Randomized Zero Forcing

RANDOMIZED ZER O F OR CING JESSE GENESON, ILL Y A HICKS, NO AH LICHTENBERG, AL VIN MOON, AND NICOLAS ROBLES Abstract. W e in troduce randomized zero forcing (RZF), a stochastic color-c hange process on di- rected graphs in whic h a white vertex turns blue with probability equal to the fraction of its incom- ing neigh bors that are blue. Unlik e probabilistic zero forcing, RZF is go v erned b y in-neigh borho o d structure and can fail to propagate globally due to directionality . The mo del extends naturally to w eighted directed graphs b y replacing neigh b or coun ts with incoming w eigh t prop ortions. W e study the expected propagation time of RZF, establishing monotonicity properties with respect to enlarging the initial blue set and increasing w eights on edges out of initially blue v ertices, as well as in v ariances that relate weigh ted and unw eighted dynamics. Exact v alues and sharp asymptotics are obtained for several families of directed graphs, including arborescences, stars, paths, cycles, and spiders, and w e deriv e tigh t extremal bounds for unw eigh ted directed graphs in terms of basic parameters suc h as order, degree, and radius. W e conclude with an application to an empirical input-output net work, illustrating ho w exp ected propagation time under RZF yields a dynamic, pro cess-based notion of cen tralit y in directed w eighted systems. 1. Introduction Color changing is a graph-theoretic concept which mo dels the spread of a prop ert y , represen ted b y the color of a vertex, through a graph ov er discrete time steps. The concept w as formalized in [1] as a deterministic process kno wn as zer o for cing . In zero forcing, a blue v ertex u of an arbitrary graph G changes the color of an adjacent white v ertex w if and only if w is the only white vertex adjacen t to u . Determining whether an initial set of blue v ertices of G will even tually turn the graph entirely blue is a central question, with extensive w ork devoted to c haracterizing such sets of minim um size and their relationship to structural and matrix-theoretic parameters of graphs [22]. The study of zero forcing on graphs was initially motiv ated by problems in b ounding the mini- m um rank of graphs [1], as well as indep enden tly arising in quantum con trol theory [28, 29]. Since its in tro duction, zero forcing has b een extended in man y directions b y modifying either the forcing rule or the class of allow able forcing vertices. P ositiv e semideﬁnite zero forcing was in tro duced to study the minim um p ositiv e semideﬁnite rank problem and diﬀers from standard zero forcing in ho w forces propagate across comp onen ts [3]. Sk ew zero forcing further relaxes the forcing rule by allo wing white vertices to force, leading to diﬀeren t extremal b eha vior and complexit y questions [14]. A complementary line of w ork fo cuses on propagation time rather than feasibility , including v ariants that study tradeoﬀs b et w een the size of an initial forcing set and the time required for complete propagation [10]. Related ideas also appear in p o wer domination, whic h mo dels monitor- ing pro cesses in net w orks and can be viewed as a forcing process with dela y ed observ ation [21]. In this setting, electrical netw orks are monitored using phasor measurement units (PMUs), and the placemen t of these devices is closely related to the zero forcing pro cess on the underlying netw ork graph. As a result, p o wer domination has b een extensiv ely studied across a v ariety of graph families and graph pro ducts [27]. In [24], Kang and Yi mo diﬁed the zero forcing deﬁnition to b e probabilistic. In this new rule, called pr ob abilistic zer o for cing (PZF), each blue vertex u of an undirected graph G attempts to indep enden tly c hange the color of an y incident white v ertex w and succeeds with probabilit y , 1 2 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES P ( u turns w blue) = n um b er of blue vertices whic h are adjacent to u degree of u . (1.1) Unlik e deterministic zero forcing, minimal forcing sets for the probabilistic color-c hange rule (1.1) are trivial: on an y connected graph, starting from any nonempty initial blue set, the pro cess colors the entire graph blue with probabilit y 1 [24]. Consequen tly , the natural parameters in probabilistic zero forcing (PZF) concern how quickly the all-blue state is reached rather than whether it can be reac hed. This motiv ates the study of the exp e cte d pr op agation time ept pzf ( G, X ), deﬁned as the exp ected num b er of up date rounds required for PZF to color all of V ( G ) blue starting from an initial blue set X . The systematic study of ept pzf w as initiated by Geneson and Hogb en [18], who established general b ounds, exact v alues on several graph families, and extremal constructions. Algorithmic and computational approac hes based on Mark o v c hains w ere developed in [13]. Subsequent w ork has sharp ened general upper b ounds for ept pzf in terms of classical graph parameters (suc h as order and radius) [25], and has dev eloped asymptotic results for additional families, including random graphs [15] and structured graphs such as grids, regular graphs, and h ypercub es [23]. F urther reﬁnemen ts and related propagation-time parameters for these families hav e also b een studied (e.g., in tight- asymptotic regimes for hypercub es and grids) [4]. Collectively , this b ody of work emphasizes that while PZF guarantees even tual propagation on connected graphs, the expected time scale captures sensitiv e structural information about the underlying top ology . In this article, we consider a v ariation of PZF on directed graphs (edges can b e bidirectional) based on a color change rule determined by the in-degree of uncolored vertices. Precisely , supp ose w is a white vertex in a directed graph G . Then, at eac h step of the color change process, the probabilit y that w changes from white to blue is given b y , P ( w turns blue) = # of incident edges to w with blue source in-degree of w . (1.2) In the case that w has indegree 0, the probabilit y that w turns blue is 0. The color c hange rule in equation (1.2) do es not imply that ev entually ev ery vertex of G will turn blue, ev en if G as an undirected graph is connected. If it is p ossible, then, just as in the case of PZF, the time it tak es to color all of G is a random v ariable determined by the initial set of blue v ertices. W e deﬁne the exp ected propagation time for this new pro cess, whic h we call r andomize d zer o for cing , to accommo date the c hance that the graph is never completely colored. W e also deﬁne randomized zero forcing on graphs with weigh ted edges b y scaling the spread probabilit y b y the relative weigh t of each incoming edge. In an input-output net w ork, this means that a no de is more likely to b e aﬀected if a sizable share of its required inputs is sourced from a neigh b or that has already b ecome infected, capturing the idea that greater economic dep endence corresp onds to greater vulnerabilit y . P ( w turns blue) = P u ∈ B − ( w ) w uw P u ∈ N − ( w ) w uw where N − ( w ) denotes the in-neigh bors of w and B − ( w ) denotes the blue in-neigh bors of v . In the case that P u ∈ N − ( w ) w uw = 0, the probability that w turns blue is 0. Deﬁnition 1.1 (Randomized zero forcing pro cess) . Let G b e a directed graph with nonnegative edge w eigh ts { w uv } , and let B 0 ⊆ V ( G ) b e the initial set of blue v ertices. F or eac h integer t ≥ 0, supp ose the curren t blue set is B t . RANDOMIZED ZERO FOR CING 3 F or eac h white vertex w / ∈ B t , deﬁne p t ( w ) :=      P u ∈ B t w uw P u ∈ V ( G ) w uw , if P u ∈ V ( G ) w uw > 0 , 0 , if P u ∈ V ( G ) w uw = 0 . A t round t + 1, each white vertex w / ∈ B t indep enden tly b ecomes blue with probability p t ( w ), and blue vertices remain blue forever. That is, B t +1 = B t ∪  w / ∈ B t : w succeeds in an independent Bernoulli trial with parameter p t ( w )  . The resulting sequence ( B t ) t ≥ 0 is a time-inhomogeneous Marko v c hain on subsets of V ( G ) with absorbing states. Deﬁnition 1.2. Let X ⊂ V G b e a subset of vertices of a directed graph G . If it is possible to color all of G blue starting with the initial set of X as the only blue vertices, then ept rzf ( G, X ), called the exp e cte d pr op agation time (for RZF) is the exp ected v alue of the n um b er of iterations it tak es to color all of G blue. If it is not p ossible to color all of G blue starting with X , then ept rzf ( G, X ) = ∞ . W e use the shorthand ept rzf ( G, v ) = ept rzf ( G, { v } ) for v ∈ V ( G ). W e let ept rzf ( G ) = min v ∈ V ( G ) ept rzf ( G, v ). If ept rzf ( G, v ) = ∞ for all v ∈ V ( G ), then ept rzf ( G ) = ∞ . Our motiv ation for this v ariation of PZF is to mo del the propagation of risk in directed production net w orks. W e view eac h v ertex as a pro duction unit whose output dep ends on inputs receiv ed from upstream suppliers, represented by directed edges. A vertex is colored blue once it exp eriences a disruptiv e even t-such as a supply failure, quality defect, or regulatory sho c k-that renders its output compromised. Crucially , vulnerability is treated as a r e cipient-side phenomenon: a pro duction unit b ecomes exposed not because a particular supplier activ ely transmits risk, but b ecause a suﬃciently large fraction of its required inputs originate from already-compromised suppliers. A simple toy mo del mak es this precise. Supp ose that at eac h discrete time step, a ﬁrm indep en- den tly samples one of its required inputs, with the probability of selecting supplier u prop ortional to the w eight of the edge ( u, w ) (for example, the share of total inputs sourced from u ). If the sampled input comes from a compromised supplier, the ﬁrm’s pro duction fails during that round, and the ﬁrm b ecomes compromised itself. Under this mo del, the probability that a white vertex w turns blue in a giv en round is exactly the fraction of its incoming weigh t that originates from blue v ertices, which is the randomized zero forcing rule in (1.2). The exp ected propagation time ept rzf then quan tiﬁes the exp ected time un til a disruption originating at a small set of ﬁrms spreads throughout the netw ork, providing a natural, pro cess-based measure of systemic fragilit y . Although highly st ylized, this mechanism captures a basic asymmetry present in many real pro duction sys- tems: risk accum ulates through dep endence on incoming inputs, rather than through outgoing inﬂuence alone. Our Con tributions. In Section 2 we c haracterize when ept rzf ( G, S ) is ﬁnite and we pro ve founda- tional monotonicity properties for w eigh ted RZF, showing that enlarging the initial blue set cannot increase ept rzf , and that increasing weigh ts on edges emanating from initially blue v ertices cannot increase ept rzf . In Section 3 we compute exact v alues and sharp asymptotics for sev eral structured families. In particular, w e show RZF propagation is deterministic on arb orescences and obtain consequences for complete k -ary trees. W e also derive closed-form expressions for weigh ted stars and establish exact propagation time on bidirected cycles with uniform incoming totals. In Section 4 w e pro v e extremal b ounds for unw eighted directed graphs. W e upper-b ound ept rzf ( G, v ) in terms of the num b er of directed edges and giv e constructions showing quadratic- order gro wth is achiev able. In Section 5 we analyze ho w ept rzf b eha v es under graph op erations for unw eigh ted directed graphs, including joining multiple graphs through a single source vertex, yielding upp er b ounds in terms of the comp onen t propagation times. 4 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES In Section 6 we return to weigh ted directed graphs, pro ving sharp upp er b ounds in terms of the minim um edge weigh t and showing sharpness via explicit weigh ted constructions. Finally , in Section 7 w e apply the method to a real-life input-output sector net work, computing singleton-start EPT v alues. W e conclude in Section 8 with op en problems and future directions. 2. Basic resul ts Unlik e probabilistic zero forcing on connected undirected graphs, randomized zero forcing on directed graphs need not propagate to all vertices, ev en when the underlying undirected graph is connected. The obstruction is purely directional: a v ertex can only turn blue if it receives p ositiv e- w eigh t input from v ertices that are already blue. This suggests that the question of whether ept rzf ( G, S ) is ﬁnite should dep end only on the existence of directed paths of p ositiv e w eigh t from the initial blue set S , rather than on more delicate sto c hastic considerations. In fact, as the next theorem shows, there is a complete and purely structural characterization: the exp ected propagation time is ﬁnite if and only if every vertex is reachable from S in the directed graph obtained b y retaining only the p ositiv e-weigh t edges of G . In particular, no additional b ottlenec ks b ey ond reac habilit y can preven t ev en tual propagation under RZF. Theorem 2.1. L et G b e a weighte d dir e cte d gr aph with nonne gative e dge weights, and let S ⊆ V ( G ) b e the initial blue set. L et G + denote the dir e cte d gr aph obtaine d fr om G by r etaining exactly those dir e cte d e dges of p ositive weight. Then ept rzf ( G, S ) < ∞ ⇐ ⇒ every vertex of G is r e achable fr om S in G + . Pr o of. W rite R for the set of v ertices reac hable from S in G + . W e ﬁrst pro ve the follo wing inv arian t: (2.1) B t ⊆ R for all t ≥ 0 , where B t is the set of blue vertices after t rounds. T o see (2.1), note that B 0 = S ⊆ R . Now ﬁx t ≥ 0 and assume B t ⊆ R . If v / ∈ R , then by deﬁnition of reachabilit y there is no vertex u ∈ R with a p ositiv e-w eight edge ( u, v ); otherwise w e w ould ha ve a directed path S ⇝ u → v in G + , forcing v ∈ R . Hence every in-edge of v with p ositiv e w eigh t has its tail in V ( G ) \ R . Since B t ⊆ R , it follo ws that the total incoming w eigh t to v from blue v ertices at time t is 0, and therefore the RZF up date probabilit y for v in round t + 1 is 0. Thus no v ertex outside R can b ecome blue, and B t +1 ⊆ R . This prov es (2.1) by induction. If R  = V ( G ), then (2.1) implies B t  = V ( G ) for every t , so the all-blue state is unreachable and hence ept rzf ( G, S ) = ∞ . Assume now that R = V ( G ). Fix a directed spanning forest F of G + ro oted at S : for eac h v ∈ V ( G ) \ S , c hoose a parent π ( v ) such that ( π ( v ) , v ) is an edge of G + (hence has positive weigh t) and such that π ( v ) lies strictly closer to S along some directed path in G + . F or eac h v / ∈ S , write α ( v ) := w π ( v ) ,v > 0 , W − ( v ) := X u ∈ V ( G ) w uv for the weigh t of the chosen paren t edge and the total incoming weigh t at v , resp ectiv ely . F or eac h v ertex v , let T ( v ) be the (random) ﬁrst round t such that v ∈ B t (so T ( v ) = 0 for v ∈ S ). Note that the propagation time to the all-blue state is τ := min { t : B t = V ( G ) } = max v ∈ V ( G ) T ( v ) . Fix v / ∈ S . On ev ery round t ≥ T ( π ( v )), v ertex π ( v ) is blue, so under the w eigh ted RZF rule the conditional probability (giv en the history up to time t ) that v turns blue at round t + 1 satisﬁes Pr  v ∈ B t +1 | F t  ≥ α ( v ) W − ( v ) =: p ( v ) , RANDOMIZED ZERO FOR CING 5 with the conv en tion p ( v ) = 1 if W − ( v ) = α ( v ). Since the random updates at v in successiv e rounds use indep enden t coin ﬂips (Deﬁnition 1.1), it follo ws that conditional on T ( π ( v )), the additional w aiting time T ( v ) − T ( π ( v )) is sto c hastically dominated b y a geometric random v ariable with success probability p ( v ). In particular, (2.2) E  T ( v ) − T ( π ( v ))  ≤ 1 p ( v ) = W − ( v ) α ( v ) < ∞ . No w ﬁx v / ∈ S and follo w the unique forest path v 0 → v 1 → · · · → v k = v in F , where v 0 ∈ S and v i − 1 = π ( v i ) for each i ≥ 1. Then T ( v ) = k X i =1  T ( v i ) − T ( v i − 1 )  . T aking expectations and applying (2.2) term-b y-term gives (2.3) E[ T ( v )] ≤ k X i =1 W − ( v i ) α ( v i ) < ∞ . Th us every v ertex has ﬁnite exp ected coloring time. Finally , since τ = max v T ( v ) ≤ P v T ( v ) p oin twise, w e obtain E[ τ ] ≤ X v ∈ V ( G ) E[ T ( v )] . The sum on the right is ﬁnite b ecause V ( G ) is ﬁnite and eac h E[ T ( v )] is ﬁnite by (2.3). Hence E[ τ ] < ∞ , i.e., ept rzf ( G, S ) < ∞ when R = V ( G ). □ Next, we show t w o basic results ab out randomized zero forcing on weigh ted directed graphs. First, w e show that adding initial blue vertices cannot increase the exp ected propagation time. Then, w e sho w that increasing edge w eigh ts for edges whic h start from blue v ertices cannot increase the expected propagation time. Using these results, w e also obtain analogous results for un w eigh ted directed graphs. Lemma 2.1. L et G b e a dir e cte d gr aph with a nonne gative weight ω ( u, v ) assigne d to e ach dir e cte d e dge ( u, v ) . L et S 1 ⊆ S 2 ⊆ V ( G ) b e two initial sets of blue vertic es, let T ⊆ V ( G ) b e an arbitr ary tar get set of vertic es, and let ℓ ≥ 0 . Deﬁne A ( S, T , ℓ ) as the event that al l vertic es in T ar e blue after ℓ r ounds of the weighte d r andomize d zer o for cing (RZF) pr o c ess starting fr om initial blue set S . Then: Pr  A ( S 1 , T , ℓ )  ≤ Pr  A ( S 2 , T , ℓ )  . Pr o of. W e construct a coupling of tw o w eigh ted RZF pro cesses, one started from S 1 (the ﬁrst pr o c ess ) and one from S 2 (the se c ond pr o c ess ), such that at ev ery time t the set of blue v ertices in the ﬁrst pro cess is con tained in the set of blue v ertices in the second pro cess. Denote by B (1) t and B (2) t the blue sets at time t in the ﬁrst and second pro cess, resp ectiv ely . Our goal is to maintain B (1) t ⊆ B (2) t for all t = 0 , 1 , . . . , ℓ. This monotone coupling immediately implies the lemma, b ecause under the coupling, whenev er all v ertices in T are blue in the ﬁrst process after ℓ rounds, they are also blue in the second pro cess. A t time 0, we hav e B (1) 0 = S 1 and B (2) 0 = S 2 . Since S 1 ⊆ S 2 b y assumption, the inclusion B (1) 0 ⊆ B (2) 0 holds. Now assume inductively that after t rounds w e ha v e B (1) t ⊆ B (2) t . W e describ e the coupling of the transitions from time t to time t + 1. 6 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES A t time t , consider an arbitrary vertex w ∈ V ( G ) that is white in at least one of the pro cesses. F or the next round, the weigh ted RZF rule sp eciﬁes that w becomes blue with a probabilit y that dep ends on its blue in-neighbors at time t . Let p (1) t ( w ) := P x ∈ N − ( w ) ∩ B (1) t ω ( x, w ) P x ∈ N − ( w ) ω ( x, w ) and p (2) t ( w ) := P x ∈ N − ( w ) ∩ B (2) t ω ( x, w ) P x ∈ N − ( w ) ω ( x, w ) . Because B (1) t ⊆ B (2) t , we ha v e p (1) t ( w ) ≤ p (2) t ( w ) for every v ertex w . T o couple the up dates, we pro ceed as follo ws. F or eac h vertex w and each time t , w e sample a single random v ariable U t ( w ), uniformly distributed in [0 , 1], and use it in both processes. W e then declare w ∈ B ( i ) t +1 ⇐ ⇒  w ∈ B ( i ) t  or  U t ( w ) ≤ p ( i ) t ( w )  for i = 1 , 2. By construction, for eac h pro cess i and eac h w , the probabilit y that w b ecomes blue at time t + 1 (given the past history) is exactly p ( i ) t ( w ), and the up dates for distinct vertices use indep enden t uniforms { U t ( w ) } w . Thus each pro cess individually has the correct w eigh ted RZF distribution. Moreo v er, since p (1) t ( w ) ≤ p (2) t ( w ) and the same U t ( w ) is used in b oth pro cesses, w e ha v e the implication U t ( w ) ≤ p (1) t ( w ) = ⇒ U t ( w ) ≤ p (2) t ( w ) . Hence, if w is blue at time t + 1 in the ﬁrst pro cess and was white at time t in b oth pro cesses, then U t ( w ) ≤ p (1) t ( w ) and therefore U t ( w ) ≤ p (2) t ( w ), so w is also blue at time t + 1 in the second pro cess. T ogether with the inductive h yp othesis B (1) t ⊆ B (2) t , this shows that B (1) t +1 ⊆ B (2) t +1 . By induction on t , it follows that B (1) t ⊆ B (2) t for all t = 0 , 1 , . . . , ℓ . In particular, in the coupled probabilit y space, whenev er every vertex in T is blue in the ﬁrst pro cess after ℓ rounds, the same is true in the second pro cess. Therefore, Pr  A ( S 1 , T , ℓ )  ≤ Pr  A ( S 2 , T , ℓ )  , as claimed. □ Corollary 2.2. L et G b e a weighte d dir e cte d gr aph and let S 1 ⊆ S 2 ⊆ V ( G ) b e two initial blue sets. Then the exp e cte d pr op agation time under r andomize d zer o for cing (RZF) satisﬁes: ept rzf ( G, S 1 ) ≥ ept rzf ( G, S 2 ) . Pr o of. W e express the exp ected propagation time as ept rzf ( G, S ) = ∞ X ℓ =0  1 − Pr  A G ( S, V , ℓ )   , since 1 − Pr( A G ( S, V , ℓ )) is the probabilit y that the propagation has not ﬁnished by time ℓ . By the previously established result, if S 1 ⊆ S 2 then Pr  A G ( S 1 , V , ℓ )  ≤ Pr  A G ( S 2 , V , ℓ )  for each ℓ . It follo ws that 1 − Pr( A G ( S 1 , V , ℓ )) ≥ 1 − Pr( A G ( S 2 , V , ℓ )) for all ℓ . Summing these inequalities o v er all ℓ gives ept rzf ( G, S 1 ) ≥ ept rzf ( G, S 2 ), as desired. □ Lemma 2.3. L et G and G ′ b e dir e cte d gr aphs on the same vertex set V , with p ossibly weighte d e dges, and let S ⊆ V b e a set of initial ly blue vertic es. Assume G ′ is obtaine d fr om G by incr e asing the weights of some dir e cte d e dges ( u, w ) with u ∈ S (i.e., al l e dges whose weights change have their tail in S ; for e ach such e dge ( u, w ) , W G ′ ( u, w ) ≥ W G ( u, w ) , and other e dge weights r emain RANDOMIZED ZERO FOR CING 7 unchange d). Her e W H ( x, w ) denotes the weight of e dge ( x, w ) in gr aph H (with W H ( x, w ) = 0 if no such e dge exists). F or any set T ⊆ V and any numb er of r ounds ℓ ≥ 1 , let A G ( S, T , ℓ ) b e the event that al l vertic es in T ar e blue after ℓ r ounds of RZF on gr aph G starting fr om initial blue set S . Then: Pr G  A G ( S, T , ℓ )  ≤ Pr G ′  A G ′ ( S, T , ℓ )  , i.e., incr e asing e dge weights out of initial ly-blue vertic es (without changing S ) c an only incr e ase the pr ob ability of eventual ly for cing al l vertic es in T blue. Pr o of. W e couple the RZF pro cesses on G and G ′ so that the blue set in G is alw a ys contained in the blue set in G ′ . F or eac h t = 0 , 1 , 2 , . . . , ℓ , let B G t and B G ′ t denote the set of blue vertices at time t in the RZF pro cesses on G and G ′ , resp ectiv ely . Note that once a vertex turns blue it remains blue forev er in either pro cess, so in particular S = B G 0 = B G ′ 0 ⊆ B G t , B G ′ t for all t ≥ 0 . A t eac h round t = 1 , 2 , . . . , ℓ and for each v ertex w ∈ V , w e dra w an indep enden t uniform random v ariable U t ( w ) ∼ Unif (0 , 1), and use the same v alue U t ( w ) for the update of w in b oth graphs. The up date rule in graph G is: w ∈ B G t ⇐ ⇒  w ∈ B G t − 1  or  U t ( w ) ≤ p G t ( w )  , where p G t ( w ) := P x ∈ V ∩ B G t − 1 W G ( x, w ) P x ∈ V W G ( x, w ) , with the conv en tion that p G t ( w ) = 0 if the denominator (total in-w eigh t into w in G ) is zero. The pro cess on G ′ is deﬁned analogously , using p G ′ t ( w ) := P x ∈ V ∩ B G ′ t − 1 W G ′ ( x, w ) P x ∈ V W G ′ ( x, w ) , again with the conv en tion p G ′ t ( w ) = 0 if the denominator is zero. Marginally , each pro cess has the correct RZF distribution, since for a ﬁxed history the probability that w b ecomes blue at time t is exactly p G t ( w ) in G and p G ′ t ( w ) in G ′ . W e no w show b y induction on t that B G t ⊆ B G ′ t for all t = 0 , 1 , . . . , ℓ . Base step ( t = 0 ). W e ha v e B G 0 = S = B G ′ 0 , so B G 0 ⊆ B G ′ 0 . Inductive step. Assume B G t − 1 ⊆ B G ′ t − 1 for some t ≥ 1. Fix a vertex w ∈ V , and let us compare its up date probabilities in G and G ′ at time t . Deﬁne b := X x ∈ V x ∈ B G t − 1 W G ( x, w ) , d := X x ∈ V W G ( x, w ) , so that (for d > 0) p G t ( w ) = b d . No w consider the in-neighbors and weigh ts in G ′ . By assumption, G ′ is obtained from G b y increasing weigh ts on edges ( u, w ) from some u ∈ S . Let ∆ := X u ∈ S  W G ′ ( u, w ) − W G ( u, w )  , 8 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES the total additional in-weigh t in to w coming from S in G ′ (note ∆ ≥ 0). Sin ce edges not from S ha v e the same w eigh t in G and G ′ , we ha v e X x ∈ V W G ′ ( x, w ) = X x ∈ V W G ( x, w ) + ∆ = d + ∆ , so that deg − G ′ ( w ) (total in-w eigh t into w in G ′ ) = d + ∆ . Eac h edge ( u, w ) whose weigh t increased has u ∈ S , and ev ery such u is blue at ev ery time step in b oth pro cesses (since S is initially blue and remains blue). In particular, for eac h such u w e hav e u ∈ B G t − 1 and u ∈ B G ′ t − 1 . Th us all the additional in-weigh t ∆ in G ′ comes from blue in-neighbors of w . Moreov er, by the inductive hypothesis B G t − 1 ⊆ B G ′ t − 1 , an y in-neigh b or of w that is blue in G at time t − 1 is also blue in G ′ at time t − 1. Therefore, the total weigh t of blue in-neigh b ors of w in G ′ at time t − 1 satisﬁes b ′ := X x ∈ V x ∈ B G ′ t − 1 W G ′ ( x, w ) ≥ b + ∆ , since it includes all of b (blue neighbors from G ) plus ∆ (the added weigh t from blue neighbors in S ). Hence (when d + ∆ > 0), p G ′ t ( w ) = b ′ d + ∆ ≥ b + ∆ d + ∆ . If d = 0, then p G t ( w ) = 0. In G ′ , either ∆ = 0 as well (so still no in-neighbor w eigh t and p G ′ t ( w ) = 0), or ∆ > 0 (meaning w has acquired some in-weigh t from blue v ertices in S ), in which case b ′ = ∆ and p G ′ t ( w ) = b ′ d +∆ = ∆ ∆ = 1. In either case w e hav e p G ′ t ( w ) ≥ p G t ( w ). If d > 0, then 0 ≤ b ≤ d , and for an y ∆ ≥ 0 we ha v e b + ∆ d + ∆ ≥ b d . Indeed, b + ∆ d + ∆ ≥ b d ⇐ ⇒ ( b + ∆) d ≥ b ( d + ∆) ⇐ ⇒ bd + ∆ d ≥ bd + b ∆ ⇐ ⇒ ∆( d − b ) ≥ 0 , whic h holds b ecause ∆ ≥ 0 and d − b ≥ 0. Com bining b oth cases, we conclude that for every v ertex w , p G ′ t ( w ) ≥ p G t ( w ) . Since we use the same U t ( w ) in b oth processes, it follo ws that U t ( w ) ≤ p G t ( w ) = ⇒ U t ( w ) ≤ p G ′ t ( w ) . Therefore, if w is blue at time t in G (and was white at time t − 1 in b oth pro cesses), then U t ( w ) ≤ p G t ( w ) implies U t ( w ) ≤ p G ′ t ( w ), so w is also blue at time t in G ′ . T ogether with the inductiv e hypothesis B G t − 1 ⊆ B G ′ t − 1 , this implies B G t ⊆ B G ′ t . By induction, B G t ⊆ B G ′ t for all t = 0 , 1 , . . . , ℓ . Finally , if all v ertices of T are blue in G after ℓ rounds (ev en t A G ( S, T , ℓ )), then T ⊆ B G ℓ ⊆ B G ′ ℓ , so A G ′ ( S, T , ℓ ) also o ccurs. Under our coupling, this means A G ( S, T , ℓ ) ⊆ A G ′ ( S, T , ℓ ) . Therefore, Pr G  A G ( S, T , ℓ )  ≤ Pr G ′  A G ′ ( S, T , ℓ )  . RANDOMIZED ZERO FOR CING 9 This completes the pro of. □ Corollary 2.4. L et G and G ′ b e weighte d dir e cte d gr aphs on the same vertex set V , and let S ⊆ V b e a set of initial ly blue vertic es. Assume G ′ is obtaine d fr om G by incr e asing the weights on some dir e cte d e dges ( u, w ) with u ∈ S (i.e., al l e dges with incr e ase d weights must originate fr om vertic es in S ). Then, we have ept rzf ( G, S ) ≥ ept rzf ( G ′ , S ) . Pr o of. W e express the exp ected propagation time as a sum o ver tail probabilities. F or any graph H on V , let B H ℓ b e the set of blue vertices after ℓ steps of the RZF pro cess on H (starting from S ), and let A H ( S, V , ℓ ) b e the even t that all vertices in V are blue within ℓ steps in H . Then: ept rzf ( G, S ) = ∞ X ℓ =0 P G  B G ℓ  = V  = ∞ X ℓ =0  1 − P G  A G ( S, V , ℓ )   , and similarly , ept rzf ( G ′ , S ) = ∞ X ℓ =0 P G ′  B G ′ ℓ  = V  = ∞ X ℓ =0  1 − P G ′  A G ′ ( S, V , ℓ )   . By the given inequalit y , for each ℓ , P G  A G ( S, V , ℓ )  ≤ P G ′  A G ′ ( S, V , ℓ )  . Subtracting from 1 yields 1 − P G  A G ( S, V , ℓ )  ≥ 1 − P G ′  A G ′ ( S, V , ℓ )  . Th us the ℓ th summand of ept rzf ( G, S ) is at least the ℓ th summand of ept rzf ( G ′ , S ). Summing ov er all ℓ gives ept rzf ( G, S ) ≥ ept rzf ( G ′ , S ), as desired. □ By restricting to the case when all edges ha v e w eight 1, w e obtain the following corollaries for un w eighted directed graphs. Corollary 2.5. L et G b e a dir e cte d gr aph. L et S 1 ⊆ S 2 ⊆ V ( G ) b e two initial sets of blue vertic es, let T ⊆ V ( G ) b e an arbitr ary tar get set of vertic es, and let ℓ ≥ 0 . Deﬁne A ( S, T , ℓ ) as the event that al l vertic es in T ar e blue after ℓ r ounds of the r andomize d zer o for cing (RZF) pr o c ess starting fr om initial blue set S . Then: Pr  A ( S 1 , T , ℓ )  ≤ Pr  A ( S 2 , T , ℓ )  . Corollary 2.6. L et G b e a dir e cte d gr aph and let S 1 ⊆ S 2 ⊆ V ( G ) b e two initial blue sets. Then the exp e cte d pr op agation time under r andomize d zer o for cing (RZF) satisﬁes: ept rzf ( G, S 1 ) ≥ ept rzf ( G, S 2 ) . Corollary 2.7. L et G and G ′ b e dir e cte d gr aphs on the same vertex set V , and let S ⊆ V b e a set of initial ly blue vertic es. Assume G ′ is obtaine d fr om G by adding some dir e cte d e dges ( u, w ) with u ∈ S (i.e., al l adde d e dges originate fr om vertic es in S ). F or any set T ⊆ V and any numb er of r ounds ℓ ≥ 1 , let A G ( S, T , ℓ ) b e the event that al l vertic es in T ar e blue after ℓ r ounds of RZF on gr aph G starting fr om initial blue set S . Then: Pr G  A G ( S, T , ℓ )  ≤ Pr G ′  A G ′ ( S, T , ℓ )  . In other wor ds, adding extr a e dges out of initial ly-blue vertic es (without changing S ) c an only incr e ase the pr ob ability of eventual ly for cing al l vertic es in T blue. Corollary 2.8. L et G and G ′ b e dir e cte d gr aphs on the same vertex set V , and let S ⊆ V b e a set of initial ly blue vertic es. Assume G ′ is obtaine d fr om G by adding some dir e cte d e dges ( u, w ) with u ∈ S (i.e., al l adde d e dges originate fr om vertic es in S ). Then, we have ept rzf ( G, S ) ≥ ept rzf ( G ′ , S ) . 10 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES 3. Special f amilies of directed graphs Before turning to sp eciﬁc graph families, it is useful to iden tify basic structural transformations that leav e the RZF pro cess unc hanged. Such in v ariances serve t wo complemen tary purp oses. First, they substan tially reduce the eﬀectiv e parameter space of weigh ted directed graphs: many distinct w eigh t assignments induce exactly the same sto c hastic dynamics, and hence the same exp ected propagation behavior. Second, they pro vide a principled wa y to normalize or simplify weigh ts without loss of generality , allowing results for un w eighted or uniformly w eigh ted graphs to b e extended immediately to broader w eigh ted settings. In particular, b ecause the RZF up date rule dep ends only on r elative incoming weigh ts at each v ertex, an y transformation that preserv es these ratios leav es the pro cess inv arian t. Lemma 3.1 formalizes this observ ation, showing that incoming w eigh ts ma y b e rescaled independently at eac h v ertex without aﬀecting the dynamics. Lemma 3.2 then identiﬁes an imp ortan t special case: when all incoming edges to a v ertex hav e equal weigh t, w eigh ted RZF coincides exactly with un weigh ted RZF on the same underlying directed graph. T ogether, these lemmas justify treating man y w eigh ted mo dels as essen tially unw eighted. Lemma 3.1. L et G b e a weighte d dir e cte d gr aph. Fix a vertex w with P u ∈ N − ( w ) w uw > 0 . If al l inc oming e dge weights to w ar e multiplie d by the same c onstant λ > 0 , then the RZF pr o c ess on G is unchange d. Pr o of. F or any blue set B , P u ∈ B λw uw P u ∈ V λw uw = P u ∈ B w uw P u ∈ V w uw . Th us the up date probability at w is unc hanged in ev ery conﬁguration, and hence the entire RZF pro cess is unc hanged. □ Lemma 3.2. L et G b e a dir e cte d gr aph with e dge weights satisfying the fol lowing pr op erty: for e ach vertex w , ther e exists a c onstant c w > 0 such that w uw = c w for al l u ∈ N − ( w ) . Then weighte d RZF on G c oincides with unweighte d RZF on the same underlying dir e cte d gr aph. Pr o of. F or any blue set B , Pr( w turns blue) = P u ∈ B − ( w ) c w P u ∈ N − ( w ) c w = | B − ( w ) | | N − ( w ) | . □ 3.1. Deterministic families. In this subsection w e describ e families of w eigh ted directed graphs for which the RZF propagation pro cess is deterministic, so that the exp ected propagation time coincides with the actual propagation time. Theorem 3.1. L et G b e a weighte d dir e cte d gr aph that has a distinguishe d ro ot vertex r with deg − ( r ) = 0 (no inc oming e dges), and every other vertex has deg − = 1 (exactly one p ar ent in G ). If al l vertic es of G ar e r e achable fr om r , then the RZF pr op agation pr o c ess starting fr om r is deterministic, and the exp e cte d pr op agation time e quals the e c c entricity of r in G (i.e., the distanc e fr om r to the furthest vertex in the dir e cte d tr e e). Pr o of. W e sho w that the pro cess is deterministic and that after round t the blue set is exactly the set of vertices at directed distance at most t from r . F or t ≥ 0, let L t := { v ∈ V ( G ) : dist( r , v ) ≤ t } , RANDOMIZED ZERO FOR CING 11 where dist( r , v ) is the length of the unique directed path from r to v in the arborescence (well-deﬁned since every non-root vertex has indegree 1). W e claim that for all t ≥ 0 we ha ve B t = L t , where B t is the set of blue vertices after t rounds. Base case t = 0: B 0 = { r } = L 0 . Inductiv e step: assume B t = L t for some t ≥ 0. Let v b e any vertex with dist( r, v ) = t + 1. Let u b e the unique in-neighbor of v (its parent). Then dist( r , u ) = t , so u ∈ L t = B t . Since v has indegree 1, its total incoming weigh t is exactly the weigh t on the edge ( u, v ), and that entire incoming weigh t comes from a blue v ertex at time t . Hence the RZF up date rule gives Pr( v ∈ B t +1 | B t ) = 1 . Th us every v ertex in L t +1 \ L t b ecomes blue at round t + 1, so L t +1 ⊆ B t +1 . Con v ersely , if dist( r , v ) ≥ t + 2, then its parent u satisﬁes dist( r, u ) ≥ t + 1, hence u / ∈ L t = B t , so v has no blue in-neighbor at time t and therefore Pr( v ∈ B t +1 | B t ) = 0 . Th us no vertex outside L t +1 can turn blue at round t + 1, giving B t +1 ⊆ L t +1 . Therefore B t +1 = L t +1 , completing the induction. It follows that the propagation time is deterministic and equals the least t suc h that L t = V ( G ), i.e. ecc( r ). Hence ept rzf ( G, r ) = ecc( r ). □ Corollary 3.3. F or the weighte d c omplete unidir e ctional k -ary tr e e of depth n , the exp e cte d pr op- agation time (starting with the r o ot blue) is exactly n . In p articular, although e ach internal no de has k childr en, the pr op agation pr o c ess is deterministic and terminates in n r ounds. Pr o of. The unidirectional k -ary tree of depth n is a directed arb orescence in which each non-ro ot v ertex has indegree 1 and the longest root-leaf path has length n . By Theorem 3.1, the propagation ﬁnishes after n rounds. (In this tree, every leaf lies at distance n from the ro ot, so the ro ot’s eccen tricit y is n .) □ 3.2. Stars. Let S m b e the bidirected star with center c and lea v es L = { ℓ 1 , . . . , ℓ m } . Assume edges ( ℓ i , c ) hav e w eigh ts a i > 0 and edges ( c, ℓ i ) hav e w eigh ts b i > 0. Remark 3.1. If v denotes the cen ter vertex of S m , then ept rzf ( ← − − − → K 1 ,n − 1 , v ) = 1. Theorem 3.2. L et S ⊆ L b e a nonempty set of le aves, and assume that c is not initial ly blue. Then ept rzf ( S m , S ) = 1 + P m i =1 a i P ℓ i ∈ S a i . In p articular, for a single le af ℓ j , ept rzf ( S m , { ℓ j } ) = 1 + P m i =1 a i a j . Pr o of. While c is white, no leaf outside S has a blue in-neighbor, so the blue set remains exactly S . In eac h round, the probability that c turns blue is p = P ℓ i ∈ S a i P m i =1 a i , whic h is constan t across rounds. Hence the time T c un til c becomes blue is geometric with E[ T c ] = 1 /p . Once c b ecomes blue, ev ery remaining leaf ℓ i / ∈ S has exactly one in-neighbor, namely c , and therefore turns blue with probabilit y 1 in the next round. Thus ept rzf ( S m , S ) = E[ T c ] + 1 = 1 + P m i =1 a i P ℓ i ∈ S a i . 12 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES □ Corollary 3.4. Among al l single-le af starting vertic es, ept rzf ( S m , { ℓ j } ) is minimize d when a j is maximal. Pr o of. Immediate from Theorem 3.2. □ Corollary 3.5. If S m has al l e dges of e qual weight and v is a le af vertex of S m , ept rzf ( S m , v ) = m + 1 . 3.3. P aths. W e next analyze the RZF propagation pro cess on w eigh ted bidirectional paths, where the linear structure p ermits an exact computation of the exp ected propagation time. Theorem 3.1. Let P b e a bidirectional path of order n with endp oin ts v 1 , v n . Supp ose that there are edges from v i to v i +1 and edges from v i +1 to v i for all 1 ≤ i ≤ n − 1. W e denote w eight on the edge from v i − 1 to v i as l i and the edge from v i +1 to v i as r i , then ept rzf ( P , v 1 ) = n − 1 X i =2 l i + r i l i + 1 and ept rzf ( P , v n ) = n − 1 X i =2 l i + r i r i + 1 . Pr o of. W e prov e the statemen t for the start v ertex v 1 ; the case v n is symmetric. F or 2 ≤ i ≤ n , let T i b e the (random) num b er of additional rounds needed to turn v i blue after v i − 1 has become blue, assuming that none of v i , . . . , v n is blue yet. W e ﬁrst justify that this describ es the pro cess: on a bidirected path starting from v 1 , at any time b efore v i turns blue, the only v ertex among { v i , . . . , v n } that can ha ve a blue in-neighbor is v i itself (its only p ossible blue in-neigh b or is v i − 1 ). Hence the pro cess colors vertices in the order v 1 , v 2 , . . . , v n almost surely . Fix 2 ≤ i ≤ n − 1. Once v i − 1 is blue and v i is still white, the blue in-weigh t into v i equals the w eigh t on ( v i − 1 , v i ), namely l i , while the total in-weigh t in to v i equals l i + r i . Therefore in each subsequen t round, indep enden tly across rounds, v i b ecomes blue with probabilit y p i = l i l i + r i . Th us T i ∼ Geom( p i ) and E[ T i ] = 1 /p i = ( l i + r i ) /l i . F or i = n , the endp oin t v n has indegree 1 (its only in-neighbor is v n − 1 ), so once v n − 1 is blue we ha v e Pr( v n turns blue next round) = 1, hence E[ T n ] = 1. By linearity of expectation, ept rzf ( P , v 1 ) = E[ T 2 + · · · + T n ] = n − 1 X i =2 l i + r i l i + 1 , as claimed. □ Corollary 3.6. If P is a bidir e ctional p ath gr aph with al l weights e qual and v is an endp oint of P , then ept rzf ( P , v ) = 2 n − 3 . 3.4. Cycles. Let ← → C n b e the bidirected cycle on v ertices v 0 , v 1 , . . . , v n − 1 (indices mo dulo n ). Assume eac h clo c kwise edge ( v i , v i +1 ) has weigh t p > 0 and each coun terclo c kwise edge ( v i , v i − 1 ) has weigh t q > 0. Th us every v ertex has total in-weigh t p + q . RANDOMIZED ZERO FOR CING 13 Theorem 3.3. L et the initial blue set c onsist of k ≥ 1 c onse cutive vertic es in ← → C n . Then the exp e cte d pr op agation time satisﬁes ept rzf ( ← → C n , B 0 ) = n − k. In p articular, the exp e cte d pr op agation time is indep endent of the weights p and q . Pr o of. Because the initial blue set is a con tiguous arc on the cycle and a white v ertex can only turn blue if it has at least one blue in-neighbor, the blue set remains a single con tiguous blo c k for all time. Let X t b e the num b er of blue vertices after round t ; then ( X t ) is a Marko v chain on { k , k + 1 , . . . , n } with absorption at n . Fix a state m with k ≤ m ≤ n − 2. Exactly tw o white vertices are adjacent to the blue blo c k (one on eac h side). Call them w L and w R . V ertex w L has exactly one blue in-neigh b or en tering it, and that in-edge has weigh t p (or q dep ending on orien tation); similarly w R has exactly one blue in-neigh b or en tering it with the opp osite w eigh t. Concretely , one boundary white v ertex turns blue with probability a := p p + q , and the other turns blue with probability b := q p + q . Under the RZF rule, distinct white vertices up date indep enden tly eac h round, so the even ts { w L turns blue } and { w R turns blue } are indep enden t. Hence, writing r := ab = pq ( p + q ) 2 , the one-step transitions from state m are: Pr( X t +1 = m ) = Pr(neither b oundary turns) = (1 − a )(1 − b ) = ab = r, Pr( X t +1 = m + 2) = Pr(both boundaries turn) = ab = r , Pr( X t +1 = m + 1) = 1 − 2 r. When m = n − 1, the unique remaining white vertex has b oth in-neighbors blue, so it turns blue with probability 1, i.e. the c hain mov es from n − 1 to n in one round deterministically . No w let T m denote the exp ected additional num ber of rounds to absorption (starting from X t = m ). Then T n = 0, T n − 1 = 1, and for k ≤ m ≤ n − 2 the law of total exp ectation giv es the standard hitting-time recurrence T m = 1 + Pr( m ) T m + Pr( m + 1) T m +1 + Pr( m + 2) T m +2 , i.e. (3.1) T m = 1 + rT m + (1 − 2 r ) T m +1 + r T m +2 . Equiv alently , (1 − r ) T m = 1 + (1 − 2 r ) T m +1 + r T m +2 . W e claim that T m = n − m for all m ∈ { k , k + 1 , . . . , n } . This is immediate for m = n and m = n − 1. F or k ≤ m ≤ n − 2, substitute T m = n − m , T m +1 = n − ( m + 1), T m +2 = n − ( m + 2) into (3.1). The right-hand side b ecomes 1 + r ( n − m ) + (1 − 2 r )( n − m − 1) + r ( n − m − 2) . Let d := n − m . Then this equals 1 + rd + (1 − 2 r )( d − 1) + r ( d − 2) = 1 +  r + (1 − 2 r ) + r  d −  (1 − 2 r ) + 2 r  = 1 + d − 1 = d, whic h matc hes the left-hand side T m = d . Th us T m = n − m satisﬁes the recurrence and boundary conditions, so it is the correct exp ected hitting time. Therefore, starting from X 0 = k we ha v e E( τ ) = T k = n − k . □ 14 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES Corollary 3.7. Consider a dir e cte d cycle gr aph on n vertic es with al l e dges oriente d c onsistently in one dir e ction (forming a dir e cte d cycle) and al l weights e qual. If one vertex is initial ly blue, then the pr op agation time is exactly n − 1 (deterministic). Corollary 3.8. Consider a bidir e ctional cycle gr aph on n vertic es with al l weights e qual. If one vertex is initial ly blue, then the pr op agation time is exactly n − 1 . 3.5. Balanced spiders. In this subsection we re strict attention to spider graphs whose legs all ha v e the same length, which allo ws a clean asymptotic analysis of the RZF propagation time. Theorem 3.2. Fix an integer k > 1 and let S b e a bidirectional spider graph with k legs each of length n and all weigh ts equal. Then ept rzf ( S ) = 2 n (1 + o (1)), where the o (1) is with resp ect to n . Pr o of. Let c b e the center. F or eac h leg i ∈ { 1 , . . . , k } , lab el its vertices b y v i, 1 , v i, 2 , . . . , v i,n , where v i, 1 is adjacent to c and v i,n is the leaf. All edges hav e weigh t 1, so a white v ertex turns blue in a round with probability equal to the fraction of its in-neigh b ors that are blue. Step 1 (a le g is a se quential pr o c ess). Assume c is blue. Fix a leg i and supp ose v i, 1 , . . . , v i,j − 1 are blue and v i,j , . . . , v i,n are white at the start of some round (where j ≥ 1). Then v i,j has a blue in-neigh b or (namely c if j = 1, and v i,j − 1 if j ≥ 2), while every v i,ℓ with ℓ > j has all in-neighbors among { v i,j , v i,j +1 , . . . , v i,n } , hence has no blue in-neigh b or. Therefore, on each leg the vertices b ecome blue in order v i, 1 , v i, 2 , . . . , v i,n almost surely . Moreov er, for 1 ≤ j ≤ n − 1 the v ertex v i,j has exactly t w o in-neigh b ors and exactly one is blue when it ﬁrst b ecomes eligible, so it turns blue in eac h subsequent round with probabilit y 1 / 2, indep enden tly across rounds. Finally v i,n has indegree 1, so once v i,n − 1 is blue it b ecomes blue in the next round with probability 1. Th us the time T i to complete leg i from the start state { c } satisﬁes T i = n − 1 X j =1 G i,j + 1 , where the G i,j are indep enden t Geom(1 / 2) random v ariables (supp orted on { 1 , 2 , . . . } ). In partic- ular, E[ T i ] = 2( n − 1) + 1 = 2 n − 1 , V ar( T i ) = ( n − 1)V ar(Geom(1 / 2)) = 2( n − 1) . Step 2 (upp er b ound fr om starting at the c enter). Let τ b e the time to turn all v ertices blue starting from { c } . Since the spider is all-blue iﬀ ev ery leg is complete, τ = max 1 ≤ i ≤ k T i , so ept rzf ( S ) ≤ ept rzf ( S, { c } ) = E[ τ ] . Fix t > 0. By Chebyshev and a union b ound, Pr( τ ≥ (2 n − 1) + t ) ≤ k X i =1 Pr( T i − (2 n − 1) ≥ t ) ≤ k X i =1 V ar( T i ) t 2 ≤ 2 k n t 2 . Using E[ τ ] = (2 n − 1) + R ∞ 0 Pr( τ ≥ (2 n − 1) + u ) du , w e obtain E[ τ ] ≤ (2 n − 1) + Z ∞ 0 min  1 , 2 k n u 2  du = (2 n − 1) + O ( √ n ) , where the implicit constant dep ends only on k . Step 3 (lower b ound). Let s b e an y starting vertex. There exists a leaf ℓ at (undirected) graph distance at least n from s (if s = c then ev ery leaf is at distance n , and if s lies on some leg then an y leaf on a diﬀeren t leg is at distance > n ). Along the unique simple path from s to ℓ , the blue set m ust adv ance across at least n edges. Each adv ance requires at least one successful round, and for the ﬁrst n − 1 adv ances the next vertex has at most one blue in-neigh b or among t wo in-neigh bors RANDOMIZED ZERO FOR CING 15 when it ﬁrst b ecomes eligible, hence succeeds with probability at most 1 / 2 eac h round; the ﬁnal adv ance (in to the leaf ) has success probabilit y 1. Therefore ept rzf ( S, s ) ≥ 2( n − 1) + 1 = 2 n − 1, and taking the minimum o v er s gives ept rzf ( S ) ≥ 2 n − 1. Com bining the b ounds yields ept rzf ( S ) = 2 n + O ( √ n ), hence ept rzf ( S ) = 2 n (1 + o (1)) as n → ∞ for ﬁxed k . □ Indeed, the last pro of implies the follo wing slightly stronger result. Theorem 3.3. Fix an integer k > 1 and let S b e a bidirectional spider graph with k legs each of length n (1 + o (1)) and all weigh ts equal. Then ept rzf ( S ) = 2 n (1 + o (1)), where the o (1) is with resp ect to n . In the case that k = 2, w e obtain the following corollary for P n . Note that P n can be considered a spider with tw o legs each of length appro ximately n/ 2. Corollary 3.9. F or bidir e ctional p aths of or der n with al l weights e qual, we have ept rzf ( ← → P n ) = n (1 + o (1)) . n ept rzf ( P n ) n ept rzf ( P n ) n ept rzf ( P n ) n ept rzf ( P n ) 4 3.0000 20 20.4216 36 37.3090 52 54.0050 5 3.6667 21 21.3519 37 38.2595 53 54.9645 6 5.2222 22 22.5494 38 39.4038 54 56.0837 7 6.0370 23 23.4836 39 40.3557 55 57.0439 8 7.4444 24 24.6711 40 41.4960 56 58.1609 9 8.3086 25 25.6086 41 42.4492 57 59.1219 10 9.6447 26 26.7875 42 43.5859 58 60.2367 11 10.5327 27 27.7279 43 44.5403 59 61.1984 12 11.8255 28 28.8993 44 45.6735 60 62.3111 13 12.7279 29 29.8421 45 46.6291 61 63.2736 14 13.9908 30 31.0069 46 47.7591 62 64.3843 15 14.9032 31 31.9520 47 48.7158 63 65.3474 16 16.1438 32 33.1109 48 49.8428 64 66.4563 17 17.0636 33 34.0579 49 50.8005 65 67.4200 18 18.2869 34 35.2115 50 51.9248 66 68.5272 19 19.2125 35 36.1603 51 52.8833 67 69.4915 T able 1. ept rzf ( P n ) for n = 4 to n = 67, obtained through computation 3.6. Other families. W e close this section b y examining a collection of additional graph families- complete graphs, stars, Sun graphs, complete bipartite graphs, and k -ary trees-that exhibit a range of asymptotic b eha viors under the RZF pro cess. Theorem 3.4. Let K 1 ,n − 1 denote the undirected unw eigh ted star graph of order n , and let ← → K n denote a bidirectional complete graph of order n with all edges of equal w eight. If c denotes the cen ter vertex of K 1 ,n − 1 , then for an y vertex v ∈ ← → K n w e hav e ept rzf ( ← → K n , v ) = ept pzf ( K 1 ,n − 1 , c ) = Θ(log n ). Pr o of. Supp ose that v in K n is colored blue and b other vertices are colored blue, so there are n − b − 1 white vertices. If w is a white vertex, then the probability that w gets colored in the next round of randomized zero forcing is b +1 n − 1 . No w supp ose that the cen ter v ertex c in K 1 ,n − 1 is colored blue and b lea ves are also colored blue, so there are n − b − 1 white lea v es. If w is a white leaf, then only c can color w in probabilistic zero forcing, so the probability that w gets colored in the next round is b +1 n − 1 . 16 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES Therefore any ﬁnite sequence 1 = a 1 ≤ a 2 ≤ · · · ≤ a k = n of n um b ers of blue v ertices by round in K n under randomized zero forcing has the same probability of o ccurrence as a 1 , . . . , a k has for the num ber of blue vertices b y round in K 1 ,n − 1 under probabilistic zero forcing. Thus the exp ected propagation times are the same. □ Theorem 3.5. F or the bidirectional Sun graph ← → S n with all edges of equal weigh t, w e hav e that ept rzf ( ← → S n ) = 1 + 3 2 ( n − 1) . Pr o of. W e pro ceed by induction on n . F or the purp ose of minimizing exp ected propagation time, it is optimal to begin at a center vertex, as starting at a leaf dela ys the ﬁrst possible forcing of any other vertex b y at least one round, whereas starting at the adjacent center forces that leaf after the ﬁrst iteration. Let ept rzf ( S ∗ n ) denote the exp ected n um b er of rounds required to force all center v ertices of ← → S n , starting from a single blue center v ertex. Note that once all center v ertices are blue, exactly one additional iteration suﬃces to force any remaining leaf v ertices. F rom a blue cen ter vertex, its attached leaf is forced deterministically in the next round, while eac h adjacen t cen ter v ertex is forced independently with probability 1 3 . Th us there is a 4 9 probabilit y that neither adjacen t center v ertex is forced, a 4 9 probabilit y that exactly one is forced, and a 1 9 probabilit y that both are forced. If exactly k ∈ { 0 , 1 , 2 } adjacen t cen ter v ertices are forced, the remaining unforced centers form S ∗ n − k . This yields the recurrence ept rzf ( S ∗ n ) = 4 9  ept rzf ( S ∗ n ) + 1  + 4 9  ept rzf ( S ∗ n − 1 ) + 1  + 1 9  ept rzf ( S ∗ n − 2 ) + 1  . A direct computation gives ept rzf ( S ∗ 3 ) = 3 and ept rzf ( S ∗ 4 ) = 9 2 . Using the recurrence, one ﬁnds ept rzf ( S ∗ 5 ) = 6, and the inductive h yp othesis ept rzf ( S ∗ n ) = 3 2 ( n − 1) satisﬁes the recurrence for all n ≥ 5. Since one additional round suﬃces to force all leaf v ertices, ept rzf ( ← → S n ) = ept rzf ( S ∗ n ) + 1 = 1 + 3 2 ( n − 1) , as claimed. □ Lemma 3.10. [16] If X is the maximum of n indep endent c opies of Ge om ( p ) , then 1 ln(1 / (1 − p )) n X j =1 1 j ≤ E( X ) ≤ 1 + 1 ln(1 / (1 − p )) n X j =1 1 j . Corollary 3.11. Fix p < 1 and supp ose that X is the maximum of n indep endent c opies of Ge om ( p ) . Then, E( X ) = Θ( 1 − p p log n ) . Deﬁne T k,n to b e the complete k -ary directed tree of depth n . This tree has n + 1 lay ers, k i v ertices in the i th la y er for each i = 0 , 1 , . . . , n , and edges in b oth directions b et w een each v ertex in lay er i and and k v ertices in lay er i + 1 for each i = 0 , 1 , . . . , n − 1. Theorem 3.6. If T k,n is a complete k -ary directed tree of depth n with all edges of equal weigh t, then ept rzf ( T k,n ) = O ( kn 2 ). Pr o of. Supp ose that we start with the depth-0 v ertex as the initial blue vertex. If the vertices in la y er i are all blue, then it takes at most O ( k i ) additional rounds for the vertices in la y er i + 1 to b e colored blue b y Corollary 3.11, where the constant in the O ( i ) bound dep ends on k . Thus, ept rzf ( T k,n ) ≤ n − 1 X i =0 O ( ki ) = O ( k n 2 ) . □ RANDOMIZED ZERO FOR CING 17 Theorem 3.7. If K m,n is the bidirectional complete bipartite graph with parts of size m and n and all edges of equal weigh t, then ept rzf ( K m,n ) = O (min( m log n, n log m )). Pr o of. Supp ose that the initial blue vertex is in the part of size m . By Corollary 3.11, it tak es at most O ( m log n ) exp ected rounds to color all the v ertices in the part of size n , and then an y remaining uncolored vertices in the graph would b e colored deterministically on the next turn. Th us, the exp ected propagation time would b e O ( m log n ). Similarly , if the initial blue vertex is in the part of size n , then the expected propagation time w ould be O ( n log m ). Thus, ept rzf ( K m,n ) = O (min( m log n, n log m )). □ Conjecture 3.12. Given a bidir e ctional c omplete bip artite gr aph K a,b with al l e dges of e qual weight wher e a > b , let A denote the p artite set with a vertic es, and B denote the p artite set with b vertic es. Given vertic es u ∈ A and v ∈ B , ept rzf ( K a,b , { v } ) < ept rzf ( K a,b , { u } ) a b ept rzf ( K a,b , { u } ) ept rzf ( K a,b , { v } ) a b ept rzf ( K a,b , { u } ) ept rzf ( K a,b , { v } ) 1 1 1.000000 1.000000 6 4 5.671967 4.756544 2 1 3.000000 1.000000 6 5 5.639959 5.233268 2 2 3.000000 3.000000 6 6 5.633242 5.633242 3 1 4.000000 1.000000 7 1 8.000000 1.000000 3 2 3.936000 3.094286 7 2 6.662118 3.367973 3 3 3.997474 3.997474 7 3 6.235815 4.191726 4 1 5.000000 1.000000 7 4 6.076556 4.784222 4 2 4.712275 3.182594 7 5 6.010218 5.250222 4 3 4.682545 4.059473 7 6 5.982660 5.637761 4 4 4.700136 4.700136 7 7 5.973384 5.973384 5 1 6.000000 1.000000 8 1 9.000000 1.000000 5 2 5.405476 3.256923 8 2 7.251845 3.408869 5 3 5.257313 4.111751 8 3 6.676058 4.222912 5 4 5.221566 4.727711 8 4 6.449886 4.810055 5 5 5.221072 5.221072 8 5 6.346981 5.268740 6 1 7.000000 1.000000 8 6 6.296968 5.647558 6 2 6.049812 3.318017 8 7 6.272747 5.973450 6 3 5.767330 4.155198 8 8 6.262249 6.262249 T able 2. Exp ected propagation time of bidirectional K a,b with all edges of equal w eigh t, by starting v ertex 4. Extremal resul ts for unweighted directed graphs In this section we study extremal b eha vior of the exp ected propagation time for randomized zero forcing on unw eigh ted directed graphs. W e obtain sharp upp er and lo w er b ounds in terms of basic graph parameters suc h as the num b er of edges, order, maxim um degree, and radius, and we c haracterize when these bounds are attained. Theorem 4.1. Let G b e an unw eigh ted directed graph with m directed edges, and assume ev ery v ertex has indegree at least d ≥ 0. Fix v ∈ V ( G ) with ept rzf ( G, v ) < ∞ . Then ept rzf ( G, v ) ≤ m − d. 18 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES Pr o of. Let τ b e the (random) propagation time of RZF on G started from { v } , so that ept rzf ( G, v ) = E[ τ ]. Because ept rzf ( G, v ) < ∞ , Theorem 2.1 implies that every v ertex of G is reac hable from v by a directed path. F or each round t ≥ 0, let B t b e the (random) set of blue vertices after t rounds, and let W t := V ( G ) \ B t b e the set of white vertices. Deﬁne the p oten tial Φ t := X x ∈ W t deg − ( x ) , the total indegree of the white vertices at time t . Note that Φ t is a nonnegative in teger-v alued random v ariable, Φ 0 = P x  = v deg − ( x ) = m − deg − ( v ), and Φ t = 0 if and only if W t = ∅ , i.e. the pro cess has completed. F or a white v ertex x ∈ W t , let b t ( x ) b e the num b er of in-neighbors of x that are blue at time t . Under un weigh ted RZF, conditional on B t , v ertex x turns blue in the next round with probability b t ( x ) / deg − ( x ), and if it turns blue then Φ decreases by exactly deg − ( x ). Therefore, conditional on B t , E  Φ t − Φ t +1   B t  = X x ∈ W t deg − ( x ) · b t ( x ) deg − ( x ) = X x ∈ W t b t ( x ) . The ﬁnal sum counts the num b er of directed edges from B t to W t (eac h suc h edge contributes 1 to exactly one term b t ( x )), hence E  Φ t − Φ t +1   B t  = e ( B t , W t ) , where e ( B t , W t ) denotes the num b er of directed edges with tail in B t and head in W t . W e claim that whenever W t  = ∅ (equiv alently , Φ t > 0), w e ha v e e ( B t , W t ) ≥ 1. Indeed, ﬁx an y w ∈ W t . Since w is reac hable from v and v ∈ B t , there exists a directed path v = u 0 → u 1 → · · · → u k = w . Let j b e the smallest index with u j ∈ W t . Then j ≥ 1 and u j − 1 ∈ B t , so the edge u j − 1 → u j is a directed edge from B t to W t . Thus e ( B t , W t ) ≥ 1. Consequen tly , for every t with Φ t > 0, E  Φ t +1   B t  = Φ t − E[Φ t − Φ t +1 | B t ] ≤ Φ t − 1 . No w deﬁne the stopping time τ := min { t : Φ t = 0 } and the pro cess M t := Φ min( t,τ ) + min( t, τ ) . The inequalit y ab o ve implies that ( M t ) t ≥ 0 is a supermartingale. T aking expectations gives E[ M t ] ≤ E[ M 0 ] = Φ 0 for all t , and so E[min( t, τ )] ≤ Φ 0 . Letting t → ∞ and using monotone con v ergence yields E[ τ ] ≤ Φ 0 = m − deg − ( v ) ≤ m − d, as claimed. □ Corollary 4.1. Over al l unweighte d dir e cte d gr aphs G with m dir e cte d e dges and over al l v ∈ V ( G ) with ept rzf ( G, v ) < ∞ , the maximum p ossible value of ept rzf ( G, v ) is m . Over al l dir e cte d gr aphs G with m dir e cte d e dges in which every vertex has inde gr e e at le ast 1 and over al l v ∈ V ( G ) with ept rzf ( G, v ) < ∞ , the maximum p ossible value o f ept rzf ( G, v ) is m − 1 . Pr o of. The upp er b ounds follo w from Theorem 4.1. T o see that the ﬁrst upp er bound is sharp, note that ept rzf ( − − − → P m +1 , v ) = m when v is the endp oin t with indegree 0. T o see that the second upp er b ound is sharp, we split in to t wo cases. When m = 2 k is ev en, we ha v e ept rzf ( ← − → P k +1 , v ) = 2 k − 1 = m − 1 when v is an endp oin t. When m = 2 k + 1 is odd, let G b e the graph obtained RANDOMIZED ZERO FOR CING 19 from ← − → P k +1 with endp oin ts u, v by adding a new v ertex with a single edge from u . Then, we hav e ept rzf ( G, v ) = 2 k = m − 1. □ Indeed, the − − − → P m +1 construction in the last pro of implies the follo wing stronger result, since the left endp oin t is the only initial blue vertex that can color the whole path. Corollary 4.2. Over al l unweighte d dir e cte d gr aphs G with m dir e cte d e dges with ept rzf ( G ) < ∞ , the maximum p ossible value of ept rzf ( G ) is m . As a corollary of Theorem 4.1, we obtain an upper b ound on the maximum p ossible exp ected propagation time for any directed graph of order n . Corollary 4.3. F or al l unweighte d dir e cte d gr aphs G of or der n and for al l v ∈ V ( G ) with ept rzf ( G, v ) < ∞ , we have ept rzf ( G, v ) = O ( n 2 ) . Pr o of. If G has order n , then G has at most n ( n − 1) directed edges, so ept rzf ( G, v ) ≤ n 2 − n − 1. □ Next, we sho w that the O ( n 2 ) b ound is sharp up to a constan t factor. Theorem 4.2. There exists an unw eighted directed graph G of order n suc h that G has only one v ertex v with forward paths to all other vertices and ept rzf ( G, v ) = Ω( n 2 ). Pr o of. Consider the directed graph D = ( V , E ) with V = { a 1 , . . . , a m , b 1 , . . . , b m +1 } whic h has edges from b i to a j for all i ≥ j and from a i to b i +1 for each i = 1 , . . . , m . The minim um randomized zero forcing set has size 1, and the only c hoice is b 1 , or else it is impossible to color b 1 . The order of the coloring must b e b 1 , a 1 , b 2 , a 2 , . . . . After b i b ecomes blue, the probabilit y that a i b ecomes blue is 1 m +2 − i , so the exp ected n um b er of rounds un til a i b ecomes blue after b i b ecomes blue is m + 2 − i . When a i b ecomes blue, b i +1 deterministically turns blue in the next round. Th us the exp ected propagation time is m + P m i =1 ( m + 2 − i ) = Ω( m 2 ). □ Corollary 4.4. Over al l unweighte d dir e cte d gr aphs G of or der n with ﬁnite ept rzf ( G ) , the maximum p ossible value of ept rzf ( G ) is Θ( n 2 ) . As another corollary of Theorem 4.1, w e determine the maximum p ossible v alue of ept rzf ( T , v ) o v er trees T . Corollary 4.5. Over al l unweighte d dir e cte d gr aphs T of or der n whose underlying undir e cte d gr aph is a tr e e and over al l v ∈ V ( T ) with ept rzf ( T , v ) < ∞ , the maximum p ossible value of ept rzf ( G, v ) is 2 n − 3 . Pr o of. A directed graph T of order n whose underlying undirected graph is a tree has at most 2( n − 1) directed edges. Th us ept rzf ( T , v ) ≤ 2( n − 1) − 1 for all v ∈ ( T ). T o see that this upp er b ound is sharp, note that ept rzf ( ← → P n , v ) = 2 n − 3 when v is an endp oin t. □ Theorem 4.3. Let G be an un w eigh ted directed graph of order n with maximum indegree at most d . If ept rzf ( G ) < ∞ , then ept rzf ( G ) ≤ dn − d ( d + 1) 2 . Pr o of. Fix v ∈ V ( G ) with ept rzf ( G ) = ept rzf ( G, v ) and run RZF from { v } . Let τ b e the propagation time and, for each t ≥ 0, let B t b e the blue set after t rounds and W t := V ( G ) \ B t the white set. F or r ∈ { 1 , 2 , . . . , n − 1 } , deﬁne σ r to b e the ﬁrst round t such that | W t | ≤ r . Then σ n − 1 = 0 and σ 0 = τ , and τ = n − 1 X r =1 ( σ r − 1 − σ r ) . W e bound E[ σ r − 1 − σ r ] for each r . 20 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES Fix r ≥ 1 and condition on the history up to time σ r . If | W σ r | = 0 there is nothing to prov e. Otherwise, since ept rzf ( G, v ) < ∞ , Theorem 2.1 implies that there exists at least one directed edge from B σ r to W σ r , and hence there exists a white v ertex w having at least one blue in-neighbor at time σ r . Let b b e the n umber of blue in-neigh b ors of w at time σ r . Since w has at least one blue in- neigh b or, w e hav e b ≥ 1. Also, b ecause | W σ r | ≤ r , at most r − 1 of the in-neighbors of w can b e white, so b ≥ deg − ( w ) − ( r − 1). Hence b ≥ max { 1 , deg − ( w ) − ( r − 1) } . Since deg − ( w ) ≤ d , we ha v e Pr( w turns blue in the next round | F σ r ) = b deg − ( w ) ≥ max { 1 , deg − ( w ) − ( r − 1) } deg − ( w ) ≥ 1 r . Indeed, if deg − ( w ) < r then max { 1 , deg − ( w ) − ( r − 1) } = 1 and 1 / deg − ( w ) ≥ 1 /r , while if deg − ( w ) ≥ r then deg − ( w ) − ( r − 1) deg − ( w ) ≥ 1 r b ecause ( r − 1)(deg − ( w ) − r ) ≥ 0. Moreo v er, as long as | W t | ≤ r and W t  = ∅ , the same reasoning applies at time t : by Theorem 2.1 there exists a directed edge from B t to W t , so there is a white v ertex with at least one blue in- neigh b or, and the conditional probabilit y that some white v ertex turns blue in the next round is at least 1 /r . Therefore, conditional on F σ r , the w aiting time to reduce the num b er of white v ertices from at most r to at most r − 1 is sto c hastically dominated by a geometric random v ariable with success probability 1 /r . Hence E[ σ r − 1 − σ r ] ≤ r . F or r ≥ d + 1 we use the trivial b ound E[ σ r − 1 − σ r ] ≤ d, b ecause some eligible white vertex has success probabilit y at least 1 /d in each round. Com bining these b ounds and using linearity of exp ectation, ept rzf ( G ) = E[ τ ] ≤ n − 1 X r = d +1 d + d X r =1 r = d ( n − d − 1) + d ( d + 1) 2 = dn − d ( d + 1) 2 , as claimed. □ Corollary 4.6. Over al l unweighte d dir e cte d gr aphs G of or der n with maximum de gr e e d and ept rzf ( G ) < ∞ , the maximum p ossible value of ept rzf ( G ) is dn − d ( d +1) 2 . Pr o of. The upp er b ound ept rzf ( G ) ≤ dn − d ( d +1) 2 follo ws from the previous theorem. T o see that this upp er b ound is sharp, consider the graph on n vertices v 1 , v 2 , . . . , v n with edges from v i to v i +1 for each 1 ≤ i ≤ n − 1 and edges from v i + j to v i for all j ≤ d − 1 and 1 ≤ i ≤ n − j . The v ertex v 1 m ust b e the initial blue vertex, since it has indegree 0. Moreov er, the vertices m ust b e colored in the order v 1 , v 2 , . . . , v n . F or each i = 1 , . . . , n − d − 1, it takes d exp ected rounds to color v i +1 after v i is colored. F or each i = n − d, . . . , n − 1, it takes n − i exp ected rounds to color v i +1 after v i is colored. Altogether, the exp ected propagation time is d ( n − d − 1) + n − 1 X i = n − d ( n − i ) = dn − d ( d + 1) 2 . □ No w, we turn to minim ums, starting with all directed graphs G of order n . RANDOMIZED ZERO FOR CING 21 Theorem 4.4. Over all unw eigh ted directed graphs G of order n > 1, the minim um possible v alue of ept rzf ( G ) is 1. Pr o of. The low er b ound ept rzf ( G ) ≥ 1 follo ws since n > 1. This b ound can be attained by K 1 ,n − 1 , so the minimum possible v alue of ept rzf ( G ) is 1. □ Next, we obtain a sharp low er b ound on ept rzf ( G ) with resp ect to rad( G ). Theorem 4.5. F or any un w eigh ted directed graph G and any v ertex v ∈ V ( G ), ept rzf ( G, v ) ≥ rad( G ). Pr o of. Let v b e any initial blue vertex in G . There m ust exist some u with forward-distance at least r = rad( G ) from v . Thus u at the earliest can only b e colored blue in round r , so the exp ected propagation time is at least r . □ The low er b ound in the last result is sharp, as evidenced by the unidirectional directed path of order r + 1. Theorem 4.6. Let G b e a directed graph and ﬁx a v ertex v ∈ V ( G ). Let rad( G, v ) denote the directed eccentricit y of v . Consider randomized zero forcing (RZF) starting from v . Then ept rzf ( G, v ) = rad( G, v ) if and only if every vertex is reach able from v and every directed cycle of G contains v . Pr o of. W e prov e eac h direction separately . ( ⇒ ) Supp ose there exists either a vertex not reachable from v , or a directed cycle C that do es not con tain v . In the ﬁrst case, ept rzf ( G ) > rad( G, v ) trivially . Thus assume every vertex is reac hable from v and that there exists a directed cycle C disjoin t from v . Let B t denote the blue set after t rounds of RZF. Because C is disjoin t from v and reac hable from v , there exists a realization of the random forcing choices (of p ositiv e probabilit y) in whic h no vertex of C is colored blue during the ﬁrst rad rounds. Th us, the propagation time is greater than rad( G, v ) on this even t. Since the ev ent has positive probabilit y , w e conclude ept rzf ( G, v ) > rad( G, v ). ( ⇐ ) Assume now that ev ery vertex is reac hable from v and that every directed cycle of G contains v . Supp ose for con tradiction that there exists a realization of the RZF pro cess and a vertex w suc h that w / ∈ B rad( G,v ) , where B rad( G,v ) denotes the blue v ertices of G after the rad( G, v ) round of RZF on this realization. Set w 0 := w . Since w 0 / ∈ B rad( G,v ) , there exists an in-neigh b or w 1 ∈ N − ( w 0 ) such that w 1 / ∈ B rad( G,v ) − 1 , as otherwise all in-neighbors of w 0 w ould b e blue b y round rad( G, v ) − 1, forcing w 0 to be blue at round rad( G, v ). Iterating this argument, w e construct vertices w 0 , w 1 , . . . , w rad( G,v ) suc h that w i +1 ∈ N − ( w i ) and w i / ∈ B rad( G,v ) − i for all i = 0 , . . . , rad( G, v ) − 1 . In particular, w rad( G,v ) / ∈ B 0 = { v } , so w rad( G,v )  = v . This yields a directed w alk w rad( G,v ) → w rad( G,v ) − 1 → · · · → w 0 = w . If all vertices in this walk are distinct, then it contains a directed path of length rad( G, v ) ending at w whose initial v ertex is not v . Since v reac hes w b y a directed path of length at most rad( G, v ), this contradicts the deﬁnition of rad( G, v ). Otherwise, some vertex rep eats along the walk. T aking the ﬁrst rep etition yields a directed cycle disjoin t from v , contradicting the assumption that ev ery directed cycle contains v . In b oth cases w e obtain a contradiction. Therefore ev ery vertex must be blue by round rad( G, v ), and th us ept rzf ( G, v ) = rad( G, v ) □ 22 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES Corollary 4.7. The only unweighte d dir e cte d gr aphs G of or der n with ept rzf ( G ) = 1 ar e those G obtaine d fr om − − − − → K 1 ,n − 1 by adding any numb er of e dges to the c enter. Ther e ar e exact l y n such G , up to isomorphism. 5. Graph opera tions for unweighted directed graphs In this section, we discuss generally ho w ept rzf could change with changes to the underlying graph. Generally , adding vertices or edges will hav e a broad eﬀect on the expected propagation time. F or example, if a white vertex v is added to an unw eigh ted directed graph G suc h that the in-degree of v is 1 and the out-degree is 0, then for any v ertex u of G not equal to v , ept rzf ( G, u ) ≤ ept rzf ( G + v , u ) ≤ ept rzf ( G, u ) + 1 . (5.1) The upp er b ound is b est p ossible, since for example, we can let G b e the directed graph with underlying undirected graph K 1 ,n ha ving edges from the center u to each leaf, and we can let v ha v e an edge from one of the leav es. In this case, ept rzf ( G, u ) = 1 and ept rzf ( G + v , u ) = 2. The lo w er b ound is also b est p ossible, since we can take the same graph G and instead let v hav e an edge from u , in which case ept rzf ( G + v , u ) = ept rzf ( G, u ) = 1. Note that the same b ounds also apply in the weigh ted case. On the other hand, for the case that a blue v ertex b is added to G with in-degree 0 and out-degree 1, we ha v e an exact formula for the exp ected propagation time starting from b . Prop osition 5.1. L et G b e any unweighte d dir e cte d gr aph. If a blue vertex b is adde d to G with in-de gr e e 0 and out-de gr e e 1 , and the incident vertex in G to b is v , then ept rzf ( G + b, b ) = d + ept rzf ( G, v ) , wher e d is the in-de gr e e of v in G + b . Pr o of. If w e color G + b starting from b , then the ﬁrst vertex in G that gets colored must b e v . It tak es d exp ected rounds un til v is colored, and then an additional ept rzf ( G, v ) rounds to color the rest of G . □ In the follo wing prop osition, w e extend this last op eration to the case that b joins multiple graphs. Prop osition 5.2. L et G 1 , . . . , G m b e unweighte d dir e cte d gr aphs. Denote by K the joine d gr aph deﬁne d in the fol lowing way: ther e is a single vertex b with out-de gr e e m and in-de gr e e 0 which is incident to a vertex v i for e ach G i . Assume further that the G i have no vertic es or e dges in c ommon as sub gr aphs of K . L et t i denote the pr op agation time for the gr aphs G i starting with the single blue no de v i . Then, ept rzf ( K, b ) ≤ max i ept rzf ( G i , v i ) +  P m i =1 V ar( t i ) 2  1 / 2 + O ( r m 1 / 2 ) . (5.2) wher e r = max { r i } and r i is the in-de gr e e of v i . Pr o of. Let t K denote the propagation time for K with starting no de b , so that ept rzf ( K, b ) = E ( t K ). By Theorem 2.1 of [2], E ( t K ) = E (max i { t i + Geom(1 /r i ) } ) (5.3) ≤ max i E ( t i + Geom(1 /r i )) + 1 √ 2  m X i =1 V ar( t i + Geom(1 /r i ))  1 / 2 . (5.4) But Geom(1 /r i ) and t i are indep enden t random v ariables for all i , and so, V ar( t i + Geom(1 /r i )) = V ar( t i ) + 1 − r − 1 i r − 2 i . (5.5) RANDOMIZED ZERO FOR CING 23 By subadditivity of the square ro ot function,  P m i =1 V ar( t i ) + Geom(1 /r i ) 2  1 / 2 ≤ ( m/ 2) 1 / 2 ( r 2 − r ) 1 / 2 +  P m i =1 V ar( t i ) 2  1 / 2 , (5.6) where r = max { r G , r H } . Lastly , it follows straigh tforw ardly from linearity of expectation that, max i E ( t i + Geom(1 /r i )) ≤ max i E ( t i ) + r . (5.7) Substituting equations (5.6) and (5.7) in to (5.4) gives the inequality . □ Corollary 5.1. Fix a c onstant k . Consider the unweighte d dir e cte d gr aph G obtaine d by taking k bidir e cte d p aths G 1 , . . . , G k (of lengths n 1 , . . . , n k r esp e ctively) and joining them at a single initial vertex x that is initial ly blue. V ertex x has dir e cte d out-e dges to the starting vertic es v 1 , . . . , v k of e ach p ath. Then, we have ept rzf ( G, x ) = 2 max i { n i } + O ( q max i { n i } ) , wher e the c onstant in the O -b ound dep ends on k . Pr o of. Let T i denote the propagation time on the path G i when v i is the initial blue vertex. By Prop osition 5.2 (applied with m = k ), w e hav e the b ound ept rzf ( G, x ) ≤ max i { E [ T i ] } + O  q max i { V ar( T i ) }  + O (1) , where the constants in the O -bounds depend on k . No w, each G i is a bidirected path with one end fed by a single source. W e show ed that E [ T i ] = 2 n i − 1 and V ar( T i ) = O ( n i ). Therefore, max i { E [ T i ] } = 2 max i { n i } − 1, and max i { V ar( T i ) } = O (max i { n i } ). Substituting these into the b ound from Prop osition 5.2 giv es ept rzf ( G, x ) ≤ 2 max i { n i } + O ( q max i { n i } ) . Finally , we hav e a lo wer b ound of ept rzf ( G, x ) ≥ 2 max i { n i } , since E [ T i ] = 2 n i − 1 and it tak es at least 1 round for v i to get colored. Hence, w e conclude that ept rzf ( G, x ) = 2 max i { n i } + O ( q max i { n i } ) , as claimed. □ In the next result, we consider the eﬀect of adding a single directed edge b et w een tw o directed graphs. Theorem 5.1. Let G, H b e un weigh ted directed graphs and supp ose that u ∈ V ( G ) and v ∈ V ( H ), where v has indegree d in H . Let K b e the directed graph obtained from G and H by adding a single edge from u to v . Then, for all w ∈ V ( G ), we ha v e ept rzf ( K, w ) ≤ 1 + d + ept rzf ( G, w ) + ept rzf ( H , v ) . Pr o of. If w is the initial blue vertex, then the exp ected num b er of rounds to color all of G is ept rzf ( G, w ). Since v has indegree d + 1 in K , it tak es d + 1 exp ected rounds for v to b e colored after u is colored. After v is colored, the exp ected n um b er of rounds to color all of H is ept rzf ( H , v ). Altogether, it takes at most ept rzf ( G, w ) + d + 1 + ept rzf ( H , v ) exp ected rounds to color all of K , starting from w . □ In the next result, w e sho w that the b ound in the last theorem is best p ossible b y exhibiting an inﬁnite family of examples whic h attain the b ound. 24 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES Theorem 5.2. F or all m and n > d , there exist unw eigh ted directed graphs G and H and v ertices u, w ∈ V ( G ) and v ∈ V ( H ) where v has indegree d in H suc h that ept rzf ( G, w ) = m , ept rzf ( H , v ) = n , and the directed graph K obtained from G and H by adding an edge from u to v satisﬁes ept rzf ( K, w ) = 1 + d + ept rzf ( G, w ) + ept rzf ( H , v ) . Pr o of. Let G = − − − → P m +1 and let H be the graph obtained from − − − → P n +1 b y adding edges from the d righ tmost vertices to the left endp oin t. Let w b e the left endp oin t of G , u b e the right endp oin t of G , and v be the left endp oin t of H . Then ept rzf ( G, w ) = m , ept rzf ( H , v ) = n , and ept rzf ( K ) = m + ( d + 1) + n . □ 6. Fur ther resul ts for weighted directed graphs On a weigh ted directed graph G , we deﬁne a fort to b e a subset of vertices such that each v ertex has at least one in-neighbor in the fort. Giv en that G is strongly connected, we note that unlik e in classical zero forcing, no subgraph can remain unforced forever. W e aim to deﬁne a RZF analogue where entry in to the fort o ccurs only through rare random even ts, allo wing the exp ected propagation time to grow arbitrarily large if the starting set of blue vertices do not intersect the fort. Theorem 6.1. On any graph G with tw o vertex disjoint forts, there exists a weigh t assignment of the edges such that ept rzf ( G ) is arbitrarily high. Pr o of. Let n = | V ( G ) | , and ﬁx an arbitrary k ∈ N . Let F 1 and F 2 b e tw o vertex-disjoin t forts in G . W e deﬁne an edge-w eigh ting as follows. Assign w eigh t ε > 0 to ev ery edge ( u, v ) with u / ∈ F 1 and v ∈ F 1 (edges entering F 1 ), and weigh t 1 to every other edge of G . W e will choose ε suﬃciently small in terms of k and n . Consider a round t suc h that F 1 ∩ B t = ∅ . In order for some vertex of F 1 to become blue in round t + 1, there must exist a blue vertex u / ∈ F 1 that forces some v ∈ F 1 along an en tering edge ( u, v ). F or an y such u , the total outgoing w eight from u to white vertices is at least 1 (since all non-en tering edges ha ve weigh t 1), while each entering edge from u to F 1 has w eigh t ε . Hence, regardless of the conﬁguration outside F 1 , Pr( u forces a vertex in F 1 | F 1 ∩ B t = ∅ ) ≤ deg + F 1 ( u ) ε 1 ≤ nε, where deg + F 1 ( u ) denotes the num b er of out-neigh b ors of u in F 1 , and we used deg + F 1 ( u ) ≤ n . Applying the union b ound ov er all v ertices u (at most n choices), w e obtain Pr(some vertex of F 1 is forced in round t + 1 | F 1 ∩ B t = ∅ ) ≤ n · ( nε ) = n 2 ε. Therefore the time T F 1 un til F 1 is ﬁrst en tered is sto c hastically dominated b elo w by a geometric random v ariable with success probability at most n 2 ε , and thus E [ T F 1 ] ≥ 1 n 2 ε . Cho osing ε < 1 kn 2 yields E [ T F 1 ] > k . Finally , for any initial blue vertex v , since F 1 and F 2 are disjoint, at least one of them do es not con tain v . Apply the abov e construction to whic hev er fort do es not intersect the initial blue set. In all cases, this pro duces an edge-weigh t assignment for whic h ept rzf ( G ) > k . Since k was arbitrary , ept rzf ( G ) can b e made arbitrarily large. □ RANDOMIZED ZERO FOR CING 25 6.1. Extremal results. W e no w derive sharp extremal b ounds for the exp ected propagation time on weigh ted directed graphs in terms of the minimum edge w eigh t and the order of the graph. Theorem 6.2. Supp ose that G is an y w eigh ted directed graph of order n with all p ositiv e edge w eigh ts at least w , and v is a v ertex with forw ard paths to all other v ertices in G . Then we hav e ept rzf ( G, v ) ≤ 1 + n − 2 w . Pr o of. Given any conﬁguration with some p ositiv e num ber of blue vertices and some p ositiv e num- b er of white vertices, there exists a white v ertex u that receives some edge from a blue vertex. Since all edge w eights are at least w , the probability that u gets colored is at least w . Therefore the probability of increasing the n um b er of blues is at least w , so the exp ected time to increase the n um b er of blues is at most 1 w . Each increase is by at least 1 and the coloring b ecomes deterministic if only one uncolored v ertex is left, so the exp ected propagation time is at most 1 + n − 2 w . □ By adding weigh ts to the construction in Theorem 4.2, we can sho w that the 1 + n − 2 w upp er b ound is sharp. Theorem 6.3. There exists a w eigh ted directed graph G of order n with all p ositiv e edge weigh ts at least w suc h that G has only one v ertex v with forw ard paths to all other v ertices and ept rzf ( G, v ) = 1 + n − 2 w . Pr o of. Consider the weigh ted directed graph on v ertices v 1 , v 2 , . . . , v n with edges from v i to v i +1 for each 1 ≤ i < n and edges from v i +1 to v i for 2 ≤ i < n . The edge from v i to v i +1 for each 1 ≤ i < n − 1 has w eigh t w , while the edge from v n − 1 to v n has weigh t 1. The initial blue v ertex m ust b e v 1 , or else it w ould nev er b e colored. Once v i has b een colored with i < n − 1, the exp ected n umber of rounds until v i +1 is colored is 1 w . Once v n − 1 has b een colored, v n is colored in the next round. Thus, ept rzf ( G, v 1 ) = 1 + n − 2 w . □ Corollary 6.1. Over al l weighte d dir e cte d gr aph G of or der n with al l p ositive e dge weights at le ast w and over al l v ∈ V ( G ) with forwar d p aths to al l other vertic es, the maximum p ossible value of ept rzf ( G, v ) is 1 + n − 2 w . 6.2. Op erations on w eigh ted directed graphs. In this subsection, w e consider some op erations on weigh ted directed graphs and their eﬀects on exp ected propagation time. Prop osition 6.1. L et G b e any weighte d dir e cte d gr aph. If a blue vertex b is adde d to G with in-de gr e e 0 and out-de gr e e 1 , the incident vertex in G to b is a vertex v with in-weight d , and the weight on the new e dge is w , then ept rzf ( G + b, b ) = d + w w + ept rzf ( G, v ) . Pr o of. As in the unw eigh ted case, if w e color G + b starting from b , then the ﬁrst v ertex in G that gets colored m ust b e v . It takes d + w w exp ected rounds un til v is colored, and then an additional ept rzf ( G, v ) rounds to color the rest of G . □ In the next result, we consider the eﬀect of adding a single directed weigh ted edge b et w een tw o w eigh ted directed graphs. Theorem 6.4. Let G, H b e w eighted directed graphs and suppose that u ∈ V ( G ) and v ∈ V ( H ), where v has in-weigh t d . Let K b e the directed graph obtained from G and H by adding a single edge from u to v with weigh t w . Then, for all x ∈ V ( G ), we ha v e ept rzf ( K, x ) ≤ d + w w + ept rzf ( G, x ) + ept rzf ( H , v ) . 26 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES Pr o of. If x is the initial blue vertex, then the exp ected n um b er of rounds to color all of G is ept rzf ( G, x ). Since the new edge has w eigh t w and the remaining in-weigh t to v is d , it takes d + w w exp ected rounds for v to b e colored after u is colored. After v is colored, the expected num ber of rounds to color all of H is ept rzf ( H , v ). Altogether, it takes at most ept rzf ( G, x ) + d + w w + ept rzf ( H , v ) exp ected rounds to color all of K , starting from x . □ In the next result, w e sho w that the b ound in the last theorem is best p ossible b y exhibiting an inﬁnite family of examples whic h attain the b ound. Theorem 6.5. F or all m, n > 0, there exist weigh ted directed graphs G and H and v ertices u, x ∈ V ( G ) and v ∈ V ( H ) such that ept rzf ( G, x ) = m , ept rzf ( H , v ) = n , v has in-w eigh t d in H , and the weigh ted directed graph K obtained from G and H by adding an edge from u to v with w eigh t w satisﬁes ept rzf ( K, x ) = d + w w + ept rzf ( G, x ) + ept rzf ( H , v ) . Pr o of. Let G = − − − → P m +1 ha v e left endp oin t x and righ t endp oin t u , with all edges of w eigh t 1. Let H b e the w eigh ted directed graph obtained from − − − → P n +1 with left endpoint v and leftmost edge ( v , v ′ ) b y adding an edge ( v ′ , v ) of weigh t d and setting all other edge weigh ts to 1. Then, ept rzf ( G, x ) = m , ept rzf ( H , v ) = n , and ept rzf ( K ) = m + d + w w + n . □ 7. RZF on economic da t a T o illustrate ho w the metho ds dev elop ed in this pap er can b e applied b ey ond stylized graph families, w e conclude with a data-driven example using the US Beaureu of Economic Analysis (BEA) input-output accounts [26]. Interpreting each sector as a no de and in ter-sectoral supply ﬂo ws as weigh ted directed edges (including self-lo ops reﬂecting internal use), w e obtain a real economic net w ork on which RZF can b e run without additional mo deling assumptions. Computing th e EPT from each singleton starting sector pro duces a propagation-time proﬁle that reﬂects the inheren t structural dependencies of the U.S. production system. Although this example is not in tended as an economic analysis, it demonstrates that RZF can be meaningfully extended to empirical netw orks, capturing asymmetries in ﬂo w structure and highligh ting sectors whose link ages mak e them faster or slo wer origins under the RZF dynamics. This suggests that RZF may b e a useful exploratory to ol for understanding propagation phenomena in complex real-w orld systems. Computation for EPT were conducted using our publicly a v ailable implementation 1 , inspired by existing w ork on using Marko v Chains to com pute exp ected propagation time in probabilistic zero forcing [13]. The implementation works as follows: the BEA input-output data is ﬁrst conv erted in to a directed graph, with no des being the 15 sectors and w eigh ted directed edges indicating the monetary v alue of commo dities supplied from one sector to another in 2024, as rep orted in the BEA input-output tables. W e next construct a Mark o v transition matrix with the transition states b eing the 2 15 p ossible colorings of our directed graph, and transition probabilities are deriv ed from the randomized zero forcing rule. Exp ected propagation times from singleton initial sets are then computed using a dynamic programming approach. F or the BEA sector netw ork, the computed ept rzf v alues v ary widely across sectors (see Fig- ure 7.1). W e ﬁrst note that the EPT v alues are extremely high across all sectors, whic h can b e attributed to the large edge weigh ts on the self-lo ops, indicating that many sectors rely mostly on themselv es rather than other sectors. Notably , the go v ernmen t sector stands out with the smallest exp ected propagation time (approximately 385), indicating that a sho c k originating in gov ernmen t w ould spread through the econom y more quic kly than one from any other sector. The utilities sec- tor is also highly central with a very low ept rzf (under 1000). In con trast, the construction sector yields by far the largest propagation time (on the order of 1 . 7 × 10 4 ), suggesting that a disturbance 1 https://github.com/noah- lichtenberg/rzf_simulation RANDOMIZED ZERO FOR CING 27 Figure 7.1. RZF exp ected propagation times on the BEA input-output sector net w ork. No des corresp ond to economic sectors and are colored b y the EPT from a singleton start; edge widths and opacities reﬂect relative in ter-sectoral supply mag- nitudes. starting in construction remains comparativ ely lo calized for a long time. Most other industries fall in an intermediate range of ept rzf (roughly 4 × 10 3 -5 × 10 3 ); for example, man ufacturing, wholesale trade, retail, information, ﬁnance, and professional services all hav e EPT v alues on the order of 4 . 6 × 10 3 -4 . 8 × 10 3 , while agriculture is mo destly higher (ab out 5 . 7 × 10 3 ). These magnitudes reﬂect the v arying degrees of connectivity and inﬂuence each sector has in the input-output netw ork. Imp ortan tly , ept rzf in this context pro vides a quan titativ e measure of p eripheralit y for each sector in the pro duction netw ork. A larger exp ected propagation time means the sector is more p eripheral - a sho c k originating there diﬀuses slo wly to the rest of the econom y - whereas a smaller ept rzf implies the sector is more structurally central, with sho c ks from that sector reac hing other sectors more rapidly . In other words, the inv erse of ept rzf can b e in terpreted as a cen tralit y index under the RZF dynamic: sectors like gov ernmen t and utilities, whic h hav e high 1 / ept rzf (due to lo w propagation times), o ccup y cen tral p ositions in the netw ork of inter-industry link ages. Con- v ersely , sectors such as construction (with a very low 1 / ept rzf due to its h uge propagation time) are situated on the net w ork’s periphery . This persp ectiv e complemen ts traditional static centralit y measures by fo cusing on dynamic risk propagation behavior. It underscores, for instance, that go v- ernmen t’s extensive link ages (e.g. pro curemen t from man y industries) and utilities’ fundamen tal input role make them piv otal for rapid con tagion, while construction’s more self-con tained supply c hain p osition renders it a slo w er conduit for propagating sho c ks. Ov erall, examining ept rzf across all nodes highlights a pronounced asymmetry in the econom y’s connectivity: some sectors serv e as fast “h ubs” for spreading disturbances, whereas others act as b ottlenec ks or dead-ends, suggesting that RZF-based analysis can b e used for iden tifying structurally critical sectors. 28 J. GENESON, I. HICKS, N. LICHTENBER G, A. MOON, AND N. ROBLES T able 3. Exp ected propagation time (EPT) from singleton initial sets for BEA sectors No de Sector EPT 1 Agriculture, forestry , ﬁshing, and hun ting 5679.87 2 Mining 4768.77 3 Utilities 933.76 4 Construction 16652.22 5 Man ufacturing 4652.18 6 Wholesale trade 4684.16 7 Retail trade 4777.49 8 T ransportation and w arehousing 4055.41 9 Information 4714.20 10 Finance, insurance, real estate, ren tal, and leasing 4667.29 11 Professional and business services 4717.40 12 Educational services, health care, and so cial assistance 4755.98 13 Arts, en tertainment, recreation, accommodation, and fo od services 4842.59 14 Other services, except gov ernmen t 4912.73 15 Go v ernmen t 385.06 8. Conclusion and future directions In this pap er we in troduced r andomize d zer o for cing (RZF), a stochastic color-c hange pro cess on directed graphs in which the probabilit y that a white vertex turns blue depends on the fraction of its in-neigh b ors that are blue, and we extended the mo del to weigh ted directed graphs b y replac- ing fractions of neighbors with fractions of incoming w eigh t. W e established basic monotonicity prop erties (with resp ect to enlarging the initial blue set, and with resp ect to increasing w eigh ts on edges out of initially blue vertices), and we iden tiﬁed w eigh t normalizations under which the w eigh ted pro cess reduces to the unw eigh ted pro cess. W e then computed exact propagation times or sharp asymptotics for a v ariet y of graph families (including arb orescences, stars, paths, cycles, and balanced spiders), and w e pro v ed extremal b ounds for un w eigh ted directed graphs in terms of parameters such as the num b er of edges, order, maximum degree, and radius, with constructions sho wing sharpness up to constant factors. Finally , we discussed ho w ept rzf b eha v es under several graph operations and illustrated ho w the weigh ted model can b e run on an empirical input-output net w ork. Our results suggest sev eral natural directions for further study . In deterministic zero forcing, throttling problems quantify the tradeoﬀ b et ween the size of the initial blue set and the time needed to force the whole graph. Because ept rzf ( G, S ) is monotone nonincreasing in S , an analogous optimization problem is w ell-p osed for RZF. F or example, one can deﬁne an RZF thr ottling numb er b y th rzf ( G ) := min S ⊆ V ( G )  | S | + ept rzf ( G, S )  , or consider constrained v arian ts such as minimizing ept rzf ( G, S ) sub ject to | S | ≤ k , or minimizing | S | sub ject to ept rzf ( G, S ) ≤ T . Throttling has previously b een inv estigated for standard zero forcing [10], skew zero forcing [14], p ositiv e semideﬁnite zero forcing [12], p o w er domination [8], the cop-v ersus-robb er game [5, 7, 11], the cop-v ersus-gam bler game [17, 20], and metric dimension [9]. It would be esp ecially interesting to determine th rzf ( G ) (or sharp b ounds for it) on the same families studied here (paths, cycles, trees, complete graphs), and to compare how the optimal tradeoﬀ dep ends on directionalit y and (in the weigh ted setting) on the distribution of incoming w eigh ts. RANDOMIZED ZERO FOR CING 29 A compelling extension for applications is to allo w reco v ery , i.e., blue v ertices can rev ert to white. One simple v arian t is: after each round of RZF updates, each blue vertex independently reverts to white with probability ρ ∈ (0 , 1) (or with a vertex-dependent rate ρ v ). This produces a ﬁnite-state Mark o v c hain with p oten tially rich b eha vior: dep ending on parameters, the all-blue state ma y no longer b e absorbing, and one ma y instead study (i) the probability of ever reac hing all-blue, (ii) exp ected hitting times conditional on reac hing all-blue, and/or (iii) the stationary distribution and mixing time. Developing techniques that replace monotone couplings, which are cen tral in the presen t pap er, would b e a key step in understanding this regime. V ertex rev ersion was inv estigated for probabilistic zero forcing in [6]. Throughout the paper w e focus on exp ectations, but several arguments (for example, the spider analysis and joined-graph b ounds) naturally raise questions ab out v ariance and higher moments. F or a giv en family , one can ask for asymptotics of V ar( τ ) where τ is the propagation time, and for tail bounds of the form P ( τ ≥ t ) or P ( | τ − E τ | ≥ t ). Suc h results w ould clarify when the exp ectation is representativ e of typical b eha vior, and could enable sharp er b ounds for graph op erations that in v olve maxima o v er indep enden t or weakly dependent sub-pro cesses. The weigh ted mo del is motiv ated b y input-output and other ﬂo w netw orks. This suggests opti- mization problems in which one mo diﬁes weigh ts or adds edges to slow or ac c eler ate propagation under constraints. F or example, given a budget to increase or decrease a small set of incoming w eigh ts, which mo diﬁcations maximize ept rzf ( G, S ) (to increase resilience) or minimize it (to mo del fast diﬀusion)? The monotonicity results in the basic section pro vide starting tools for such com- parativ e statics, but a broader theory for general weigh t perturbations (not restricted to edges out of S ) w ould b e esp ecially useful. Man y real-world net works ev olv e ov er time, with edges or weigh ts that change in resp onse to the state of the system. Extending RZF to temp oral graphs or adaptiv e-weigh t mo dels, where w eigh ts resp ond dynamically to the coloring pro cess itself, could signiﬁcan tly broaden its applicability . In such settings, exp ected propagation time ma y interact non trivially with feedback mec hanisms, creating new phenomena absent in static graphs. The BEA example illustrates that ept rzf can be computed on real w eighted directed net works and used as a dynamic, pro cess-based notion of cen tralit y . 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