Finding the Mule in the NetworkJohn L. Pfaltz

Dept. of Computer ScienceUniversity of Virginia

Email: jlp@virginia.edu

Abstract—There exist a variety of procedures foridentifying clusters in large networks. This paper fo-cuses on finding the connections between such clusters.

We employ the concept of closed sets to reduce anetwork down to its fundamental cycles. These cyclesbegin to capture the global structure of the network byeliminating a great deal of the fine detail. Nevertheless,the reduced version is completely faithful to theoriginal. No connection in the reduced version existsunless it was in the original network; connectivity ispreserved.

Reductions of as much as 80% can be observed inreal networks. Just reducing the size makes compre-hension of the network much easier.

I. INTRODUCTION

A network, or graph, is a set P of n points,or nodes, or vertices, or individuals together withan n × n relation A defining the associations, oredges, between the points or individuals. Figure 1is a representative example. Each non-zero entry,

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Fig. 1. An adjacency representation of a small 23 node network.

such as (1, 8) ∈ A, denotes an edge, or connection,between the node, or point, 1 and the node 8. Forthe purposes of this paper we will assume that Ais symmetric, or equivalently, that the network isundirected.

Networks play an important role in the studyof many phenomena. Social networks describe theinteractions between individuals [2], [3], [21]; theinternet is a familiar feature of our every daycomputer life [11], [13]; genetic expression can bemodeled as proteins in a regulatory network [4], [9].In these examples, and others, the structure of these

networks plays a vital role. But what is a network’s“structure”?

Frequently, some of the nodes form “clusters”, or“communities”, with many intra-cluster connections.The identification of such clusters has been thefocus of considerable research; there is an immenseamount of literature in disparate fields, of which [1],[5], [15], [17], [18] is but a small fraction. The focusof our research, however, is the identification ofthose nodes, or connections, that “go between” dif-ferent clusters. They are often harder to discover. Inthe drug trade, a “mule” is a go-between that bringsdrugs from a cluster of producers to a community ofusers. Hence the term “mule” in our title. In real life,mules are equally hard to discover. To find mules wemust understand the network structure.

Possibly the most effective approach to under-standing the overall structure of a network, whenit works, is to graphically represent it [6]. Manytimes this allows a visual comprehension of itsessential features. Figure 2 is one such represen-tation of Figure 1. Readily, this particular rendition

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Fig. 2. A graphic representation of Figure 1.

yields little insight! One can improve the pictureconsiderably by iteratively moving nodes and theirconnections, so as to shorten the connecting edgesand thus cluster related nodes. We’ll see such areorganized drawing later as Figure 8 in Section III.Unfortunately however, most interesting networksare “large”, far too large to effectively organizein such an iterative fashion without some priorunderstanding of the underlying structure.

A common approach is to try to spatially clusterthose highly interconnected nodes and separate them

2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

978-0-7695-4799-2/12 $26.00 © 2012 IEEE

DOI 10.1109/ASONAM.2012.111

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2012 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining

978-0-7695-4799-2/12 $26.00 © 2012 IEEE

DOI 10.1109/ASONAM.2012.111

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in the image. Then one should be able to spotthe connecting edges and outlier points. But, assuggested by much of the literature cited above,clustering is not an automatic process. For exam-ple, many techniques must first input the estimatednumber of clusters to function effectively.

The approach described in this paper involvesreducing the network to its “fundamental cycles”.These are cycles of length greater than 3 with nocross connections. The reduction process and thenature of fundamental cycles is explored in SectionIII. The network shown in figures 1 and 2 will beused as a running example in this section. But, it isfar to small to be considered realistic. In Section IVwe explore this reduction process as it is applied tolarger, real life networks. In Section V, we examinesome of the underlying mathematics.

II. NEIGHBORHOOD CLOSURE

A collection of sets C = {Yi} is a closure systemon a set P provided (a) P ∈ C, and (b) Yi ∩ Yk ∈C for all i, k. We use the neighborhood concept toestablish a closure system on a network, or graph,N = (P,A). Non-zero entries (x, z) in A denotesome form of connection between x and z. Theyconstitute edges in a graphical representation of Nsuch as figures 2 or 3.

Let Y ⊆ P , the neighborhood of Y , denotedY.η, is Y.η = {z ∈ P−Y | (yi, z) ∈ A for someyi ∈ Y }. That is, Y.η consists of all points not in Ythat are connected to at least one element of Y . Bythe region dominated by Y , denoted Y.ρ, we meanY.ρ = Y ∪ Y.η, or Y together with its surroundingneighborhood. Finally, we define the closure of Y ,denoted Y.ϕη , by Y.ϕη = {z ∈ Y.η | z.η ⊆ Y.ρ},that is, those neighbors z whose own neighborhoodis completely dominated by Y . A set Y is closed ifY.ϕη = Y . The collection of sets {Yi.ϕη}, Yi ⊆ Pconstitute a closure system. Readily, P.ϕη = P andit is not hard to show that Yi.ϕη ∩ Yk.ϕη is closed.

In Figure 3, {a}.η = {b, c, d}. In our work, all

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Fig. 3. An undirected graph or network.

the elements are sets; thus we regard the “point” ato be a singleton set {a}. But, repeated use of thedelimiters {, } becomes tedious, so we will normallyelide them and write a.η = bcd, unless we want tospecifically emphasize the set nature.

The region dominated by d, or d.ρ, is abcefh.Readily h.η ⊆ d.ρ, so h ∈ d.ϕη . Similarly, a.η ⊆c.ρ, so a ∈ c.ϕη. In the network of Figure 3, c.ϕη =ac, d.ϕη = dh, and x.ϕη = x for all x �= c ord. Consequently, we see that the individual pointsof a network may, or may not, be closed. If everysingleton set (i.e. point) is closed, the network issaid to be irreducible.

There is a much more detailed treatment of neigh-borhood closure in [20] where it is used to definecontinuous network change. But, we will not needit here. Instead we focus solely on those nodes zfor which there exist a singleton node y such thatz ∈ {y}.ϕη. We say z is subsumed by y.

III. NETWORK REDUCTION

In classical mathematics, open sets are used toestablish the topology of continuous manifolds. Wefind closed sets are more valuable tools for under-standing discrete structures like a network.

If y subsumes z in N , then any closed setstructure involving z must equally include y, soz contributes little to our understanding of N . Wecan delete z and all incident edges with almost noloss of information. If the subsumed points a, h andincident edges are removed from Figure 3 we obtainFigure 4. The deleted nodes and edges are indicated

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by dashed lines. Each of the remaining nodes, e.g.{b}, is a closed set.

We have automated this rather simple processof removing subsumed points from a network. Thecore of the reduction process, which we will denoteby ω, is the following loop: it iteratively removes

for_each y in P{for_each z in y.nbhd

{if (z.nbhd contained_in y.region

{ // z is subsumed by yfor_each x in z.nbhd

remove edge (x, z)remove z from network}

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Fig. 5. Key loop in reduction process.

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subsumed points and edges until all singleton sets(i.e. nodes) are closed and the network is reduced.

Applied to the network of Figure 2, we obtainthe network of Figure 6 with only 11 nodes and 13edges.

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A. Fundamental Cycles

The striking characteristic of the irreducible sub-networks of figures 4 and 6 is that the remain-ing nodes and connecting edges form cycles. Thetwo cycles of Figure 4 are < b, c, d, e, b > and< c, d, f, g, c >. A cycle < y0, y1, . . . , yn = y0 >of length n ≥ 4 is said to be a fundamental cycleif yi+1 ∈ yi.η, for 0 < i < n, and yk ∈ yi.ηimplies k = i + 1. That is, there are no “crossconnections”. The two cycles of length 4 in Figure4 are fundamental. A graph/network composed onlyof fundamental cycles is in many ways the antithesisof “triadic closure” in social theory [12], [16] and“chordal graphs” in graph theory [10], [14]. Thelatter are graphs with no chordless cycles of lengthgreater than 3.

The cyclic structure of Figure 6 is more evidentwhen redrawn as Figure 7. Here, the fundamental

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Fig. 7. The fundamental cycles of Figure 6.

cycle < 16, 15, 4, 20, 10, 12, 23, 18, 16 > of length8 is quite prominent. Longer fundamental cyclesconvey more information about the global structureof the network.

Our program keeps a count of the numbers ofnodes subsumed directly, or indirectly, by each node.The numbers in angle brackets, < n >, indicate this.The number, < 5 >, of subsumed nodes associated

with node 20, and < 4 > associated with node 18suggest that there may be small clusters attached tothe cycle at these points. The count of subsumednodes, < 1 > and < 2 >, associated with nodes 6and 9 are less convincing.

In Figure 8 we have reconstituted the originalnetwork around the two fundamental cycles of Fig-ure 7. It is similar to one found in [17]. This is

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Fig. 8. Figure 2 reconstituted around its fundamental cycles.

accomplished by iteratively expanding each point ona fundamental cycle. For example, nodes 13 and 21were the two points subsumed by node 9. Node 6subsumed 22.

In Figure 8 we have circled the small clusterswith dashed lines. In the program they are simplydenotes as sets of nodes. It is fairly obvious that 4and 12 are the “mules” in this small network. Thecross connection between 16 and 18 may also be ofinterest.

IV. NETWORK REDUCTION IN REAL LIFE

A more striking example is the reduction ofthe 379 node network of collaborating scientistsconstructed by M.E.J. Newman [18]. The readeris encouraged to view an annotated version atwww.umich.edu/mejn/centrality.

After 3 iterations identifying and removing sub-sumed points, our program produced the graph ofFigure 9. There are 65 points in this reduced net-work. This network is much more comprehensible,both visually and algorithmically, than the original.Values < n > denote the number of points sub-sumed by each node.

Our graph reduction algorithm removes the pen-dant chordal subgraphs that were present in theoriginal. Basically, only the key fundamental cyclesremain, together with those triangles all of whosepoints lie on some chordless cycle of length ≥ 4.But the information loss is minimal. As seen above,the subsumed portions can be regenerated if neces-sary. Of more significance to us are those retainednodes, and paths in Figure 9, which “connect” thecommunities of intellectual activity.

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Fig. 9. The reduced version of Newman’s 379 node collaboration network.

In his book [8], Malcolm Gladwell talks of con-nectors which are the central nodes of communities,or clusters, comprised of many connections. Whatwe see in Figure 9 are those singular “connectors”between clusters which are rarer, and harder to findin A by more usual numerical methods. For exam-ple, there is a connection/edge between the nodeat the top, denoting a cluster of < 23 > individualswith a node representing < 18 > persons. Similarly,there is a path towards the lower left through nodeslabeled < 1 > and < 2 > to a community of< 17 > individuals. These two nodes are likelycandidates to be mules.

Because of the arbitrary order in which nodes aresubsumed, which was exhibited to some extent inthe expansion of Figure 8, the values < n > shouldnot be taken too literally. For example, one couldargue that the nodes encircled with the dashed lineactually comprise a single community. But, the node< 18 > would be a natural candidate for automaticre-expansion, subject to whatever independent com-munity criteria is appropriate for the application.

A. Process Performance

We have not had the opportunity to processenough large networks to accurately measure theperformance of our reduction algorithm. But, itappears to be essentially linear in the number, n,of nodes of the network.

First, all operations are performed on sets, or col-lections of sets. We use a C++ software package thatrepresents every set, or collection, as a simple bitstring. Individual points and edges are only involvedin conversion to/from singleton sets and neighbor-hoods during input and output. Consequently, anoperation such as

if (z.nbhd contained_in y.region)

involves only the intersection (logical and) of twobit strings followed by a test for equality. The codedoes not loop over either set.

Next, we observe that the process of Figure 5loops over all n nodes, y, of the network. And thenthe inner loop runs over its neighborhood, y.η. If Nis a complete graph then |y.η| = n−1 and the orderof ω would appear to be n2 in this worst case. But,since nodes are removed from P within the loop,all of the nodes of Y.η will be subsumed and theouter loop will execute but once. It still appears tobe linear.

In practice, the size of neighborhoods is bounded,say |y.η| �= k, and average neighborhood size isquite a bit smaller. Consequently execution of thereduction loop in Figure 5 is bounded by k · n.

However, the subsumption loop of Figure 5 mayhave to be iterated. A point z may not be subsumeduntil after other points in its neighborhood, z.η,have been subsumed and removed. This is dependenton the order in which the points y of the outerloop are accessed. We are convinced, but have noexample, that there exists a worst case ordering ofthe points such that only one node is subsumed ineach iteration. Thus the worst case behavior couldbe of order n2.

However, in practice, no network reduction hasrequired more than 3 iterations, and each iteration isover a set P of decreasing size. As indicated above,we believe this process is effectively linear and thusscalable.

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V. MATHEMATICAL DETAILS

Although network reduction is a process thatlargely speaks for itself, its mathematical underpin-nings can be of interest. We explore these detailshere.

A graph, or subgraph, is said to be chordalif it contains no cycles of length greater than 3without a chord (edge) joining two of its points[10], [14]. Any complete graph, Kn, is chordal.Every tree is chordal. In fact, chordal graphs can beregarded as tree-like structures of point connected,or edge connected, complete graphs Kn. A cycleC =< y0, . . . yn = y0 > is chordless if no subsetof its points {yi, . . . yk}, 0 ≤ i, k ≤ n forms a cycle.We will call a chordless cycle, such as C, an n-cycle, where n denotes both the number of pointsand number of edges. Thus, a graph is chordal if itcontains no n-cycles, n ≥ 4.

An n-cycle C is fundamental if for all yi ∈ C,{yi} is closed. The key elements of Figure 8 areits fundamental 4-cycle < 2, 6, 9, 16, 2 > and its8-cycle < 16, 15, 4, 20, 10, 12, 23, 18, 16 >. Thesefundamental cycles define the topology of the net-work in much the same manner that 1-cycles can beused to define the topological structure of manifolds[7].

Proposition 5.1: Let G be a finite network andlet G′ = G.ω be a reduced version, then G′ isirreducible.Proof: Suppose {y} is not closed. Then ∃z ∈ y.ϕη im-plying z.ρ ⊆ y.ρ or that z is subsumed by y contradictingtermination of the reduction code.

Proposition 5.2: Let G be a finite network withG′ = G.ω an irreducible version. If y ∈ G′ is notan isolated point then either

(1) there exists a fundamental n-cycle C, n ≥ 4such that y ∈ C, or

(2) there exist fundamental n-cycles C1, C2 eachof length ≥ 4 with x ∈ C1 z ∈ C2 and y lies on apath from x to z.

Proof: Let y1 ∈ PG′ . Since y1 is not isolated,let y0 ∈ y1.η, so (y0, y1) ∈ A. With out lossof generality, we may assume y0 ∈ C1 a cycleof length ≥ 4. Since y1 is not subsumed by y0,∃y2 ∈ y1.η, y2 �∈ y0.η, and since y2 is not subsumedby y1, ∃y3 ∈ y2.η, y3 �∈ y1.η. Since y2 �∈ y0.η,y3 �= y0.Suppose y3 ∈ y0.η, then < y0, y1, y2, y3, y0 >constitutes a n-cycle n ≥ 4, and we are done.Suppose y3 �∈ y0.η. We repeat the same path exten-sion. y3.η �⊆ y2.η implies ∃y4 ∈ y3.η, y4 �∈ y2.η. Ify4 ∈ y0.η or y4 ∈ y1.η, we have the desired cycle.If not ∃ y5, . . . and so forth. Because G is finite,this path extension must terminate with yn ∈ yi.η,where 0 ≤ i ≤ n− 3. Let x = y0, z = yn.

The points of those chordal subgraphs still remain-ing in Figure 8 such as the triangle < 9, 15, 16 >,are all elements of other fundamental cycles aspredicted by Proposition 5.2.

Proposition 5.3: Let G′ = G.ω be an irreducibleversion of a finite network G. A path (possibly oflength 0) exists between w′ = w.ω and x′ = x.ω inG′ if and only if there exists a path between w andx in G.

Proof: First, let there be a path from w to x inG. If z is on some path between w and x, and zhas been subsumed by y in the reduction process,then z.η ⊆ y.η implies there exists a path betweenw and x through y.Now, assume there exists a path between w′ = w.ωand x′ = x.ω in G′. Suppose w′ = x′ then w andx have been subsumed by some single point, say y.Since subsumption, of say w, requires w ∈ y.η, andsimilarly for x, y must lie on a path between w andx in G.Now, suppose w′ �= x′. The reduction process neveradds an edge, so the edges in the path between w′

and x′ in G′ are edges in G, and there exists a pathfrom w in the tree-like chordal graphs that mappedonto w′ and similarly a path from x in the tree thatmapped on to x′; thus a path between w and x inG.

Corollary 5.4: If a finite network G is connectedthen its reduced network G′ = G.ω is connected.

Note, however, that the reduced graph G′ may bea single point. In particular, this will be the casewhenever G is a tree, or a chordal graph.

If x and y can mutually subsume each other in P ,then x.η = y.η. But the converse need not be true asillustrated by Figure 10. The graph (a) is irreducible,

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Fig. 10. Mutual subsumption illustrated by two graphs.

all singleton points z are closed. The graph (b) willbecome irreducible with the deletion of either x or y.In graph (b) x.ρ = y.ρ, the necessary and sufficientcondition for mutual subsumption.

Two graphs, or networks, G = (P,A) and G′ =(P ′,A′) are said to be isomorphic, or G ∼= G′, ifthere exists a bijection, i : P → P ′ such that for allx, y ∈ p, (i(x), i(y)) ∈ A′ if and only if (x, y) ∈ A.That is the mapping i precisely preserves the edgestructure, or equivalently its neighborhood structure,that is i(y) ∈ i(x).η′ if and only if y ∈ x.η.

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As indicated by Figure 10, the order in whichpoints, or more accurately the singleton subsets,of G are encountered can alter which points aresubsumed and subsequently deleted. Nevertheless,the reduced graph G.ω will be unique, upto isomor-phism.

Proposition 5.5: Let G.ω and G.ω′ be irreducibleimages of a finite network G, then G.ω ∼= G.ω′.

Proof: Let y0 ∈ G.ω, y0 �∈ G.ω′. Then y0 issubsumed by some point y1 in G.ω′ and y1 �∈ G.ωelse because y0.ρ ⊆ y1.ρ implies y0 ∈ {y1}.ϕη soG.ω would not be irreducible.Similarly, since y1 ∈ G.ω′ and y1 �∈ G.ω, thereexists y2 ∈ G.ω such that y1 is subsumed by y2.Now we have two possible cases; either y2 = Y0,or not.Suppose y2 = y0 (which is most often the case),then y0.ρ ⊆ y1.ρ and y1.ρ ⊆ y0.ρ or y0.η = y1.η.Hence i(y0) = y1 is part of the desired isometry, i.Now suppose y2 �= y0. There exists y3 �= y1 ∈ G.ω′

such that y2.ρ ⊆ y3.ρ, and so forth. Since G is finitethis construction must halt with some yn. The points{y0, y1, y2, . . . yn} constitute a complete graph Yn

with {yi}.ρ = Yn.ρ, for i ∈ [0, n]. In any reductionall yi ∈ Yn reduce to a single point. All possibilitieslead to mutually isomorphic maps.

Proposition 5.6: Let G′ = G.ω be an irreducibleimage of a finite network G. For all Y ⊆ PG,Y.ϕη.ω ⊆ Y.ω.ϕη

′.Proof: Let z ∈ Y.ϕη. We must show that either

z ∈ Y or z ∈ Y.ω or z ∈ Y.ω.ϕη′.

If z �∈ Y , then ∃y0 ∈ Y, z ∈ y0.η. If z �∈ Y.ω, then∃x ∈ z.η, x �∈ y0.η so ∃y1 ∈ Y, x ∈ y1.η. Whethery0, z, x, y1 are distinct in G.ω or not, by Prop. 5.3,x ∈ Y.ω.η′ so z ∈ Y.ω.ϕη

′.An operator, such as ω, satisfying the containment

property of Proposition 5.6 is said to be continuous[19], [20]. Continuous operations on discrete spaceshave a number of interesting properties; but theyare not relevant to this paper. Nevertheless, that thisreduction process is a continuous transformation ofthe network N onto its irreducible version N.ω isquite satisfying.

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