Collective Influence of Multiple Spreaders Evaluated by Tracing Real Information Flow in Large-Scale Social Networks
aa r X i v : . [ phy s i c s . s o c - ph ] N ov Collective Influence of Multiple Spreaders Evaluatedby Tracing Real Information Flow in Large-ScaleSocial Networks
Xian Teng , Sen Pei , Flaviano Morone , and Hern ´an A. Makse Levich Institute and Physics Department, City College of New York, New York, NY 10031, USA Department of Environmental Health Sciences, Mailman School of Public Health, Columbia University, New York,NY 10032, USA * [email protected] ABSTRACT
Identifying the most influential spreaders that maximize information flow is a central question in network theory. Recently,a scalable method called “Collective Influence (CI)” has been put forward through collective influence maximization. In con-trast to heuristic methods evaluating nodes’ significance separately, CI method inspects the collective influence of multiplespreaders. Despite that CI applies to the influence maximization problem in percolation model, it is still important to examineits efficacy in realistic information spreading. Here, we examine real-world information flow in various social and scientificplatforms including American Physical Society, Facebook, Twitter and LiveJournal. Since empirical data cannot be directlymapped to ideal multi-source spreading, we leverage the behavioral patterns of users extracted from data to construct “virtual”information spreading processes. Our results demonstrate that the set of spreaders selected by CI can induce larger scaleof information propagation. Moreover, local measures as the number of connections or citations are not necessarily the de-terministic factors of nodes’ importance in realistic information spreading. This result has significance for rankings scientistsin scientific networks like the APS, where the commonly used number of citations can be a poor indicator of the collectiveinfluence of authors in the community.
Introduction
Identification of the most influential nodes in social networks has broad applications in a variety of network dynamics.
Forexample, in viral marketing, advertising a small group of influential customers to adopt a new product can inexpensively triggera large scale of further adoption; in epidemics control, the immunization of structurally important persons can efficientlyhalt global epidemic outbreaks in contact networks; and in biological systems like brain networks, some significant nodesare responsible for broadcasting information and therefore locating and protecting them are crucial for the whole informationprocessing system. Given its practical significance, the problem of finding the optimal set of influencers in a given networkhas attracted much attention in network science.
For a long time, researchers have developed numerous heuristic measures as predictors of nodes’ importance in informationspreading. Among the most frequently used topological properties are the number of connections (degree), betweenness and eigenvector centralities, PageRank, k-core, etc. All of them are established in the non-interacting setting, wherenodes’ significance is evaluated by taking them as isolated agents. As a result, these ad-hoc approaches, designed for findingsingle superspreaders, fail to provide the optimal solution for the general case of multiple influencers. To address this many-body issue, a rigorous theoretical framework based on collective influence (CI) theory has recently been presented. Witha broader notion of influence – collective influence, the CI method pursues the goal of maximizing the overall influence ofmultiple spreaders. Such explicit optimization objective enables CI to give the minimal set of spreaders.Although CI exhibits good performance with scalability in the optimal percolation model, more validation work regard-ing its efficacy in real-world information spreading still needs to be done. Previously, the lack of real data of informationdiffusion has led to the mainstream adoption of artificial spreading models to simulate spreading dynamics. However, theover-simplified spreading models usually neglect such important factors as activity frequency, connection strength and be-havioral preferences, thus fail to reproduce some observed characteristics of real information spreading. More importantly,different models may produce model-dependent contradictory results. Therefore, it is necessary to evaluate CI’s perfor-mance empirically through realistic information diffusion before applying it to real-world applications like marketing andadvertising.ere, we address this problem by tracking and analyzing the real-world information flows in a wide range of socialmedia: journals of American Physical Society (APS), an online social network Facebook.com (Facebook), a microbloggingservice Twitter.com (Twitter) and a blog website LiveJournal.com (LiveJournal). Rather than tracking the spreading rangeof single spreaders, we intend to investigate the overall spreading range, i.e., the collective influence, of multiple spreaders.To achieve this, the most straightforward idea is to extract and examine the real instances of information diffusion that aretriggered by multiple spreaders. Unfortunately, such ideal multi-source spreading instances in which spreaders send out thesame piece of message at the same time rarely exist in reality. Even though we can find such instances, the initial spreaders arehardly the same as the set of nodes selected by CI or other heuristic strategies, making the comparison between those methodsimpossible.To overcome the aforementioned difficulties, we construct “virtual” multi-source spreading processes by following users’behavioral patterns in the data. In particular, under the assumption that users will maintain their personal preferences inspreading processes, we measure the strength of directed social ties shown in historical diffusion records to represent theinfluence strength of a user imposing on another. For a node under influence of several spreaders, the overall influence on itis defined as the highest influence strength. In this way, we are able to quantify the collective influence imposing on the entirenetwork, corresponding to the collective spreading range of virtual processes initiated by any given set of seeds. Throughcomparisons with competing heuristic methods, including high degree (HD), adaptive high degree (HDA), PageRank(PR) and k-core, we find that the set of spreaders selected by CI can exert larger collective influence on the populationwith the same number of initial seeds. This provides a direct empirical validation of CI’s good performance in real informationspreading. In addition, some individual properties such as the number of connections and citations, which were previouslyregarded as reliable predictors of influence, are found to be invalid in the context of collective influence. This in turn reflectsthat it is the interplay between spreaders that determines the collective influence rather than individual features. Results
Introduction of Datasets
In the following empirical study, four datasets are examined: the journals of American Physical Society (APS), an online socialnetwork Facebook.com (Facebook), a microblogging service Twitter.com (Twitter), and a blog website LiveJournal.com(LiveJournal). All datasets are available at kcore-analytics.com . During the period of data collection, people not onlymaintain social relations with their friends but also interact with others to spread and receive information. Certainly, there arediverse manifestations with respect to the social relation and interaction in distinct platforms. For instance, in the academicdata of APS, authors show their social relations, i.e. coauthorship, through jointly publishing articles, and they reveal theirinteractions and information transmission by citing others’ papers. While in the online social media like Facebook, Twitterand LiveJournal, users reflect their social relations by becoming “cyber friends”, and they interact with each other by creating,receiving, and transmitting messages. With the collection of such information, we can obtain the full network structure aswell as the empirical information flows. Details about these data are explained as follows. • The American Physical Society (APS) is the world’s largest organization of physicists. APS data contains the infor-mation of all the scientific papers published on APS journals until 2005, including Physical Review A, B, C, D, E andPhysical Review Letters. From the author lists and references of scientific publications, we can obtain the informa-tion about collaborations and citations. In total, there are 299,996 articles and 230,521 authors in the data, along with2,356,525 records of citations. We construct the underlying collaboration network according to their coauthorship. Iftwo authors have published one article together, one undirected edge is built between them. Beyond that, we trace theinformation diffusion based on the reference flows. If a scientist i cites one paper written by j , then we can say thatinformation spreads from j to i . • Facebook is an online social networking service. In Facebook, each registered user maintains a friend list, which is agood representation of actual social relationships. Users can exchange messages, post status updates and photos, sharevideos, and browse the posts published by their friends. The Facebook data contains the friend lists and the entirerecords of wall posts from the New Orleans regional network, over a period of two years from September 26th, 2006 toJanuary 22nd, 2009. This data contains 63,731 users and 838,092 wall posts in total. The social network is extractedfrom the friend lists. If user j is added into user i ’s friend list (or i is in j ’s friend list), we assume that they are friendsso that we build an undirected edge between them. According to the wall posts, we can infer the information diffusionflows. If user i makes comments on user j ’s page, we presume that i has gained information from j to motivate him/herto write comments. • Twitter is a microblogging service that enables users to send and read short word-limited messages called “Tweets”. Inthe 2016 election year, Donald Trump, who is the presumptive nominee of the Republican Party for President of the nited States, has become one of the most popular topics being discussed in Twitter. From February 10, 2016 to March14, 2016, we collect approximately 670,000 Tweets that contain the key word “Donald Trump” or “Trump”. In thecollection of Tweets, we extract four kinds of Tweets: mention, replies, retweet and quote. A mention is a Tweet thatcontains another user’s @username anywhere in the body of the Tweet. A reply is a response to another user’s Tweetthat begins with the @username of the person you’re replying to. Replies are also considered as mentions. Besides,a retweet is a re-posting of someone else’s Tweet, in which such character RT@username appears at the beginningto indicate that users are re-posting others’ content. A quote is a special form of retweet that users can write theirown comments when they are re-posting. We consider the mention (and also reply) relationship as a representative ofstrong social ties and use them to construct the network structure. Meanwhile, we use retweets (and quotes) to obtaininformation flows. If user i retweets a Tweet from user j , we assume information diffuses from user j to user i . • LiveJournal is a blog-sharing website where users can maintain friend lists, keep a blog, journal or diary. Our datacontains the friend lists for all users and their blog posts published from February 14th, 2010 to November 21st, 2011,which involves 9,573,127 users and 3,462,504 records of blog reference. Similar to Facebook, we depend on the friendlist to build the underlying network topology. More importantly, LiveJournal users usually add URL links pointingto other relevant blogs when they refer them. As a result, we could use the URL reference to trace the informationdiffusion among users.The originally constructed network is indicated by ¯ G = { ¯ V , ¯ E } in which ¯ V stands for the set of nodes and ¯ E the set ofedges. In the raw datasets of online social platforms including Facebook and LiveJournal, we find many inactive users whoneither spread nor receive messages in network. Actually, they just register an account but do nothing during the period oftime we collect data. Considering that no contributions are made by those inactive nodes to the information diffusion process,we exclude them from the original networks ¯ G and construct an active network ¯ G A = { ¯ V A , ¯ E A } . Different from the onlinesocial platforms, APS has no such inactive nodes as all the authors have to publish papers and cite others’ work. However,APS data contains a minority of articles ( ∼ . G = { V , E } . Properties of the original andtruncated networks are provided in Table 1. Construction of Virtual Information Spreading
In order to decide which strategy to use to locate the most influential nodes in networks, we intend to evaluate the collectiveinfluence exerted by the same number of influencers. The one that achieves the largest collective influence would be our firstchoice. To this end, the most straightforward idea is to compare the spreading range of multi-source spreading processestriggered by a fixed number of seeds selected by different methods. However, the multi-source spreading is an ideal process.In the ideal setting, multiple sources should be activated by the same piece of message at the same time. While in reality, suchideal situation rarely exists because of the intrinsic properties of real data. Users are interested in a wide range of topics, andthey are receiving and delivering multifarious messages from time to time. It is unlikely that we can find enough real instancesin which the spreaders happen to send out a same piece of message at the same time. Therefore, rather than enforcing realdata to match the ideal expectation, we propose an alternative way - to construct a virtual multi-source spreading process.The main idea behind the virtual multi-source spreading processes is that users are expected to follow the behavioralpatterns expressed in real data. For user i with k i neighbors who have chances to access information from i , the closely-tiedneighbors interested in user i ’s publications or posts would be more likely to inherit messages from i . On the contrary, thoseweakly-tied friends would occasionally be influenced by the information released from i . To reflect this effect, we proposea notion named the strength of directed ties r . For a directed link from i to j , the strength r ( i , j ) is defined by the numberof messages, e.g., publications or posts, passed from i to j . By definition, the strength of directed tie r ( i , j ) from i to j isnot generally equal to r ( j , i ) from j to i . Figure 1a reveals that the strength of directed tie follows a power-law distribution.We assume that, in the virtual processes, people would continue to maintain such behavioral patterns. In this way, we canapproximate the multi-source information diffusion and obtain the collective influence as follows.In the virtual processes, suppose a q -percentage of initial spreaders are activated at the beginning, denoted by S = { s i | i = , , ..., n , n = N · q } . We introduce a quantity I u ( s ) ∈ [ , ] to represent the single influence strength that node u is affected byspreader s . Correspondingly, we employ I u to indicate the collective influence strength enforced by all seeds S . Both of theircalculations can rely on the above mentioned strength of directed ties (shown in Figure 1b). For an arbitrary spreader s , theinfluence strength I g ( s ) from s to its neighbor g depends on the strength of directed tie r ( s , g ) , or in other words, depends onthe tendency of g to receive information from s . Assume that, during one period of time, s has totally sent out r ( s ) pieces ofmessages and g has accepted r ( s , g ) of them [ r ( s , g ) ≤ r ( s ) ]. The proportion of acceptance r ( s , g ) / r ( s ) can be viewed as aproxy of influence strength from s to g , i.e. I g ( s ) = r ( s , g ) / r ( s ) . Next, g might affect its neighbor g = s in the same way. hen we follow the spreading paths, multiply the proportions together and then acquire the influence strength s enforcing onits l -step neighbor g l , say I g l ( s ) = l (cid:213) k = r ( g k − , g k ) / r ( g k − ) , (1)where g = s . Figure 1b gives an example with l =
2. As none of messages can spread infinitely, we set a number L as themaximum layer of spreading, so that the influence range, denoted by R s , could be approximated by a ball around s with theradius L (shown in Figure 1c). Within each R s , we have I g ( s ) = s , then the value decreases as l becoming larger, and I g l ( s ) = ( l > L ) for any external node. The schematic diagram regarding the distribution of influencestrength within R s can be seen in Figure 1c. For APS and LiveJournal data, we know more information about references, thedetailed calculation of influence strength is shown in Methods .To obtain the collective influence I u for node u , we apply I u = max ni = I u ( s i ) . (2)Referring to Figure 1b,c, it is straightforward to understand when node u does not belong to any influence range, I u ( s i ) = i , in which case the collective influence should be zero. For the case that node u is only influenced by one spreader,for example I u ( s i ) > I u ( s j ) = j = i , the collective influence should be chosen as the positive (largest) one I u = I u ( s i ) . More generally, if node u lies within the overlapping areas of more than one influence ranges, i.e. it is affectedby more than one sources, we ought to choose the largest potential influence to be its collective influence during the virtualspreading process. Finally, we sum up all the { I u | u = , , ..., N } together to obtain the collective influence that spreadersimpose on the entire system through Q ( q ) = N (cid:229) u = I u / N . (3)Since 0 ≤ I u ≤
1, we have 0 ≤ Q ( q ) ≤
1, which corresponds to the collective spreading range for the virtual process (seeFigure 1c).In general, the virtual process of multi-source spreading constructed here is an approximation of real information diffusion.We take advantage of real data to extract users’ behavioral patterns, base on which, we can calculate the single influence andcollective influence that spreaders impose on each node. Given that, we can finally compute the collective influence exertedby all influencers on the entire network.
Comparison of Different Methods
In this section, we compare CI algorithm with four other widely-used heuristic measures, including adaptive high-degree(HDA), high-degree(HD), PageRank (PR) and k-core (details about methods are shown in Methods ). Recall that,our first step is to identify the q -percentage of initial spreaders according to different methods. Secondly, we construct a virtualmulti-source spreading process. Finally, we compare the virtual spreading range Q ( q ) , i.e. the collective influence of thoseinitial influencers.Figure 2a,c,e,g show the virtual collective influence scores obtained by CI, HDA, HD, PR and k-core for the four networks– APS, Facebook, Twitter and LiveJournal. It can be seen that for a certain value of q , the set of nodes selected by CIcan diffuse the information to a larger scale of populations than those obtained by other methods. CI’s good performanceis more prominent for APS and Facebook data as their diffusion instances are relatively abundant. To clearly distinguishthe performances of different methods, we also present the ratios between CI’s collective influence score and those of otherapproaches (Figure 2b,d,f,h). It reveals that the ratios are always larger than one (indicated by the baseline at 1) for alldatasets. Besides, the ratio is relatively large when q is small. As q increases, it would decline accordingly, suggesting that ifwe select a larger amount of influencers, the collective influence score obtained by all methods would become similar. Amongthe competing heuristic methods, HDA can be viewed as a special case of CI with the calculation radius being zero (see Methods ). However, HDA’s capability in locating influencers is limited by the lack of knowledge of the surrounding nodes,so it is a strategy obtained from the non-interacting point of view. K-core method, a good predicator for locating single“superspreaders”, whereas fails to identify multiple spreaders in the multi-source spreading process. This is because theselected influential nodes tend to cluster together in the core shells which induces large overlapping of their influence areas.Besides, we also investigate the characteristics of influencers that CI has identified. Figure 3a shows the degree comparisonof nodes ranked by CI and HD (from the most influential to the least). Unlike HD finding influencers just relying on degree,CI’s most important nodes contain not only hubs but also many weakly-connected nodes. Besides, some of the most connected odes turn out to be moderate influencers. It confirms the former conclusion that collective influence is determined by theinterplay of all the influencers. Under certain circumstances, some low-degree nodes surrounded by hierarchical coronasof hubs have larger contributions to collective influence than those high-degree nodes connecting to peripheral leaves. Inaddition, we have also examined the correlation between CI ranking and the number of citations in Figure 4. The number ofcitation for each user is defined as how many times other people have accepted or inherited information from him/her directly.We acquire such information through checking the citations (APS), comments (Facebook), retweets (Twitter) as well as URLsreference (LiveJournal). Except for Twitter, the other datasets show us that the most influential nodes are not necessarilythose with the largest number of citations. The uniqueness of Twitter might be explained by considering the mechanism ofnetwork formation and the way of data collection. Twitter platform facilitates users arbitrarily following others, making itpossible that super hubs with millions of followers emerge and hold significant influence; Besides, Twitter is gathered byfocusing on a popular topic ”Donald Trump”, the topic-based data might easily detect those extremely popular users whoalso play important role in spreading. Therefore, the phenomenon shown in APS, Facebook and LiveJournal suggest practicalimplications for academic rankings. When evaluating a researcher’s scientific impact within a field, his/her number of citationis not the determinative factor.
It also reminds us that influence is an emergent property arising from interactions ratherthan an evaluation by viewing nodes individually.
Discussion
It is of importance to search for the most influential nodes in social networks. For a long time, heuristic approaches havebeen widely used to find superspreaders, yet without an ultimate solution for finding multiple influencers. Recently, a rigorousframework called collective influence (CI), along with a scalable algorithm, has been put forward to resolve the many-bodyproblem. Even though CI has been shown to be effective in percolation model, we still need to verify its performanceparticularly in the real case of information diffusion. To achieve this, we collect data from four social media – APS journals,Facebook, Twitter as well as LiveJournal platforms. Different from the situation of finding single superspreaders where wecheck each node’s spreading range, under the circumstance of multiple spreaders, we should examine the collective spreadingrange. Given the difficulty that ideal multi-source spreading processes triggered by same messages at the same time are scarcein real-world diffusion, we propose a virtual multi-source spreading according to users’ behavioral patterns to approximatethe ideal process. Finally, by comparing the collective influence, i.e. the spreading ranges in virtual process, we find that CI iseffective in finding multiple influencers.Moreover, our finding indicates that quantities from a non-interacting viewpoint, such as degree and the number of cita-tions, are not reliable in measuring nodes’ importance in collective influence. Our investigation for influencers’ propertiesconfirms that influence is an effect of cooperation in multi-source spreading. Our results can be transformed into an effec-tive way to rank scientist in academic communities according to their collective influence rather than on the commonly usedlocal connectivity metric, like the number of citations or collaborations in the H-index (Hirsch number). Using the numberof citations, as shown in Fig. 4, can be a poor indicator of the collective influence of a researcher on other researchers inthe community. A global quantity like the Collective Influence that takes into account the optimization of influence of allresearchers at once, provides a meaningful ranking of researchers according to the maximization of their influence. Morestudies will follow to elaborate on this particular point.
Methods
Collective Influence Method
Collective Influence (CI) Algorithm . CI is an optimization algorithm that aims to find the minimal set of nodes thatcould fragment the network in optimal percolation. In percolation theory, if we remove nodes randomly, the network wouldundergo a structural collapse at a critical fraction where the probability that the giant connected component exists is G = q c to achievethe result G ( q c ) =
0. Let the vector n = ( n , n , ..., n N ) represent whether a node is removed ( n i =
0) or not ( n i = v = ( v , v , ..., v N ) represent whether a node belongs to the giant connected component ( v i =
1) or not ( v i = n and v can be derived in locally tree-like networks using message passing (MP) approach: v i → j = n i [ − (cid:213) k ∈ ¶ i \ j ( − v k → i )] , (4)where v i → j indicates the probability of i being in the giant component when j is absent, and ¶ i \ j is the neighbors of i besides j . The equation’s possible solution v i → j = i → j is associated with the special situation where the giant connectedcomponent is absent; therefore, to obtain G ( q ) =
0, the stability of this solution must be guaranteed. As a matter of fact, the tability of v i → j = l ( n ; q ) of the linear operator ˆ M , which is defined on the directededges of networks as M k → l , i → j ≡ ¶ v i → j ¶ v k → l | { v i → j = } . (5)It can be expressed as M k → l , i → j = n i B k → l , i → j , (6)where B k → l , i → j is the non-backtracking matrix of the network. B stores the topological interconnections of network whoseelement B k → l , i → j = l = i , j = k . So far, the original optimal percolation problem has been rephrased as a mathematicalstatement: finding the optimal configuration of n ∗ with size q c that achieves the critical threshold: l ( n ∗ ; q c ) = . (7)The eigenvalue l ( n ; q ) can be calculated according to power method: l ( n ) = lim l → ¥ (cid:20) | w l ( n ) || w | (cid:21) / l . (8)At a finite l , | w l ( n ) | is the cost energy function of influence that needs to be minimized. Take Equation 8 as a starting point,the problem of finding the optimal set of influencers can be solved by minimizing the following cost function: E l ( n ) = N (cid:229) i = ( k i − ) (cid:229) j ∈ ¶ Ball ( i , l ) (cid:213) k ∈ P l ( i , j ) n k ! ( k j − ) , (9)where Ball ( i , l ) is the set of nodes inside the ball of radius i around the central node i , and P l ( i , j ) is the shortest path of length l connecting i and j . To minimize the energy function of a many-body system, an adaptive method is developed with the mainidea of removing the nodes causing the biggest drop in the energy function - CI algorithm. In general, CI algorithm can bestated as follows. Firstly, it considers the nodes at the frontier j ∈ ¶ Ball ( i , l ) and assigns to node i a collective influence valueat the level of l asCI l ( i ) = ( k i − ) (cid:229) j ∈ ¶ Ball ( i , l ) ( k j − ) . (10)Starting with the node with the highest CI l , CI adaptively removes nodes and after each removal, it recalculates CI l for all therest nodes in the system. From the calculation we know that CI has richer topological contents and its performance will beimproved as l increases, but no larger than the network diameter because this case amounts to random identification. In ouranalysis, we adopt the parameter L = l =
0, we have CI ( i ) = ( k i − ) . Under this situation, CI algorithm is reduced tothe High-degree adaptive (HDA) method . For l ≥
1, CI also considers the surrounding neighborhoods and the interactionsamong nodes; meanwhile, it is an easily-implemented algorithm as it only needs local topological structure within the ballof the radius l instead of the whole network structure. More importantly, its computational complexity is O ( N log N ) , whichguarantees its application for large real networks. Heuristic Methods k-core . In k-core method, nodes are ranked based on their k S values, which are calculated during the process of k -shelldecomposition. In k -shell decomposition, nodes are removed iteratively. Firstly, nodes with k = k S =
1. Similarly, the next k-shells with index k S > k S values. Actually, in k -shell composition, all the nodes are divided into different shells according to their relative locationsin networks. Compared with the peripheral nodes, the core nodes have higher probabilities to cause large-scale diffusions.This method has been revealed to perform well in searching for single spreaders who can yield large influence areas. However,it has a poor performance when being used to optimizing the collective spreading caused by multiple spreaders. Becausek-core would select a bunch of nodes within or near the network core, so their influence areas would heavily overlap andproduce a bad collective outcome. ageRank(PR) . PageRank algorithm was firstly proposed by S. Brin and L. Page and used by Google in order to rankwebsites. It extends the idea in academic citation that the number of citations or backlinks give some approximation of a page’simportance, by not counting links equally but normalizing by the number of links on a page. Its calculation is as follows: ifpage A has pages T , ..., T N citations with the associated PageRank as PR ( T ) , ..., PR ( T N ) , then the PageRank of A is given byPR ( A ) = ( − d ) + d (cid:18) PR ( T ) C ( T ) + ... + PR ( T N ) C ( T N ) (cid:19) , (11)in which C ( A ) is defined as the number of links going out of page A . PageRank outputs a probability distribution used to rep-resent the likelihood that a person randomly clicking on links will arrive at any particular page. The higher the probability, thehigher the PR value of this page. In practice, PageRank can be calculated using a simple iterative algorithm and correspondingto the principal eigenvector of the normalized link matrix of the web network. High-Degree(HD) . HD method ranks nodes directly according to the number of connections. Compared with othermethods requiring global network structures like k-core and PageRank, HD only needs local information and is easily imple-mented. However, it cannot deal with the circumstance in which hubs form tight community such that their spreading areaswould heavily overlap.
High-Degree Adaptive(HDA) . HDA is the refined adaptive version of HD method. To help mitigate the above mentionedsituation, HDA recalculates the degrees after each removal. It can also be viewed as a special case of CI algorithm at l = Data Processing
Analyzing APS and LiveJournal . In terms of APS, we know the specific article pairs ( a , b ) , which means paper a citespaper b , In other word, the authors A b of b spread their scientific discoveries to the authors A a of a . Therefore, for an arbitraryauthor s , we can know his or her journal set J ( s ) = { J i | i = , , ..., n s } in which J i indicates each piece of paper and n s standsfor the number of papers published by s . By tracking the spreading for each paper J i through citation flows, we can determineits influence range R s ( J i ) containing all people who have cited this paper J i . For each receiver u ∈ R s = { R s ( J i ) | i = , , ..., n s } ,we calculate the individual influence strength by I u ( s ) = ( (cid:229) n s i = d u ∈ R s ( J i ) ) / n s where d u ∈ R s ( J i ) = u ∈ R s ( J i ) . Largevalues of I u ( s ) means that u is more likely to cite the work of s than other peers. Next, the collective influence strength fromall sources can be obtained by I u = max ni = I u ( s i ) . In LiveJournal, we know information about blog references. So, we canfollow the similar method as in APS to process LiveJournal data. References Valente, T. W. & Davis, R. L. Accelerating the diffusion of innovations using opinion leaders.
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Sci. Rep. cknowledgements This work was supported by NIH-NIGMS 1R21GM107641, NSF-PoLS PHY-1305476 and ARL Cooperative AgreementNumber W911NF-09-2-0053, the ARL Network Science CTA. We thank Lev Muchnik for providing the data on LiveJournal.
Author contributions statement
H.A.M. designed research; X.T., S.P. , F.M., H.A.M. analyzed data, prepared figures and wrote the main manuscript text; Allauthors reviewed the manuscript.
Additional information
Competing financial interests
The authors declare no competing financial interests.
Figure 1.
Construction of virtual spreading based on people’s interactions. a, Distribution of directed tie strength for realnetworks. The power law distribution demonstrates the heterogeneity of interactions between nodes. b, Calculation forinfluence strength. Nodes s and s are two distinct spreaders, the maximum spreading layer is set as L =
2. Node u isinfluenced by two seeds with the strength I u ( s ) and I u ( s ) . We select the largest value to indicate the collective influenceenforcing on it. c, An illustration of single influence strength I u ( s ) along with collective influence strength I u . The threecircle-like areas represent the corresponding influence ranges R s , R s , R s for distinct spreaders s , s , s , and the contourlines indicate the levels of influence strength I u . When projecting it onto 2-dimensional space, we have the correspondingdistribution. The collective outcome I u (indicated by gray curve) is obtained by combining the single influence strengths ofall the spreaders.Networks ¯ N ¯ M ¯ N A ¯ M A N M h k i h k d i q c APS 230,521 1,607,305 230,521 1,607,305 190,161 1,582,710 16.4 37.4 20%Facebook 63,731 817,090 45,746 703,924 45,459 703,803 31.0 18.8 45%Twitter 311,334 151,654 311,334 151,654 29,463 143,220 9.7 5.1 6%LiveJournal 9,573,126 188,240,039 304,858 19,785,460 290,362 19,783,730 136.3 7.7 46%
Table 1.
Properties of the original and processed networks ¯ G , ¯ G A , G in this article. In the table, ¯ N ( ¯ M ) is the number of nodes(edges) in the original networks ¯ G , ¯ N A ( ¯ M A ) represents the number of nodes (edges) in the active network ¯ G A , N ( M ) indicatesthe number of nodes (edges) in the network G . h k i is the average degree of network G . h k d i denotes the average out-degree ofdiffusion graph, i.e. the average number of messages which have been sent out. Besides, q c indicates CI’s minimal fractionof influencers to fragment the networks in optimal percolation. All datasets are available at kcore-analytics.com . .01 0.05 0.1 0.2 q s p r ead i ng r ange CIHDAHDPRkcore 0.01 0.05 0.1 0.15 0.2 q r a t i o baselineCI/HDACI/HDCI/PRCI/kcore0.01 0.05 0.1 0.2 0.3 0.5 q s p r ead i ng r ange CIHDAHDPRkcore 0.01 0.1 0.2 0.3 0.4 0.5 q r a t i o baselineCI/HDACI/HDCI/PRCI/kcore APSFacebook FacebookAPS bac d q s p r ead i ng r ange CIHDAHDPRkcore 0.02 0.04 0.06 0.08 0.1 q r a t i o baselineCI/HDACI/HDCI/PRCI/kcore0.01 0.05 0.1 0.2 0.4 q s p r ead i ng r ange CIHDAHDPRkcore 0.01 0.1 0.2 0.3 0.4 q r a t i o baselineCI/HDACI/HDCI/PRCI/kcore LiveJournalTwitterTwitter fge h
LiveJournal
Figure 2.
Performance of CI in large-scale real social networks. The datasets contain APS ( a,b ), Facebook ( c,d ), Twitter( e,f ) and LiveJournal ( g,h ). We compare the virtual spreading ranges of different methods in a,c,e,f . With a fixed fraction q ofseeds, CI’s virtual spreading range is larger than all the heuristic approaches. Besides, we also show the ratios of spreadingranges between CI and others in b,d,f,h . It reveals that the ratios are always larger than 1 (higher than the baseline), implyingthat CI is an effective strategy in locating multiple spreaders. We set L = h k d i , and L = h k d i . We care about the results when q is small, so welimit q within the range of small value. As q increases, the performances of all the strategies become similar. igure 3. Degree versus ranking. We show the degrees of nodes ranked (from highest to lowest) by CI and HD for APS ( a ),Facebook ( b ), Twitter ( c ) and LiveJournal ( d ). It shows that CI can find those previously neglected weak nodes to emergeamong most significant influencers. Meanwhile, some most connected nodes are ranked as moderate influencers by CI,indicating that such weak node effect is a consequence of collective influence in the case of multiple spreaders. This resulthas important consequences for ranking of researchers in scientific networks. igure 4. The number of citations versus CI ranking. We present the number of citations (comments, reposts or references)of nodes ranked by CI strategy for APS ( a ), Facebook ( b ), Twitter ( c ) and LiveJournal ( d ). Despite that in Twitter data, themost influential user is exactly the one with the largest amount of citations, the overall results still prove that large number ofcitations is not necessarily a reliable measure for identification of top-ranking influencers. This fact has meaning especiallyfor academic rankings for physicists in community like APS. CI takes into account the maximization of influence in thewhole network of each scientist rather than just the local information given by the number of citations. Thus a highly citedauthor may not have a large impact in the community if he/she is isolated in the periphery. An optimal measure as CI shouldrank such a scientist lower in the scientific community. This result calls for a revision of rankings based solely on the localinformation rather than the collective influence in the entire network community. We elaborate more on this problem insubsequent publications.). Despite that in Twitter data, themost influential user is exactly the one with the largest amount of citations, the overall results still prove that large number ofcitations is not necessarily a reliable measure for identification of top-ranking influencers. This fact has meaning especiallyfor academic rankings for physicists in community like APS. CI takes into account the maximization of influence in thewhole network of each scientist rather than just the local information given by the number of citations. Thus a highly citedauthor may not have a large impact in the community if he/she is isolated in the periphery. An optimal measure as CI shouldrank such a scientist lower in the scientific community. This result calls for a revision of rankings based solely on the localinformation rather than the collective influence in the entire network community. We elaborate more on this problem insubsequent publications.