user.

Fig. 4: (Color Online) The characteristics of the interaction decay in the decayed and alive websites of the Stack

Exchange websites. The ﬁgures show the probability distributions of diﬀerent types of interactions in these websites.

Markers with bold boarders are decayed websites, µis the mean, and σis the standard deviation. From the ﬁgures

it is clear that the decayed networks have diﬀerent distribution properties from the other alive networks.

Algorithm 1 depicts the steps we followed in our experiments. Line 4 initializes the initial leave

proability π0

v, which is a design decision and we selected values from 0.0005 to 0.045 with an 0.0005

increase step. For each of these values, the model runs and simulates Equation 4. The update step

in line 13 simulates Equation 5. The result of the algorithm is a set of graphs that are used for the

analysis. The output of this algorithm results in a large number of graphs. For example in the case

of the Startup Business website we have analyzed more than 200kgraphs with 250 runs for each

probability to get more conﬁdence of the results. The tie strength was a normalized edge weight

where the weight is the frequency of the interaction between two nodes.

(a) (b)

Node coreness over time

Node degrees over time

(c)

Network density over time

Fig. 5: (Color Online) Macro properties of the real networks under decay for the Startup business site. Figures 5.a, 5.b, 5.c

show the degrees of the nodes, the node coreness, and the network density over time.

In Figure 5 we show the macro properties of the real networks of the Startup Buisness website over

time. The network evolution shows a clear decay that is represented as a decrease in the number

of the nodes. This decrease was associated with a decrease in the average degrees of the nodes

over time and also with a decrease of the node’s coreness [7]. Another macro measure we used is

the network density. Figure 5c shows an increase in the density over time. This increase is due to

early leave of the nodes with less degrees, i.e., the nodes that are part of dense subgraphs seem

to leave the network late. Now, we will show the results of the model simulation. Figure 6a shows

the number of components in the network over simulation for diﬀerent values of π0

v. The number

of components start to increase to a maximum value before it start to decrease. The reason is that

at the beginning the model starts with a one-connected component graph and after each step some

nodes are removed due to the leave probability. The leave of some nodes results in a disconnected

graph with more components. The number of these disconnected components increases until these

disconnected components are composed of only triples or simple edges. As a result, a node that leaves

8 Mohammed Abufouda and Katharina A. Zweig

from these triples or from these edges will not increase the number of the components anymore.

Figure 6b and Figure 6c show a similar behaviour for the average degree and the average coreness

over time, respectively. The more nodes are being removed from the network, the less edges remain

and thus the average degree and the average coreness decrease uniformly over time. This behavior

of the model is similar to the real data presented in Figure 5. The last global measure that we use

is the network density as shown in Figure 6d. The density of the simulated networks increases over

time for the same reason stated for the real networks in Figure 5. These results show that the model

provides a real-like behaviour of the networks under decay.

Algorithm 1 Model simulation

1: Input: Graph G0

2: Output: Graphs= {G0, G1,··· , Gn−1}where Gnis an empty graph

3: for all v∈V(G0)do

4: initialize π0

v

5: t= 0, Gt=G0, Graphs.add(Gt)

6: while Gtis not empty do

7: LeftNodest=∅

8: t=t+ 1

9: for v∈V(Gt)do

10: if Leave(v,πt

v) is T rue then

11: LeftNodest.Add(v)

12: for all u /∈LeftNodes & Γt−1

u6=∅do

13: update(πt

u,Γt−1

u)

14: remove LeftNodestfrom Gt

15: Graphs.add(Gt)

5 Discussion

There are diﬀerent applications where the model can be utilized. 1. Social network resilience: the

resilience against huge disruptions in social networks is not well-studied. We think that the model

provides a ﬁrst step towards engineering a resilient social network via understanding the decay

dynamics of a network. 2. Leave cascade detection: the leave of one member is not as harmful as

a cascade of leaves for the networks that seek growth. The model captures the dynamics of leave

cascades by observing the leave probabilities of the nodes and their increase. 3. Maximizing the

leave eﬀect: for a network where a dissolving process is required, like criminal social networks, the

model is able to provide a viable disruption maximization (thanks to the submodularity property of

the model) to the network with insights about the inﬂuential members and the eﬀect of the leave.

6 Conclusion

In this work, we presented an empirical analysis of the social decay dynamics of the closed Stack

Exchange websites. The closed websites showed an inactivity, which might have caused their decay.

We model these interactions between the members of these websites as a network that enabled us to

Stochastic Modeling of the Decay Dynamics of Online Social Networks 9

Fig. 6: (Color Online) The results of multiple global measures of the model. Figure 6a, Figure 6b, Figure 6c, and

Figure 6d show the number of components, the average degree, the average coreness, and the density of the network

over time for diﬀerent values of initial leave probability π0

v, respectively. The model started with G0as the input

network and simulates the decay over it.

build a model to understand the decay dynamics. Then, we have presented a model for capturing the

decay dynamics in social networks. The model is a probabilistic model that assumes that the leave

of social network members aﬀects the leave of their neighbors. In this work we have also presented

some mathematical properties and proved them. We proved that the model’s main equations are

submodular, which entails doing optimization of the model in a feasible way. Also, we presented

the macro network properties of real networks under decay and compared these results with the

results of the model simulation. The results of the model and the real networks under decay showed

a similar behavior that supports the potential of the model for diﬀerent usages. In the future, we

will design the optimization algorithms and study the applicability of the model and also provide

more empirical validation of its properties.

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