Modeling group formation in society:The networked seceder model
January 14, 2003
Petter Holme &Andreas GrönlundDepartment of Physics,Umeå University, SwedenNORDITA, Copenhagen,Denmark
e-print cond-mat/0312010
LARGE SCALE SOCIAL NETWORK DATA
Affiliation networks
Actor collaborations. N = 449913, M = 25516482 (Watts)
Scientific collaborations. N = 52909− 1520251, M = 245300− 11803064 (Newman)
Music collaborations. N = 1275 (Gleiser & Danon, de Lima e Silva et al.)
Board of directors. N = 7673, M = 55392 (Davis et al.)
Online interaction
E-mails. N = 1700− 59912, M = 15640− 86300 (Ebel et al., Newman et al., Guimerà et al.)
Instant messaging. N = 50259, M = 239452 (Smith)
Internet communities. N = 29341, M = 115684 (Holme et al.)
Phone calls. N = 47000000, M = 80000000 (Aiello et al.)
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Large scale social network data (continued)
Interviews
Student romance. N = 573, M = 477 (Bearman et al.)
Student friendship. N = 417 (Fararo et al.)
Sexual contacts. N = 2810 (Liljeros et al.)
Contact tracing. N ∼ 300 (Potterat et al.)
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STRUCTURE OF SOCIAL NETWORKS
Clustering coefficient
High in affiliation networks and interview based acquaintance networks. Neutral (= unbiased) inInternet communities, romantic and sexual networks.
Degree distribution
Power-law in telephone graphs, one e-mail study, sexual contacts. Truncated power-law / stretchedexponential in Internet communities, Instant messaging, one e-mail studies and all affiliation data.Exponential, Gaussian, Poissonian Two e-mail studies, acquaintance networks based on interviews.
Degree-degree correlations
Positive in affiliation networks and acquaintance networks. Neutral in Internet communities, In-stant messaging and one e-mail study.
Community (group) structure
Strong in affiliation networks, acquaintance networks, and (?) all other networks.
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NETWORK MODELS OF GROUP FORMATION
Skyrms and Freemantle, PNAS 97, 9340 (2000). Weighted network. The weight increase the moreactors interact. Actors with higher weights are more likely to interact.
Jin, Girvan and Newman, PRE 64, 046132 (2001). The degree cannot increase beyond a certain thresh-old. People with mutual friends are likely to get acquainted.
Motter, Nishikawa and Lai, PRE 68, 036105 (2003). The traits of a person are given by a n-dimesionalvector. Each dimension represent a characteristic social feature. Each dimension is hierarchically orga-nized. Two vertices that are closer than a threshold value (in a certain metric) get an edge.
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THE SECEDER MODEL
(Dittrich et al., 2000.) N individuals with a real number s(i) representing the traits of individual i.The algorithm is then to repeat the following steps:
1. Select three individuals i1, i2 and i3 with uniform randomness.
2. Pick the one (we call it i) of these whose s-value is farthest away from the average [s(i1) + s(i2) +s(i3)]/3.
3. Replace the s-value of a uniformly randomly chosen agent with s(i) + η, where η is a randomnumber from the normal distribution with mean zero and variance one.
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OUR NETWORKED SECEDER MODEL
Starting from any graph with N vertices and M edges we iterate the following steps:
(a) Select three different vertices i1, i2 and i3 with uniform randomness.
(b) Pick the one i of these that is least central in the following sense: If the graph is connected verticesof highest eccentricity are the least central. If the graph is disconnected the most eccentric verticeswithin the smallest connected subgraph are the least central. If more than one vertex is leastcentral, let i be a vertex in the set of least central vertices chosen uniformly randomly.
(a)
i1
i3i2
(b) i’
j
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Our networked seceder model (continued)
(c) Choose a vertex j by uniform randomness. If deg j 6 deg i + 1, rewire j’s edges to i and a randomselection of i’s neighbors. If deg j > deg i + 1, rewire j’s edges to i, i’s neighborhood and (ifdeg j > deg i + 1) to deg j − deg i− 1 randomly selected other vertices.
(d) Go through j’s edges once more and rewire these with a probability p to a randomly chosen vertex.
(d) i’jj i’(c)
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THE GROUP STRUCTURE
One realization of the networked seceder model. The model parameters are N = 50, M = 150 andp = 0. The indicated groups are identified with Newman’s clustering algorithm (cond-mat/0309508).This realization have modularity Q = 0. 575, clustering coefficient C = 0. 530, and assortative mixingcoefficient r = 0. 0456.
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The group structure (continued)
originalrandomized(a)
(b)
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1p
Q ′ = ∑s∈S
(ess − a2s )
where S is the set of subnetworks at a specific it-eration of the algorithm and ess′ is the fraction ofedges that goes between a vertex in s and a vertexin s′, and as = ∑s′ ess′.
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The group structure (continued)
originalrandomized
1000 2000 3000500
b
N
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The number of groups b as a function of the systems size N . The other parameter values are M = 3Nand p = 0. 1. The line is a fit to a power-law abβ.
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OTHER STRUCTURAL STATISTICS
k0 5 10 15 20 25
secederinitial
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k()
P
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Degree distribution of the networked seceder model. The model parameters are N = 1800, M =5400 and p = 0. 1. The squares indicate the degree distribution of a random graph with the sizes (Nand M ), i.e., the initial network before the iterations of the seceder model commence.
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Other structural statistics (continued)
−0.01
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(a) Clustering coefficient is finite. (b) Degree-degree correlations are positive.
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TIME EVOLUTION OF THE COMMUNITIES
+1
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Communities at consecutive time steps are identified by maximizing the overlap.
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Time evolution of the communities (continued)
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5075
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(e)
r (a)C (b)
Q (c)
y (d)
t
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