war pact network model : generative model of networks that shrink - PowerPoint PPT Presentation

war pact network model : generative model of networks that shrink Lovro ˇ joint work with Subelj Luka Nagli´ c University of Ljubljana Faculty of Computer and University of Zagreb Information Science Faculty of Science EUSN ’19 1/15

network models ( soa ) network models as baseline , explanation & generation ( existing ) majority for static or growing networks [ ER59 , Pri76 ] ( missing ) generative models of shrinking networks [ KNB08 ] ? ? ? static network growing network shrinking network [ ER59 ] Erd˝ os & R´ enyi (1959) On random graphs I. Publ. Math. Debrecen 6 , 290-297. [ Pri76 ] Price (1976) A general theory of bibliometric and other cumulative. . . J. Am. Soc. Inf. Sci. 27 (5), 292-306. [ KNB08 ] Kejˇ zar et al. (2008) Probabilistic inductive classes of graphs. J. Math. Sociol. 32 (2), 85-109. 2/15

shrinking models ( intuition ) entities / nodes often merge in real world / network ( which ) merged nodes / entities are random , hubs , isolates etc. two entities merged entity ( wars ) nations / alliances form pact or one occupies other • ( trade ) countries form alliance or companies after merger ( Bitcoin ) cryptocurrency addresses owned by same user ( Internet ) autonomous systems merge their traffic 3/15

war pact model ( model ) shrinking network with n nodes & m edges initial network first step second step final network ( initialize ) create perfect matching on 2 m nodes ( select ) select nodes at random, preferentially etc. ( shrink ) merge nodes by rewiring their edges ( loop ) continue until network has n nodes 4/15

model details ( shrink ) merging nodes at distance d creates d -cycle edge with d = 1 self-edge path of length d = 2 parallel edges path of length d = 3 triangle ( model ) war pact is parameter-free except n nodes & m edges ( initialize ) create perfect matching , random graph or tree ◦ ( select ) select nodes at random , by degree or degree − 1 • 5/15

model pseudocode input nodes n & edges m output graph G 1: H ← empty map ⊲ map of nodes’ hashes 2: G ← empty graph ⊲ empty war pact graph 3: for i ∈ [1 , m ] do H ( i ) ← i & H ( m + i ) ← m + i ⊲ map nodes to hashes 4: add nodes H ( i ) & H ( m + i ) to G ⊲ add nodes to graph 5: add edge { H ( i ) , H ( m + i ) } to G ⊲ add edges to graph 6: 7: while G has > n nodes do h ← random ( H ) ⊲ select random node 8: i ← random ([ 1 , 2m ]) ⊲ select node by degree 9: if h � = H ( i ) & edge { h , H ( i ) } / ∈ G then 10: merge nodes h & H ( i ) in G ⊲ merge selected nodes 11: H ( i ) ← h ⊲ unify nodes’ hashes 12: 13: return G 6/15

model networks ( layout ) node selection impacts ( modular ) structure [ Pei18 ] ( left ) both nodes are selected by degree ( middle ) nodes selected by degree & degree − 1 ( right ) nodes selected by degree & at random [ Pei18 ] Peixoto (2018) Bayesian stochastic blockmodeling. e-print arXiv:1705.10225v7 , 1-44. 7/15

model selection ( structure ) node selection impacts scale-free / small-world Distance distribution Clustering coefficient Degree distribution 10 2 1 10 0 Average clustering coefficient C(k) KK model KK model KK model 0.9 KR model Probability density function p d KR model Probability density function p k KR model KI model KI model KI model 10 0 0.8 RR model RR model RR model 10 -1 0.7 p k ∼ k -1.55 10 -2 0.6 10 -2 0.5 10 -4 0.4 0.3 10 -3 10 -6 0.2 0.1 10 -4 10 -8 0 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 10 4 3 4 5 6 7 Node degree k Node degree k Node distance d ( KK model ) both are nodes selected by degree ( KR model ) nodes selected by degree & at random ( KI model ) nodes selected by degree & degree − 1 ( RR model ) both nodes are selected at random 8/15

model initialization ( structure ) model initialization has no apparent impact Degree distribution Degree distribution Degree distribution 10 0 10 0 10 0 KK model KK model KK model Probability density function p k Probability density function p k Probability density function p k KR model KR model KR model KI model KI model KI model 10 -1 RR model 10 -1 RR model 10 -1 RR model p k ∼ k -1.55 p k ∼ k -1.68 p k ∼ k -1.68 10 -2 10 -2 10 -2 10 -3 10 -3 10 -3 10 -4 10 -4 10 -4 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 10 4 10 0 10 1 10 2 10 3 10 4 Node degree k Node degree k Node degree k ( left ) networks initialized by perfect matching ( middle ) networks initialized by random graph ( right ) networks initialized by random tree 9/15

model evolution ( structure ) model evolution when increasing node degree � k � Largest connected component Clustering coefficient Degree mixing 0.35 0.2 Largest connected component LCC KK model 1 Average clustering coefficient � C � KK model KR model KR model 0.3 0.15 0.95 KI model Degree mixing coefficient r KI model RR model 0.9 RR model 0.1 0.25 0.85 0.05 KK model 0.2 0.8 KR model 0 0.75 KI model 0.15 RR model 0.7 -0.05 0.65 0.1 -0.1 0.6 0.05 -0.15 0.55 0.5 0 -0.2 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 Average node degree � k � Average node degree � k � Average node degree � k � ( left ) emergence of giant component LCC when increasing � k � ( middle ) increasing node clustering � C � when increasing � k � ( right ) “ fixed ” degree mixing r when changing � k � 10/15

model comparison ( network ) international trade (i.e. food import & export) ( models ) war pact ≫ small-world , scale-free & random graphs ( left ) simplified D -measure [ SCDPMR17 ] ( right ) portrait divergence P [ BB19 ] [ SCDPMR17 ] Schieber et al. (2017) Quantification of network structural dissimilarities. Nat. Commun. 8 , 13928. [ BB19 ] Bagrow & Bollt (2019) An information-theoretic, all-scales approach to comparing. . . Appl. Netw. Sci. 4 , 45. 11/15

model validation ( networks ) national wars , Bitcoin transactions & Internet map ( models ) war pact ≫ small-world , scale-free & random graphs ( measure ) portrait divergence P [ BB19 ] [ BB19 ] Bagrow & Bollt (2019) An information-theoretic, all-scales approach to comparing. . . Appl. Netw. Sci. 4 , 45. 12/15

model structure ( size ) model reproduces nodes n & edges m by design ( connectivity ) model well reproduces giant component LCC ( distance ) model well reproduces distance � d � & diameter d max n m � k � LCC � C � � d � d max 41 54 2 . 63 87 . 8% 2 . 58 8 0 . 28 Correlates of war 41 54 2 . 63 90 . 2% 0 . 06 2 . 64 7 130 3 730 57 . 38 100 . 0% 0 . 50 2 . 24 5 International trade 130 3 730 57 . 38 100 . 0% 0 . 53 2 . 17 5 1 288 6 236 9 . 68 98 . 8% 0 . 33 2 . 83 9 Bitcoin transactions 1 288 6 236 9 . 68 98 . 0% 0 . 13 3 . 08 7 3 213 11 248 7 . 00 100 . 0% 3 . 77 9 0 . 18 Autonomous systems 3 213 11 248 7 . 00 98 . 3% 0 . 03 3 . 62 9 ( clustering ) model often underestimates node clustering � C � 13/15

model conclusions ( novel ) simple model of networks that shrink ( others ) in contrast to classic static & growing models ( networks ) model well reproduces structure except clustering ( question ) growing or shrinking models more “ reasonable ”? ( future ) combined model , other networks & analytical results 14/15

thank you! arXiv: 1909.00745v1 c & ˇ Nagli´ Subelj (2019) War pact model of shrinking networks. PLoS ONE , under review. joint work with Lovro ˇ Subelj Luka Nagli´ c University of Ljubljana University of Zagreb lovro.subelj@fri.uni-lj.si lu.naglic@gmail.com http://lovro.lpt.fri.uni-lj.si http://www.pmf.unizg.hr 15/15