Structural transition in social networks: the role of homophily Y. Murase, H.-H. Jo, J. Torok, J. Kertesz, K. Kaski RIKEN R-CCS, APCTP, BME, CEU, Aalto Univ. ref. Y. Murase et al., Scientific Reports 9, 4310 (2019)

"Stylized facts" in social networks • echo chamber • segregation • perception bias • spread of fake news A B 1 100 10 • (overlapping, hierarhical) communities • the strength of weak ties (Granovetterian structure) • homophily • bursty dynamics dynamics on networks J. P. Onnela et al., Proc. Nat. Acad. Sci, 104, 7332 (2007)

Basic tie-formation mechanisms in social networks Homophily: The tendency of “like to associate with like” is one of the most striking and robust empirical regularities of social life. All tend to be more similar to each other with respect to a variety of dimensions, including race, age, gender, socioeconomic status, and education. It is related to segregation, inequality, social mobility. Cyclic closure: The tendency of nodes in social networks to make links with a topologically close nodes, i.e., people often form social ties with a person sharing common friends. a key inducing factor of community structures Kossinets & Watts "Origins of Homophily in an Evolving Social Network", Amer. J. Soc. (2009) Kossinets & Watts "Empirical Analysis of an Evolving Social Network", Science (2006)

choice homophily & observed homophily "The dynamic interplay of choice homophily and induced homophily, compounded over many “generations” of biased selection of similar individuals to structurally proximate positions, can amplify even a modest preference for similar others, via a cumulative advantage–like process, to produce striking patterns of observed homophily." "To what extent can observed patterns of homophily be attributed to individual preferences and structural constraints? A thorough answer to this question would require the use of simulation models, in which choice homophily as well as focal and cyclic closure biases could be systematically varied." Kossinets & Watts "Origins of Homophily in an Evolving Social Network", Amer. J. Soc. (2009) sim ilar similar

Weighted Social Network (WSN) model J. M. Kumpula et al., Phys. Rev. Lett., 99, 228701 (2007) (1) Local Attachment (2) Global Attachment (3) Link Deletion +δ +δ w0 +δ +δ +δ Node i chooses one of its neighbors j with probability proportional to wij . Then, node j chooses one of its neighbors except i, say k, with probability proportional to wjk . If node i and k are not connected, create a new link with pΔ. Weight of these links are increased by δ. with probability pr , node i is connected to a randomly chosen node. w0 With probability pd , a link is deleted. undirected weighted network of N nodes. The links in the networks are updated by the following three rules.

a minimal model to reproduce Granovetterian community structure

homophilic WSN model Each node i has a vector of F components. Each feature can take q different values. 1 i , 2 i , . . . , F i AAACI3icbZDLSgMxFIYzXmu9VV26CRahgpSZKiiuioK4rGAv0BmHTJqZhmYuJGeEMvRd3PgqblwoxY0L38X0AtbWHwIf/zknyfm9RHAFpvllLC2vrK6t5zbym1vbO7uFvf2GilNJWZ3GIpYtjygmeMTqwEGwViIZCT3Bml7vZlRvPjGpeBw9QD9hTkiCiPucEtCWW7iyBfOhhG3Fg5C4/NE6/eWK5k4MasbKbgfYljzowolbKJplcyy8CNYUimiqmlsY6ttoGrIIqCBKtS0zAScjEjgVbJC3U8USQnskYG2NEQmZcrLxjgN8rJ0O9mOpTwR47M5OZCRUqh96ujMk0FXztZH5X62dgn/pZDxKUmARnTzkpwJDjEeB4Q6XjILoayBUcv1XTLtEEgo61rwOwZpfeREalbJ1Vq7cnxer19M4cugQHaESstAFqqI7VEN1RNEzekXv6MN4Md6MofE5aV0ypjMH6I+M7x8eu6NZ {1, . . . , q} AAAB9HicbVDLSsNAFL2pr1pfVZdugkVwUUpSBV0W3bisYB/QhDKZTtuhk0k6c1Mood/hxoUibv0Yd/6N0zYLbT0wcDjnHu6dE8SCa3Scbyu3sbm1vZPfLeztHxweFY9PmjpKFGUNGolItQOimeCSNZCjYO1YMRIGgrWC0f3cb02Y0jySTziNmR+SgeR9TgkayfdSt+z1ItTlsTfrFktOxVnAXiduRkqQod4tfpksTUImkQqidcd1YvRTopBTwWYFL9EsJnREBqxjqCQh0366OHpmXxilZ/cjZZ5Ee6H+TqQk1HoaBmYyJDjUq95c/M/rJNi/9VMu4wSZpMtF/UTYGNnzBuweV4yimBpCqOLmVpsOiSIUTU8FU4K7+uV10qxW3KtK9fG6VLvL6sjDGZzDJbhwAzV4gDo0gMIYnuEV3qyJ9WK9Wx/L0ZyVZU7hD6zPHzt5kcA= F: social complexity of the population: the larger F implies the greater number of cultural characteristics that are attributable to each individual. q: heterogeneity of the population: The larger q means the greater number of cultural options in the society. We use same q for all features for simplicity. c.f. Axelrod model (e.g. gender, ethnicity, language, political stance, religion, etc.) Y. Murase et al., Sci.Rep.(2019)

(2,3,1) ….. (a) Global Attachment i (*,3,*) (2,3,1) i (1,3,1) (3,3,2) (2,1,3) j (2,3,2) k (3,1,2) (b) Local Attachment The feature values or traits of the nodes are initially chosen to be uniformly random and then kept unchanged. (c.f. Axelrod model, bounded confidence model) The network is updated similarly to the simple WSN model but the features of nodes are taken into account. A feature f of the focal node is randomly chosen from F features and it can make links only to the nodes sharing the same trait for the feature f.

Segregated phase & overlapping phase segregation occurs even without updating features when F is low (c.f. opinion dynamics models, such as Axelrod model, bounded confidence model) F=2 F=4 http://yohm.github.io/p5js_simulations/wsn_homophily/

Network properties 15 20 25 30 35 40 45 50 55 1 3 5 7 9 11 13 average degree F q = 10 7 5 3 2 WSN 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 3 5 7 9 11 13 clustering coefficient F q = 10 7 5 3 2 WSN (a) (b) An anomaly is observed at Fc , indicating a second-order like phase transition. 1 2 3 4 5 6 7 8 1 3 5 7 9 11 13 # of communities per node F q=10 7 5 3 2

15 20 25 30 35 40 45 50 55 1 3 5 7 9 11 13 average degree F q = 10 7 5 3 2 WSN (a) ( Spreading on networks 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 q=5 F=8 F=2 fraction of infected nodes time simulation of SI model

feature overlap 0 0.2 0.4 0.6 0.8 1 1 3 5 7 9 11 13 null model for q=5 without LA wr = 0 feature overlap F q=10 7 5 3 2 oF (i, j) = 1 F F X f=1 ( f i , f j ) 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

(a) (b) (c) (d) ego-centric networks q < qc AAAB7nicbVA9SwNBEJ2LXzF+RS1tFoNgFe6ioIVF0MYygvmA5Ah7m7lkyd7eZXdPCCE/wsZCEVt/j53/xk1yhSY+GHi8N8PMvCARXBvX/XZya+sbm1v57cLO7t7+QfHwqKHjVDGss1jEqhVQjYJLrBtuBLYShTQKBDaD4d3Mbz6h0jyWj2acoB/RvuQhZ9RYqTkiN2TUZd1iyS27c5BV4mWkBBlq3eJXpxezNEJpmKBatz03Mf6EKsOZwGmhk2pMKBvSPrYtlTRC7U/m507JmVV6JIyVLWnIXP09MaGR1uMosJ0RNQO97M3E/7x2asJrf8JlkhqUbLEoTAUxMZn9TnpcITNibAllittbCRtQRZmxCRVsCN7yy6ukUSl7F+XKw2WpepvFkYcTOIVz8OAKqnAPNagDgyE8wyu8OYnz4rw7H4vWnJPNHMMfOJ8/TbCO5A== q > qc AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKexGQU8S9OIxgnlAsoTZSW8yZHZ2MzMrhJCP8OJBEa9+jzf/xkmyB00saCiquunuChLBtXHdbye3tr6xuZXfLuzs7u0fFA+PGjpOFcM6i0WsWgHVKLjEuuFGYCtRSKNAYDMY3s385hMqzWP5aMYJ+hHtSx5yRo2VmiNyQ0Zd1i2W3LI7B1klXkZKkKHWLX51ejFLI5SGCap123MT40+oMpwJnBY6qcaEsiHtY9tSSSPU/mR+7pScWaVHwljZkobM1d8TExppPY4C2xlRM9DL3kz8z2unJrz2J1wmqUHJFovCVBATk9nvpMcVMiPGllCmuL2VsAFVlBmbUMGG4C2/vEoalbJ3Ua48XJaqt1kceTiBUzgHD66gCvdQgzowGMIzvMKbkzgvzrvzsWjNOdnMMfyB8/kDUMCO5g== q ⇡ qc AAAB9HicbVBNSwMxEJ31s9avqkcvwSJ4KrtV0GPRi8cK9gPapWTTbBuaTdIkWyylv8OLB0W8+mO8+W9M2z1o64OBx3szzMyLFGfG+v63t7a+sbm1ndvJ7+7tHxwWjo7rRqaa0BqRXOpmhA3lTNCaZZbTptIUJxGnjWhwN/MbI6oNk+LRjhUNE9wTLGYEWyeFQ9TGSmn5hIYd0ikU/ZI/B1olQUaKkKHaKXy1u5KkCRWWcGxMK/CVDSdYW0Y4nebbqaEKkwHu0ZajAifUhJP50VN07pQuiqV2JSyaq78nJjgxZpxErjPBtm+WvZn4n9dKbXwTTphQqaWCLBbFKUdWolkCqMs0JZaPHcFEM3crIn2sMbEup7wLIVh+eZXUy6XgslR+uCpWbrM4cnAKZ3ABAVxDBe6hCjUgMIRneIU3b+S9eO/ex6J1zctmTuAPvM8fZF+R2g== F = 4 AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KolU9CIUBfFYwX5AG8pmu2mXbjZhdyKU0N/gxYMiXv1B3vw3btsctPXBwOO9GWbmBYkUBl3321lZXVvf2CxsFbd3dvf2SweHTROnmvEGi2Ws2wE1XArFGyhQ8naiOY0CyVvB6Hbqt564NiJWjzhOuB/RgRKhYBSt1Lgj16TaK5XdijsDWSZeTsqQo94rfXX7MUsjrpBJakzHcxP0M6pRMMknxW5qeELZiA54x1JFI278bHbshJxapU/CWNtSSGbq74mMRsaMo8B2RhSHZtGbiv95nRTDKz8TKkmRKzZfFKaSYEymn5O+0JyhHFtCmRb2VsKGVFOGNp+iDcFbfHmZNM8rXrVy8VAt127yOApwDCdwBh5cQg3uoQ4NYCDgGV7hzVHOi/PufMxbV5x85gj+wPn8AT8Zjas=

phase diagram ’-’ 1 3 5 7 9 F 100 101 102 q 0 0.2 0.4 0.6 0.8 1 segregated phase overlapping phase

0 10 20 30 40 50 100 101 102 103 104 105 =N/qF average degree N/qF N=5000, F=3 N=50000, F=3 N=50000, F=4 ’-’ 1 3 5 7 9 F 100 101 102 q 0 0.2 0.4 0.6 0.8 1 segregated phase overlapping phase ¯ N ⇡ N/qF AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrOVEGXRUFclQr2AZ2xZNK0Dc1MYpIRyzAbf8WNC0Xc+hnu/BvTdhbaeuDC4Zx7ufeeQDCqtON8W7mFxaXllfxqYW19Y3PL3t5pKB5LTOqYMy5bAVKE0YjUNdWMtIQkKAwYaQbDy7HffCBSUR7d6pEgfoj6Ee1RjLSROvaeFyCZVFPoISEkf4RVeAzv7646dtEpORPAeeJmpAgy1Dr2l9flOA5JpDFDSrVdR2g/QVJTzEha8GJFBMJD1CdtQyMUEuUnkwdSeGiULuxxaSrScKL+nkhQqNQoDExniPRAzXpj8T+vHeveuZ/QSMSaRHi6qBczqDkcpwG7VBKs2cgQhCU1t0I8QBJhbTIrmBDc2ZfnSaNcck9K5ZvTYuUiiyMP9sEBOAIuOAMVcA1qoA4wSMEzeAVv1pP1Yr1bH9PWnJXN7II/sD5/ACP2lXQ= scaling relation number of nodes having identical set of features Each node has a strong tendency to connect to matching nodes as long as a matching node can be found. The nodes compromise with partially matching nodes as few matching nodes exist in the whole population.

key ingredients of the phase transition 0 0.2 0.4 0.6 0.8 1 1 3 5 7 9 11 13 null model for q=5 without LA wr = 0 feature overlap F q=10 7 5 3 2 null model: links are randomly created between nodes i and j with a probability proportional to oF (i, j). hoF i = 1 q + 1 F ✓ 1 1 q ◆ 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 no phase-transition model without link reinforcement / model without cyclic closure : no phase-transition homophilic link formation + local attachment mechanism key ingredients of the phase-transition

Local attachment amplifies homophily LA promotes a much stronger preference in link formation between matching pairs than predicted by the null model. +δ +δ w0 +δ +δ +δ higher feature overlap links are more likely to be reinforced by LA triadic closures are more likely to be occur between matching nodes strong links are even more likely to be reinforced The above effect is significantly weakened in overlapping phase. differ in 1 feature differ in 1 feature may differ in 2 features

implications • What is the relevance of this finding, considering that in reality every person has a large F? • There are cultures and situations, in which a few features get extreme importance, e.g. sharpening political situations, turmoils, wars, or online social groups, where people get into contact based on very few features. • "Effective" F can be quite low, leading to segregation and echo- chambers. • Perceiving the complexity of the world (increasing F) or allowing for more subtle opinions (increasing q) can break up these bubbles. Photo by roya ann miller on Unsplash Photo by Charles Deluvio on Unsplash

Conclusion • We have generalized the weighted social network model to incorporate homophily. The model shows a phase transition: segregated phase (FFc ). • Social segregation may be intensified by a joint effect between choice homophily, cyclic closure and link reinforcement. (unlike previous studies on Schelling model or opinion dynamics models). • Increasing effective F or q can be a key to resolve segregation in society.

Y. Murase et al., Scientific Reports 9, 4310 (2019)