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Scientific RepoRts | 7:42956 | DOI: 10.1038/srep42956

www.nature.com/scientificreports

Coupling eect of nodes popularity

and similarity on social network

persistence

Xiaogang Jin

, Cheng Jin

, Jiaxuan Huang

& Yong Min

Network robustness represents the ability of networks to withstand failures and perturbations. In

social networks, maintenance of individual activities, also called persistence, is signicant towards

understanding robustness. Previous works usually consider persistence on pre-generated network

structures; while in social networks, the network structure is growing with the cascading inactivity of

existed individuals. Here, we address this challenge through analysis for nodes under a coevolution

model, which characterizes individual activity changes under three network growth modes: following

the descending order of nodes’ popularity, similarity or uniform random. We show that when nodes

possess high spontaneous activities, a popularity-rst growth mode obtains highly persistent networks;

otherwise, with low spontaneous activities, a similarity-rst mode does better. Moreover, a compound

growth mode, with the consecutive joining of similar nodes in a short period and mixing a few high

popularity nodes, obtains the highest persistence. Therefore, nodes similarity is essential for persistent

social networks, while properly coupling popularity with similarity further optimizes the persistence.

This demonstrates the evolution of nodes activity not only depends on network topology, but also their

connective typology.

Network robustness is one of the core issues in network science

1–4

. Early research mainly focuses on static robust-

ness, i.e. the resilience of network connectivity to random errors or targeted attacks of components like nodes

or edges

5–7

. Furthermore, dynamics robustness or network persistence, which concerns about the ability of a

network to maintain certain states or functions, received major attention

. For example, in high voltage networks,

cascading failure is a common eect, where a single point of failure on a fully loaded or slightly overloaded system

results in a sudden spike across nearly all nodes of the system

9,10

. In food-webs, extinction of a certain tropic spe-

cies may cause a threat to the balance of the food-web, and the persistence of food-webs can be measured as the

fraction of initial species remaining at the end of a perturbation

11–14

. Similarly, in social networks, maintaining

individual activity is a key issue towards the overall welfare of the community

e real social networks possess three particular features, each of which may aect persistence of individual

activity. Firstly, diverse network growth modes. Social networks may grow under various modes like following

celebrities

16,17

, sharing the same hobbies

18,19

or uniformly random ways. Attraction comes from nodes popular-

ity or similarity

, and the heterogeneous node and link types may change network persistence even under the

same network structure

21,22

. Secondly, the cascading eects, i.e. inactivation of a node may cause its neighbors

turn inactive. Concepts like k-core and k-core decomposition are adopted to understand the function of network

structure on cascading eects in social networks

23–25

; besides, models initially describing cascading failures

power grids

7,9,27

or epidemic spreading

28,29

are reformed to understand dynamic cascading processes in social

networks, including rumor cascades

30,31

, inuence maximization

32–34

, viral marketing

35,36

, etc. irdly, unlike

power grids or stable food webs, the total number of nodes in social networks is in growth and the node state is

also changing

37,38

; however, the eect of the coevolution process brings to network persistence remains unclear.

erefore, social network persistence is an interesting and open issue.

Here, based on the three features in social networks, we integrate Papadopoulos’s

network growth model and

k-core

based cascading processes, which can characterize the coevolution of network growth with diverse

modes and the cascading process of node states. To display multiple growth modes of social networks, the

extended model contains four steps (See Fig.1). (1) generate n nodes under polar coordinates each with two

Institute of Articial Intelligence, College of Computer Science, Zhejiang University, Hangzhou, 310027, China.

College of Computer Science, Zhejiang University of Technology, Hangzhou, 310023, China. Correspondence and

requests for materials should be addressed to Y.M. (email: myong@zjut.edu.cn)

received: 04 October 2016

accepted: 17 January 2017

Published: 21 February 2017

OPEN

www.nature.com/scientificreports/

Scientific RepoRts | 7:42956 | DOI: 10.1038/srep42956

properties, the popularity of nodes (polar radius

==…riinln ,1,2,,

) and the similarity with other nodes

(random polar angle

θπ∈ [0,2 )

). (2) Initially, the network is empty and the adding order of nodes is dened,

which corresponds to the descending order of nodes’ popularity or nodes’ similarity, or is uniformly random (in

the following, we call them popularity-rst, similarity-rst and random network growth modes). (3) at time t, a

node v in order t is picked up and joins the network. (4) Node v connects to a subset of existing nodes S, consist-

ing of nodes with hyperbolic distance

θ=++≤Xrrrln(/2)

vs vs vs v

, which co-consider the pair nodes’ pop-

ularity and similarity with the central angle

θ ππθθ=− −−

vs vs

, and v should connect to no more than m

existing nodes that obtain the smallest X

. Here, high popularity stands for small r and high similarity means a

small angle to

θ =

. In this way, we inherit the connection rule introduced by Papadopoulos

, that is connec-

tions link to nodes with small hyperbolic distance, but we split the binding between node popularity and node

join time. e generated network still follows the preferential attachment mechanism, but the network growth

modes can be very exible.

To describe the coevolution of state cascading with the network growth, we introduce a cascading process

inspired from the denition of k-core. When joining in the network, each node has an initial setup of spontaneous

activity, say h, which is node’s ability to maintain active unconditionally; aer the spontaneous time, for each time

step, nodes with at least c active neighbors could keep active. Otherwise, they would turn inactive permanently.

For the demonstration in Fig.1, h is the length of time steps for nodes to remain unconditional active. For the

following results, for simplicity, h is a relative value of a total number of nodes and h ∈ (0%, 100%]. e idea of

active neighbor threshold c dates back to k-core or k-degenerate graph in graph theory. e network evolves until

all nodes are added, and states for each node never changes, and the persistence of complex social networks can

be quantied by the nal active ratio of nodes (FAR). Since active nodes satisfy active neighbor threshold c, FAR

can be seen as a variant measure of k-core in dynamic processes in which k = c.

We analyze the response of the proportion of durable, active nodes with dierent network growth modes, and

further study the coupling eects of popularity and similarity on the promotion of network persistence. e cou-

pling eect in this work means by a combined consideration of nodes popularity and similarity; the network may

obtain an enhanced or dierent persistence. Since the single growth modes (popularity-rst or similarity-rst)

based on one dimension of attraction reveal unique characters, we introduce a weighted score S

for nodes popu-

larity and similarity to combine the two preferential growth modes together to nd whether an optimized mixing

Figure 1. Geometric interpretation of network formation by nodes popularity and similarity, and

evolution of individual activity. e process is shown in polar coordinates. At each time step t, a node

= (r

,θ

) with the active state (red point) is added to the network and can connect to at most m existing nodes

(here m = 2) whose hyperbolic distance from v

is less than r

(i.e. within the green areas). If there are m′ > m

existing nodes within the green area of node v

, then v

preferentially links to the m nearest nodes regarding

hyperbolic distance. e hyperbolic distance between current node v

and an existing node v

θ++

()

rrln /2

ij ij

. An active node can remain in the active state unconditionally within h time steps (we call

this the spontaneous activity) since it was added. Aer the spontaneous activity, the node remains in the active

state on the condition that it has at least c active neighbours (we call this active neighbour thresholds, e.g. node 1

at time steps t = 3 and 4); otherwise, it changes to the inactive state (e.g. blue node 2 at time step t = 4) and

remains in this state indenitely thereaer. e order in which nodes are added can be freely chosen. In this

illustration, we use a demonstration of ve nodes to show the network growths and activity changes under

popularity-rst, similarity-rst and random growth modes. We set spontaneous activity h = 2 and active

neighbour threshold c = 2 both as absolute values here. But the following discussion adopts the relative value

within range [0%, 100%] for simplication, and h is a relative value of total number of nodes, c is a relative value

of m.

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