When is complex too complex? Graph Energy, Proactive Complexity Management, and the First Law of Systems Engineering Prof. Olivier de Weck Massachusetts Institute of Technology [email protected]1 Joint work with Dr. Kaushik Sinha and Narek Shougarian
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Fig. 2.1 Biphenylene H is a typical conjugated hydrocarbon. Its carbon–atom skeleton isrepresented by the molecular graph G. The carbon atoms in the chemical formula H and thevertices of the graph G are labeled by 1; 2; : : : ; 12 so as to be in harmony with Eqs. (2.2) and(2.3)
In the HMO model, the wave functions of a conjugated hydrocarbon with ncarbon atoms are expanded in an n-dimensional space of orthogonal basis functions,whereas the Hamiltonian matrix is a square matrix of order n, defined such that
ŒH!ij D
8ˆ<
ˆ:
˛ if i D j
ˇ if the atoms i and j are chemically bonded
0 if there is no chemical bond between the atoms i and j :
The parameters ˛ and ˇ are assumed to be constants, equal for all conjugatedmolecules. Their physical nature and numerical value are irrelevant for the presentconsiderations; for details see [76, 101, 503].
For instance, the HMO Hamiltonian matrix of biphenylene is
H D
2
666666666666666666664
˛ ˇ 0 0 0 ˇ 0 0 0 0 0 0
ˇ ˛ ˇ 0 0 0 0 0 0 0 0 ˇ
0 ˇ ˛ ˇ 0 0 0 0 0 0 ˇ 0
0 0 ˇ ˛ ˇ 0 0 0 0 0 0 0
0 0 0 ˇ ˛ ˇ 0 0 0 0 0 0
ˇ 0 0 0 ˇ ˛ 0 0 0 0 0 0
0 0 0 0 0 0 ˛ ˇ 0 0 0 ˇ
0 0 0 0 0 0 ˇ ˛ ˇ 0 0 0
0 0 0 0 0 0 0 ˇ ˛ ˇ 0 0
0 0 0 0 0 0 0 0 ˇ ˛ ˇ 0
0 0 ˇ 0 0 0 0 0 0 ˇ ˛ ˇ
0 ˇ 0 0 0 0 ˇ 0 0 0 ˇ ˛
3
777777777777777777775
(2.2)
which can be written also as
2.1 Huckel Molecular Orbital Theory 13
H D ˛
2
666666666666666666664
1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1
3
777777777777777777775
C ˇ
2
666666666666666666664
0 1 0 0 0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0 1
0 1 0 1 0 0 0 0 0 0 1 0
0 0 1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 1 0 0 0 0 0 0
1 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 1
0 0 0 0 0 0 1 0 1 0 0 0
0 0 0 0 0 0 0 1 0 1 0 0
0 0 0 0 0 0 0 0 1 0 1 0
0 0 1 0 0 0 0 0 0 1 0 1
0 1 0 0 0 0 1 0 0 0 1 0
3
777777777777777777775
: (2.3)
The first matrix on the right-hand side of Eq. (2.3) is just the unit matrix of ordern D 12, whereas the second matrix can be understood as the adjacency matrix of agraph on n D 12 vertices. This graph is also depicted in Fig. 2.1 and in an obviousmanner corresponds to the underlying molecule (in our example, to biphenylene).
From the above example, it is evident that also in the general case withinthe HMO model, one needs to solve the eigenvalue–eigenvector problem of anapproximate Hamiltonian matrix of the form
H D ˛ In C ˇA.G/ (2.4)
where ˛ and ˇ are certain constants, In is the unit-matrix of order n, and A.G/ isthe adjacency matrix of a particular graph G on n vertices that corresponds to thecarbon–atom skeleton of the underlying conjugated molecule.
As a curiosity, we mention that neither Huckel himself nor the scientists whodid early research in HMO theory were aware of the identity (2.4), which was firstnoticed only in 1956 [139].
As a consequence of Eq. (2.4), the energy levels Ej of the !-electrons are relatedto the eigenvalues "j of the graph G by the simple relation
Ej D ˛ C ˇ "j I j D 1; 2; : : : ; n:
In addition, the molecular orbitals, describing how the !-electrons move within themolecule, coincide with the eigenvectors j of the graph G.
In the HMO approximation, the total energy of all !-electrons is given by
E! DnX
jD1gj Ej
where gj is the so-called occupation number, the number of !-electrons that movein accordance with the molecular orbital j . By a general physical law, gj mayassume only the values 0, 1, or 2.
12 2 The Chemical Connection
Fig. 2.1 Biphenylene H is a typical conjugated hydrocarbon. Its carbon–atom skeleton isrepresented by the molecular graph G. The carbon atoms in the chemical formula H and thevertices of the graph G are labeled by 1; 2; : : : ; 12 so as to be in harmony with Eqs. (2.2) and(2.3)
In the HMO model, the wave functions of a conjugated hydrocarbon with ncarbon atoms are expanded in an n-dimensional space of orthogonal basis functions,whereas the Hamiltonian matrix is a square matrix of order n, defined such that
ŒH!ij D
8ˆ<
ˆ:
˛ if i D j
ˇ if the atoms i and j are chemically bonded
0 if there is no chemical bond between the atoms i and j :
The parameters ˛ and ˇ are assumed to be constants, equal for all conjugatedmolecules. Their physical nature and numerical value are irrelevant for the presentconsiderations; for details see [76, 101, 503].
For instance, the HMO Hamiltonian matrix of biphenylene is
Fig. 2.1 Biphenylene H is a typical conjugated hydrocarbon. Its carbon–atom skeleton isrepresented by the molecular graph G. The carbon atoms in the chemical formula H and thevertices of the graph G are labeled by 1; 2; : : : ; 12 so as to be in harmony with Eqs. (2.2) and(2.3)
In the HMO model, the wave functions of a conjugated hydrocarbon with ncarbon atoms are expanded in an n-dimensional space of orthogonal basis functions,whereas the Hamiltonian matrix is a square matrix of order n, defined such that
ŒH!ij D
8ˆ<
ˆ:
˛ if i D j
ˇ if the atoms i and j are chemically bonded
0 if there is no chemical bond between the atoms i and j :
The parameters ˛ and ˇ are assumed to be constants, equal for all conjugatedmolecules. Their physical nature and numerical value are irrelevant for the presentconsiderations; for details see [76, 101, 503].
For instance, the HMO Hamiltonian matrix of biphenylene is
H D
2
666666666666666666664
˛ ˇ 0 0 0 ˇ 0 0 0 0 0 0
ˇ ˛ ˇ 0 0 0 0 0 0 0 0 ˇ
0 ˇ ˛ ˇ 0 0 0 0 0 0 ˇ 0
0 0 ˇ ˛ ˇ 0 0 0 0 0 0 0
0 0 0 ˇ ˛ ˇ 0 0 0 0 0 0
ˇ 0 0 0 ˇ ˛ 0 0 0 0 0 0
0 0 0 0 0 0 ˛ ˇ 0 0 0 ˇ
0 0 0 0 0 0 ˇ ˛ ˇ 0 0 0
0 0 0 0 0 0 0 ˇ ˛ ˇ 0 0
0 0 0 0 0 0 0 0 ˇ ˛ ˇ 0
0 0 ˇ 0 0 0 0 0 0 ˇ ˛ ˇ
0 ˇ 0 0 0 0 ˇ 0 0 0 ˇ ˛
3
777777777777777777775
(2.2)
which can be written also as
2.1 Huckel Molecular Orbital Theory 13
H D ˛
2
666666666666666666664
1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1
3
777777777777777777775
C ˇ
2
666666666666666666664
0 1 0 0 0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 0 0 0 0 1
0 1 0 1 0 0 0 0 0 0 1 0
0 0 1 0 1 0 0 0 0 0 0 0
0 0 0 1 0 1 0 0 0 0 0 0
1 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 1
0 0 0 0 0 0 1 0 1 0 0 0
0 0 0 0 0 0 0 1 0 1 0 0
0 0 0 0 0 0 0 0 1 0 1 0
0 0 1 0 0 0 0 0 0 1 0 1
0 1 0 0 0 0 1 0 0 0 1 0
3
777777777777777777775
: (2.3)
The first matrix on the right-hand side of Eq. (2.3) is just the unit matrix of ordern D 12, whereas the second matrix can be understood as the adjacency matrix of agraph on n D 12 vertices. This graph is also depicted in Fig. 2.1 and in an obviousmanner corresponds to the underlying molecule (in our example, to biphenylene).
From the above example, it is evident that also in the general case withinthe HMO model, one needs to solve the eigenvalue–eigenvector problem of anapproximate Hamiltonian matrix of the form
H D ˛ In C ˇA.G/ (2.4)
where ˛ and ˇ are certain constants, In is the unit-matrix of order n, and A.G/ isthe adjacency matrix of a particular graph G on n vertices that corresponds to thecarbon–atom skeleton of the underlying conjugated molecule.
As a curiosity, we mention that neither Huckel himself nor the scientists whodid early research in HMO theory were aware of the identity (2.4), which was firstnoticed only in 1956 [139].
As a consequence of Eq. (2.4), the energy levels Ej of the !-electrons are relatedto the eigenvalues "j of the graph G by the simple relation
Ej D ˛ C ˇ "j I j D 1; 2; : : : ; n:
In addition, the molecular orbitals, describing how the !-electrons move within themolecule, coincide with the eigenvectors j of the graph G.
In the HMO approximation, the total energy of all !-electrons is given by
E! DnX
jD1gj Ej
where gj is the so-called occupation number, the number of !-electrons that movein accordance with the molecular orbital j . By a general physical law, gj mayassume only the values 0, 1, or 2.
επ = nα + β hiσ ii=1
n
∑ ≤ nα + β hii=1
n
∑⎛⎝⎜⎞⎠⎟
n
σ ii=1
n
∑⎛⎝⎜⎞⎠⎟
E ( A)
∴επ ≤ nα + n2β E( A)n
⎛⎝⎜
⎞⎠⎟
Introduceanotionofofconfiguration energy:
Ξ := nαC1
+ mβC2
E( A)
n⎛⎝⎜
⎞⎠⎟
C3
= C1 +C2C3
C = C1 +C2C3
= α ii=1
n
∑ + βijj=1
n
∑i=1
n
∑⎛
⎝⎜⎞
⎠⎟E( A)
n⎛⎝⎜
⎞⎠⎟= α i
i=1
n
∑ + βijj=1
n
∑i=1
n
∑⎛
⎝⎜⎞
⎠⎟γ E( A)
Use the above functional form to measure the complexityassociated to the system structure – Structural Complexity ofthe system where α’s stand for component complexity while β’sstand for interface complexity:
Fig. 2.1 Biphenylene H is a typical conjugated hydrocarbon. Its carbon–atom skeleton isrepresented by the molecular graph G. The carbon atoms in the chemical formula H and thevertices of the graph G are labeled by 1; 2; : : : ; 12 so as to be in harmony with Eqs. (2.2) and(2.3)
In the HMO model, the wave functions of a conjugated hydrocarbon with ncarbon atoms are expanded in an n-dimensional space of orthogonal basis functions,whereas the Hamiltonian matrix is a square matrix of order n, defined such that
ŒH!ij D
8ˆ<
ˆ:
˛ if i D j
ˇ if the atoms i and j are chemically bonded
0 if there is no chemical bond between the atoms i and j :
The parameters ˛ and ˇ are assumed to be constants, equal for all conjugatedmolecules. Their physical nature and numerical value are irrelevant for the presentconsiderations; for details see [76, 101, 503].
For instance, the HMO Hamiltonian matrix of biphenylene is
Number of components [Bralla, 1986] ! Component development
(count-based measure) "
Number of interactions [Pahl and Beitz, 1996] ! Interface development
(count-based measure) "
Whitney Index [Whitney et al., 1999] ! Components and interface
developments "
Number of loops, and their distribution [] " Feedback effects "
Nesting depth [Kerimeyer and
Lindemann, 2011] " Extent of hierarchy "
Graph Planarity [Kortler et al., 2009] ! Information transfer
efficiency "
CoBRA Complexity Index [Bearden, 2000] ! Empirical correlation in
similar systems "
Automorphism-based Entropic Measures
[Dehmer et al., 2009] "
Heterogeneity of network structure, graph reconfigurability
!
Matrix Energy / Graph Energy ! Graph Reconstructabality !
• Graph Energy stands out as both computable and satisfies Weyuker’s criteria andestablishes itself as a theoretically valid measure (i.e., construct validity) of complexity.
• The P point on graph energy – density plot: Phase transition for complxity
• At densities higher than P point, structural complexity increases but that does notbuy much improvement in terms of performance measures (e.g., network diameter)
P-point is critical, becausehere DSM reaches full rank
• Can compare systems at same level of abstraction in this space• Use equivalent random networks (Erdős–Rényi) as background (red curve)• P-point has E(A) equivalent to fully connected system, critical for design• If we go beyond the P-point in System Design will have diminishing returns
checking follows because, for the cases in which thesolution is determined by a combination of fa and fr, itcan be shown that if the optimal solution is a two-peakdistribution, then equality holds in (7), while if the opti-mal solution is a three-peak distribution, equality (1) alsoholds in addition to (7). Therefore the potentially optimalvalues of pk! and pkm can always be expressed in terms ofthe candidate k!.
In this Letter we have shown that the network con-figurations that maximize the percolation thresholdunder attack and/or random failures have at most threedistinct node degrees. From a practical point of view, bothengineered and naturally occurring networks have a di-versity of factors influencing and constraining theirultimate configuration. Nonetheless, the optimal con-figurations we present provide a standard againstwhich the robustness of real networks can be com-pared and act as an intuitive guide for network-robustnessengineering.
A. S. deeply appreciates the guidance and support ofProfessor David Nelson and Professor X. Sunney Xie, as
well as the financial support of NSF (DMR-0231631) andHMRL (DMR-0213805).
*Electronic address: [email protected][1] D. J.Watts and S. H. Strogatz, Nature (London) 393, 440–
442 (1998).[2] L. A. N. Amaral, A. Scala, M. Barthelemy, and H. E.
Stanley, Proc. Natl. Acad. Sci. U.S.A. 97, 11149 (2000).[3] M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comput.
Commun. Rev. 29, 251 (1999).[4] R. Albert, H. Jeong, and A. Barabasi, Nature (London)
406, 378 (2000).[5] H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and
A. Barabasi, Nature (London) 407, 651 (2000).[6] J. Hasty, D. McMillen, and J. J. Collins, Nature (London)
420, 224 (2002).[7] R.V. Sole and J. M. Montoya, Proc. R. Soc. London,
Ser. B 268, 2039 (2001).[8] S. Strogatz, Nature (London) 410, 268 (2001).[9] R. Albert and A. Barabasi, Rev. Mod. Phys. 74, 47
(2002).[10] X. F. Wang, Int. J. Bifurcation Chaos Appl. Sci. Eng. 12,
885 (2002).[11] R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan,
D. Chklovskii, and U. Alon, Science 298, 824 (2002).[12] Z. N. Oltvai and A. Barabasi, Science 298, 763 (2002).[13] Some problems are best modeled with the removal of
links rather than nodes. The mathematical analysis forsuch cases parallels closely that for the case of noderemoval. See, for example, Ref. [26].
[14] B. Shargel, H. Sayama, I. R. Epstein, and Y. Bar-Yam,Phys. Rev. Lett. 90, 068701 (2003).
[15] M. E. J. Newman, S. Strogatz, and D. Watts, Phys. Rev. E64, 026118 (2001).
[16] M. Molloy and B. Reed, Combinatorics Probab. Comput.7, 295 (1998).
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[20] D. J. Watts, Proc. Natl. Acad. Sci. U.S.A. 99, 5766 (2002).[21] A. E. Motter and Y. C. Lai, Phys. Rev. E 66, 065102
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Rev. Lett. 86, 3682 (2001).[24] Although km for the power grid was 19, only 1.5% of the
nodes had a degree above 8. Similarly, only 3% of theinternet nodes had a degree above 8. Because of this, weconsidered that imposing a maximum of km " 8 in ouroptimal peaked networks would make the comparisonfairer. Had we imposed km " 19 or 20, the robustnessdifference between our peaked networks and the internetand power grid networks would only increase.
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FIG. 4 (color online). Simultaneous optimization against in-tentional attacks and random failures. We take #k‘; km$ " #1; 8$.To each combination of desired minimal network percolationthresholds, fa under attack and fr under random failures,corresponds an optimal network, i.e., one that also minimizeshki. (a) These optimal networks can be divided into differentqualitative classes, illustrated using different colors: A—Robustness to these #fa; fr$ pairs is not attainable due to thekm constraint. B—fr is the limiting constraint. There are twonode degrees present in these networks, k‘ and km. C—fa is thelimiting constraint. There are at most two distinct node degreesin these networks, k‘ and k!. D—Both fa and fr affect theoptimal degree distribution. These networks still have just twodistinct node degrees, k‘ and k! (i.e., the potential third degree,km, turns out to have zero frequency). E —As in D, both fa andfr affect the optimal degree distribution but there are nowthree distinct node degrees in the network, k‘, k!, and km.(b) Contour plot of hki for the optimal networks. The hki "km " 8 contour represents the maximum achievable robustness.For comparison, the #fr; fa$ robustness thresholds of two realnetworks were plotted: *, Western United States power grid(exponential network); %, Internet router (power-law network).For the power grid hki " 2:7 and for the internet hki " 2:5.Note how the points fall below the respective optimal hkicontours.
P H Y S I C A L R E V I E W L E T T E R S week ending19 MARCH 2004VOLUME 92, NUMBER 11
'I&! g&g! 1+6105-3,! -F5-! P* !.350F3, Pmax !5,! n / m !.5-1*! 1+0.35,3,&! P5.23! n / m !.5-1*! 1+6105-3,! ,F5.83.! 83.A*.E5+03! 251+! X1-F! #3--3.! 0*E8H3Z1-$!F5+6H1+2mE5+523E3+-!0585#1H1-$&!?H,*!5!,E5HH3.! k !0*E#1+36!X1-F!H5.23.! n / m .5-1*!H356,!-*!F12F3.!0*E8H3Z1-$!H373H(! C* !5-!E5Z1E5H!75H/3!A/+0-1*+&!!
• Once we set a complexity budget, there are different ways to distribute this totalstructural complexity, C into its three components {C1, C2, C3} : IsoComplexity Surface
Iso-complexity surface: n = 20 components, assuming,c1 in [10,60]; c2 in [12,40] and C = 100.
• Tradeoff between (i) complexcomponents and simple architecture, or(ii) simpler components and morecomplex architecture.
• Choice can be made depending oncomplexity handling capabilities of thedevelopment organization. E.g.o Excellent component designerso Systems integrators