Matrix Energy as a Measure of Topological Complexity of a Graph

📝 Abstract

The complexity of highly interconnected systems is rooted in the interwoven architecture defined by its connectivity structure. In this paper, we develop matrix energy of the underlying connectivity structure as a measure of topological complexity and highlight interpretations about certain global features of underlying system connectivity patterns. The proposed complexity metric is shown to satisfy the Weyuker criteria as a measure of its validity as a formal complexity metric. We also introduce the notion of P point in the graph density space. The P point acts as a boundary between multiple connectivity regimes for finite-size graphs.

💡 Analysis

The complexity of highly interconnected systems is rooted in the interwoven architecture defined by its connectivity structure. In this paper, we develop matrix energy of the underlying connectivity structure as a measure of topological complexity and highlight interpretations about certain global features of underlying system connectivity patterns. The proposed complexity metric is shown to satisfy the Weyuker criteria as a measure of its validity as a formal complexity metric. We also introduce the notion of P point in the graph density space. The P point acts as a boundary between multiple connectivity regimes for finite-size graphs.

📄 Content

1   Matrix Energy as a Measure of Topological Complexity of a Graph Kaushik Sinha1 and Olivier L. de Weck1 1Engineering  Systems  Division,  Massachusetts  Institute  of  Technology,     77  Massachusetts  Avenue,  Cambridge,  MA,  USA,  {sinhak@mit.edu;   deweck@mit.edu}  

ABSTRACT

  The  complexity  of  highly  interconnected  systems  is  rooted  in  the  interwoven   architecture  defined  by  its  connectivity  structure.  In  this  paper,  we  develop  matrix   energy   of   the   underlying   connectivity   structure   as   a   measure   of   topological   complexity  and  highlight  interpretations  about  certain  global  features  of  underlying   system  connectivity  patterns.    The  proposed  complexity  metric  is  shown  to  satisfy   the  Weyuker’s  criteria  as  a  measure  of  its  validity  as  a  formal  complexity  metric.  We   also  introduce  the  notion  of  P  point  in  the  graph  density  space.  The  P  point  acts  as  a   boundary  between  multiple  connectivity  regimes  for  finite-­‐size  graphs.    

  AMS  classification:  05C50   Keywords: Matrix Energy, Topological Complexity, Weyuker’s Criteria, Architectural Regime, P point. 1 INTRODUCTION

  In   the   context   of   complex   interconnected   systems,   the   quantification   of   complexity   has   increasingly   gained   importance   over   the   last   few   years.   While   working   with   large,   complex   systems,   the   challenge   of   quantifying   complexity   is   central   and   rigorous,   formalized   framework   to   compute   and   compare   their   respective  complexities  and  aid  decision-­‐making.    

  In   particular,   the   consideration   of   connectivity   structure   attracts   attention   because   they   affect   system   behavior.   The term “structure” is directly linked to the definition of a system. In general, the term “structure” is understood as the network formed by dependencies between components of any system [9].  One  emerging  area  of   application  is  in  the  characterization  and  impact  of  complex  system  architectures   that  are  fast  becoming  highly  networked  and  distributed  in  nature  [17,  19,  21].  

  The  concept  of  network  dimension  can  be  used  to  determine  the  underlying   network  structure  and  its  function.  A  network  with  higher  dimension  is  said  to  be   more   complex   than   one   with   a   lower   dimension.   Here,   we   focus   on   the   spectral   dimension  of  the  binary  adjacency  matrix  that  represents  the  connectivity  structure   of  the  system.  Recently,  the  idea  of  spectral  dimension  has  been  used  to  estimate  the   reconstructability  of  networks  [13,  19].    

  In   this   paper,   we   propose   matrix  energy   of   the   underlying   binary   adjacency  

  2   matrix  of  the  networked  complex  system  as  a  measure  of  topological  complexity  of   the  system.  We  explore  the  properties  of  matrix  energy,  including  important  bounds   for  general  adjacency  matrices  that  are  asymmetric.  The  matrix  energy  is  shown  to   satisfy  the  Weyuker’s  criteria  [20]  and  is  therefore  a  mathematically  valid  construct   for  measuring  complexity.     2 MATRIX ENERGY AND TOPOLOGICAL COMPLEXITY

  Topological   complexity   originates   from   interaction   between   elements   and   depends  on  the  combinatorial  nature  of  such  connectivity  structure.  The  topological   complexity   is   defined   as   the   matrix   energy   of   the   adjacency   matrix   [6].   The   adjacency  matrix   A ∈Mnxn of  a  network  is  defined  as  follows:  

  where   Λ  represents   the   set   of   connected   nodes   and   n   being   the   number   of   components   in   the   system.   The   diagonal   elements   of   A   are   zero.   The   associated   matrix  energy  [6,  14,  16]  of  the  network  is  defined  as  the  sum  of  singular  values  of   the  adjacency  matrix:  

  This   definition   extends   the   applicability   of   the

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