A Modified Measure of Covert Network Performance


 Mitchell Garrett
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1 A Modified Measure of Covert Network Performance LYNNE L DOTY Marist College Deartment of Mathematics Poughkeesie, NY UNITED STATES Abstract: In a covert network the need for secrecy is at odds with the desire for easy transmission of information Lindelauf et al [7] have defined several measures that can be used to evaluate the total erformance of a covert social network by using a roduct of individual measures of secrecy and of information transmission As one of their simlest measures of secrecy, Lindelauf et al use a modified version of the idea of neighbor connectivity Using this measure, the otimal network structure is a star grah In this aer we modify the Lindelauf measure of secrecy to include information about the connectedness (not just the order of the survival subgrah using a measure based on ideas related to Chvátal s toughness arameter We determine an uer bound on erformance of trees and conjecture that a secific class of sider grahs achieves maximum erformance We also describe several oortunities for further research Key Words: covert networks, neighbor connectivity, sider grahs, toughness Introduction In a covert network the need for secrecy is at odds with the desire for easy transmission of information A hallmark of any covert network is the idea that the discovery of one erson (one vertex in the network renders that erson and all members of the network known to him useless to the network This means the closed neighborhood of the vertex is removed from the grah roducing what has been called the survival subgrah Gunther, Hartnell, and Novakowski studied survival subgrahs from the standoint of their ability to remain connected using neighbor connectivity Lindelauf et al [7], [8] have defined several measures that can be used to evaluate the total erformance of a covert social network by using a roduct of individual measures of secrecy and of information transmission They use a standard social network notion (total distance to measure ease of information transmission As one of their simlest measures of secrecy, Lindelauf et al use a modified version of the idea of neighbor connectivity To measure a network s secrecy they calculate for each vertex the ratio of the order of the survival subgrah when this vertex is discovered (subverted to the order of the original grah, and then sum these ratios over all vertices of the grah Using this measure, the otimal network structure is a star grah The discovery of any vertex in the star, however, results in a grah that is totally disconnected or emty The urose of this aer is to modify the Lindelauf measure of secrecy to include information about the connectedness (not just the order of the survival subgrah Using this modified measure, cycles erform better than aths or stars, as seems more intuitively aealing We determine an uer bound for this new erformance measure for trees We conjecture that the otimal tree is a double star a secial tye of sider grah and we give some evidence and analysis to suort the conjecture Hartnell and Gunther [6] have identified these double star grahs as best ossible using different covert network measures thereby roviding some validation for the usefulness of the new measure introduced in this aer Definitions and Examles In order to exlain these ideas more recisely, we introduce some definitions All standard notation and terminology can be found in [0] Suose G is a grah with vertex set V For any vertex u of V, N[u] = {u} {v V v is adjacent to u} is called the closed neighborhood of u The next two definitions essentially follow Gunther and Hartnell [], [] and Gunther, Hartnell, Nowakowski [4] To subvert a vertex v of G means to remove all elements of N[v] from G The resulting induced subgrah, called the survival subgrah, is exactly the subgrah of G induced by V \ N[v] Lindelauf et al use the term discover instead of subvert, but the concet is the same The secrecy of a grah G is defined as a sum of se EISSN: Issue 9, Volume, Setember 0
2 crecy values for each vertex in the grah In the scenario where we assume each vertex is equally likely to be discovered or subverted, and whenever a vertex is discovered all its neighbors are detected and therefore must be removed from the grah, the secrecy measure of Lindelauf et al is defined as follows For any grah G with vertices {v, v,, v }, TS(G = ( N[vi ] = ( N[v i ] ( This secrecy measure is combined with a standard measure for ease of information sharing, the normalized recirocal of total distance For any grah G with vertices {v, v,, v }, we use d G (v i, v j to denote the distance in G between v i and v j, suressing the subscrit whenever the grah being referred to is clear from context The total distance of G, TD(G, is defined by TD(G = d(v i, v j v i v j The information erformance, IP(G, is defined by IP(G = ( TD(G Note that information erformance is normalized with resect to K and 0 IP(G with a higher value indicating better erformance Finally, the Lindelauf measure of network erformance of grah G is simly the roduct TS(G IP(G To define a secrecy measure that includes information about the order and number of comonents of the survival subgrah, we use a modified version of toughness of a grah, an idea first defined by Chvátal [] For each vertex v i of grah G, we use c i to denote the number of comonents in the survival subgrah G N[v i ] For the rest of the aer we use the following total secrecy measure of a grah G Definition Let G be a grah with vertices {v, v,, v } The total secrecy of grah G, TS(G, is defined by TS(G = ( N[vi ] c i = ( N[vi ] c i ( This secrecy measure modifies each term of the sum in equation ( For the subversion of each vertex, the order of the survival subgrah is relaced by the average order of each comonent of the survival subgrah Using this measure we have the following definition of erformance of a grah G, Definition Let G be a grah with vertices {v, v,, v } The erformance of grah G, PERF(G, is defined by PERF(G = IP(G TS(G ( = TD(G ( N[vi ] c i ( ( N[vi ] c i = TD(G ( Note that larger values of erformance occur when the exected average order of comonents in the survival subgrah is larger and when the total distance of the original grah is smaller This is the measure of erformance that will be used for the remainder of the aer Examle To illustrate these definitions we comute the secrecy, information, and erformance for three basic grahs on vertices: cycle, ath, and star For the cycle, C, TS(C = For the ath, P, TS(P = ( N[vi ] = c i ( N[vi ] c i = = [ ( + ( + ( 4 ] = + 8 For the star, K,, TS(K, = ( N[vi ] c i = [ 0 + ( ] = EISSN: Issue 9, Volume, Setember 0
3 By comaring these formulas, one can see that for all 4, TS(C TS(P TS(K, The following total distance formulas for the cycle, ath, and star are easily calculated: TD(C = { 4, if is even 4, if is odd TD(P = ( and TD(K, = ( Combining these formulas and recalling that PERF(G = IP(G TS(G = ( TD(G TS(G leads to the following erformance measures for the cycle, ath, and star PERF(C = { 4( 4( PERF(P = = 4( (, if is even = 4( (, if is odd; ( ( ( + 8 = ( + 8 ( +, PERF(K, = ( ( = With a bit more algebra it can be shown that for all 5, PERF(C PERF(P PERF(K,, a result that seems to align well with other measures of network survivability Main Result The main result of this aer is the determination of a bound for erformance of trees with vertices As we analyze the erformance of trees, the following notation and definitions will be useful We denote the degree of vertex v in grah G by deg G (v We also use ts G (u to abbreviate ( N[vi ] and td G (u to abbreviate v V d(u, v In all these abbreviations we suress the subscrit whenever the grah being referred to is clear from context A uv twig in a tree is a u v ath in which v has degree at least three, u has degree one and all other vertices on the ath have degree two We call the vertex v the base of the srig A ktwig is a twig of length k For any vertex v with deg G (v and which is adjacent to deg G (v leaves, we define a vsrig to be the subgrah induced by v and its adjacent leaves, and we call the vertex v the base of the srig Observation The basic leverage for establishing the bound is the observation that for M, whenever a M, it follows that a + c b + d M b M and c d For a given tree T, we look at the terms contributing to ts(v and to td(v for an individual vertex, or for a grou of vertices, V, and we show v V ts(v M td(v v V Then for the entire vertex set, V, of T v V ts(v M td(v v V Another imortant observation is that for any vertex u in a tree of order with c u 6, we immediately know = N[u] More c u 6 over, for vertex u in any tree excet K,, we know td(u Hence for all vertices with c u 6, td(u /6 to show that 6 For c u, it is straightforward td(u 6, and when c u =, to show ( + 5 When c u =, however, the td(u 6 roofs are more comlex, and so are resented searately Lemma Let T be a tree of order 0 that is not K, and u be a vertex of T If c u, then td(u 6 If c u = and d T (u, then td(u 6 If c u = and deg T (u =, then td(u 6 ( + 5 Proof: Let u be a vertex of tree T, not K,, of order 0 Let c u For ease of notation we use du as an abbreviation for deg T (u Since all vertices of T \ N[u] are at least distance from u, td(u du + ( du = du Combining this with the fact that = du and recalling that c u, we have td(u du c u ( du = du c u du ( 6 c u Now let c u = If du, then there are exactly vertices that are exactly distance from u, and at least c u EISSN: Issue 9, Volume, Setember 0
4 du that are distance at least from u Hence td(u du ( du = du 5 When du, we have td(u du ( du 5 = 6 + du 6 When du =, the revious equation becomes td(u ( 7 = 6 + 6( 7 ( ( + 5 = ( + 5, 6 6 since we have We now begin the analysis of for a vertex td(u u with c u = Note that for any such vertex u in a tree, T, u is adjacent to deg T (u leaves and one vertex with degree Hence u is the base of a srig and is adjacent to exactly one other vertex of degree, or u is the leaf on a ktwig with k, or u is the vertex adjacent to the leaf of a ktwig for k These ossibilities are shown in Figure We begin by bounding the value of for the base of a srig td(u u z Figure : Vertex u with c u = Lemma 4 Let T be a tree of order 0 and u be a vertex of T that is the base of a srig and with c u = Let z be the vertex of T adjacent to u with deg(z = Define V = N[u] \ {z} Then V V td(u du du + (4 6du + { /6, if du 4; /5, if du = where du = deg(u Proof: Let u i, i du, be the leaves in N[u], so V = {u i i du } {u} Then = du and ts(u i = du Thus V = u ( du + (du = du For du the distance calculation, note that since c u =, there is exactly one vertex with distance from u (and from each u i So there are du vertices each of which is at least distance from u (and distance 4 from each u i Hence td(u ( du +du+ = du 4, and td(u i 4( du + (du ++ = 4 du 6 Combining these, we have V td(u = ( du 4 + (du (4 du 6 = du + (4 6du + Hence V V td(u du du + (4 6du + (4 To comlete the roof we analyze the quadratic in the denominator Since du <, the minimum value of this quadratic is realized at the min imum value of du Substitution into the fraction in equation (4 gives the values listed in the lemma We next establish a bound for the remaining two cases when vertex u has c u =, namely when ( u is the leaf of a ktwig for k, and ( u is the vertex adjacent to the leaf of a ktwig for k u z u u* w z u u* z Case Case Case Figure : Lemma 5 Cases Lemma 5 Let T be a tree of order 0 and let u be a vertex of T with c u = that is not the base of a srig Then u is on a ktwig with base z For k =,, let V be the set of vertices on the twig, not including z For k 4, let V contain the leaf and the single vertex adjacent to the leaf of the ktwig Then V V td(u 4 Proof: For vertex u with c u = that is not the base of a srig, u is on a ktwig with base z We use cases deending on k The cases are illustrated in Figure Case : Assume vertex u is on a twig Let z be the base of the twig, and label the other vertex on the twig w Let V = {u, w} Again we use to abbreviate deg(z Then = and ts(w = Thus V ts(v = ( + For distance, note that there are vertices distance from u (distance from w, and there are vertices each of distance at least 4 from u (at least EISSN: Issue 9, Volume, Setember 0
5 from w So td(u 4( + ( + = 4 8, and td(w ( + ( + = 6 Combining we have V td(v = 7 4 Hence V ts(v V td(v Some straightforward algebra shows ( ( (5 The minimum value of the quadratic (for occurs at the maximum value of, and substituting into the quadratic yields 9+4 which is clearly nonnegative for all Hence the last inequality in equation (5 is true Case : Assume vertex u is on a twig Then u may be the leaf or the vertex adjacent to the leaf Since we will grou these two vertices we let u be the leaf and u the vertex adjacent to it Let z be the base of the twig, and label the other vertex on the twig w Let V = {u, u, w} In this case, =, ts(u =, and ts(w = Thus V ts(v = ( + ( + Analyzing distances as in Case, we have td(u 5( + 4( + 6 = 5, td(u 4( + ( + 4 = 4, and td(w ( +( +4 = 7 Combining we have V td(v = Hence V ts(v 5 + td(v V Some straightforward algebra shows ( (4 4 + (6 The minimum value of this quadratic occurs at the extreme values of, and Substituting each of these values into the quadratic it is easy to verify that the quadratic is nonnegative for all 0 Hence the last inequality in equation (6 is true Case : Assume vertex u is on a ktwig, k 4 As in the revious case, u may be the leaf or the vertex adjacent to the leaf Since we will grou these two vertices we let u be the leaf and u the vertex adjacent to it Let z be the base of the twig Let V = {u, u } In this case, = and ts(u = Thus V ts(v = 5 For distances, note that there are k vertices each with distance at least k + from u Thus we have td(u (k+( k + k i = k + k+ Note that the distance between u and the other vertices is less than the distance between u and these other vertices So td(u k + k +, and we have V ts(v V td(v 5 k + ( k + Again straightforward algebra shows 5 k + ( k k + ( k (7 Note that k, and so the minimum value of the quadratic occurs at the minimum value of k We substitute k = 4 and verify that the quadratic is nonnegative for all 0 Hence the inequality in equation (7 is true We are now ready to rove the main result Main Theorem Let T be a tree of order 6 Then PERF(T 4 (8 Proof: Using the list of trees of order 0 or less in Aendix of [5], we have verified for all trees T with order, 6 9 that PERF(T 4 by direct calculation Now let T be a tree of order at least 0 We consider vertices, or grous of vertices, according to the value of c u For any vertex u with c u, we have td(u 6, by Lemma If c u = and u is the base of a srig, then by Lemma 4, V V td(u 5, where V is the set of srig vertices If c u = and u is not the base of a srig, then by Lemma 5, V V td(u 4, for the grou of vertices V described in the lemma Using Observation, it follows that ( PERF(T = V ts(v ( V td(v 4 EISSN: Issue 9, Volume, Setember 0
6 4 A Family of Almost Otimal Trees The trees that we conjecture have best erformance are sider grahs in which all legs, excet ossibly one, have length two Secifically, for all 7 we define the double star as follows Definition 6 The double star with 7 vertices, denoted DS, is the grah with vertex set V = {v 0, v,, v } and edge set E where the recise nature{ of E deends on the arity of If } is odd, E = {v 0, v i }, {v i, v i+ } i If is { even, E = {{v, v }} {v 0, v i }, {v i, v i+ } i } Thus the double star grah DS has one vertex, v 0, with degree together with aths of length two (twigs if is odd, or aths of length two (twigs and one ath of length three ( twig if is even See Figure = u V w = u V w d G (u, v + d G (v, w d G w (u, v + d G (v, w = td G w (v + d G (v, w Using this information we have TD(G = v V = = ( v V w ( = v V w v V w td G (v td G (v + td G (w (td G w (v + d G (v, w td G w (v + v V w = TD(G w + td G (w, + td G (w d G (v, w + td G (w as desired Now we calculate the erformance measures for the double stars and give a lower bound for the measures that is indeendent of arity of Theorem 8 Let DS be the double star with 7 Figure : Double stars for = 9, 0 Hartnell and Gunther [6] have shown, in informal terms, that these double stars rovide the best defense against otimal subversion strategies when one considers order of the remaining covert network and the order of the remaining comonents Thus our conjecture can be seen as being an extension of their work in the sense that we are measuring the information erformance of the network as well as its survivability as measured by secrecy erformance Before calculating the erformance of this grah, we establish useful fact for calculating total distance of any grah Lemma 7 Let G be a grah If w is a leaf of G, then TD(G w = T D(G td G (w Proof: Let G be a grah with leaf w, and use V and V w to denote the vertex sets of G and G w, resectively First we note that since w is a leaf, d G (v, u = d G w (v, u, for all u, v V w Thus td G (v = u V d G (u, v (a If is odd, PERF(DS = + ( 5 (b If is even, PERF(DS = ( ( ( + ( 8 (c For all, PERF(DS ( ( Proof: For art (a, we assume is odd Note that ts(v 0 = since DS N[v 0 ] is ( isolated vertices Next observe that for i, ts(v i = since each comonent of DS N[v i ] has exactly two vertices Finally, for each leaf ( i (, ts(v i = since the survival subgrah is connected Hence ts(v i = ts(v 0 + ts(v i + ( = + ( = + + i= ( ts(v i = + ( EISSN: Issue 9, Volume, Setember 0
7 So for odd, TS (DS = + (9 To calculate TD (DS we artition the vertices in the same way that we did to calculate secrecy TD (DS = td(v i = td(v 0 + Hence IP(DS = td(v i + i= td(v i = ( + ( ( ( ( + 7 = ( (8 + 6( = ( ( 5 ( TD(DS =, and so 5 PERF(DS = IP(DS TS(DS = + ( 5 when is odd For art (b, assume even, then the double star has one twig So for all calculations we grou the vertices on each twig For secrecy we have ts(v i = ts(v ( ts(v i + ts(v i+ ts(v + ts(v + ts(v ( ( 4 = ( + ( + ( = ( ( = 6 ( ( 4 + ( + + ( 5 = So for even, + TS (DS = ( (0 To calculate TD (DS when is even, we note that DS = DS v Since v is a leaf we use the revious lemma together with the formula for total distance of the double star with an odd number of ( vertices First we note that td DS (v = = ( 8 From the revious lemma we have TD(DS = TD(DS + ts(v So we have and IP(DS = = (( (( 5 + ( 8 = ( + ( 8 ( TD(DS = ( ( + ( 8 PERF(DS = IP(DS TS(DS = ( ( ( + ( 8 when is even For art (c, straightforward algebra verifies that the bound in art (c is less than or equal to each of the bounds in arts (a and (b It is reasonable to conjecture that a best erforming tree has only one large degree ( vertex More large degree vertices would increase the distance between eriheral vertices At the same time, if such vertices are not the bases of srigs, they would contribute smaller (comared to degree and degree vertices terms to the total secrecy sum Within this restriction, we can see that the double star has a large number of vertices that contribute the maximum ossible term to total secrecy, namely This tree however does not have the best secrecy even among sider grahs A sider grah with legs of length (twigs has aroximately of its vertices that contribute either or Note that once the length of the leg (or twig increases beyond, each additional vertex on the leg contributes only So it would seem that legs of length or would rovide the best secrecy erformance The tradeoff comes, however, when total distance (used to measure information is considered Although, as the following roosition establishes, a sider grah with all legs of length has better secrecy than the double star of the same order, the larger total distance (worse information erformance offsets, just barely, the better secrecy erformance The total erformance is quite close, and so if secrecy were of higher imortance, one could make the case that the sider grahs with legs of length could be a better choice in certain covert network oerations EISSN: Issue 9, Volume, Setember 0
8 Proosition 9 Let S be the sider grah with order If S has = k + and k legs of length, then TS (S = ( + and IP(S = 4( 7 Further, for k, TS (S > TS (DS, but PERF(DS > PERF(S Proof: Let G be the sider grah with order = k + and v 0 be the one large degree vertex Label the vertices on each leg of the sider, in order, as v i+, v i+, v i+ with v i+ adjacent to v 0, and v i+ as the leaf, 0 i k To calculate TS (S, note first that ts(v 0 = since each comonent of S N[v 0 ] has exactly two vertices Next ts(v i+ = k and ts(v i+ = Hence k ts(v i = ts(v 0 + (ts(v i+ + ts(v i+ ( = + k + k( 5 k ( = + ( + ( 5 = 4 + So TS (S = ( + ( To calculate TD (S we artition the vertices in the same way that we did to calculate secrecy Note first that td(v 0 = 6k, recalling that there are 4k = legs on the sider Next td(v i+ = 4+9(k, td(v i+ = 4+(k, and td(v i+ = 6+5(k Thus TD (S = td(v i k = 6k + (4 + 6(k = k( (k = k(6k 6 = (( 6 4( ( 7 = Hence IP(S = ( TD(S =, and so 4( 7 PERF(S = IP(S TS(S = + ( 7 ( when is divisible by To see that TS (S > TS (DS when divides, we comare TS(S (see Equation with TS(DS (see Equations 9 and 0 Simle algebra shows ( + > +, for all 9, and ( + >, for all 8 Finally, to verify that DS has better overall erformance that S, we comare Theorem 8(c with Equation and note that it is easy to verify 6 ( ( + 5 > + ( 7, ( for all Direct calculation shows that PERF (DS 0 > PERF (S 0 It is interesting to note that the advantage in secrecy of these sider grahs over the double stars, TS(S TS(DS, asymtotically increases to 6, while the erformance advantage of the double stars over these sider grahs, PERF(DS PERF(S < 4, goes to 0 as increases Note that the formula for PERF(DS (from Theorem 8(c in equation drastically underestimates the actual erformance of DS for small values of Hence the erformance advantage is also underestimated Table shows secrecy and erformance measures for DS and S for selected small values of in order to give a more comlete understanding of the true relationshis TS PERF S DS S DS Table : Secrecy & Performance for DS and S EISSN: Issue 9, Volume, Setember 0
9 5 Further Research Since the family of double star grahs does not achieve the bound established in the main result and we have not been able to discover any trees of order at least 7 (the smallest order for which the double star is not simly a ath that erform better than the double stars, we conjecture that the bound can be imroved to the erformance of the double star of any articular order A quick review of Lemma and the lower bound for double star erformance shows that the any vertex u with c u tends to kee the erformance of a tree less than that of the double star As noted in the discussion receding Proosition 9, the vertices that can hel a tree achieve erformance larger than 6 are those with c u =, and so any attemt to lower the bound to around 6 should focus on showing that the existence of such vertices forces the existence of other vertices with larger values of In articular, the td(u bound on the contribution from vertices on srigs and twigs are rime candidates for further analysis Another aroach to showing double stars are otimal could focus on showing that under certain conditions trees with exactly one vertex of degree greater than erform better than trees with more such large degree vertices Once the tree conjecture is settled one could attemt to find high erformance grahs by focusing, erhas, on grahs that have neighbor connectivity at least For these grahs the total secrecy will be high simly because c u = for all vertices u Moreover, whenever c u = for all vertices in G, TS(G has a articularly simle form that should make analysis of overall erformance easier Secifically, TS(G = ( N[v i ] = ( N[v i] = [] G Gunther and B Hartnell, On minimizing the effects of betrayals in a resistance movement, Proc 8th Manitoba Conf on Numerical Mathematics and Comuting, Winnieg, Manitoba, Canada, 978, [] G Gunther and B Hartnell, Otimal ksecure grahs, Discrete Al Math, 980, 5 [4] G Gunther, B Hartnell, and R Nowakowski, Neighborconnected grahs and rojective lanes, Networks 7, 987, 4 47 [5] F Harary, Grah Theory, AddisonWesley, Reading, MA 969 [6] B Hartnell and G Gunther, Security of underground resistance movements, in Mathematical Methods in Counterterrorism, N Memon, JD Farley, DL Hicks and T Rosenorn, eds, Sringer Verlag, Wien 009, [7] R Lindelauf, P Borm, and H Hamers, The influence of secrecy on the communication structure of covert networks, Social Networks, 009, 0 7 [8] R Lindelauf, P Borm, and H Hamers, On heterogeneous covert networks, in Mathematical Methods in Counterterrorism, N Memon, JD Farley, DL Hicks and T Rosenorn, eds, Sringer Verlag, Wien 009, 5 8 [9] N Memon, J D Farley, D L Hicks and T Rosenorn, eds, Mathematical Methods in Counterterrorism, N Memon, JD Farley, DL Hicks and T Rosenorn, eds, Sringer Verlag, Wien 009, [0] D B West, Introduction to Grah Theory, second ed Prentice Hall, Providence, RI 00 ( deg(v i = ( q, where q is the number of edges This form also suggests that work on characterizing networks with fixed number of vertices and bounds on the number of edges could be fruitful Memon et al [9] contains several articles focusing on the descrition of grah models that have been alied successfully to study covert networks These articles rovide a good introduction to alications of grah theory in the study of covert networks References: [] V Chvátal, Tough grahs and hamiltonian circuits, Discrete Math 5, 97, 5 8 EISSN: Issue 9, Volume, Setember 0
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