aa r X i v : . [ m a t h . C O ] M a y Propagation time for weighted zero forcing
P.A. CrowdMathMay 18, 2020
Abstract
Zero forcing is a graph coloring process that was defined as a tool for bounding theminimum rank and maximum nullity of a graph. It has also been used for studyingcontrol of quantum systems and monitoring electrical power networks. One of theproblems from the 2017 AIM workshop
Zero forcing and its applications was to exploreedge-weighted probabilistic zero forcing, where edges have weights that determine theprobability of a successful force if forcing is possible under the standard zero forcingcoloring rule.In this paper, we investigate the expected time to complete the weighted zero forc-ing coloring process, known as the expected propagation time, as well as the time forthe process to be completed with probability at least α , known as the α -confidencepropagation time. We demonstrate how to find the expected and confidence propa-gation times of any edge-weighted graph using Markov matrices. We also determinethe expected and confidence propagation times for various families of edge-weightedgraphs including complete graphs, stars, paths, and cycles. Zero forcing is a graph coloring process that was defined in [1] as a tool for bounding theminimum rank and maximum nullity of a graph, and it was later used for studying controlof quantum systems [6, 20]. Zero forcing is also used in another graph coloring process calledpower domination, which is a graph theoretic model of the phase measurement unit (PMU)placement problem in electrical power networks. Zero forcing has also been applied to graphsearching [22].Zero forcing is defined in terms of a color change rule, where there is an initial set ofvertices that are colored blue, with all other vertices in the graph colored white. If u is ablue vertex that has exactly one white neighbor v , then the color of v is changed to blue onthe next step. With zero forcing, researchers have investigated the minimum possible sizeof a set of vertices that can be used to color the whole graph G with repeated iterations ofthe color change rule (the zero forcing number Z( G )), and the minimum time that it takesfor the graph to be colored by a minimum zero forcing set (the propagation time pt( G ))[15, 21, 7]. 1any variants of zero forcing have been studied with different color change rules, eachwith their own variation of propagation time, including positive semidefinite zero forcing[3, 8, 11], skew zero forcing [16, 18, 14, 10], and probabilistic zero forcing [17, 12, 9]. Inthis paper, we investigate a new variant of zero forcing that was proposed at the 2017AIM workshop Zero forcing and its applications [2]. At the AIM workshop, they asked thefollowing question.
Question 1.1.
What can we say about edge-weighted probabilistic zero forcing, i.e. edgeshave weights and if standard zero forcing says we can force we do so with probability of theedge weight?
We call the resulting graph coloring process weighted zero forcing . Specifically we startwith an edge-weighted graph G with edge weights in (0 ,
1) and an initial set B of blue vertices.Whenever a blue vertex u can color a white vertex v using the standard zero forcing rule, wedo so with probability equal to the weight of the edge between u and v . If the whole graph G can eventually be colored blue with weighted zero forcing starting with only the vertices of B blue, we say that B is a weighted zero forcing set of G . The weighted zero forcing number of G is the minimum possible size of a weighted zero forcing set of G .Note that the weighted zero forcing number of G is just the standard zero forcing numberof the unweighted graph G ′ obtained from G by removing the edge weights, so we use thesame notation Z( G ) as the standard zero forcing number to denote the weighted zero forcingnumber of G . Also observe that the weighted zero forcing sets of G are the zero forcing setsof G ′ .The propagation time of a nonempty set B of vertices of an edge-weighted graph G ,ptw( G, B ), is a random variable that reflects the time (number of the round) at which thelast white vertex turns blue when applying a weighted zero forcing process starting with theset B blue. For an edge-weighted graph G and a set B ⊆ V ( G ) of vertices, the expectedpropagation time of B for G is the expected value of the propagation time of B , i.e.,eptw( G, B ) = E[ptw(
G, B )] . The expected propagation time of an edge-weighted graph G is the minimum of the expectedpropagation time of B for G over all minimum weighted zero forcing sets B of G , i.e.,eptw( G ) = min { eptw( G, B ) : | B | = Z( G ) } . We also define the α -confidence propagation time of B for G , denoted cptw( G, B, α ),as the minimum t for which the probability is at least α that G is fully colored after t steps starting with the vertices of B blue. The α -confidence propagation time of G , denotedcptw( G, α ), is the minimum of cptw(
G, B, α ) over all minimum weighted zero forcing sets B of G , i.e., cptw( G, α ) = min { cptw( G, B, α ) : | B | = Z( G ) } . In Section 2 we explain how to find eptw( G ) for any edge-weighted graph G using Markovmatrices. We also explain how to find cptw( G, α ). In Section 3 we determine eptw( G ) exactlywhen G = K ,n (a star of order n + 1), G = K n (a complete graph of order n ), G = P n (a2ath of order n ), G = C n (a cycle of order n ). In Section 4 we determine cptw( G, α ) forthe same families of graphs. In Section 5, we discuss some further directions for research onweighted zero forcing.
Given an initial set B of blue vertices, the weighted zero forcing coloring process of an edge-weighted graph G is a Markov chain. Each possible state of the Markov chain represents apossible set of blue vertices. Each state corresponds to some subset C ⊂ V ( G ) for which B ⊂ C , so there are a total of s = 2 | V ( G ) |−| B | states.More precisely we say that a properly ordered state list for B , denoted by S = ( S , . . . , S s ),is an ordered list of all states for B in which S is the initial state (where exactly the verticesin B are blue), S s is the final state (where all vertices are blue), and | S i | < | S j | implies i < j .We construct an s × s matrix M where the ( i, j ) entry is the probability of transitioningfrom state S i to state S j in one time step. This matrix is upper-triangular since the cardi-nality of states is non-decreasing. The probability that all vertices are blue after round r is( M r ) s , so we obtain a formula for the expected propagation time of B like the formula forprobabilistic zero forcing in [12].eptw( G, B ) = ∞ X r =1 r (cid:0) ( M r ) s − ( M r − ) s (cid:1) . As with probabilistic zero forcing in [9], it is possible to obtain a simpler formula foreptw(
G, B ). Theorem 2.1.
Suppose that G is a graph, B ⊂ V ( G ) is nonempty, S is a properly orderedstate list for B with s states, and M = M ( G, S ) . Then eptw( G, B ) = (( M − e sT − I ) − ) s + 1 , where = [1 , . . . , T and e s = [0 , . . . , , T . Given the formulas for eptw(
G, B ) we can determine eptw( G ) for any graph G by calcu-lating the minimum of eptw( G, B ) over all minimum weighted zero forcing sets B of G . Forcptw( G, α ), we can compute powers M r for each minimum weighted zero forcing set B untilwe find an r such that ( M r ) s ≥ α . The first such r among all of the minimum zero forcingsets B gives the value of cptw( G, α ). In this section, we calculate the expected propagation times of several families of edge-weighted graphs. These include complete graphs, complete bipartite graphs, paths, andcycles. We start with complete graphs. 3 heorem 3.1. If G = K n with edge weights w { i,j } ∈ (0 , , then eptw( G ) = − W where W is the minimum possible value of Q j = i (1 − w { i,j } ) over all ≤ i ≤ n .Proof. Let Z be the minimum forcing set consisting of vertices v , v , . . . , v n − and let w i,n be the weight of the edge from v i to v n . As only v n remains to be forced, eptw( G, Z )is the reciprocal of the probability that v n is forced during the first step or − W where W = Q j = n (1 − w { n,j } ).Next we find the expected propagation time of stars with edge weights, before findingthe expected propagation time of complete bipartite graphs. Theorem 3.2. If G = K ,n with edge weights w , . . . , w n ∈ (0 , , then eptw( G ) is theminimum over ≤ i ≤ n of w i + − W , where W = Q j = i (1 − w j ) .Proof. For the star graph K ,n let v , v , . . . , v n be the outer vertices with weights w , w , . . . , w n towards the center v . Let Z be the minimum zero forcing set consisting of the first n − Z = Z ∪ v . For simplicity, let W = Q n − j =1 (1 − w j )be the probability that v is not forced during the first time step. Then eptw( G, Z ) = W · eptw( G, Z ) + (1 − W ) · eptw( G, Z ) + 1. Also eptw( G, Z ) = w n since the only remainingwhite vertex is v n which is forced with probability w n . Therefore eptw( G ) = w n + − W . Theorem 3.3. If G = K a,b for a, b ≥ with parts A , A , . . . , A a and B , B , . . . , B b suchthat p i,j denotes the weight of edge A i B j , then eptw( G ) is the minimum over ≤ x ≤ a and ≤ y ≤ b of − p x + − p y − − p x p y , where p x = (1 − p x, )(1 − p x, ) . . . (1 − p x,b ) and p y = (1 − p ,y )(1 − p ,y ) . . . (1 − p a,y ) .Proof. Suppose that A x and B y are the only white vertices. Then the probability that A x isnot colored on any given step is p x = (1 − p x, )(1 − p x, ) . . . (1 − p x,b ) and the probability that B y is not colored on any given step is p y = (1 − p ,y )(1 − p ,y ) . . . (1 − p a,y ). If T is the randomvariable for the number of steps in the coloring process, we obtain E ( T ) = P ∞ k =1 P ( T ≥ k ) = P ∞ k =1 (1 − P ( T < k )) = P ∞ k =1 (1 − (1 − p k − x )(1 − p k − y )) = − p x + − p y − − p x p y . Then we pick x and y to minimize this quantity.In the next result, we find the expected propagation time of edge-weighted paths, beforeusing a similar method for edge-weighted cycles. Theorem 3.4. If G = P n with edge weights w , . . . , w n − in order, then eptw( G ) = P n − k =1 1 w k .Proof. The minimum forcing set of a path is a single vertex located at one of its endpoints.Let Z k denote the forcing set of a path consisting of the first k vertices in the forcing chainof one of the endpoints of P n . Given k forced vertices, the probability that the ( k + 1) st vertex is forced in the next time step is w k . Therefore eptw( P n , Z k ) = w k · eptw( P n , Z k +1 ) +(1 − w k ) · eptw( P n , Z k ) + 1 so eptw( P n , Z k ) = eptw( P n , Z k +1 ) + w k for all 1 ≤ k ≤ n − P n , Z n ) = 0, we obtain eptw( P n ) = eptw( P n , Z ) = P n − k =1 1 w k .4bserve that the expected propagation time of cycles can be calculated recursively in asimilar manner. Consider the graph C with weights w , w , w for edges { , } , { , } , { , } respectively. Suppose that our minimum zero forcing set consists of the vertices 1 ,
3. Let Z a,b denote the zero forcing set consisting of a consecutive vertices in the forcing chainof vertex 1 and b consecutive vertices in the forcing chain of vertex 3. Furthermore, leteptw( C , Z a,b ) = 0 whenever a + b ≥ C , Z , ) = w (1 − w ) · eptw( C , Z , )+(1 − w ) w · eptw( C , Z , )+(1 − w )(1 − w ) · eptw( C , Z , ) + 1 so our expected propagation for C is eptw( C , Z , ) = w + w − w w since eptw( C , Z , ) = eptw( C , Z , ) = 0. Now define E ( w , w ) = w + w − w w .Using analogous notation for C , we haveeptw( C , Z , ) = w (1 − w ) · eptw( C , Z , ) + (1 − w ) w · eptw( C , Z , )+ (1 − w )(1 − w ) · eptw( C , Z , ) + w w · eptw( C , Z , ) + 1Notice that eptw( C , Z , ) = E ( w , w ) since we only have a single vertex 3 left to force,a state identical to that of C . Similarly eptw( C , Z , ) = E ( w , w ), so eptw( C , Z , ) = w + w − w w ( w (1 − w ) · E ( w , w ) + (1 − w ) w · E ( w , w ) + 1).In general, if we define E n ( w , w , . . . , w n − ) = eptw( C n , Z n , ), the expected propaga-tion time of a C n with consecutive weights w , w , . . . , w n − along the forcing path, wehave eptw( C n , Z n , ) = w (1 − w n − ) · eptw( C n , Z n , ) + (1 − w ) w n − · eptw( C n , Z n , ) + (1 − w )(1 − w n − ) · eptw( C n , Z n , ) + w w n − · eptw( C n , Z n , ) + 1 = w + w n − − w w n − ( w (1 − w n − ) · eptw( C n , Z n , ) + (1 − w ) w n − · eptw( C n , Z n , ) + w w n − · eptw( C n , Z n , ) + 1) = w + w n − − w w n − ( w (1 − w n − ) · E n − ( w , w , . . . , w n − )+(1 − w ) w n − · E n − ( w , w , · · · , w n − )+ w w n − · E n − ( w , w , . . . , w n − ) + 1).With the recurrence, we can find eptw( C n , Z n , ). To find eptw( C n ) simply take theminimum over all cyclic placements of Z n , . In this section, we determine confidence propagation times for special families of graphsincluding complete graphs, stars, paths, and cycles. We start with complete graphs andstars.
Theorem 4.1. If G = K n with edge weights w { i,j } ∈ (0 , , then cptw( G, α ) is ⌈ log P (1 − α ) ⌉ ,where P is the minimum of Q j = i (1 − w { i,j } ) over all ≤ i ≤ n .Proof. Suppose v i is the white vertex. Then the probability that v i is colored at any givenmoment is 1 − Q j = i (1 − w { i,j } ) = 1 − P , so the probability that it takes m steps or less is thecomplement of the probability that it takes more than m steps, which is just 1 − P m . Thenminimize P over all choices of v i , say at P , and solve for the minimum m with 1 − P m ≥ α ,i.e., m = ⌈ log P (1 − α ) ⌉ . 5 heorem 4.2. If G = K ,n with edge weights p ≤ p ≤ · · · ≤ p n , then cptw( G, α ) is theminimum value of m for which p n (1 − P ) − Pm − P − − (1 − pn ) mpn P − (1 − p n ) ≥ α .Proof. We first find the probability that the coloring process completes in exactly s steps ifwe initially color all vertices except the center and vertex i . We let P = Q j = i (1 − p j ). Atsome point, we must have two forces, which have probabilities 1 − P and p i of occurring. Wepick some number of the remaining s − p i (1 − P ) · s − X j =0 P j (1 − p i ) s − − j = p i (1 − P ) · P s − − (1 − p i ) s − P − (1 − p i ) , which is maximized when p i is maximized, or i = n .Then the probability that K ,n is all blue after at most m steps is p n (1 − P ) · m X s =2 P s − − (1 − p n ) s − P − (1 − p n ) = p n (1 − P ) − P m − P − − (1 − p n ) m p n P − (1 − p n ) . In the next two results, we evaluate confidence propagation times for paths and cycleswhere all of the edge weights are equal. We use generating functions in both proofs.
Theorem 4.3. If G = P n with edge weights all equal to p ∈ (0 , , then cptw( G, α ) is theminimum value of m for which P ms = n − (cid:0) ms (cid:1) p s (1 − p ) m − s ≥ α .Proof. The weighted zero forcing of G is equivalent to the following process. We start withthe number 1. Each step, we add 1 with probability p , and 0 with probability 1 − p . We willdetermine the probability that we reach n in m steps or less.Consider the generating function ( px + (1 − p )) m , where the coefficient of x s is theprobability that the number is s + 1 after m steps. The coefficient of x s is (cid:0) ms (cid:1) p s (1 − p ) m − s ,and the probability in question is m X s = n − (cid:18) ms (cid:19) p s (1 − p ) m − s . Then pt ( P n , α ) is the least m for which the sum is at least α . Theorem 4.4. If G = C n with edge weights all equal to p ∈ (0 , , then cptw( G, α ) is theminimum value of m for which P ms = n − (cid:0) ms (cid:1) p s (1 − p ) m − s ≥ α .Proof. A minimum zero forcing set of G is two adjacent vertices, so we can consider theequivalent coloring process on P n with both endpoints blue. We start with the number 2.Each step, we add 0, 1, or 2 with probability (1 − p ) , p (1 − p ) , p respectively. We willdetermine the probability that we reach n in m steps or less.6hen we proceed as usual: consider the generating function ( p x + 2 p (1 − p ) x + (1 − p ) ) m = ( px + (1 − p )) m . The coefficient of x s , or (cid:18) ms (cid:19) p s (1 − p ) m − s is the probability that our number is s + 2 after m steps. So the probability that we reach n in m steps or less is m X s = n − (cid:18) ms (cid:19) p s (1 − p ) m − s . In the next result, we evaluate confidence propagation times for paths with any distinctedge weights in (0 , Theorem 4.5. If G = P n +1 with distinct edge weights w , w , . . . , w n ∈ (0 , in path order,then cptw( G, { } , α ) is the minimum value of t with w w . . . w n P ni =1 (1 − w i ) n − − (1 − w i ) t w i W i ≥ α , where W i = Q j = i ( w j − w i ) .Proof. Consider a path P n +1 with n + 1 vertices, and n edges with weights w , w , . . . , w n in path order. Suppose that the first blue vertex is the endpoint of the w edge. Let W i = Q j = i ( w j − w i ) for 1 ≤ i ≤ n , let A s,t denote the set of s -tuples of non-negative integersthat add to t , and for any a ∈ A s,t let a i denote the i th term in a . Then we claim that theprobability the entire graph is blue after t turns is w w . . . w n " n X i =1 (1 − w i ) n − − (1 − w i ) t w i W i . To prove this, we use the fact that if X n,i = Q j = i ( x i − x j ), then X a ∈ A n,k x a x a . . . x a n n = n X i =1 x k + n − i X n,i for all integers n ≥ k ≥ − n . Note that for the entire graph to be blue, theremust have been at least n forces, each with probability w , w , . . . , w n , and for the remaining t − n turns, there must have been no forces, and these non-forcing turns had probabilities of1 − w , − w , . . . , − w n .Thus, the probability that the whole graph is blue after exactly t turns is w w . . . w n X a ∈ A n,t − n (1 − w ) a (1 − w ) a . . . (1 − w n ) a n . Using the identity, this sum reduces to w w . . . w n n X i =1 (1 − w i ) t − W i .
7o the probability that all vertices in the graph are blue after at most t turns is w w . . . w n t X s = n n X i =1 (1 − w i ) s − W i = w w . . . w n " n X i =1 (1 − w i ) n − − (1 − w i ) t w i W i . Thus, cptw(
G, α ) is the minimum value of t such that the last sum is at least α . In this paper, we defined propagation time for weighted zero forcing and found formulasusing Markov matrices for the expected propagation times and confidence propagation timesof all edge-weighted graphs with edge weights in (0 , k -partite graphs, hypercubes,grids, spiders, caterpillars, and complete binary trees.Besides zero forcing number and propagation time, another related parameter of graphsis the throttling number, which is the minimum of k + pt k ( G ) over all positive integers k , where pt k ( G ) is the minimum possible propagation time of G with standard zero forcingusing k initial blue vertices. Throttling has been studied for standard zero forcing [7], positivesemidefinite zero forcing [8, 11], skew zero forcing [10], probabilistic zero forcing [12, 19], andpursuit evasion games [5, 4, 13]. It is natural to define and investigate throttling for weightedzero forcing, both for expected propagation time and confidence propagation time.Some other directions for future research are to define weighted positive semidefinite zeroforcing, weighted skew zero forcing, and variants of probabilistic zero forcing that depend onedge weights. For each of the variants of edge-weighted zero forcing, it would be interestingto investigate properties and applications of their Markov matrices. References [1] AIM Minimum Rank Special Graphs Work Group (F. Barioli, W. Barrett, S. Butler,S.M. Cioaba, D. Cvetkov´ıc, S. M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkel-son, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanov´ıc, H. van der Holst, K.Vander Meulen, and A. Wangsness). Zero forcing sets and the minimum rank of graphs.Lin. Alg. Appl. 428 (2008) 1628-1648 82] 2017 AIM Workshop: Zero forcing and its applications, Problem List. http://aimpl.org/zeroforcing [3] F. Barioli, W. Barrett, S. Fallat, H.T. Hall, L. Hogben, B. Shader, P. van den Driessche,H. van der Holst. Zero forcing parameters and minimum rank problems.
Linear AlgebraAppl. , 433 (2010), 401–411.[4] A. Bonato, J. Breen, B. Brimkov, J. Carlson, S. English, J. Geneson, L. Hog-ben, K. Perry, C. Reinhart, Cop throttling number: Bounds, values, and variants.arXiv:1903.10087 (2019)[5] J. Breen, B. Brimkov, J. Carlson, L. Hogben, K.E. Perry, C. Reinhart. Throttling forthe game of Cops and Robbers on graphs.
Discrete Math. , 341 (2018) 2418–2430.[6] D. Burgarth, V. Giovannetti. Full control by locally induced relaxation. Phys. Rev. Lett.PRL 99 (2007), 100501.[7] S. Butler, M. Young. Throttling zero forcing propagation speed on graphs.
Australas.J. Combin. , 57 (2013), 65–71.[8] J. Carlson, L. Hogben, J. Kritschgau, K. Lorenzen, M.S. Ross, S. Selken, V. ValleMartinez. Throttling positive semidefinite zero forcing propagation time on graphs.
Discrete Appl. Math. , in press, https://doi.org/10.1016/j.dam.2018.06.017 .[9] Y. Chan, E. Curl, J. Geneson, L. Hogben, K. Liu, I. Odegard, M. Ross, Using Markovchains to determine expected propagation time for probabilistic zero forcing. To appearin Electronic Journal of Linear Algebra (2020)[10] E. Curl, J. Geneson, L. Hogben, Skew throttling. arXiv:1909.07235 (2019)[11] J. Geneson, Throttling numbers for adversaries on connected graphs. arXiv:1906.07178(2019)[12] J. Geneson, L. Hogben, Propagation time for probabilistic zero forcing.arXiv:1812.10476 (2018)[13] J. Geneson, C. Quines, E. Slettnes, S. Tsai, Expected capture time and throttlingnumber for cop versus gambler. arXiv:1902.05860 (2019)[14] C. Grood, J.A. Harmse, L. Hogben, T. Hunter, B. Jacob, A. Klimas, S. McCathern,Minimum rank of zero-diagonal matrices described by a graph. Electron. J. Linear Al-gebra, 27 (2014) 458-477.[15] L. Hogben, M. Huynh, N. Kingsley, S. Meyer, S. Walker, M. Young. Propagation timefor zero forcing on a graph. Discrete Appl. Math., 160 (2012) 1994-2005916] IMA-ISU research group on minimum rank (M. Allison, E. Bodine, L.M. DeAlba, J.Debnath, L. DeLoss, C. Garnett, J. Grout, L. Hogben, B. Im, H. Kim, R. Nair, O.Pryporova, K. Savage, B. Shader, A. Wangsness Wehe). Minimum rank of skewsym-metric matrices described by a graph. Linear Algebra and its Applications, 432 (2010)2457-2472[17] C. Kang, E. Yi. Probabilistic zero forcing in graphs. Bull. Inst. Combin. Appl. 67 (2013)9-16.[18] N.F. Kingsley. Skew propagation time. Thesis (Ph.D.), Iowa State University (2015)[19] S. Narayanan, A. Sun, Bounds on expected propagation time of probabilistic zero forc-ing. CoRR abs/1909.04482 (2019)[20] S. Severini. Nondiscriminatory propagation on trees. J. Physics A, 41 (2008) 482-002(Fast Track Communication)[21] N. Warnberg. Positive semidefinite propagation time.