E-Companion to The Evolution of Beliefs over Signed Social Networks

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1 OPERATIONS RESEARCH INFORMS E-Companion to The Evolution of Beliefs over Signed Social Networks Guodong Shi Research School of Engineering, CECS, The Australian National University, Canberra ACT 000, Australia Alexandre Proutiere, Mikael Johansson ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 0044, Sweden John S. Baras Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 074, USA Karl H. Johansson ACCESS Linnaeus Centre, School of Electrical Engineering, Royal Institute of Technology, Stockholm 0044, Sweden Proof of Lemma The Triangle Lemma We assume n 5. Generality is not lost by making this assumption because for n = 3 and n = 4, some tedious but straightforward analysis on each possible G leads to the desired conclusion. i. There are two cases: i 0, i } E pst, or i 0, i } E neg. We prove the desired conclusion for each of the two cases. Without loss of generality, we assume that x i0 0 < x i 0. Let i 0, i } E pst. If x i 0 [ 3 x 4 i x 4 i 0, x 4 i x 4 i 0 ], we have J i i 0 J 4 i 0 i 0. Thus, the desired conclusion holds for δ =, arbitrary Z > 0, and any node pair 4 sequence over 0,,..., Z for which i 0, i, i are never selected. On the other hand suppose x i 0 / [ 3 x 4 i x 4 i 0, x 4 i x 4 i 0 ]. Take log d = if α, α 4, if α =. EC. If i 0, i } is selected for 0,,..., d, we obtain J i0 i d J 4 i 0 i 0 which leads to [ 5 x i d 8 x 0 + i0 8 x 0, 3 i 8 x ] i0 8 x 0 i ; x i d = x i 0. This gives us J i i d 8 J i 0 i 0.

2 e Let i 0, i } E neg. If x i 0 / [ x i x i 0, x i x i 0 ], we have J i i 0 J i 0 i 0. The conclusion holds for δ =, arbitrary Z > 0, and any node pair sequence over 0,,..., Z for which i 0, i, i are never selected. On the other hand let x i 0 [ x i x i 0, x i x i 0 ]. Take d = log +β 4. Let i 0, i } be selected for 0,,..., d. In this case, x i0 s and x i s are symmetric with respect to their center x i x i 0 for all s = 0,..., d, and J i0 i d 4J i0 i 0. Thus we have x i d = x i 0, and x i d x i x i 0 + x i 0 x i0 0 = 3 x i x i 0. EC. We can therefore conclude that J i i d J i0 i 0. In summary, the desired conclusion holds for δ = 8 and Z = max log +β 4, log α 4 if α log +β 4, if α =. EC.3 ii. We distinguish the cases i 0, i } E pst and i 0, i } E neg. Without loss of generality, we assume that x i0 0 < x i 0. Let i 0, i } E pst. If x i 0 / [ x i x i 0, x i x i 0 ], we have J i i 0 J i 0 i 0. The conclusion holds for δ =, arbitrary Z > 0, and any node pair sequence 0,,..., Z for which i 0, i, i are never selected. Now let x i 0 [ x i x i 0, x i x i 0 ]. We write x i 0 = ςx i0 0 + ςx i 0 with ς [, 3 ]. Let i 0, i } be the node pair selected for 0,,..., d with d defined by EC.. Note that according to the structure of the update rule, x i0 s and x i s will be symmetric with respect to their center ς x i ς x i 0 for all s = 0,..., d, and J i0 i d 4 J i 0 i 0. This gives us x i d = x i 0 and which implies [ x i d ς x i ς x i 0 8 x i 0 x i0 0, ς x i ς x i 0 + ] 8 x i 0 x i0 0 [ = 3ς 8 x i ς 8 x i 0, 5ς 8 x i ς ] 8 x i 0, EC.4 J i i d 5ς 8 J i 0 i 0 6 J i 0 i 0. EC.5

3 e3 Let i 0, i } E neg. If x i 0 / [ x i x i 0, x i x i 0 ], the conclusion holds for the same reason as in the case where i 0, i } E pst. Now let x i 0 [ x i x i 0, x i x i 0 ]. We continue to use the notation x i 0 = ςx i0 0 + ςx i 0 with ς [, 3 ]. Let i 0, i } be the node pair selected for 0,,..., d where d = log +β 4. In this case, x i0 s and x i s are still symmetric with respect to their center ς x i ς x i 0 for all s = 0,,..., d, and J i0 i d 4J i0 i 0. This gives us x i d = x i 0 and which implies x i d ς x i ς x i 0 + x i 0 x i0 0 = 5ς x i ς x i 0 EC.6 J i i d 5ς J i 0 i 0 4 J i 0 i 0. EC.7 In summary, the desired conclusion holds for δ = 6 with Z defined in EC.3.. Proof of Theorem Let ω / C. Then there exists an initial value x 0 R n from which lim sup Xkω > 0. k EC.8 According to Lemma, EC.8 implies that which implies PC + PD =. P lim sup Xk = C c = P D C c =, EC.9 k With PC +PD =, D is a trivial event as long as C is a trivial event. Therefore, for completing the proof we just need to verify that C is a trivial event. We first show that C = lim k W k... W 0 = U }. In fact, if lim sup k max i,j x i k x j k = 0 under x 0 R n, then we have lim k xk = n x 0 because the sum of the beliefs is preserved. Therefore, we can restrict the analysis to x 0 = e i, i =,..., n and on can readily see that C = limk W k... W 0 = U }. Next, we apply the argument, which was originally introduced in Tahbaz-Salehi and Jadbabaie 008 for establishing the weak ergodicity of product of random stochastic matrices with positive diagonal terms, to conclude that C is a trivial event. A more general treatment to zero-one laws of random averaging algorithms can be found in Touri and Nedić 0. Define a sequence of event C s = lim k W k... W s = U } for s =,,.... We see that

4 e4 PC s = PC for all s =,,... since W k, k = 0,,..., are i.i.d. C s+ C s for all s =,,... since lim k W k... W s+ = U implies lim k W k... W s = U due to the fact that UW s U. Therefore, we have s= C s is a tail event within the tail σ-field s= σg s, G s+,.... By Kolmogorov s zero-one law, s= C s is a trivial event. Hence PC = lim s PC s = P s= C s is a trivial event, and the desired conclusion follows. 3. Proof of Lemma 3 Let x ave = i V x i0/n be the average of the initial beliefs. We introduce V k = n i= x ik x ave = I Uxk. The evolution of V k follows from } } E V k + xk = E xk + I U xk + xk } a = E xk W ki UW kxk xk b = E xk I U [ W ki UW k ] } I Uxk xk c λ max EW ki UW k} I Uxk d = λ max EW k} U V k, EC.0 where a is based on the facts that W k is symmetric and the simple fact I U = I U, b holds because I UW k = W ki U always holds and again I U = I U, c follows from Rayleigh-Ritz theorem cf. Theorem 4.. in Horn and Johnson 985 and the fact that W k is independent of xk, d is based on simple algebra and W ku = UW k = U. We now compute EW k. Note that I αei e j e i e j = I α αei e j e i e j ; I + βei e j e i e j = I + β βei e j e i e j. EC. This observation leads to P W k = I α αe i e j e i e j P W k = I + β + βe i e j e i e j As a result, we have = p ij + p ji, i, j} E pst ; n = p ij + p ji, i, j} E neg. n EW k} = I α αl pst + β + βl neg. EC. Consequently, we have 0 < γ := λ max EW k U < for all β satisfying β + β < λ L pst α α. λ max L neg EC.3

5 e5 Since gβ = β + β is nondecreasing, we conclude from EC.0 that E V k + xk } < γv k EC.4 with 0 < γ < for all 0 β < β. This means that V k is a supermartingale as long as 0 γ Durrett 00, and V k converges to a limit almost surely by the martingale convergence theorem Theorem 5..9, Durrett 00. Next we show that this limit is zero almost surely if 0 γ <. Let ɛ > 0 and 0 γ <. We have: a P V k > ɛ infinitely often = P b P ɛ c γ P ɛ k=0 P V k + > ɛ xk = k=0 k= E V k + xk } = V k =, EC.5 where a is straightforward application of the Second Borel-Cantelli Lemma Theorem in Durrett 00, b is from the Markov s inequality, and c holds directly from EC.4. Observing that EV k} γ k V 0 γ V 0 <, EC.6 γ k= k= we obtain P γ ɛ k= V k = = 0. Therefore, we have proved that P V k > ɛ infinitely often = 0, or equivalently, Plim k V k = 0 =. Finally, observe that: V k = n x i k x ave x ρ k x ave + x ρ k x ave x ρ k x ρ k = X k, i= where ρ and ρ are chosen such that x ρ k = X min k, x ρ k = X max k. Hence Plim k V k = 0 = implies Plim k Xk = 0 =. This completes the proof. 4. Proof of Lemma 4 Suppose X0 > 0. We have: J ij k + = α J ij k, if G k = i, j} E pst β + J ij k, if G k = i, j} E neg. EC.7 Thus, Xk > 0 almost surely for all k as long as X0 > 0. As a result, the following sequence of random variables is well defined: ζ k = Xk +, k = 0,,.... EC.8 Xk

6 e6 The proof is based on the analysis of ζ k. We proceed in three steps. Step. In this step, we establish some natural upper and lower bounds for ζ k. First of all, from EC.7, it is easy to see that: and P ζ k < P one link in E pst is selected. Xk + P ζ k = α = EC.9 Xk On the other hand let i 0, j 0 } G neg. Suppose i and j are two nodes satisfying J i j = X0. Repeating the analysis in the proof of Theorem by recursively applying the Triangle Lemma, we conclude that there is a sequence of node pairs for time steps 0,,..., n Z which guarantees J i0 j 0 n Z δn X0 EC.0 where δ = /6 and Z = max log +β 4, log α } are defined in the Triangle Lemma. For the 4 remaining of the proof we assume that β is sufficiently large so that log +β 4 log α 4, which means that we can select Z = log α independently of β. 4 Now take an integer H 0. Continuing the previous node pair sequence, let i 0, j 0 } be selected at time steps n Z,..., n Z + H 0. It then follows from EC.7 and EC.0 that Xn Z + H 0 J i0 j 0 n Z + H 0 β + H 0δ n X0. EC. Denote Z H0 = n Z + H 0. This node sequence for 0,,..., Z H0, which leads to EC., is denoted S i0 j 0 [0, Z H0. Step. We now define a random variable Q ZH0 0, associated with the node pair selection process in steps 0,..., Z H0, by α Z H 0, if at least one link in E pst is selected in steps 0,,..., Z H0 ; β+ Q ZH0 0 = H 0 δ n, if node sequence S i0 j 0 [0, Z H0 is selected in steps 0,,..., Z H0 ;, otherwise. EC. In view of EC.9 and EC., we have: P Z H 0 k=0 ζ k = XZ H 0 X0 Q ZH0 0 =. EC.3 From direct calculation based on the definition of Q ZH0 0, we conclude that } p ZH0 E log Q ZH0 0 log β + H 0δ n + p n n E 0Z H0 log α Z H 0 := C H0 EC.4

7 e7 where p = maxp ij + p ji : i, j} E} and E 0 = E pst denotes the number of positive links. Since Z does not depend on β, we see from EC.4 that for any fixed H 0, there is a constant β H 0 > 0 with log +β 4 log α guaranteeing that 4 β > β H 0 C H0 > 0. Step 3. Recursively applying the analysis in the previous steps, node pair sequences S i0 j 0 [sz H0, s+ Z H0 can be found for s =,,..., and Q ZH0 s, s =,,... can be defined associated with the node pair selection process following the same definition of Q ZH0 0. Since the node pair selection process is independent of time and node states, Q ZH0 s, s = 0,,,..., are independent random variables not necessarily i.i.d since S i0 j 0 [sz H0, s + Z H0 may correspond to different pair sequences for different s. The lower bound established in EC.4 holds for all s, i.e., E log Q ZH0 s } C H0, s = 0,,.... EC.5 Moreover, we can prove as EC.3 was established that: P tz H 0 k=0 ζ k = XtZ H 0 X0 t s=0 Q ZH0 s, t = 0,,,... =. EC.6 It is straightforward to see that V log Q ZH0 s }, s = 0,,... is bounded uniformly in s. Kolmogorov s strong law of large numbers for a sequence of mutually independent random variables under Kolmogorov criterion, see Feller 968 implies that: P lim t t t s=0 Using EC.5, EC.7 further implies that: log Q ZH0 s E log Q ZH0 s } = 0 =. EC.7 P lim inf t t t log Q ZH0 s C H0 =. EC.8 s=0 The final part of the proof is based on EC.6. With the definition of ζ k, EC.6 yields: P log X t+z H0 t + Z H0 log X 0 = log ζ k which together with EC.8 gives us: k=0 t s=0 log Q ZH0 s, t = 0,,,... =, P lim inf X t + Z H0 = =. EC.9 t We can further conclude that: P lim inf X k = = EC.30 k

8 e8 since P X k α Z H 0 X k Z H0 ZH0 = in view of EC.9. Therefore, for any integer H 0, we have proved that belief divergence is achieved for all initial condition satisfying X0 > 0 if β > β H 0. Define β := inf H 0 β H 0. With this choice of β, the desired conclusion holds. 5. Proof of Proposition 5 Note that there exist i s, j s } G pst, s =,,..., T with T such that W + i T j T W + i j = U EC.3 if and only if for any y0 = y 0 = y 0... y 0 n, the dynamical system yk = W + i k j k yk, k =,..., T EC.3 drives yk = y k,..., y n k to yt = avey0 where avey0 = n i= y0 i /n. Thus we may study the matrix equality EC.3 through individual node dynamics, which we leverage in the proof. The claim follows from an induction argument. Assume that the desired sequence of node pairs with length T k = k k exists for m = k. Assume that G pst has a subgraph isomorphic to an m + dimensional hypercube. Without loss of generality we assume V has been rewritten as 0, } k+ following the definition of hypercube. Now define V 0 := i i k+ V : i k+ = 0}; V := i i k+ V : i k+ = }. It is easy to see that each of the subgraphs G V and G 0 V contains a positive subgraph isomorphic with an m-dimensional hypercube. Therefore, for any initial value of y0, the nodes in each set G V s, s = 0, can reach the same value, say C 0 y0 and C y0, respectively. Then we select the following k edges for updates from G: i i k 0, i i k } : i s 0, }, s =,..., k. After these updates, all nodes reach the same value C 0 y0 + C y0/ which has to be avey0 since the sum of the node beliefs is constant during this process. Thus, the desired sequence of node pairs exists also for m = k +, with a length This proves the desired conclusion. T k+ = T k + k = k k + k = k + k.

9 e9 6. Proof of Proposition 6 The requirement of α = / is obvious since otherwise W + ij is nonsingular for all i, j} E pst, while ranku =. The necessity of m = k for some k 0 was proved in Shi et al. 04 through an elementary number theory argument by constructing a particular initial value for which finite-time convergence can never be possible by pairwise averaging. It remains to show that G pst has a perfect matching. Now suppose Eq. EC.3 holds. Without loss of generality we assume that Eq. EC.3 is irreducible in the sense that the equality will no longer hold if any one or more matrices are removed from that sequence. The idea of the proof is to analyze the dynamical system EC.3 backwards from the final step. In this way we will recover a perfect matching from i, j },..., i T, j T } }. We divide the remaining of the proof into three steps. Step. We first establish some property associated with i T, j T }. After the last step in EC.3, two nodes i T and j T reach the same value, avey 0, along with all the other nodes. We can consequently write y it T = avey 0 + h T y 0, y jt T = avey 0 h T y 0, where h T is a real-valued function marking the error between y it T, y jt T and the true average avey 0. Indeed, the set y 0 : h T y 0 = 0} is explicitly given by } y 0 : 0... }}... }}... 0W + i T j T W + i j y 0 = 0, i T th j T th which is a linear subspace with dimension n recall that the equation W + i T j T... W + i j = U is irreducible. Thus there must be h T y 0 0 for some initial value y 0. Step. If there are only two nodes in the network, we are done. Otherwise i T, j T } = i T, j T }. We make the following claim. Claim. i T, j T / i T, j T }. Suppose without loss of generality that i T = i T. Then y jt T = y jt T = y it T = y it T = avey 0 + h T y 0. While on the other hand y jt T = avey 0 for all y 0. The claim holds observing that as we just established, h T y 0 0} is a nonempty set. We then write: y it T = avey 0 + h T y 0, y jt T = avey 0 h T y 0

10 e0 where h T is again a real-valued function and h T y 0 0 for some initial value y 0 applying the same argument as for h T y 0 0. Note that y 0 : h T y 0 0 } y 0 : h T y 0 0 } = y 0 : h T y 0 = 0 } y 0 : h T y 0 = 0 } c is nonempty because it is the complement of the union of two linear subspaces of dimension n in R n. Step 3. Again, if there are only four nodes in the network, we are done. Otherwise, we can define: } T := max τ : i τ, j τ } i T, j T, i T, j T } EC.33 We emphasize that T must exist since Eq. EC.3 holds. As before, we have i T, j T / i T, j T, i T, j T } and h T y 0 can be found with h T y 0 = 0} being another n -dimensional subspace such that y it T = avey 0 + h T y 0, y jt T = avey 0 h T y 0. We thus conclude that this argument can be proceeded recursively until we have found a perfect matching of G pst in i, j },..., i T, j T } }. We have now completed the proof. References Durrett, R. 00 Probability Theory: Theory and Examples. 4th ed. Cambridge University Press: New York. Feller, W. 968 An Introduction to Probability Theory and Its Applications. 3rd ed. New York: Wiley. Horn, R. A. and Johnson, C. R. 985 Matrix Analysis. Cambridge University Press. Shi, G., Li, B., Johansson, M., and Johansson, K. H. 04 When do gossip algorithms converge in finite time? in The st International Symposium on Mathematical Theory of Networks and Systems MTNS, Groningen, The Netherlands. Tahbaz-Salehi, A. and Jadbabaie, A. 008 A necessary and sufficient condition for consensus over random networks. IEEE Trans. on Autom. Control, 53: Touri, B. and Nedić, A. 0 On ergodicity, infinite flow and consensus in random models. IEEE Trans. on Automatic Control, 56: