Influence of Reciprocal links in Social Networks
Yu-Xiao Zhu, Xiao-Guang Zhang, Gui-Quan Sun, Ming Tang, Tao Zhou, Zi-Ke Zhang
aa r X i v : . [ phy s i c s . s o c - ph ] J un epl draft Influence of Reciprocal links in Social Networks
Yu-Xiao Zhu , , Xiao-Guang Zhang , Gui-Quan Sun , , Ming Tang , Tao Zhou and Zi-Ke Zhang
Web Sciences Center, University of Electronic Science and Technology of China, Chengdu 610051, P. R. China Beijing Computational Science Research Center, Beijing 100084, P. R. China Department of Mathematics, North University of China, Taiyuan 030051, P. R. China Institute of Information Economy, Hangzhou Normal University, Hangzhou 310018, P. R. China
PACS – Computer science and technology
PACS – Networks and genealogical trees
PACS – Social and economic systems
Abstract. - In this Letter, we empirically study the influence of reciprocal links, in order to un-derstand its role in affecting the structure and function of directed social networks. Experimentalresults on two representative datesets,
Sina Weibo and
Douban , demonstrate that the reciprocallinks indeed play a more important role than non-reciprocal ones in both spreading informationand maintaining the network robustness. In particular, the information spreading process canbe significantly enhanced by considering the reciprocal effect. In addition, reciprocal links arelargely responsible for the connectivity and efficiency of directed networks. This work may shedsome light on the in-depth understanding and application of the reciprocal effect in directed onlinesocial networks.
Introduction. –
Nowadays, the emergence of socialnetworks and affiliated applications have triggered an in-creasing attention from various disciplines, ranging fromstudying the social interactions and spreading patterns insocial sciences [1,2] to uncovering the underlying structureand dynamics in mathematics and physics [3, 4]. Gen-erally, social networks can be classified into two typicalclasses according to the edge properties: undirected anddirected. Undirected social networks, such as
Flick [5] and
Okut [5], do not allow two users to be connected unless therelation is mutually confirmed, hence, they are normallyregarded as equivalent individuals in graph theory. Com-paratively, directed social networks, such as
Twitter [5]and
Epinions [5], contain both unidirectional and bidi-rectional links, which consequently build up a so-called follower/followee structure [6–8]. An online user is con-sidered as a follower once s/he collects some other usersas friends (followees), and puts close attention to them viaautomatically receiving their real-time information, as wellas online activities [9]. A considerable fraction of thosefollowees would also give positive feedback and add someof their followers with similar interests as online neigh-bors. Subsequently, such intermediate directed structure (a)
Corresponding author: [email protected] property, namely reciprocity [10], facilitates a great deal ofattention from the scientific community. Nowak and Sig-mund discussed that the indirect reciprocity would help inbuilding reputation systems, judging morality and even-tually promote the cooperation level [12] and benefit theevolution of natural selection [11] in both social environ-ment [13, 14] and supply networks [15]. Pereira et al. ex-perimentally discussed that negative reciprocity, becauseof lower cost and less effort, was somehow more favoredthan the positive reciprocity [16]. Moreover, the power ofreciprocity [17] does not only play a vital role in socialeconomic systems [18, 19] and human social organizations[20,21], but also has been found wide applications in char-acterizing the property [22, 23], maintaining the structure[24,25], and uncovering the underlying function of directedsocial networks [26, 27].Typically, the simplest definition of reciprocity, r , canbe quantified as the ratio of the number of bidirectionallinks, L ↔ , to the total number of links L [28, 29], r = L ↔ L . (1)For the extreme cases, r = 0 represents an absolute di-rected network where all links are unidirectional, and r = 1 stands for a complete undirected network wherep-1.-X. Zhu et al. all links are reciprocal. Therefore, the value of r measuresthe probability that two nodes of a given link are mutu-ally connected. However, Garlaschelli and Loffredo [22]argued that Eq. (1) failed to precisely describe the fullnetwork information, For example, the network densityand self-loops can significantly affect the final measure-ment of mutual connections. Alternatively, they proposeda new measure of reciprocity considering the ordering ofdifferent networks according to their actual degree of reci-procity, denoted as ρ = L ↔ L − ¯ a − ¯ a = r − ¯ a − ¯ a , (2)where ¯ a = L/N ( N −
1) measures the ratio of observedlinks to all possible directed links (namely link density).Based on this improved measure, Zlati´c et al. [23] reportedthat the reciprocity of Wikipedia [5] could be very similarto other directed networks, but having a stronger reci-procity than the networks of associations and dictionaryterms, and smaller than that of World Wide Web. Be-sides that, they found that such a measure is quite stablefor different scales of Wikipedia networks, hence is veryimportant for describing the structure and evolution ofwiki-based networks. Bogu˜n´a et al. [30] found that recip-rocal connections played a crucial role in constructing thegiant connected component and possibly affecting the Webnavigability. Futhermore, Serrano et al. [31] provided anin-depth study of the effect of reciprocal links on degree-degree correlations and clustering. They found that recip-rocal links indeed organized the local subgraphs of theWorld Wide Web network by forming start-like struc-tures, as well as cliques and communities, which containedhighly interconnected pages. What’s more, Gorka et al. [32] argued that the reciprocity was largely dependent ondegree-degree correlation, which, consequently could par-tially reveal the underlying hierarchical structure of net-works. Zlati´c and ˇStefanˇci´c [33] discussed the influence ofreciprocity on vertex degree distribution and degree cor-relations. They found that networks driven by recipro-cal mechanisms are significantly different from static net-works.In this Letter, we shall provide a specific empirical studyof the reciprocity influence on the structure and function ofsocial networks. In particular, we apply a widely used epi-demic spreading model [34–36] to observe the effect of reci-procity on information spreading. Numerical results showthat reciprocal links can noticeably enhance the speed ofinformation spreading. In addition, we show how recip-rocal links affect the structure robustness as percolationcatalysts in maintaining the global connectivity by inves-tigating the avalanche of giant components, the networksusceptibility and the network distance [37, 38].
Data and Analysis. –
In this Letter, we con-sider two representative directed social networks: (i)
Sina Weibo [39]: the largest Chinese microblogging web-site, where a user ( follower ) can add others as his/her friends ( followee ) and automatically receive their postsand events. In addition, users can forward, comment orshare their followees’ news on their own post walls; (ii)
Douban [40]: the largest Chinese website for reviewingonline movies, books, and music. Besides users’ generallyproactive contribution,
Douban also provides services viaits recommendation mechanism, which can suggest itemsof users’ potential interests by mining their personalizedpreferences. Similar with
Sina Weibo , users in
Douban canalso build follower-followee relationship with each other.Table 1: Basic statistics of the two observed data sets. N = | V | and M = | E | are the total number of nodes andlinks, respectively, ρ is the network reciprocity denotedby Eq. (2), and S = M/N ( N −
1) denotes the networksparsity.Data sets
N M ρ S
Sina Weibo × − Douban × − −4 −2 k in p ( k i n ) −4 −2 k out p ( k ou t ) −4 −2 k in p ( k i n ) −4 −2 k out p ( k ou t ) DoubanSina Weibo Sina WeiboDouban
Fig. 1: In-degree (left) and out-degree (right) distributionsof the two observed data sets.Consequently, such relationship can be represented by adirected network G ( V, E ), where V is the set of nodes and E is the set of edges. Each node represents a user, and onelink from user i to user j indicates i is followed by j , thatis to say, i is the followee of j , and j is one of i ’s follower .Table 1 summarizes the basic statistics of the observeddatasets. In addition, Fig. 1 shows the out-degree distri-butions which power-law p ( k out ) ∝ k − λout with exponents λ =1.366 and 1.958, for Sina Weibo and
Douban , respec-tively. This common feature suggests that most users areordinary beings who have relative small number of follow-ers and keep only a small fraction of celebrities. Compar-atively, the in-degree distribution of the two datasets doesp-2nfluence of Reciprocal links in Social Networksnot exhibit the same phenomenon. The in-degree distri-bution of
Douban still keep power-law shape with expo-nent 2.387, but
Sina Weibo has a cut-off around k in = 20.One possible reason is that Sina Weibo only allows a cer-tain number of followers for each free account. It mightalso suggest the different mechanisms driving the growthof two sites: information diffusing automatically in mi-croblogging system of
Sina Weibo , comparing with the in-formation filtering by recommendation-related techniquein
Douban . Similar difference between passive and auto-matic patterns was also empirically reported in bipartiteand hypergraph networks [41, 42]. In addition, we furtherinvestigate the average number of common follower andfollowees (see Table 2). Compared to non-reciprocal nodepairs, reciprocal ones tend to have more common follow-ers and followees, which is in accordance with previouswork [27].Table 2: Comparisons of the average number of commonfollowees ( N CI ) and followers ( N CF ) for reciprocal andnon-reciprocal node pairs, respectively.Sina Weibo Douban N CI N CF N CI N CF Reciprocal
Non-reciprocal 0.274 0.295 0.029 0.083
Methods and Results. –
To better understand theinfluence of link reciprocity in social networks, in the fol-lowing, we shall evaluate its effects on information spread-ing and network robustness from the perspectives of thenetwork function and structure, respectively.
Effect on Information Spreading.
Information spread-ing [43] is one of the most important functions of socialnetworks, where the information (messages, tweets, com-ments, etc.) can distribute at a remarkably fast speedthrough the whole online society via frequent interactionsamong users, although its structure is not designed onpurpose for spreading news [44]. Up to now, there is aconsiderable number of theoretical models to study in-formation diffusion on social networks [45–49]. In thisLetter, in order to understand the underlying mechanismsand possible factors that would result in the informationoutbreaks, we adopt the classic epidemic spreading model,
Susceptive-Infected (SI) model [34], to evaluate the effectof reciprocal links in the two aforementioned social net-works. The diffusion process is described as following, • Initially, user i publishes an information item, I , inthe corresponding social network. I could be about apiece of news, a photo, a comment, etc; • All i ’s followers will automatically receive I accordingto the follower-followee directed network structure.Then an arbitrary fraction of those followers mightnotice I , and forward it on their own homepages if they find it interesting. We consider this forwardingwillingness as the transmission probability , denotedby p ; • The above step will be repeated to the followers of i ’sfollowers, and eventually diffuses to the all achievablenetwork nodes.Note that, the main difference between the directed SI(DSI) and classical SI model is that the link direction istaken into account. In the proposed DSI model, the in-formation only can be transmitted from the followee to itsown followers along with the direction of edges. There-fore, the final fraction of influenced nodes, ρ I , is deter-mined by such a structure. In order to observe the ef-fects of reciprocal links on information diffusion, we quan-tify the influence according to an edge percolation process[38,51–53]. Obviously, if one reciprocal link is more impor-tant than two separate non-reciprocal links, the informa-tion diffusion results will be affected significantly when weremove the same fraction of reciprocal and non-reciprocallinks. Fig. 2 compares the information coverage of remov-ing the two types of links. Compared with removing non-reciprocal links, ρ I decays much faster when we removethe same amount of reciprocal links. Analogously, it alsocan be seen from Fig. 3 that the diffusion speed is affectedmuch remarkably when removing reciprocal links. There-fore, it demonstrates that reciprocal links indeed play amore important role in the information diffusion processon directed social networks. Effect on Structural Robustness.
In conventional com-plex network theory, it is wildly agreed that the networkfunction is largely influenced by its specific structure [50].Therefore, to give solid and comprehensive understandingof the aforementioned results, we adopt the a dynamicalremoving process to measure the effects of reciprocal linkson maintaining the structural robustness of networks [38].For comparison, we apply three metrics to quantify thecorresponding performance. (i) R GSCC : the relative sizeof the strongly connected giant component. A sudden de-cline of R GSCC will be observed if the network disinte-grates after deleting a certain fraction of edges; (ii) thenetwork susceptibility ( ˜ S ): defined as˜ S = X s ∈ E,i = j d , (4)p-3.-X. Zhu et al. Fig. 2: (Color online) The fraction of influenced nodes as the function of the fraction of removed links f . In eachsubgraph, the red and green curves correspond to removing reciprocal and non-reciprocal links, respectively. Theexperimental results are averaged over 30 independent realizations.Fig. 3: (Color online) The fraction of influenced nodes as the function of observed time-step t , where f is the fraction ofremoved links. The red and green curves correspond to removing reciprocal links and non-reciprocal links, respectively.The experimental results are averaged over 30 independent realizations.p-4nfluence of Reciprocal links in Social Networkswhere d is the distance from node i to j . d isset to N when there is no directed path from node i to j . Clearly, the smaller h d i is, the better connectivity andmore efficient the network will be.Fig. 4 and Fig. 5 show the corresponding results of thethree examined matrices. In Fig. 4, it shows different dy-namical patterns of removing reciprocal and nonreciprocallinks, respectively. The size of strongly connected giantcomponent ( R GSCC ) decreases more sharply when remov-ing reciprocal links than deleting non-reciprocal ones. Ac-cordingly, the network susceptibility ( ˜ S ) result shows apercolation phenomenon when removing reciprocal links.Comparatively, this phenomenon is not observed when re-moving non-reciprocal links. In addition, Fig. 5 showsthat the average network distance ( h d i ) increases muchfaster when removing reciprocal links than deleting thenonreciprocal ones. In a word, different dynamical resultsindicate that reciprocal links play a more important rolein both maintaining the connectivity and keeping the ef-ficiency of directed networks than non-reciprocal links. Italso strongly supports the results in the previous sectionthat reciprocity can much promote the speed of informa-tion diffusion, as it takes a more significant responsibilityfor the robustness of directed networks. R G S CC reciprocalnon−reciprocal 0 0.1 0.2 0.30.40.60.81 f S reciprocalnon−reciprocal0 0.1 0.2 0.3 0.400.10.20.30.4 f R G S CC reciprocalnon−reciprocal 0 0.1 0.2 0.3 0.40.40.60.811.21.4 f S reciprocalnon−reciprocal(b) Douban(a) Sina Weibo Fig. 4: (Color online) The fraction of giant component size( R GSCC ) and the susceptibility ( ˜ S ) as the function of thefraction of removed links f on the two observed datasets,(a) Sina Weibo and (b)
Douban . In each subgraph, the redand green curves correspond to the results of reciprocaland non-reciprocal links, respectively. The experimentalresults are averaged over 30 independent realizations.
Conclusions and Discussion. –
In this Letter, wehave studied the influence of reciprocal links of directednetworks from two perspectives: (i) information spread-ing; (ii) structural robustness. Experimental results ontwo representative directed social networks,
Sina Weibo f < d > reciprocalnon−reciprocal 0 0.1 0.2 0.3 0.401234 x 10 f < d > reciprocalnon−reciprocalSina Weibo Douban Fig. 5: (Color online) The average network distance( h d i ) as the function of removed links f on the two ob-served datasets, (left panel) Sina Weibo and (right panel)
Douban . The red and green curves correspond to the re-sults of reciprocal and non-reciprocal links, respectively.The experimental results are averaged over 30 indepen-dent realizations.and
Douban , show that reciprocal links indeed play a moreimportant role than non-reciprocal ones. In particular, theresults of information spreading show that reciprocity cansignificantly enhance the spreading speed. In addition, thecorresponding observations on the two examined datasetsshow that the reciprocity is also largely responsible formaintaining the connectivity and keeping the efficiency ofdirected networks, which suggests its significant impact ininformation spreading on networks.The findings of this work may have a wide-range ap-plication in studying the role and influence of reciprocallinks. Firstly, the topic of community detection has beenwell discussed [56], however, the progress on directed net-works [57] is relatively slow. The main reason is that themodularity [58] of directed networks is rather difficult tobe precisely defined. Secondly, most studies on epidemicspreading and information diffusion [59] focus on study-ing the corresponding dynamics on undirected networks,the in-depth theoretical understanding of the underlyingspreading mechanism on directed networks [60] still re-mains to be solved. Finally, the area of information fil-tering [61] confronts a huge challenge as more and moredirected social services are provided in the informationera. The present work just provides a start point to seethe preliminary effects of reciprocal links, a more com-prehensive and in-depth understanding of reciprocity stillneed further efforts to discover. ∗ ∗ ∗
This work was partially supported by the National Nat-ural Science Foundation of China (Grant Nos. 11105024,11105025, 1147015 and 11205040). ZKZ acknowledgesthe Zhejiang Provincial Natural Science Foundation ofChina (Grant Nos. LY12A05003 and LQ13F030015), thestart-up foundation and Pandeng project of HangzhouNormal University. ZYX acknowledges the Fundamen-tal Research Funds for Central Universities (Grant No.A03008023401042).p-5.-X. Zhu et al.
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