Layer 1 Layer 2 Layer 3 Layer 4 Interconnected Economies Eugene Neduv, Xiao Xu, Kehan Lu IEOR, School of Engineering and Applied Science US Equity Market: InterconnecIons between largest market cap US companies References 1. “The mul+plex dependency structure of financial markets”, Nicol ́o Musmeci, Vincenzo Nicosia, Tomaso Aste, Tiziana Di MaDeo,and Vito Latora 2. “Centrality in Interconnected Mul+layer Networks” Manlio De Domenico, Albert Sol ́e-Ribalta, Elisa Omodei, Sergio G ́omez,and Alex Arenas 3. “A Graph-theore+c perspec+ve on centrality” Stephen P. BorgaV , Mar+n G. EvereD Multilayered Networks: Layers represent various types of relationships Information carried by each layer is sufficiently different: correlation between layers and unique edge weight quantify the differences . Interlayer links weights define relative importance of information by type. Tensor algebra works for multiplex networks measures. Node degree – number of neighbors, weighted degree sum of neighboring edge weights are basic node measures. Further Research Clustering of companies based on mul+ple layers of connec+ons. Iden+fy groups with similar characteris+cs. Can be used as a recommenda+on engine for porZolio selec+on Centrality: Represents node influence in the system Centrality types: Weighted Degree, PageRank, Closeness, Betweenness. Chose appropriate centrality for the process being modeled. • Strength Tensor • Eigenvector Centrality • Diffusion EquaIon • Laplacian Trends in business dominance over the past 10 years Correlation Centrality by Sector MulIplex Centrality by Sector CompeIIon Centrality by Sector Supply-Customer Centrality by Sector Information diffusion can be generalized to multilayered network to account for interlayer diffusion. Interlayer Degree Correlation ! " =$ " % & % ' [%,"] = ∑ , (! , % − ! % )(! , " − ! " ) ∑ , (! , % − ! % ) 0 (! , " − ! " ) 0 Fraction of Edges unique to Layers 1 [%] = 2 03 % 4 ,,5 6 ,5 % 7 "8% (2 − 6 ,5 ["] ) Average Edge Overlap 9= 2 03 4 ,,5 46 ,5 [%] Degree Evolution by Layer :; " < = :> =? " < = %< @ ; %< @ > −? " < = %< @ 1 %< @ A '< B ; " < = > :; " < = :> = −C " < = %< @ ; %< @ (D) ; " < = (>) = ; %< @ (E)F GC " < = %< @ > Random walks on mul+layered network include interlayer jumps. H " < = >+2 = −C " < = %< @ H %< @ D, J " < = %< @ = = " < = %< @ −C " < = %< @ Eigenvectors of Laplacian tensor can be viewed as a centrality measure. Unfolding of a tensor into a supra matrix generalizes eigenvector problem. C " < = %< @ K %< @ = LK " < = K " < = =M 2 G2 ? " < = %< @ K %< @ • Bonacich Eigenvector Centrality K %< @ • Unfolded rank-1 supra eigenvector Using tensor algebra all monoplex measures can be generalized to mul+layered networks. Studying spectral proper+es of graph Laplacian can be used to understand the dynamics of informa+on diffusion. Acknowledgments DSI Summer Research Funding, Lab for Intelligent Asset Allocation

Summer-Scholar-Poster EN 20181120 xx edited · Layer 1 Layer 2 Layer 3 Layer 4 Interconnected Economies Eugene Neduv, Xiao Xu, Kehan Lu IEOR, School of Engineering and Applied Science

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Layer 1 Layer 2 Layer 3 Layer 4

Interconnected EconomiesEugene Neduv, Xiao Xu, Kehan Lu IEOR, School of Engineering and Applied Science

US Equity Market: InterconnecIons between largest market cap US companies

References1. “The mul+plex dependency structure of financial markets”, Nicol o Musmeci, Vincenzo Nicosia, Tomaso Aste, Tiziana Di MaDeo,and Vito Latora2. “Centrality in Interconnected Mul+layer Networks” Manlio De Domenico, Albert Sol e-Ribalta, Elisa Omodei, Sergio G omez,and Alex Arenas3. “A Graph-theore+c perspec+ve on centrality” Stephen P. BorgaV , Mar+n G. EvereD

Multilayered Networks: Layers represent various types of relationshipsInformation carried by each layer is sufficiently different: correlation between layers and unique edgeweight quantify the differences . Interlayer links weights define relative importance of information bytype. Tensor algebra works for multiplex networks measures. Node degree – number of neighbors,weighted degree sum of neighboring edge weights are basic node measures.

Further ResearchClustering of companies based on mul+ple layers of connec+ons. Iden+fy groups with similarcharacteris+cs. Can be used as a recommenda+on engine for porZolio selec+on

Centrality: Represents node influence in the systemCentrality types: Weighted Degree, PageRank, Closeness, Betweenness.Chose appropriate centrality for the process being modeled.

• Strength Tensor • Eigenvector Centrality

• DiffusionEquaIon

• Laplacian

Trends in business dominance over the past 10 years

Correlation Centrality by Sector

MulIplex Centrality by Sector CompeIIon Centrality by Sector

Supply-Customer Centrality by Sector

Information diffusion can be generalized to multilayered network to account for interlayer diffusion.

Interlayer Degree Correlation!" = $"

%&%

'[%,"] =∑,(!,

% − ! % )(!," − ! " )

∑,(!,% − ! % )0(!,

" − ! " )0

Fraction of Edges unique to Layers

1[%] =2

03 %4,,5

6,5% 7

"8%

(2 − 6,5["])

Average Edge Overlap

9 =203

4,,5

46,5[%]

Degree Evolution by Layer

:;"<=:>

= ?"<=%<@;%<@ > −?"<=

%<@ 1%<@A'<B;"<= >:;"<=:>

= −C"<=%<@;%<@(D) ;"<=(>) = ;%<@(E)F

GC"<=%<@>

Random walks on mul+layered network include interlayer jumps.H"<= > + 2 = −C"<=

%<@H%<@ D , J"<=%<@ = ="<=

%<@ − C"<=%<@

Eigenvectors of Laplacian tensor can be viewed as a centrality measure. Unfolding of a tensor into asupra matrix generalizes eigenvector problem.

C"<=%<@K%<@ = LK"<=

K"<= = M2G2?"<=

%<@K%<@• Bonacich Eigenvector Centrality

K%<@• Unfolded rank-1 supra eigenvector

Using tensor algebra all monoplex measures can be generalized to mul+layered networks.Studying spectral proper+es of graph Laplacian can be used to understand the dynamics ofinforma+on diffusion.

Acknowledgments DSI Summer Research Funding, Lab for Intelligent Asset Allocation