Deep Learning Exploration of Agent- Based Social Network Model Parameters Yohsuke Murase, RIKEN Center for Computational Science Collaborators: Hang-Hyun Jo, The Catholic University of Korea Janos Torok, Budapest University of Technology and Economics Janos Kertesz, Central European University Kimmo Kaski, Aalto University

stylized facts in social networks J. P. Onnela et al., Proc. Nat. Acad. Sci, 104, 7332 (2007) • Broad distribution of degree, strength, and weight • Assortative mixing • high clustering and high modularity • Granovetter-type community structure H.-H. Jo, Y. Murase, J. Torok, J. Kertesz, K. Kaski Physica A (2018) V. Palchykov et al.,Sci. Rep, 2, 370(2012)

modeling networks static models dynamic models random networks with certain constraints • Erdos-Renyi random graph • configuration model • exponential random graph model • stochastic block model nodes and links are dynamically updated (agent-based models) • Barabasi-Albert model • weighted social network model analytically solvable serve as null models to test hypothesis getting insight into how and why the observed networks have been generated predicting the possible evolution of real networks

difficulties in dynamic models • In general, dynamic models are analytically unsolvable and computationally demanding. • Dynamic models are usually designed as simple as possible to identify the most important mechanism. However, simplistic models are not suitable for quantitative comparison and prediction. • When models are extended to incorporate aspects of reality, understanding their behavior becomes a formidable task since the parameter space is high-dimensional and non-trivial interactions between the parameters may occur. Use simulation results as training data We are going to overcome this difficulty by a large-scale computing & meta-modeling. Simulation Results Input Parameters Simulation Model Regression Model Unknown Input Parameters Prediction Input Parameters Input Parameters Input Parameters Input Parameters Input Parameters Simulation Results Simulation Results Simulation Results Simulation Results Simulation Results useful for understanding model behavior, sensitivity analysis, and parameter tuning

Weighted Social Network model (1) Local Attachment (2) Global Attachment (3) Node 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 , node i loses all of its links. J. M. Kumpula et al., Phys. Rev. Lett., 99, 228701 (2007) undirected weighted network of N nodes. The links in the networks are updated by the following three rules. a minimal agent-based model for social networks

WSN model reproduces stylized facts Intra-communy links are strong while Inter-community links are weak 100 101 102 10 10 6 10 4 10 2 100 100 102 104 106 108 10 10 12 10 10 10 8 10 6 10 4 10 2 vi vj Oij=0 Oij=1/3 Oij=1 Oij=2/3 A B w , b 0.1 0.15 0.2 C D Degree k Link weight w (s) P(k) P(w) A B 1 100 10 J. P. Onnela et al., Proc. Nat. Acad. Sci, 104, 7332 (2007) The strength of weak ties (M. Granovetter, 1973) Hypothesis about the relation between link topology and weight. “The stronger the tie between A and B, the larger proportion of individuals S to whom both are tied.” • assortative mixing • high clustering and community structure • decreasing C(k) • Granovetterian community structure WSN model empirical network WSN model without link reinforcement

extensions of the WSN model Layer 1 Layer 2 Layer 1+2 PLoS ONE (2011) “Multiplex Modeling of the Society" J. Kertesz et al. (2017) Plos One (2015) Scientific Reports (2019) multi-layer network temporality termination of links homophily

extension of the WSN model (1) In reality, the termination of a relationship may occur in various ways. https://flic.kr/p/6kkouX breaking up https://flic.kr/p/5NCHBx leaving https://flic.kr/p/dfnh18 fading How do such differences affect the emergent network properties? Node Deletion (ND) With probability pnd , node i loses all of its links. Link Deletion (LD) With probability pld , a link is removed. Link Aging (Aging) At each time step, weights are multiplied by f. A link is removed if weight is less than wth . termination of links

(a) ND (b) LD (c) aging all these models show the Granovetterian structure lower modularity excessively high modularity even lower modularity weak and strong links coexist in communities weak and strong links coexist in each community homogeneous link weights in each community Link Deletion Node Deletion Link Aging Q=0.93 Q=0.56 Q=0.99 The link deletion model reproduces the empirical networks the best, implying that sudden break up may play a major role in our society

extension to the WSN model (2) Homophily: The tendency of forming ties to similar ones. (2,3,1) (2,3,1) i (2,1,3) (2,3,2) (3,1,2) ….. (a) Global Attachment (b) Local Attachment i (*,3,*) j AAACEXicbVC7TsMwFHV4lvIKMLJYVEhdqBJAwFhggLFI9CE1IXJctzW1E8t2kKoov8DCr7AwgBArGxt/g9tmgJYjWTo6515dnxMKRpV2nG9rbn5hcWm5sFJcXVvf2LS3thsqTiQmdRyzWLZCpAijEalrqhlpCUkQDxlphoPLkd98IFLROLrVQ0F8jnoR7VKMtJECuyyClN5nd6knObw6z6AnZCx0DGWuH3iIiT7KArvkVJwx4Cxxc1ICOWqB/eV1YpxwEmnMkFJt1xHaT5HUFDOSFb1EEYHwAPVI29AIcaL8dJwog/tG6cBuLM2LNByrvzdSxJUa8tBMcqT7atobif957UR3z/yURiLRJMKTQ92EQZN4VA/sUEmwZkNDEJbU/BXiPpIIa1Ni0ZTgTkeeJY3DintSObo5LlUv8joKYBfsgTJwwSmogmtQA3WAwSN4Bq/gzXqyXqx362MyOmflOzvgD6zPHx8jndc= pGA ij / r ↵ ij AAAB63icbVBNSwMxEJ2tX7V+VT16CRZBEMquFfVY9OKxgv2AdinZNNuGJtklySpl6V/w4kERr/4hb/4bs+0etPXBwOO9GWbmBTFn2rjut1NYWV1b3yhulra2d3b3yvsHLR0litAmiXikOgHWlDNJm4YZTjuxolgEnLaD8W3mtx+p0iySD2YSU1/goWQhI9hk0tlTX/XLFbfqzoCWiZeTCuRo9MtfvUFEEkGlIRxr3fXc2PgpVoYRTqelXqJpjMkYD2nXUokF1X46u3WKTqwyQGGkbEmDZurviRQLrScisJ0Cm5Fe9DLxP6+bmPDaT5mME0MlmS8KE45MhLLH0YApSgyfWIKJYvZWREZYYWJsPCUbgrf48jJpnVe9y2rt/qJSv8njKMIRHMMpeHAFdbiDBjSBwAie4RXeHOG8OO/Ox7y14OQzh/AHzucP1d+OHQ== +wr AAAB63icbVBNSwMxEJ2tX7V+VT16CRZBEMquFfVY9OKxgv2AdinZNNuGJtklySpl6V/w4kERr/4hb/4bs+0etPXBwOO9GWbmBTFn2rjut1NYWV1b3yhulra2d3b3yvsHLR0litAmiXikOgHWlDNJm4YZTjuxolgEnLaD8W3mtx+p0iySD2YSU1/goWQhI9hk0tlTX/XLFbfqzoCWiZeTCuRo9MtfvUFEEkGlIRxr3fXc2PgpVoYRTqelXqJpjMkYD2nXUokF1X46u3WKTqwyQGGkbEmDZurviRQLrScisJ0Cm5Fe9DLxP6+bmPDaT5mME0MlmS8KE45MhLLH0YApSgyfWIKJYvZWREZYYWJsPCUbgrf48jJpnVe9y2rt/qJSv8njKMIRHMMpeHAFdbiDBjSBwAie4RXeHOG8OO/Ox7y14OQzh/AHzucP1d+OHQ== +wr AAAB8XicbVBNS8NAEN3Ur1q/qh69LBbBU0lU1GNRDx4r2A9sQ9lsJ+3SzSbsToQS+i+8eFDEq//Gm//GbZuDVh8MPN6bYWZekEhh0HW/nMLS8srqWnG9tLG5tb1T3t1rmjjVHBo8lrFuB8yAFAoaKFBCO9HAokBCKxhdT/3WI2gjYnWP4wT8iA2UCAVnaKWHpJd1b0Aim/TKFbfqzkD/Ei8nFZKj3it/dvsxTyNQyCUzpuO5CfoZ0yi4hEmpmxpIGB+xAXQsVSwC42eziyf0yCp9GsbalkI6U39OZCwyZhwFtjNiODSL3lT8z+ukGF76mVBJiqD4fFGYSooxnb5P+0IDRzm2hHEt7K2UD5lmHG1IJRuCt/jyX9I8qXrn1dO7s0rtKo+jSA7IITkmHrkgNXJL6qRBOFHkibyQV8c4z86b8z5vLTj5zD75BefjG7VAkPM= p (3,3,2) (1,3,1) j l {1, . . . , q} AAAB9HicbVDLSsNAFL2pr1pfVZdugkVwUUpSBV0W3bisYB/QhDKZTtuhk0k6c1Mood/hxoUibv0Yd/6N0zYLbT0wcDjnHu6dE8SCa3Scbyu3sbm1vZPfLeztHxweFY9PmjpKFGUNGolItQOimeCSNZCjYO1YMRIGgrWC0f3cb02Y0jySTziNmR+SgeR9TgkayfdSt+z1ItTlsTfrFktOxVnAXiduRkqQod4tfpksTUImkQqidcd1YvRTopBTwWYFL9EsJnREBqxjqCQh0366OHpmXxilZ/cjZZ5Ee6H+TqQk1HoaBmYyJDjUq95c/M/rJNi/9VMu4wSZpMtF/UTYGNnzBuweV4yimBpCqOLmVpsOiSIUTU8FU4K7+uV10qxW3KtK9fG6VLvL6sjDGZzDJbhwAzV4gDo0gMIYnuEV3qyJ9WK9Wx/L0ZyVZU7hD6zPHzt5kcA= (2,3,1) i Each node has a feature vector of F components. Each feature can take q different values. sim ilar similar homophily & cyclic closure may jointly amplify the similarity between neighbors homophily Scientific Reports (2019)

low F/low q: segregated phase high F/high q: overlapping phase (a) (b) (c) (d) (a) (b) (c) (d) 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 What induces the structural transition is joint effect of homophilic interaction & triadic closure. When people are focusing only on a few features (such as political stance during political turmoil), the society tends to get segregated. second-order like transition

extensions cause non-trivial consequences While these extensions have been independently studied for simplicity, these are not exclusive and work simultaneously in reality. We study "Generalized Weighted Social Network (GWSN) model", which incorporates all these extensions.

Generalized Weighted Social Network model Dependence on the parameters are highly non-linear and non-trivial. It is not easy to fully explore the parameter- space.

meta-modeling Use simulation results as training data Simulation Results Input Parameters Simulation Model Predictive Model Unknown Input Parameters Prediction Input Parameters Input Parameters Input Parameters Input Parameters Input Parameters Simulation Results Simulation Results Simulation Results Simulation Results Simulation Results meta-model, surrogate model, emulators Meta-modeling: An engineering method to approximate the input/output relation of the simulation model. A meta-model is useful for parameter tuning, parameter-space exploration, sensitivity analysis, and what-if analysis. supercomputer Fugaku By Hiroko Hama - Own work, CC BY-SA 4.0, https:// commons.wikimedia.org/w/index.php?curid=102551957 • 4k nodes (192k CPU cores) • OpenMP & MPI parallelization • 8 hours • 500k simulation runs • We used multi-layer perceptron

regression results (a) (b) (d) (g) (e) (h) (c) (f) (i) meta-model prediction simulation output

interactive plot https://yohm.github.io/GWSN_metamodeling/

sensitivity analysis (a) (b) (d) (g) (e) AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqewWUY9FLx4r2A9ol5JNs21sNlmSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvTAQ31vO+UWFtfWNzq7hd2tnd2z8oHx61jEo1ZU2qhNKdkBgmuGRNy61gnUQzEoeCtcPx7cxvPzFtuJIPdpKwICZDySNOiXVSq6dHqj/ulyte1ZsDrxI/JxXI0eiXv3oDRdOYSUsFMabre4kNMqItp4JNS73UsITQMRmyrqOSxMwE2fzaKT5zygBHSruSFs/V3xMZiY2ZxKHrjIkdmWVvJv7ndVMbXQcZl0lqmaSLRVEqsFV49joecM2oFRNHCNXc3YrpiGhCrQuo5ELwl19eJa1a1b+s1u4vKvWbPI4inMApnIMPV1CHO2hAEyg8wjO8whtS6AW9o49FawHlM8fwB+jzB6Gsjyw= ⇢k AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbVqEciF4+QyCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoJRdea3nlBpHssHM07Qj+hA8pAzaqxUr/aKJbfszkFWiZeREmSo9Ypf3X7M0gilYYJq3fHcxPgTqgxnAqeFbqoxoWxEB9ixVNIItT+ZHzolZ1bpkzBWtqQhc/X3xIRGWo+jwHZG1Az1sjcT//M6qQlv/QmXSWpQssWiMBXExGT2NelzhcyIsSWUKW5vJWxIFWXGZlOwIXjLL6+S5kXZuy5f1q9KlbssjjycwCmcgwc3UIF7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AJiBjM8= C AAAB6HicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY9BL95MwDwgWcLspDcZMzu7zMwKIeQLvHhQxKuf5M2/cZLsQRMLGoqqbrq7gkRwbVz328mtrW9sbuW3Czu7e/sHxcOjpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGtzO/9YRK81g+mHGCfkQHkoecUWOl+n2vWHLL7hxklXgZKUGGWq/41e3HLI1QGiao1h3PTYw/ocpwJnBa6KYaE8pGdIAdSyWNUPuT+aFTcmaVPgljZUsaMld/T0xopPU4CmxnRM1QL3sz8T+vk5rw2p9wmaQGJVssClNBTExmX5M+V8iMGFtCmeL2VsKGVFFmbDYFG4K3/PIqaVbK3mW5Ur8oVW+yOPJwAqdwDh5cQRXuoAYNYIDwDK/w5jw6L86787FozTnZzDH8gfP5A6pfjNo= O AAAB8HicbVDLSgNBEOz1GeMr6tHLYBA8hd0g6jHoxWME85BkCbOT2WTIPJaZWSUs+QovHhTx6ud482+cJHvQxIKGoqqb7q4o4cxY3//2VlbX1jc2C1vF7Z3dvf3SwWHTqFQT2iCKK92OsKGcSdqwzHLaTjTFIuK0FY1upn7rkWrDlLy344SGAg8kixnB1kkPXT1UvUw9TXqlsl/xZ0DLJMhJGXLUe6Wvbl+RVFBpCcfGdAI/sWGGtWWE00mxmxqaYDLCA9pxVGJBTZjNDp6gU6f0Uay0K2nRTP09kWFhzFhErlNgOzSL3lT8z+ukNr4KMyaT1FJJ5ovilCOr0PR71GeaEsvHjmCimbsVkSHWmFiXUdGFECy+vEya1UpwUanenZdr13kcBTiGEziDAC6hBrdQhwYQEPAMr/Dmae/Fe/c+5q0rXj5zBH/gff4AS8eQvQ== ⇢ow (c) AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKewGUY9BLx4jmIckS5idzCZD5rHMzAphyVd48aCIVz/Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNaJMornQnwoZyJmnTMstpJ9EUi4jTdjS+nfntJ6oNU/LBThIaCjyULGYEWyc99vRI9TMynvbLFb/qz4FWSZCTCuRo9MtfvYEiqaDSEo6N6QZ+YsMMa8sIp9NSLzU0wWSMh7TrqMSCmjCbHzxFZ04ZoFhpV9Kiufp7IsPCmImIXKfAdmSWvZn4n9dNbXwdZkwmqaWSLBbFKUdWodn3aMA0JZZPHMFEM3crIiOsMbEuo5ILIVh+eZW0atXgslq7v6jUb/I4inACp3AOAVxBHe6gAU0gIOAZXuHN096L9+59LFoLXj5zDH/gff4AJ0OQpQ== ⇢ck (f) (h) (i) AAACA3icbZDLSsNAFIZP6q3WW9WdbgaLUBeWREVdFt24rGAv0MQwmU7aoZNJmJkIpRTc+CpuXCji1pdw59s4bbPQ1h8GPv5zDmfOHyScKW3b31ZuYXFpeSW/Wlhb39jcKm7vNFScSkLrJOaxbAVYUc4ErWumOW0lkuIo4LQZ9K/H9eYDlYrF4k4PEupFuCtYyAjWxvKLe2XnOPTJPT5yORZdTlEfuXJCfrFkV+yJ0Dw4GZQgU80vfrmdmKQRFZpwrFTbsRPtDbHUjHA6KripogkmfdylbYMCR1R5w8kNI3RonA4KY2me0Gji/p4Y4kipQRSYzgjrnpqtjc3/au1Uh5fekIkk1VSQ6aIw5UjHaBwI6jBJieYDA5hIZv6KSA9LTLSJrWBCcGZPnofGScU5r5zenpWqV1kcediHAyiDAxdQhRuoQR0IPMIzvMKb9WS9WO/Wx7Q1Z2Uzu/BH1ucPLDyWnA== (1 fa c )hki AAACA3icbZDLSsNAFIYn9VbrLepON4NFqAtLoqIui25cVrAXaGKYTCbt0MkkzEyEEgpufBU3LhRx60u4822cpllo6w8DH/85hzPn9xNGpbKsb6O0sLi0vFJeraytb2xumds7bRmnApMWjlksuj6ShFFOWooqRrqJICjyGen4w+tJvfNAhKQxv1OjhLgR6nMaUoyUtjxzr2Yfhx6+D44chnifETiEjsjJM6tW3coF58EuoAoKNT3zywlinEaEK8yQlD3bSpSbIaEoZmRccVJJEoSHqE96GjmKiHSz/IYxPNROAMNY6McVzN3fExmKpBxFvu6MkBrI2drE/K/WS1V46WaUJ6kiHE8XhSmDKoaTQGBABcGKjTQgLKj+K8QDJBBWOraKDsGePXke2id1+7x+entWbVwVcZTBPjgANWCDC9AAN6AJWgCDR/AMXsGb8WS8GO/Gx7S1ZBQzu+CPjM8fMP6Wnw== (1 fd c )hki AAAB+3icbZDLSsNAFIZPvNZ6i3XpZrAIrkqioi6LblxWsBdoQplMJ+3QySTMTMQS+ipuXCji1hdx59s4SbPQ1h8GPv5zDufMHyScKe0439bK6tr6xmZlq7q9s7u3bx/UOipOJaFtEvNY9gKsKGeCtjXTnPYSSXEUcNoNJrd5vftIpWKxeNDThPoRHgkWMoK1sQZ2zeNYjDhFE+TJgqoDu+40nEJoGdwS6lCqNbC/vGFM0ogKTThWqu86ifYzLDUjnM6qXqpogskEj2jfoMARVX5W3D5DJ8YZojCW5gmNCvf3RIYjpaZRYDojrMdqsZab/9X6qQ6v/YyJJNVUkPmiMOVIxygPAg2ZpETzqQFMJDO3IjLGEhNt4spDcBe/vAyds4Z72Ti/v6g3b8o4KnAEx3AKLlxBE+6gBW0g8ATP8Apv1sx6sd6tj3nrilXOHMIfWZ8/90aTwA== hki AAAB+3icbZDLSsNAFIZP6q3WW6xLN4NFcFUSFXVZdOOygr1AE8pkOm2HTiZhZqKW0Fdx40IRt76IO9/GSZqFtv4w8PGfczhn/iDmTGnH+bZKK6tr6xvlzcrW9s7unr1fbasokYS2SMQj2Q2wopwJ2tJMc9qNJcVhwGknmNxk9c4DlYpF4l5PY+qHeCTYkBGsjdW3qx7HYsQpekSezKnSt2tO3cmFlsEtoAaFmn37yxtEJAmp0IRjpXquE2s/xVIzwums4iWKxphM8Ij2DAocUuWn+e0zdGycARpG0jyhUe7+nkhxqNQ0DExniPVYLdYy879aL9HDKz9lIk40FWS+aJhwpCOUBYEGTFKi+dQAJpKZWxEZY4mJNnFlIbiLX16G9mndvaif3Z3XGtdFHGU4hCM4ARcuoQG30IQWEHiCZ3iFN2tmvVjv1se8tWQVMwfwR9bnDwnxk8w= hwi variance-based sensitivity analysis Sensitivity analysis is a statistical method to evaluate which input parameters are important and which are not. The variance-based method is effective for non-linear and non-additive systems. Saltelli et al., "Global Sensitivity Analysis" first-order index: total effect index:

conclusion • We studied the formation of social network using an agent-based GWSN model, which incorporates several realistic extensions. • traditional approach: build first a simple model and add elements one by one to study more realistic models. • our approach: started from a generalized model and then studied its behavior using large- scale computation and meta-modeling. • Although GWSN model is another simplistic model considering the complexity of the reality, we opened a way for more realistic quantitative agent-based modeling.

reference https://doi.org/10.3389/fdata.2021.739081