MAE 298, Lecture 7Jan 30, 2008

“More important network measures”

Today

• Complete discussion of network epidemiology (filling in details)

• More basic network measures/features

• Partitioning networks:– spectral methods– community structure

Classic mathematical epidemiologyMean field/complete graph

Recall SIR model, and Tc the epidemic threshold

• dSdt = −βIS,

• dIdt = βIS − γI,

• dRdt = γI.

Tc = βS0γ

(Can set γ = 1 with out loss of generality.)

Percolation/epidemic threshold on power law randomgraphs, P (k) ∼ k−γ for 2 < γ ≤ 3

• Network robustness and fragility: Percolation on randomgraphs, Callaway, Newman, Strogatz and Watts, Phys. Rev.Lett., 85 (2000).

• Epidemic Spreading in Scale-Free Networks, R. Pastor-Satorras and A. Vespignani Phys. Rev. Lett. 86 (2001).

• The Epidemic Threshold in Scale-Free Graphs, N. Berger,C. Borgs, J. Chayes, and A. Saberi, Symposium on DiscreteAlgorithms (SODA), (2005).

Find Tc = 0, in other words absence of epidemiological threshold.

For all β > 0 and S0 > 0, the steady-state result is that somenon-zero, fraction of the population has the disease.

Results from Callaway et al

• Degree dist, pk ∼ k−γe−k/C

• Let qk be probability that a vertex of degree k is “infected”.They analyze qk = q (independent of k).

• Then pkq is probability of having degree k and being infected.

• Calculate 〈s〉, the mean cluster size. Find (via generatingfunctions ... details omitted here) that

〈s〉 = q +q2 〈k〉

1− (q 〈k2〉 / 〈k〉)

• 〈s〉 → ∞ when denominator 1− q⟨k2⟩/ 〈k〉 = 0, i.e.,

qc = 〈k〉〈k2〉

qc versus C, the cutoffVOLUME 85, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 DECEMBER 2000

0 10 20 30 40

size of cluster s

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FIG. 1. Probability Ps that a randomly chosen vertex belongsto a cluster of s sites for k ! 10, t ! 2.5, and q ! 0.65 fromnumerical simulation on systems of 107 sites (circles) and ourexact solution (solid line). Inset: the percolation threshold qcfrom Eq. (12) (solid lines) vs computer simulations with t !1.5 (circles), 2.0 (squares), and 2.5 (triangles).

A result equivalent to this one has been derived previouslyby Cohen et al. [2] by different means.

In the language of disease propagation qc is the pointat which an epidemic of the disease first occurs. In thelanguage of network robustness, it is the point at which thenetwork achieves large scale connectivity and can thereforefunction as an effective distribution network. Conversely,if we are approaching the transition from values of q aboveqc it is the point at which a sufficient number of individualsare immune to a disease to prevent it from spreading, or thepoint at which a large enough number of nodes have beendeleted from a distribution network to prevent distributionon large scales.

The inset in Fig. 1 shows the behavior of the percola-tion threshold with the cutoff parameter k for a variety ofvalues of t. Note that as the values of k become large,the percolation threshold becomes small, indicating a highdegree of robustness of the network to random deletionof nodes. For t ! 2.5 (roughly the exponent for the In-ternet data [8]) and k ! 100, the percolation thresholdis qc ! 0.17, indicating that one can remove more than80% of the nodes in the network without destroying the gi-ant component — the network will still possess large-scaleconnectivity. This result agrees with recent studies ofthe Internet [1,2] which indicate that network connectiv-ity should be highly robust against the random removal ofnodes.

Another issue that has attracted considerable recent at-tention is the question of robustness of a network to non-random deletion targeted specifically at nodes with highdegree. Albert et al. [1] and Broder et al. [3] both lookedat the connectivity of a network with power-law distributedvertex degrees as the vertices with highest degree wereprogressively removed. In the language of our percolationmodels, this is equivalent to setting

qk ! u!kmax 2 k" , (13)

where u is the Heaviside step function [21]. This re-moves (unoccupies) all vertices with degree greater thankmax. To investigate the effect of this removal, we cal-culate the size of the giant component in the network, ifthere is one. Above the percolation transition the genera-ting function H0!x" gives the distribution of the sizes ofclusters of vertices which are not in the giant component[17], which means that H0!1" is equal to the fraction ofthe graph which is not occupied by the giant component.The fraction S which is occupied by the giant componentis therefore given by

S ! 1 2 H0!1" ! F0!1" 2 F0!u" , (14)

where u is a solution of the self-consistency condition

u ! 1 2 F1!1" 1 F1!u" . (15)

In cases where this last equation is not exactly solvable wecan evaluate u by numerical iteration starting from a suit-able initial value. In Fig. 2 we show the results for S fromthis calculation for graphs with pure power-law degree dis-tributions as a function of kmax for a variety of values of t.(The removal of vertices with high degree regularizes thecalculation in a similar way to the inclusion of the cutoffk in our earlier calculation, so no other cutoff is needed inthis case.) On the same plot we also show simulation re-sults for this problem, and once more agreement of theoryand simulation is good.

Opinions appear to differ over whether networks suchas this are robust or fragile to this selective removal ofvertices. Albert et al. [1] point out that only a small frac-tion of the highest-degree vertices need be removed todestroy the giant component in the network and henceremove all long-range connectivity. Conversely, Broderet al. [3] point out that one can remove all vertices withdegree greater than kmax and still have a giant componenteven for surprisingly small values of kmax. As we showin Fig. 2, both viewpoints are correct: they are merelydifferent representations of the same data. In the upperframe of the figure, we plot giant component size as afunction of the fraction of vertices removed from the net-work, and it is clear that the giant component disappearswhen only a small percentage are removed — just 1% forthe case t ! 2.7—so that the network appears fragile. Inthe lower frame we show the same data as a function ofkmax, the highest remaining vertex degree, and we see thatwhen viewed in this way the network is, in a sense, robust,since kmax must be very small to destroy the giant compo-nent completely — just 10 in the case of t ! 2.7.

To conclude, we have used generating function methodsto solve exactly for the behavior of a variety of percolationmodels on random graphs with any distribution of vertexdegrees, including uniform site, bond and site/bond

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Following Pastor-Satorras and A. Vespignani

• ρk(t) is density of infected nodes of degree k at time t.(Hence [1− ρk(t)] is probability a node of degree k is NOTinfected.)

• λ = β/γ, the effective spreading rate. Set γ = 1.

• The time evolution:

dρk(t)dt

= −ρk(t) + λk [1− ρk(t)] Θ(ρ(t))

• First term: nodes recover with unit rate (γ = 1)

• Second term: Infection rate λ, times number of neighbors k,times prob node of degree k is healthy, times prob of beingconnected to an infected node Θ(ρ(t)).

• Prob of pointing to a node of degree k is k pk, so

Θ(ρ(t)) =∑k k pk ρk(t)∑

k k pk=

1〈k〉∑k

k pk ρk(t)

• Steady state, dρkdt = 0 implies:

ρk =λ k Θ

1 + λ Θ

• Inserting into expression for Θ:

Θ =1〈k〉∑k

k pkλ k Θ

1 + λ Θ

(Note Θ = 0 always satisfies, but is quite dull! ... ρk = 0)

Searching for more solutions to last equation,in interval 0 < Θ ≤ 1

• Taking derivative w.r.t. Θ of both sides of last equation:

d

dΘ

[1〈k〉∑k

k pkλ k Θ

1 + λ Θ

]∣∣∣∣∣Θ=0

= 1, at λ = λc

• solving this:1〈k〉∑k

k pk λc k =

⟨k2⟩

〈k〉= 1

λc = 〈k〉〈k2〉

If⟨k2⟩→∞ but 〈k〉 finite, then λc→ 0.

Last three slides, actually pieced from three papers

• Epidemic Spreading in Scale-Free Networks, R. Pastor-Satorras and A. Vespignani Phys. Rev. Lett. 86 (2001).

• Epidemic dynamics in finite size scale-free networks, R Pastor-Satorras, A Vespignani Physical Review E (2002).

• Immunization of complex networks, R Pastor-Satorras, AVespignani Physical Review E (2002).

Final points

From SIS to SIR

• PSV’01 and BBCS’05 consider SIS.

• R. M. May, A. L. Lloyd “Infection dynamics on scale-freenetworks” Phys. Rev. E, (2001). Show similar results hold forSIR. (Lord Robert May, founder of theoretical ecology/population biology/evolutionary

game theory... great wiki entry)

Immunization

• Many subsequent papers on immunization by knocking outnodes.

• But recall from last time, the recovery rate depends on otherattributes of node (age, medical history....) and can overridenetwork structure. (i.e., less overall infected or less overallfatalities important?)

Whew! ... On to something more fun

Basic network metrics

• Number of vertices, N , and edges, E.

• Average degree, 〈k〉.

• Degree distribution, p(k).– often power law.– or power law with exponential tail.– in general “heavy-tailed”.

• Clustering coefficient, ci.– For an individual node ci.– Average over the network 〈ci〉

What else???

A classic example from Social Network Analysis (SNA)

[http://www.fsu.edu/∼spap/water/network/intro.htm]

The “Kite Network”

Who is important and why?

The Kite Network

• Degree – Diane looks important (a “hub”).

• Betweenness – Heather looks important (a “connector”/“broker”).

• Closeness – Fernando and Garth can access anyone via ashort path.

• Boundary spanners – as Fernando, Garth, and Heather arewell-positioned to be “innovators”.

• Peripheral Players – Ike and Jane may be an importantresources for fresh information.

Betweenness Centrality

[Freeman, L. C. “A set of measures of centrality based onbetweenness.” Sociometry 40 1977]

A measure of how many shortest paths between all othervertices pass through a given vertex.

Betweenness (formal definition)

For a given vertex i:

B(i) =∑

s 6=t 6=iσst(i)σst

• Where σst is the number of shortest geodesic paths betweens and t.

• And σst(i) are the number of those passing through vertex i.

Betweenness and eigenvalues(bottlenecks)

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R = 4.61τ[med] = 604 to

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R = 5.315τ[max] = 5314 to

• Bottlenecks have large betweenness values.

• In social networks betweenness is a measure of a nodes“centrality” and importance (could be a proxy for influence).

• In a road network, high betweenness could indicate wherealternate routes are needed.• Also a measure of the resilience of a network (remove high

betweenness nodes and destroy connectivity). More on this at end!!

Other measures for SNA

• Structural Equivalence- determine which nodes play similarroles in the network

• Cluster Analysis- find cliques and other densely connectedclusters

• Structural Holes- find areas of no connection between nodesthat could be used for advantage or opportunity

• External/Internal Ratio- find which groups in the network areopen or closed to others

• Small Worlds- find node clustering, and short path lengths, thatare common in networks exhibiting highly efficient small-worldbehavior

Structural Equivalence

Narrow definition: Two vertices in a network are structurallyequivalent if they have all the same neighbors.

Broader definition: identifying groups of nodes that are similarin their patterns of ties to all other nodes.

How to determine?

Measures of similarity and structural equivalence

• Valued relations– Pearson correlations covariances and cross-products– Euclidean, Manhattan, and squared distances

• Binary relations– Matches: Exact, Jaccard, Hamming

http://www.faculty.ucr.edu/∼hanneman/nettext/

and in particular

http://www.faculty.ucr.edu/∼hanneman/nettext/C13 %20Structural Equivalence.html#measure

Some SNA resources

• International Network for Social Network Analysis(http://www.insna.org/)

• InFlow 3.1 - Social Network Mapping Software(http://www.orgnet.com/inflow3.html)– Network centrality, cluster analysis, structural equivalence,prestige/influence....

• UCI net (http://www.analytictech.com/ucinet.htm)

• S. Wasserman and K. Faust, “Social Network Analysis:Methods and Applications”, Cambridge University Press,Cambridge, UK, 1994

Mixing

• In almost all networks, nodes of different types (e.g., gender,race, function).

• Does probability of connection between two vertices dependon their types?

• In other words: Mixing by scalar characteristics.

Example: food web

• Types of nodes: plants, herbivores and carnivores.

• Many links between plants and herbivores.

• Many links between herbivores and carnivores.

• Almost no plant-plant or herbivore-herbivore edges.

Assortative mixing

Instead consider a case with many liketype-liketype edges.Classic example is mixing by race in a social network:

Measure of Assortativity

• Define j different classes/types

• Let Eij be the number of edges connecting types i and j.

• Let ||E|| be the total number of edges between all classes.

• Define the mixing matrix with matrix elements:mij = Eij/E

Getting a scalar quantity from mij

The assortativity coefficient:

r = (Trm− ||m2||)/(1− ||m2||)

• r = 0 for a randomly mixed network.

• r = 1 for a perfectly assortative one.

[ MEJ Newman, “Assortative mixing in networks”, Phys RevLett. 89(20): 2002]

Degree correlation

What is the scalar quantity of interest is the degree?

Found that social networks are assortative, while technologicaland biological ones are dissortative.

Why? Just an observation for now.

182

M.E

.J.N

EWM

AN

Table 3.1 Basic statistics for a number of published networks. The properties measured are as follows: total number of vertices n; total number of edges m;mean degree z; mean vertex–vertex distance !; type of graph, directed or undirected; exponent " of degree distribution if the distribution follows apower law (or “–” if not; in/out-degree exponents are given for directed graphs); clustering coe!cient C(1) from (3.3); clustering coe!cient C(2)

from (3.6); degree correlation coe!cient r, section 3.6. The last column gives the citation for the network in the bibliography. Blank entries indicateunavailable data.

Network Type n m z ! " C(1) C(2) r Ref(s).

Soci

al

film actors undirected 449 913 25 516 482 113.43 3.48 2.3 0.20 0.78 0.208 [20, 415]company directors undirected 7 673 55 392 14.44 4.60 – 0.59 0.88 0.276 [105, 322]math coauthorship undirected 253 339 496 489 3.92 7.57 – 0.15 0.34 0.120 [107, 181]physics coauthorship undirected 52 909 245 300 9.27 6.19 – 0.45 0.56 0.363 [310, 312]biology coauthorship undirected 1 520 251 11 803 064 15.53 4.92 – 0.088 0.60 0.127 [310, 312]telephone call graph undirected 47 000 000 80 000 000 3.16 2.1 [8, 9]email messages directed 59 912 86 300 1.44 4.95 1.5/2.0 0.16 [136]email address books directed 16 881 57 029 3.38 5.22 – 0.17 0.13 0.092 [320]student relationships undirected 573 477 1.66 16.01 – 0.005 0.001 !0.029 [45]sexual contacts undirected 2 810 3.2 [264, 265]

Info

rmat

ion WWW nd.edu directed 269 504 1 497 135 5.55 11.27 2.1/2.4 0.11 0.29 !0.067 [14, 34]

WWW Altavista directed 203 549 046 2 130 000 000 10.46 16.18 2.1/2.7 [74]citation network directed 783 339 6 716 198 8.57 3.0/– [350]Roget’s Thesaurus directed 1 022 5 103 4.99 4.87 – 0.13 0.15 0.157 [243]word co-occurrence undirected 460 902 17 000 000 70.13 2.7 0.44 [119, 157]

Tec

hnol

ogic

al

Internet undirected 10 697 31 992 5.98 3.31 2.5 0.035 0.39 !0.189 [86, 148]power grid undirected 4 941 6 594 2.67 18.99 – 0.10 0.080 !0.003 [415]train routes undirected 587 19 603 66.79 2.16 – 0.69 !0.033 [365]software packages directed 1 439 1 723 1.20 2.42 1.6/1.4 0.070 0.082 !0.016 [317]software classes directed 1 377 2 213 1.61 1.51 – 0.033 0.012 !0.119 [394]electronic circuits undirected 24 097 53 248 4.34 11.05 3.0 0.010 0.030 !0.154 [155]peer-to-peer network undirected 880 1 296 1.47 4.28 2.1 0.012 0.011 !0.366 [6, 353]

Bio

logi

cal metabolic network undirected 765 3 686 9.64 2.56 2.2 0.090 0.67 !0.240 [213]

protein interactions undirected 2 115 2 240 2.12 6.80 2.4 0.072 0.071 !0.156 [211]marine food web directed 135 598 4.43 2.05 – 0.16 0.23 !0.263 [203]freshwater food web directed 92 997 10.84 1.90 – 0.40 0.48 !0.326 [271]neural network directed 307 2 359 7.68 3.97 – 0.18 0.28 !0.226 [415, 420]

More basic measures, summary

From Social Network Analysis:

• Betweenness (betweenness centrality)

• Structural Equivalence

From network theory:

• Mixing patterns

Next time:

• Partitioning networks