Weighted reciprocity in human communication networks∗

Nitesh V. Chawla3,4, David Hachen1,4, Omar Lizardo1,4, Zoltan Toroczkai2,3,4, Anthony Strathman2,4, andCheng Wang1,4

1Deparment of Sociology, University of Notre Dame, 810 Flanner Hall, Notre Dame, IN, 465562 Deparment of Physics, University of Notre Dame, 225 Nieuwland Science Halll, Notre Dame, IN, 46556

3Deparment of Computer Science and Engineering, College of Engineering, University of Notre Dame, 384 Fitzpatrick Hall, NotreDame, IN, 46556

4Interdisciplinary Center for Network Science and Applications (ICenSA), 384E Nieuwland Science Hall, Notre Dame, IN 46556

August 18, 2011

Abstract

In this paper we define a metric for reciprocity appropriate for weighted networks in order to investi-gate the distribution of reciprocity in a large-scale social network composed of billions of communicationattempts across millions of actors. We find that relations in this network are characterized by muchlarger degrees of non-reciprocity than we would expect if persons kept only those relationships that ex-hibit balance. We point to two structural features of human communication behavior and relationshipformation—the division of contacts into strong and weak ties and the tendency to form relationships withsimilar others—that either help or hinder the ability of persons to obtain balance in their relationships.Finally, we examine the extent to which deviations from reciprocity in the observed network are partiallytraceable to these characteristics.

1 Introduction

Reciprocity has been recognized to be one of the mostimportant properties of the connections linking enti-ties in networked systems [12, 2, 33, 36, 31]. Thestudy of dyadic reciprocity began in the sociometricand social network analysis tradition as a way to char-acterize the relative behavioral or cognitive “balance”in social relationships [16, 17, 18, 7, 22, 21, 27]. Thesestudies defined reciprocity in a very simple but funda-mentally limited way. A dyad was reciprocal if bothpartners nominated one another as friends, or—in thetradition of “balance theory” [19, 29, 28, 6, 8]—if itwas found that the relationship had the same valence(positive or negative) for both participants. Dyadswere viewed as non-reciprocal either when one part-ner reported considering the other one a friend or aclose associate and the other did not, or if one part-ner displayed positive sentiments towards a partnerwho felt negatively towards him or her. The funda-

mental hypothesis of balance concerned a dynamicprediction: over time ties that were imbalanced wereexpected either to become balanced or to dissolve[16, 29, 28, 8].

This definition of “reciprocity” fit very well withthe representation of social networks in early graphtheory as consisting of binary (1,0) edges connectingtwo nodes [33]. Analysts can then establish the levelof reciprocity in the network via the so-called “dyadiccensus.” This presupposes a binary adjacency matrixA, where aij = 1 if actor i chooses actor j as a neigh-bor and aij = 0 otherwise. Three types of dyads canthen be defined: asymmetrical—sometimes also re-ferred to as “non-reciprocal” (aij = 1 and aji = 0 oraij = 0 and aji = 1), symmetrical (aij = aji = 1)and null (aij = aji = 0), otherwise known as theUMAN classification [5]. The phenomenon of dyadicreciprocity at the level of the whole network hasbeen studied by comparing the relative distributionof asymmetric and mutual dyads in a graph [22, 12].

∗Research was sponsored in part by the Army Research Laboratory and was accomplished under Cooperative AgreementNumber W911NF-09-2-0053 and in part by the National Science Foundation (NSF) Grant BCS-0826958. The views andconclusions contained in this document are those of the authors and should not be interpreted as representing the officialpolicies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government isauthorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon.

1

arX

iv:1

108.

2822

v2 [

cs.S

I] 1

7 A

ug 2

011

We can consider levels of non-reciprocity to be highif the proportion of asymmetric dyads is larger thanwould be obtained by chance in a graph with simi-lar topological properties (for instance a graph withthe same number of nodes and edges). This tradi-tional definition of reciprocity has been extended anddeveloped for the analysis of reciprocity in complexsystems (social, technological, biological, etc.) orga-nized as networks [12].

The binary classification of dyads into three typesmisses one of the most important features of a dyadicrelationship: the relative frequency of contact be-tween the two partners [18, 10, 9]. This is a dimensionof dyadic relationships that has always been consid-ered crucial in previous treatments of the dynamicsand static correlates of dyadic ties [18, 24, 11], butwhich has not been treated in depth in the existing lit-erature, mainly due to lack of reliable behavioral dataon repeated social interactions among humans in nat-ural environments [10]. Furthermore, it should alsobe clear that our intuitive notions of what reciprocityis requires information about the relative “balance”not of static mutual nominations or sentiments, but ofrepeated behavioral interactions, exchanges or flowsin a dyad [34, 3, 4]. This means that a more empir-ically accurate definition of reciprocity can only beobtained in the context of a weighted graph [1, 35, 20].In this representation, instead of a tie being thoughtof as simply being present or absent, the adjacencymatrix is now defined by weights (aij = wij) whichindicate the relative flow strength of the arc (e.g., thecount of the number of interactions initiated by i anddirected towards j).

In this context, what have traditionally been con-sidered “non-reciprocal” dyads—e.g., one partner inthe dyads nominates the other but not vice versa—[5]can be better thought of as the limiting case of com-plete one-way non-reciprocity in which one partner inthe relationship initiates all contact attempts and re-ceives no reciprocation from the other member of thedyad. Intuitively it is doubtful whether we can callthis a relationship in the first place. In the very sameway, what have been traditionally conceived of as “re-ciprocal” (e.g. “mutual” or “symmetric”) dyads canexhibit high levels of weighted non-reciprocity withmost of the interaction being one-way. Consider forinstance a dyadic relationship in which one partnerinitiates 90% of the total number of one-way commu-nication attempts. It is not very intuitive to call thisdyad “reciprocal,” but that is precisely what tradi-tional measurement methods (with binary ties) wouldforce us to do.

In sum, advancing research on dyadic reci-

procity requires that we define dyadic reciprocity forweighted graphs. In this paper we do that, whileoffering an empirical account of the distribution andcorrelates of dyadic reciprocity in a weighted commu-nication network. Finally, we compare the observeddistribution to that obtained when we vary key di-mensions of the networks that we argue are importanttopological and structural drivers of reciprocity.

2 Weighted reciprocity

2.1 Weighted reciprocity metric

We seek to define a measure of dyadic reciprocitythat has the following properties: it should be at aminimum when the weight of the directed arc goingfrom vertex i to vertex j approaches the weight ofthe directed arc going from vertex j to vertex i andit should increase monotonically with the weight dif-ference between the two directed arcs, but it shouldadjust for the different communicative propensitiesof each vertex and it should not depend on the ab-solute magnitude of the respective weights. Finally,the measure should be the same irrespective of direc-tionality (Rij = Rji).

One measure that satisfies these conditions is:

Rij = |ln(pij)− ln(pji)| (1)

With,

pij =wij

wi+(2)

Where wij is the weight corresponding to the di-rected i → j arc, and wi+ is the strength of the ith

vertex as given by [1]:

wi+ =∑

j∈N(i)

wij (3)

Because pij is the instantaneous probability thatif i makes a communication attempt it will be di-rected towards j (and viceversa for pji), a substantiveinterpretation of this measure is that a dyad is recip-rocal when two persons have the same probability ofcommunicating with one another, and a dyad is non-reciprocal when the probability of one person direct-ing a communication towards another differs substan-tially from the probability of that person returningthat communication. Factors that affect this proba-bility, such as the number of neighbors connected toeach vertex, the relative communicative propensities

2

(a) (b) (c)

Figure 1

Figure 2

of each vertex or the dispersion of edge-weights acrossneighbors for each vertex, should thus be implicatedin moving each dyad closer or farther away from theideal of full reciprocity.

2.2 Some special cases

The characterization of reciprocity given above al-lows us to outline some idealized conditions underwhich we should expect full reciprocity and underwhich we should expect systematic deviations fromthe reciprocity ideal. To build some intuition it helpsto rewrite 1 as:

Rij =

∣∣∣∣ln [wij

wji

wj+

wi+

]∣∣∣∣ (4)

The first idealized condition that we can con-

sider is an equidispersion regime. Under this con-dition, persons distribute their communicative activ-ity equally across partners, with the only constraintbeing the number of partners (ki) and their commu-nicative propensity (wi+). It is easy to see that underthis regime all directed weights are given by:

wij =wi+

kouti

(5)

Substituting 5 into 2 we find that the expectedpij under this regime is simply:

pij =1

kouti

(6)

Finally, substituting 6 into 1 shows that in this

3

Table 1

Assortativity Non-assortativity

Equidispersion Robsij Rrw

ij

Non-equidispersion Robsij Rrw

ij

case the reciprocity equation simplifies to:

Rij = |ln(koutj )− ln(kouti )| (7)

Because vertex strength drops out of the pictureunder the equidispersion constraint, if persons dis-perse their calls equally across neighbors, and havethe same number of outgoing arcs (kouti = koutj ), thenreciprocity is assured. This is the situation depictedin Figure 1a.

Conversely, when equidispersion obtains, devia-tions from reciprocity are solely traceable to the mag-nitude of the degree-differences across the two ver-tices in a dyad. This is in spite of the fact thatthe two actors dedicate (proportionally) the sameamount of energy and time to interacting with eachof their neighbors. As shown in Figure 1b, the rea-son for this is that the green vertex divides her en-ergy over a larger number of neighbors than the redvertex, reducing the outgoing probability of commu-nication in relation to the incoming probability fromthe less popular neighbor. This implies that, hold-ing all else equal, degree-assortativity in social net-works (the existence of more same-degree dyads thanwe would expect by chance) should drive the averagereciprocity of a random dyad towards the reciprocitypoint (Rij = 0). Non-assortativity (or negative assor-tativity) should move dyads towards less reciprocalrelations.

Finally, as shown in Figure 1c, deviations fromthe ideal of reciprocity can be produced even whenpersons share the same number of neighbors and havethe same communicative propensities but they do notdistribute their communicative activity equally acrosscontacts. In the example shown above, the green ver-tex follows the equidispersion rule but the red vertexdoes not. Instead the red vertex concentrates hercommunicative activity on the green vertex at theexpense of her other neighbors. Setting wi+ = wj+

in 4, gives us the expected reciprocity for this case:

Rij = |ln(wij)− ln(wji)| (8)

In other words, when vertices have the samestrength and have the same number of neighbors, but

Rij 6= 0, we can be sure that either: (1) at least oneof the vertices is investing more in that relationshipthan in his or her other relationships; or (2) at leastone of the vertices is investing in that relationship lessthan he or she does in his other relationships. Nat-urally, both things could be happening at the sametime (one partner under-invests while the other oneover-invests).

A fourth case that would produce systematic non-reciprocity according to 4 would be one in whichthe directed weights for each arc in the dyad match(wij = wji), but the vertex strength of the partnersis different. In this case, the level of non-reciprocityfor that dyad is given by:

Rij = |ln(wj+)− ln(wi+)| (9)

Note that the case of equal weight but non-equalvertex strength is redundant since it is implies thateither one partner is under-investing or another part-ner is over-investing in the relationship; this is there-fore another version of the non-equidispersion storyshown in 1c. This is intuitive since, as we saw above,when both vertices disperse their communicative ac-tivity equally across neighbors Rij is independent ofvertex strength differences. Thus, any dependence ofthe expected value of Rij on either wi+ or wj+ whenwij = wji can only be produced by deviations fromequidispersion.

In a real communication network, we should ex-pect the values of wij to vary across i ’s j neighbors:equidispersion is an ideal that will usually fail tobe met in real social networks. Empirical evidenceindicates that persons typically divide their neigh-borhood into core and peripheral members, directingstrong (large weight) ties toward core members andkeeping only weak (small weight) ties with peripheralmembers [15, 23]. Arcs that are considered strong tiesin ego’s neighborhood have should have much largerweights than those that are considered weak ties.Non-reciprocity obtains when they are mismatchesin the directional tie strength between two vertices:one member of the dyad considers a strong tie whatfrom the point of the view of the other member is a

4

(a) Local configuration beforerewiring

(b) Local configuration afterrewiring

Figure 3

weak tie. Thus, holding all else equal, deviations fromthe equidispersion ideal should move the average dyadaway from the reciprocity point (Rij = 0).

3 The empirical distribution ofweighted reciprocity

The data that we will consider in what follows con-sist of a weighted graph of a human communicationnetwork constructed from trace-logs of over 1 billioncellular telephone voice calls made by 8 million sub-scribers of a single cellular telephone provided in aEuropean country over a two-month period in 2008.Among these 8 million subscribers there are over 34million directed arcs, that is instances in which a sub-scriber made at least one call to another subscriber.Of these 34 million arcs, about 16.8 million (49%)are asymmetric dyads, meaning that the directed arcis not reciprocated. The remaining 17.2 million sym-metric arcs are in 8.6 million mutual dyads consistingof two arcs, indicating that each person in the dyadmade at least to the other person during this timeperiod. The focus below is on these 8.6 million mu-tual dyads given that reciprocity is only defined forthese types of dyads. We define the weight (wij) ofthe incoming and outgoing arcs for each vertex asthe number of calls either received from or made toeach neighbor (respectively) during the time periodin question.

Figure 2 depicts the observed reciprocity distri-bution of reciprocity computed according to equa-tion 1. We divide the observed dyads into threeclasses: reciprocal dyads are those in which the com-munication probability ratio (taking the largest prob-ability as the numerator) ranges from 1.0 to 1.5. Par-tially reciprocal dyads are those in which the com-

munication probability ratio is larger than 1.5 butsmaller than 9.0. Finally, non-reciprocal dyads arethose with a probability ratio exceeding 9.0. We findthat a substantial minority (28%) of dyads belong tothe reciprocal class, about 58% of dyads can be con-sidered partially reciprocal, and a non-trivial minor-ity of dyads (14%) exhibit extreme non-reciprocity,with one partner being more than nine times morelikely to contact the other than being contacted bythat partner.

The most surprising finding to emerge from thisanalysis is how relatively common large levels of non-reciprocity are in this network. Had we confined our-selves to the purely binary definition of reciprocityas mutuality or symmetry, we would have missed thelarge levels of communicative imbalance hidden be-neath this surface. This result indicates that thereare systematic features of human communicative be-havior that drive dyads towards non-reciprocity inspite of often noted preferences for balance in humanrelationships [14]. We investigate some of the forcesthat lead networks towards and away from reciprocitynext.

3.1 Comparing the observed distribu-tion to alternative regimes

To what extent are the patterns of reciprocity ob-served in this social network deviations from what wewould expect by chance? To answer this question wecompare the observed reciprocity distribution to thatobtained from three-alternative regimes, correspond-ing to the three out of the four different configurationsin a two-dimensional space defined by the presence orabsence of degree assortativity, versus the presence orabsence of a tendency toward equidispersion. This isshown in shown in Table 1.

5

Figure 4

As we have already noted, the observed social net-work is located in the lower-left corner of the table(Robs

ij ). This is a network displaying positive degreeassortativity. The Pearson correlation coefficient be-tween the degree sequences of each of the two ver-tices across linked dyads in the observed network ispositive: rkikj = 0.33, which is a value typical forhuman social networks [30]. However, this network—not surprisingly—also exhibits non-equidispersion ofweights across neighbors.

We proceed to create three alternative compari-son networks, all of which preserve the most relevanttopological and statistical features of the original net-work (number of vertices the number of links andthe degree-distribution), but which either remove as-sortativity, impose equidispersion in the distributionof directed weights across neighbors for all vertices,or do both. We remove assortativity in the origi-nal network using the Maslov-Sneppen local rewiringalgorithm [26] illustrated in Figures 3a and and 3b.It is easy to verify that this procedure preserves thenumber of edges attached to each vertex, but makesthe vertex-to-vertex connections independent of de-gree. We can verify that the algorithm is success-ful by computing the degree-correlation after reshuf-fling. The resulting network is indeed non-assortative(rkikj

= 0).

Accordingly, the assortative-equidispersed net-work (upper-left corner) is just like the originally ob-served network, except that now the number of callsacross partners are redistributed and forced to be

same; here reciprocity is given by Robsij ). The non-

assortative, equidispersed network (upper-right cor-ner) is just like this last network, except that nowthe links are reshuffled to remove degree-assortativityaccording to the procedure described above; here reci-procity is given by Rrw

ij ). Finally, the non-assortativenon-equidispersed network (lower-right hand corner)is just like this last network, except that the distri-bution of calls across neighbors matches that of theoriginal data set; here reciprocity is given by Rrw

ij .

Because assortativity and non-equidispersion pullin different directions with respect to reciprocity, weshould observe that Robs

ij < Robsij due to the non-

equidispersion effect; that is reciprocity in the ob-served network is farther away from zero than in anetwork with similar characteristics where personsdistributed calls equally across partners. Followingthe same line of reasoning, we should observe thatRobs

ij < Rrwij due to the assortativity effect; that is

reciprocity in the observed network is closer to zerothan in a network with the same characteristics wherethere is no tendency for persons of similar degree tobe connected to one another. Finally, due to the non-equidispersion effect, we should find that Rrw

ij < Rrwij .

That is even in a network without assortativity, onein which persons distribute calls equally across neigh-bors should have reciprocity values closer to zero thanone where this condition does not obtain. If thesethree inequalities hold, then we should find the fol-lowing partial ordering of expected (average) non-

6

reciprocity across the four networks:

Robsij <

[Rrw

ij ≤≥ Robsij

]< Rrw

ij (10)

The most reciprocal network should be the onewhich has both assortativity and equidispersion, andthe least reciprocal network should be one withoutassortativity and without equidispersion. Note thatthe ordering of the expected values of Rrw

ij and Robsij

cannot be predicted a priori, since the question ofwhich force is greater, (1) the ability of assortativityto drive reciprocity towards zero or (2) the ability ofnon-equidispersion to move the same quantity awayfrom zero, is an empirical issue. We can however becertain that reciprocity in these two networks shouldfall in between the two extremes described above,since they are positive in a factor that lowers reci-procity and negative on a factor that increases it.If assortativity is a stronger factor in driving non-reciprocity towards zero that non-equidispersion is indriving it away from zero, then we should find thatRrw

ij > Robsij . If the opposite is the case, then we

should find that Rrwij < Robs

ij .

Figure 4 summarizes the differences in the relativedistribution of reciprocity across all four networks.The results largely agree with expectations regard-ing the two regimes that should fall at the lower andhigher extremes of weighted reciprocity. Thus, thenetwork without assortativity and without equidis-persion displays empirically extreme levels of non-reciprocity. While only about 14% of dyads in the ob-served network exhibit extreme non-reciprocity (e.g.one partner being nine times more likely to initiatea communication attempt than than the other), thisproportion more than doubles once we remove the as-sortativity bias but keep everything the same (30%).Meanwhile while about 28% of dyads enjoy some levelof reciprocity this number drops to 16% in the non-assortative version of the same network. Also as ex-pected, the network displaying values closest to thezero (full reciprocity) level is the one that has bothassortativity and equidispersion. Here the proportionof reciprocal dyads is 44% (in comparison to 28% inthe original data, and the proportion of extremelynon-reciprocal dyads is only 1%.

The results also provide an answer to our origi-nal question of which of the two tendencies observedin human communication networks—assortativity ornon-equidispersion—contributes more to system levelreciprocity. The answer is clear: adding equidisper-sion to the least reciprocal network results in a muchmore dramatic move towards reciprocity than does

adding assortativity to the same network. In this re-spect, while assortativity keeps human communica-tion networks from resembling the least reciprocal ofour baseline networks, the tendency to disperse com-munication activity inequitably across contacts is re-sponsible for the bulk of the observed non-reciprocity.

Accordingly, the final ordering of expected reci-procity (with smaller values indicating more reci-procity) all four networks is as follows:

Robsij < Rrw

ij < Robsij < Rrw

ij

4 Discussion

In this paper we have defined a metric for reciprocityapplicable to weighted networks. Under this concep-tualization, reciprocity is defined as balance in thenumber of communications flowing from one partnerto another, normalized by the communicative activityof each person. This yields a notion reciprocity inter-pretable as a matching of the probabilities that thetwo vertices in a dyad will initiate directed contactattempts towards each other. When persons matchin overall communicative propensity, reciprocity re-duces to the (absolute value of the logged) ratio ofthe weights of the incoming and outgoing arcs. Whenthe weights of the arcs are the same, reciprocity sim-plifies to the (absolute value of the logged) ratio ofthe strength of the vertices. The most revealing spe-cial case obtains when vertices disperse their commu-nication attempts equally across neighbors. In thiscase reciprocity simplifies to the (absolute value ofthe logged) ratio of the number of neighbors (outde-gree) of each vertex.

We examined the distribution of reciprocity as de-fined here in an actual social network composed ofcell-phone communications between individuals dur-ing a two month period. We found that these rela-tionships exhibit varying levels of balance, with themajority of relationships exhibiting moderate to ex-treme large imbalances. In this respect, while reci-procity might certainly be a communicative prefer-ence across persons, there are systematic features ofhuman communication behavior and network topol-ogy that prevent it from becoming a statistical“norm” as would be predicted by social-scientific the-orists [14]. We found that one such bias consistsprecisely in the propensity of persons to divide theirneighbors into strong and weak ties, thus concentrat-ing their communicative activity on a few partners

7

at the expense of others, since imposing equidisper-sion on the observed network moves it closer to theideal of full-reciprocity. In addition, we showed thatone systematic feature that differentiates social net-works from other networks—namely, the tendency oflike to associate with like as manifested in the degree-assortativity property—actually makes the observedcommunication network more reciprocal, by facili-tating the matching of probabilities across incomingand outgoing arcs. When we remove assortativityfrom the observed network, we observe that non-reciprocity increases dramatically. It is worth noting,however, that the effect of non-equidispersion in mov-ing reciprocity away from the ideal of zero is strongerthan the effect of assortativity in moving this quan-tity closer to zero.

These results have important implications for howwe think of reprocicity in human and other networkedsystems. For instance, smaller groups or dense com-munities that impose homogeneity in most topologi-cal characteristics (including the number of neighborsas in fully-connected cliques) should exhibit moreweighted reciprocity than social systems that inducelarge inequalities in connectivity across partners (e.g.,social systems characterized by “popularity tourna-ment” dynamics) [25, 32, 13]. Networked systemsthat induce anti-correlation in the number of neigh-bors of each vertex in a dyad should—all else be-ing equal—be characterized by high-levels of non-reciprocity. In the same way, positive correlations be-tween other relevant characteristics (e.g. average out-going arc weight or vertex strength) should move so-cial relationships towards the reciprocity ideal, whilemismatches in these vertex-level traits should in-crease non-reciprocity. In this respect, observed ten-dencies for persons to match in these traits may bethe indirect result of an underlying tendency to se-lect the most reciprocal relationships available in thenetwork and remove the least reciprocal than a directpreference to be concordant on these surface features.

References

[1] A. Barrat, M. Barthelemy, R. Pastor-Satorras,and A. Vespignani. The architecture of complexweighted networks. Proceedings of the NationalAcademy of Sciences, 101(11):3747–3752, 2004.

[2] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez,and D.U. Hwang. Complex networks: Structureand dynamics. Physics Reports, 424(4-5):175–308, 2006.

[3] S.P. Borgatti. Centrality and network flow. So-cial Networks, 27(1):55–71, 2005.

[4] S.P. Borgatti, A. Mehra, D.J. Brass, andG. Labianca. Network analysis in the social sci-ences. Science, 323(5916):892, 2009.

[5] K.M. Carley and D. Krackhardt. Cognitive in-consistencies and non-symmetric friendship. So-cial Networks, 18(1):1–27, 1996.

[6] J. A. Davis. The davis/holland/leinhardt stud-ies: an overview. In P.W. Holland and S. Lein-hardt, editors, Perspectives on Social NetworkResearch., pages 51–62. Academic Press, NewYork, 1979.

[7] James A Davis. Structural balance, mechanicalsolidarity, and interpersonal relations. AmericanJournal of Sociology, 68:444–462, 1963.

[8] P. Doreian. Event sequences as generatorsof social network evolution. Social Networks,24(2):93–119, 2002.

[9] N. Eagle, A.S. Pentland, and D. Lazer. Infer-ring friendship network structure by using mo-bile phone data. Proceedings of the NationalAcademy of Sciences, 106(36):15274, 2009.

[10] N. Eagle, A.S. Pentland, D. Lazer, andMA Cambridge. Mobile phone data for inferringsocial network structure. Social Computing, Be-havioral Modeling, and Prediction, pages 79–88,2008.

[11] Scott L. Feld. The focused organization of so-cial ties. American journal of sociology, 86:1015–1035, 1981.

[12] D. Garlaschelli and M.I. Loffredo. Patterns oflink reciprocity in directed networks. PhysicalReview Letters, 93(26):268701, 2004.

[13] Roger V. Gould. The origins of status hier-archies: A formal theory and empirical test.American Journal of Sociology, 107(5):1143–1178, 2002.

[14] Alvin W. Gouldner. The norm of reciprocity:A preliminary statement. American SociologicalReview, 25(2):161–178, 1960.

[15] Mark S. Granovetter. The strength of weak ties.American Journal of Sociology, 78(6):1360–1380,1973.

[16] Maureen T. Hallinan. The process of friendshipformation. Social Networks, 1:193–210, 1978.

8

[17] M.T. Hallinan and E.E. Hutchins. Structural ef-fects on dyadic change. Social Forces, 59(1):225–245, 1980.

[18] Muriel Hammer. Implications of behavioral andcognitive reciprocity in social network data. So-cial Networks, 7:189201, 1985.

[19] F. Heider. The Psychology of Interpersonal Re-lations. Wiley, New York, 1958.

[20] G. Kossinets and D.J. Watts. Empirical anal-ysis of an evolving social network. Science,311(5757):88–90, 2006.

[21] D. Krackhardt. Cognitive social structures. So-cial Networks, 9(2):109–134, 1987.

[22] M. Mandel. Measuring tendency towards mu-tuality in a social network. Social Networks,22(4):285–298, 2000.

[23] Peter V. Marsden. Core discussion networksof americans. American Sociological Review,52(1):122–131, 1987.

[24] P.V. Marsden and K.E. Campbell. Measuringtie strength. Social Forces, 63:482–501, 1984.

[25] John Levi Martin. Social Structures. PrincetonUniversity Press, Princeton, 2009.

[26] S. Maslov and K. Sneppen. Specificity and sta-bility in topology of protein networks. Science,296(5569):910–913, 2002.

[27] Theodore M. Newcomb. Interpersonal balance.In R.P. Abelson, E. Aronson, W.J. McGiure,T.M. Newcomb, M.J. Rosenberg, and O.H. Tan-nenbaum, editors, Theories of Cognitive Con-sistency. Holt, Rinchart & Winston, New York,1968.

[28] Theodore M. Newcomb. Reciprocity of interper-sonal attraction: a non-confirmation of a plau-sible hypothesis. Social Psychology Quarterly,42:299–306, 1979.

[29] T.M. Newcomb. The acquaintance process. Holt,Rinehart and Winston, New York, 1961.

[30] M. E. J. Newman and J. Park. Why social net-works are different from other types of networks.Physical Review E, 68(3):36122, 2003.

[31] J. Skvoretz and F. Agneessens. Reciprocity, mul-tiplexity, and exchange: Measures. Quality andQuantity, 41(3):341–357, 2007.

[32] Willard Waller. The rating and dating com-plex. American Sociological Review, 2(5):727–734, 1937.

[33] Stanley Wasserman and Katherine Faust. So-cial Network Analysis: Methods and Applica-tions. Cambridge University Press, Cambridge,1994.

[34] Barry Wellman and Scot Wortley. Differentstrokes from different folks: Community ties andsocial support. American Journal of Sociology,96:558–588, 1990.

[35] SH Yook, H. Jeong, A.L. Barabasi, and Y. Tu.Weighted evolving networks. Physical ReviewLetters, 86(25):5835–5838, 2001.

[36] Gorka Zamora-Lopez, Vinko Zlatic, ChangsongZhou, Hrvoje Stefancic, and Jurgen Kurths.Reciprocity of networks with degree correlationsand arbitrary degree sequences. Physical Re-view E (Statistical, Nonlinear, and Soft MatterPhysics), 77(1):016106, 2008.

9